Research ArticleExact Boundary Derivative Formulation forNumerical Conformal Mapping Method

Wei-Lin Lo1 Nan-Jing Wu2 Chuin-Shan Chen1 and Ting-Kuei Tsay1

1Department of Civil Engineering National Taiwan University Taipei 10617 Taiwan2Department of Civil and Water Resources Engineering National Chiayi University Chiayi City 60004 Taiwan

Correspondence should be addressed to Chuin-Shan Chen dchenntuedutw

Received 10 October 2015 Accepted 11 January 2016

Academic Editor Ivano Benedetti

Copyright copy 2016 Wei-Lin Lo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Conformal mapping is a useful technique for handling irregular geometries when applying the finite difference method to solvepartial differential equations When the mapping is from a hyperrectangular region onto a rectangular region a specific length-to-width ratio of the rectangular region that fitted the Cauchy-Riemann equations must be satisfied In this research a numericalintegral method is proposed to find the specific length-to-width ratio It is conventional to employ the boundary integral method(BIEM) to perform the conformal mapping However due to the singularity produced by the BIEM in seeking the derivativeson the boundaries the transformation Jacobian determinants on the boundaries have to be evaluated at inner points instead ofdirectly on the boundaries This approximation is a source of numerical error In this study the transformed rectangular propertyand the Cauchy-Riemann equations are successfully applied to derive reduced formulations of the derivatives on the boundaries forthe BIEM With these boundary derivative formulations the Jacobian determinants can be evaluated directly on the boundariesFurthermore the results obtained are more accurate than those of the earlier mapping method

1 Introduction

The finite difference method (FDM) is a conventionalnumerical method commonly used in computational sci-ence because partial differential equations can be directlydiscretized [1ndash3] However when the computational regionis irregular boundary values must be evaluated throughinterpolation extrapolation or both which results in greatercomputation costs and decreases accuracy [4 5]

A common solution to circumvent this problem is totransform the irregular region into a rectangular regionand solve partial differential equations on the rectangularregion [6ndash10] According to a study by Thompson et al [11]a system of elliptic partial differential equations can guar-antee a one-to-one transformation between the physical andcomputational domains When the transformation is con-formal the number of additional terms introduced by thetransformation of the governing equation is the minimumThus inaccuracy can be avoided

By applying the conformal mapping method to build aneffective grid-generation system Tsay et al [12] employed

the boundary integral equation method (BIEM) (also calledthe boundary element method BEM) to solve the Laplaceequations The BIEM is preferable to the Laplace-equation-governed transformation process for a number of reasons(1) it is not necessary to discretize the entire domain (2)solutions can be evaluated exactly (3) derivatives of variablescan be evaluated directly without difference schemes and (4)once the solutions at all the discretized boundary nodes areobtained at any given point in the domain the solution aswell as the partial derivatives can be easily found withoutremeshing When the conformal mapping is performed thegoverning equation of the physical domain will be trans-formed into a formulation in terms of derivatives with respectto the coordinates Although the derivatives in the domaincan be directly evaluated by using the BIEM it suffers fromthe fact that the derivatives cannot be evaluated exactly onthe boundaries [13 14] This drawback makes one be onlyable to evaluate the derivatives close to the boundaries as theboundary derivatives

A limitation of the Laplace-equation-governed transfor-mation is that the region to be transformed should be a

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5072309 18 pageshttpdxdoiorg10115520165072309

2 Mathematical Problems in Engineering

quadrilateral with right-angled corners which is known asa hyperrectangular region [12] When a hyperrectangularregion is transformed into a rectangular region by using con-formal mapping the length-to-width ratio of the rectangularregion is a particular value [15] To find this ratio Seidl andKlose [15] converted the Cauchy-Riemann equations intoa set of two Laplace equations and proposed an iterativeprocedure with a finite difference solution In this researchan explicit integral method was proposed to find this ratiowithout the time consumption of solving a system of linearequations

Although the applicability of the conformal transforma-tion used by Tsay et al [12] to arbitrary irregular regionscould be extended by using a sequence of 119885

119899 transformations[16] the applicability of the grid-generation system proposedby Tsay et al [12 14 16] is still limited due to the above-mentioned problem of inaccurate boundary partial deriva-tivesThis problem not only precludes the derivatives and thetransformation Jacobian determinants from being evaluatedexactly on the boundaries but also decreases the accuracy ofthe finite difference computation after the transformation

In this paper by applying the rectangular properties ofthe transformed region and the Cauchy-Riemann equationsthe derivatives and the Jacobian determinants of the transfor-mation can be evaluated on the boundaries This evaluationsignificantly improves the transformation accuracy

2 Mathematical Description ofthe Coordinate Transformation

Based on the Cauchy-Riemann equations the forward con-formal transformation from the physical domain (119909-119910 coor-dinates) to the computational domain (120585-120578 coordinates) canbe reduced to a problem governed by two Laplace equations

1205972

120585

1205971199092+

1205972

120585

1205971199102= 0 (1a)

1205972

120578

1205971199092+

1205972

120578

1205971199102= 0 (1b)

with the boundary conditions

n sdot nabla120578 = 0 on 120585 = 0 120585 = 1205850 (2a)

n sdot nabla120585 = 0 on 120578 = 0 120578 = 1205780 (2b)

where n is the outward normal vector of the boundariesand nabla is the gradient operator Here the range of either 120585

or 120578 can be chosen arbitrarily while the range of the othercoordinate is restricted by the satisfaction of the Cauchy-Riemann equations

120597120585

120597119909=

120597120578

120597119910 (3a)

120597120585

120597119910= minus

120597120578

120597119909 (3b)

That is when satisfying the Cauchy-Riemann equationsthere is always a specific value of 120578

0that corresponds to a

given 1205850 and vice versa To find this specific value Seidl and

Klose [15] proposed an iterative scheme Besides Wu et al[17] showed that the transformation is not conformal but stillorthogonal if one simply chooses both values of 120585

0and 120578

0

arbitrarily In Section 4 we will propose a numerical integralmethod to find the specific ratio of 120585

0to 1205780which satisfies the

Cauchy-Riemann equationsBy applying the divergence theorem (1a) and (1b) can be

represented as an integral equation [13]

120572Φ (119875) = intΓ

[Φ (119876)120597119903

120597119899minus ln 119903

120597

120597119899Φ (119876)] 119889120591 (4)

where Φ represents 120585 or 120578 Γ is the bounding contour of theproblem domain 119863 119899 is the out-normal direction of Γ 119875

represents a source point 119876 represents a boundary point119903 is the distance from 119875 to 119876 120572 = 2120587 when 119875 is locatedinside the domain and 120572 is 1205872 or 120587 when 119875 is locatedon the corners or the straight boundaries respectively asshown in Figure 1 An elementwise local coordinate system(119904-120582) is adopted to discretize the geometry as linear elementsas shown in Figure 2 where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element and thesubscript 119894 represents the 119894th source point After discretizingthe boundary and choosing the source points as the nodalpoints of the elements the unknown function values Φand their normal derivatives 120597Φ120597119899 can be formulated assimultaneous equation system and then be solved [13 16] Toease the explanation the above solution for the unknowns ofboundary elements is named boundary-unknown solving stepin this paper

The inverse conformal transformation is also governed byLaplace equations whose function values are 119909 and 119910 withindependent variables of 120585 and 120578

1205972

119909

1205971205852+

1205972

119909

1205971205782= 0 (5a)

1205972

119910

1205971205852+

1205972

119910

1205971205782= 0 (5b)

and the following Cauchy-Riemann equations should besatisfied

120597119909

120597120578= minus

120597119910

120597120585 (6a)

120597119910

120597120578=

120597119909

120597120585 (6b)

Since the values of 119909 and 119910 on the boundary are alreadyknown the boundary condition of the inverse transformationis theDirichlet typeThe formulation of integral equations for(5a) and (5b) is the same as (4) but Φ represents 119909 or 119910 Asketch of the transformation is illustrated in Figure 3

After Φ and 120597Φ120597119899 on the boundary are solved Φ

inside the domain can be obtained by using (4) explicitlywhere 119875 is now located either inside the domain or onthe boundary In this research a hyperrectangular region istransformed into a rectangular region It is easy to generate

Mathematical Problems in Engineering 3

Γ

Q

r

DP

120572

(a)Γ

P

r

Q

D

120572

(b)

Figure 1 Configuration for 119875 119876 119903 and 120572 associated with the problem domain 119863 with the bounding contour Γ (a) 119875 is enclosed by Γ (b) 119875

is located on Γ

y

x

120579

s

Pi

ri

sej

120582ei

n120595

sej+1

Q(s 120582)

eth element

Figure 2 Definition of the elementwise local coordinates s-120582 coordinates The 119909-119910 coordinates are global coordinates The 119890th element119904119895

minus 119904119895+1

is a local element with outward normal vector n and an included angle 120595 with the global 119909-119910 coordinates The local coordinate 120582 ispositive if 120582 and n are in the same direction 119903

119894is the distance from the 119894th source point 119875

119894to a boundary point 119876(119904 120582)

1205972x

1205971205852+1205972x

1205971205782= 0

1205972y

1205971205852+1205972y

1205971205782= 0

1205972120585

120597x2+

1205972120585

120597y2= 0

1205972120578

120597x2+

1205972120578

120597y2= 0

120585 = 0

120597120578

120597n= 0

120597120585

120597n= 0

120578 = 1205780

120585 = 1205850

A

B C

D

A998400

B998400 C998400

D998400

120578

120578 = 0 120585

120597120578

120597n= 0

120597120585

120597n= 0x

y

Figure 3 A sketch of the orthogonal transformation from the physical domain (119909-119910 coordinates) to the computational domain (120585-120578coordinates)

4 Mathematical Problems in Engineering

120597x

120597120585(directly solved)

120597y

120597120585(directly solved)

120597x

120597120578= minus

120597y

120597120585

120597y

120597120585= minus

120597x

120597120578

120597y

120597120578=

120597x

120597120585

120597x

120597120585=

120597y

120597120578

120597x

120597120578(directly solved)

120597y

120597120578(directly solved)120585

120578

Figure 4 An illustration for the solution of the partial derivatives on the boundaries The values of 120597119909120597120585 and 120597119910120597120585 on the top and bottomboundaries can be solved for directly using the BIEMwith the improved formulation and then the values of 120597119909120597120578 and 120597119910120597120578 can be obtainedusing theCauchy-Riemann equations Similarly the values of 120597119909120597120578 and 120597119910120597120578 on the right and left boundaries can be solved for directly usingthe BIEM with the improved formulation and then the values of 120597119909120597120585 and 120597119910120597120585 can be obtained using the Cauchy-Riemann equations

an orthogonal grid in the rectangular region by using (4)where Φ represents the coordinate of the rectangular regionThen each grid node is mapped onto the hyperrectangularregion by using (4) again where Φ represents the coordinateof the hyperrectangular region now The above evaluationof grid nodes is named grid-generation step in this paperThe regular orthogonal grid in the rectangular region is veryconvenient for applying FDM To perform FDM on the gridin the rectangular region the governing equation usually hasto transform into a formulation with Jacobian determinants[18ndash20] For example the Biharmonic equation nabla4120601

(119909119910)= 0

has to transform into nabla2

(119869(119909119910)

nabla2

120601(120585120578)

) = 0 where 119909 and 119910

are coordinates of the hyperrectangular region 120585 and 120578 arecoordinates of the rectangular region 119869

(119909119910)is the Jacobian

determinant for the forward transformationThe Jacobian determinants for the forward transforma-

tion and inverse transformation are defined respectively asbelow

119869(119909119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597120585

120597119909

120597120578

120597119909

120597120585

120597119910

120597120578

120597119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597120585

120597119909

120597120578

120597119910minus

120597120585

120597119910

120597120578

120597119909 (7a)

119869(120585120578)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597119909

120597120585

120597119910

120597120585

120597119909

120597120578

120597119910

120597120578

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597119909

120597120585

120597119910

120597120578minus

120597119909

120597120578

120597119910

120597120585 (7b)

With the help of Chain rule and these Jacobian determinantsthe relationships between the partial derivatives of 119909-119910 and120585-120578 can be built as follows

120597119909

120597120585=

1

119869(119909119910)

120597120578

120597119910 (8a)

120597119909

120597120578= minus

1

119869(119909119910)

120597120585

120597119910 (8b)

120597119910

120597120585= minus

1

119869(119909119910)

120597120578

120597119909 (8c)

120597119910

120597120578=

1

119869(119909119910)

120597120585

120597119909 (8d)

120597120585

120597119909=

1

119869(120585120578)

120597119910

120597120578 (8e)

120597120585

120597119910= minus

1

119869(120585120578)

120597119909

120597120578 (8f)

120597120578

120597119909= minus

1

119869(120585120578)

120597119910

120597120585 (8g)

120597120578

120597119910=

1

119869(120585120578)

120597119909

120597120585 (8h)

After substituting (8a)ndash(8d) into (7b) or substituting (8e)ndash(8h) into (7a) the relation between 119869

(119909119910)and 119869

(120585120578)can be

derived as follows

119869(119909119910)

119869(120585120578)

= 1 (9)

3 The Derivatives and Jacobian Determinantson the Boundaries

The Jacobian determinant is a combination of partial deriva-tives Since these derivatives are evaluated by the BIEMsingularities occur when the unknown derivatives are locatedon the boundaries A previous research has suggested thatthese boundary derivatives can be approximated by evalu-ating the inner points that are very close to the boundaries[16] In this study the derivative formulations of the BIEMare carefully investigated By applying the Cauchy-Riemannequations a method of evaluating the boundary derivativesis proposed Unlike conventional methods that just calculate

Mathematical Problems in Engineering 5

the derivative close to the boundary as succedaneums thismethod can produce accurate numerical results exactly onthe boundaries

The partial derivatives of Φ can be found by differenti-ating (4) and then the following formulations are obtained[13]

120572120597

1205971199091

Φ (119875) = intΓ

[Φ (119876)120597

1205971199091

(1

119903

120597119903

120597119899)

minus120597

1205971199091

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10a)

120572120597

1205971199092

Φ (119875) = intΓ

[Φ (119876)120597

1205971199092

(1

119903

120597119903

120597119899)

minus120597

1205971199092

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10b)

The function value and its normal derivative on the boundaryΦ(119876) and (120597120597119899)Φ(119876) are already known after the evaluationof (4) 119909

1and 119909

2represent 119909 and 119910 respectively when the

function value Φ is 120585 or 120578 Conversely 1199091and 119909

2represent 120585

and 120578 respectively when the function value Φ is 119909 or 119910 Afterthe geometry discretization the following derivative termscan be represented with the local coordinate system (119904-120582)

120597119903

120597119899=

120582

119903 (11a)

120597120582

1205971199091

= sin120595 (11b)

120597119903

1205971199091

= minus119904 cos120595 minus 120582 sin120595

119903 (11c)

120597120582

1205971199092

= minus cos120595 (11d)

120597119903

1205971199092

= minus120582 cos120595 + 119904 sin120595

119903 (11e)

By applying (11a)ndash(11e) (10a) and (10b) can be representedwith the following formulations respectively

120597

1205971199091

Φ (119875) =1

120572intΓ

[Φ sin120595

1199032

+2Φ120582 (119904 cos120595 minus 120582 sin120595)

1199034

+119904 cos120595 minus 120582 sin120595

1199032

120597Φ

120597119899] 119889120591

(12a)

120597

1205971199092

Φ (119875) =1

120572intΓ

[minusΦ cos120595

1199032

+2Φ120582 (120582 cos120595 + 119904 sin120595)

1199034

+120582 cos120595 + 119904 sin120595

1199032

120597Φ

120597119899] 119889120591

(12b)

Liggett and Liu [13] have indicated that these cannot beevaluated on the boundaries because singularities will occurHowever they are actually applicable on the boundariesin the conformal mapping process through the followinginvestigation

The discretization of these boundary integral equations isperformed by linear approximations and the potential and itsnormal derivative are subsequently approximated by

Φ =

[(Φ119895+1

minus Φ119895) 119904 + (119904

119895+1Φ119895

minus 119904119895Φ119895+1

)]

(119904119895+1

minus 119904119895)

120597Φ

120597119899=

[(120597Φ120597119899)119895+1

minus (120597Φ120597119899)119895] 119904 + [119904

119895+1(120597Φ120597119899)

119895minus 119904119895(120597Φ120597119899)

119895+1]

(119904119895+1

minus 119904119895)

(13)

for 119904119895

le 119904 le 119904119895+1

where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element Aftersubstituting (13) into (12a) and integrating each term in thelocal coordinate system (119904-120582) the discretized formulationof the integral equation for the derivative in the horizontaldirection can then be represented as

120597

1205971199091

Φ (119875)

=1

120572

119899

sum

119890=1

[119860 119861] [

Φ119895

Φ119895+1

] + [119862 119863][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(14)

where 119890 represents the 119890th discretized element on the bound-ary 119899 is the number of the elements and

119860 = minus1198681199091

11minus 1198691199091

11+ 1198701199091

11+ 119904119895+1

(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15a)

119861 = 1198681199091

11+ 1198691199091

11minus 1198701199091

11minus 119904119895(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15b)

119862 = minus1198681199091

21+ 1198691199091

21+ 119904119895+1

(1198681199091

22minus 1198691199091

22) (15c)

119863 = 1198681199091

21minus 1198691199091

21minus 119904119895(1198681199091

22minus 1198691199091

22) (15d)

in which

1198681199091

11=

sin120595

119904119895+1

minus 119904119895

times1

2ln

119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16a)

6 Mathematical Problems in Engineering

1198681199091

12=

sin120595

119904119895+1

minus 119904119895

times1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16b)

1198691199091

11=

cos120595

119904119895+1

+ 119904119895

[

120582119894119904119895

119904119895

2 + 120582119894

2minus

120582119894119904119895+1

119904119895+1

2 + 120582119894

2

+ (tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16c)

1198691199091

12=

120582119894cos120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16d)

1198701199091

11=

120582119894

2 sin120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16e)

1198701199091

12=

sin120595

119904119895+1

minus 119904119895

[minus

119904119895

119904119895

2 + 120582119894

2+

119904119895+1

119904119895+1

2 + 120582119894

2

+1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16f)

1198681199091

21=

cos120595

119904119895+1

minus 119904119895

[119904119895+1

minus 119904119895

minus 120582119894(tanminus1

119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16g)

1198681199091

22=

cos120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16h)

1198691199091

21=

120582119894sin120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16i)

1198691199091

22=

sin120595

(119904119895+1

minus 119904119895)

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16j)

During the grid-generation step when 119875 is located onthe top or bottom boundary of the transformed rectangularregion where 120582

119894= 0 singularity will occur in (16b) and

(16f) However at the same time the value of 120595 is either 0or 120587 which makes sin120595 = 0 This zero value of sin120595 makesthe singularities of (16b) and (16f) vanish Besides it shouldbe emphasized that the value of the term (tanminus1(119904

119895+1120582119894) minus

tanminus1(119904119895120582119894)) is also zero when 120582

119894= 0 [14] After substituting

120582119894= 0 sin120595 = 0 and (tanminus1(119904

119895+1120582119894) minus tanminus1(119904

119895120582119894)) = 0 into

(16a)ndash(16j) the original discretized equations are reduced tothe following formulations

1198681199091

11= 1198681199091

12= 1198691199091

11= 1198691199091

12= 1198701199091

11= 1198701199091

12= 1198691199091

21= 1198691199091

22= 0 (17a)

1198681199091

21=

1 120595 = 0

minus1 120595 = 120587

(17b)

1198681199091

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 0

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 120587

(17c)

This result not only is relatively simple but can also bedetermined except at the source point (119904

119895= 0 or 119904

119895+1=

0) Similarly (12b) the derivative formulation in the verticaldirection can be discretized to the following equation

120597

1205971199092

Φ (119875)

=1

120572

119899

sum

119890

[119864 119865] [

Φ119895

Φ119895+1

] + [119866 119867][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(18)

where

119864 = 1198681199092

11minus 1198691199092

11minus 1198701199092

11+ 119904119895+1

(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19a)

119865 = minus1198681199092

11+ 1198691199092

11+ 1198701199092

11minus 119904119895(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19b)

119866 = minus1198681199092

21minus 1198691199092

21+ 119904119895+1

(1198681199092

22+ 1198691199092

22) (19c)

119867 = 1198681199092

21+ 1198691199092

21minus 119904119895(1198691199092

22+ 1198681199092

22) (19d)

Notice that the formulation of 1198681199092

11 1198681199092

12 1198691199092

11 1198691199092

12 1198701199092

11 1198701199092

12

1198681199092

21 1198681199092

22 1198691199092

21 and 119869

1199092

22can be obtained by changing sin120595 to

cos120595 and cos120595 to sin120595 in (16a)ndash(16j) and similar reducedformulations can be derived when 119875 is located on the left orright boundary of the transformed rectangular region

1198681199092

11= 1198681199092

12= 1198691199092

11= 1198691199092

12= 1198701199092

11= 1198701199092

12= 1198691199092

21= 1198691199092

22= 0 (20a)

1198681199092

21=

1 120595 =1

2120587

minus1 120595 =3

2120587

(20b)

1198681199092

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

1

2120587

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

3

2120587

(20c)

The Jacobian determinant of the inverse transformationfrom the rectangular region to the irregular region defined as(7b) is a combination of partial derivatives in the rectangularregion On the top and bottom boundaries of the rectangularregion the horizontal partial derivatives 120597119909120597120585 and 120597119910120597120585

can be evaluated using (14)ndash(17c) Instead of using the BIEMdirectly high-accuracy vertical derivatives 120597119909120597120578 and 120597119910120597120578

can be evaluated by applying the Cauchy-Riemann equationsfor inverse transformation (6a) and (6b) Similarly on theright and left boundaries the vertical derivatives 120597119909120597120578 and120597119910120597120578 can be derived using (18)ndash(20c) and the horizontalpartial derivatives 120597119909120597120585 and 120597119910120597120585 can be subsequently

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

quadrilateral with right-angled corners which is known asa hyperrectangular region [12] When a hyperrectangularregion is transformed into a rectangular region by using con-formal mapping the length-to-width ratio of the rectangularregion is a particular value [15] To find this ratio Seidl andKlose [15] converted the Cauchy-Riemann equations intoa set of two Laplace equations and proposed an iterativeprocedure with a finite difference solution In this researchan explicit integral method was proposed to find this ratiowithout the time consumption of solving a system of linearequations

Although the applicability of the conformal transforma-tion used by Tsay et al [12] to arbitrary irregular regionscould be extended by using a sequence of 119885

119899 transformations[16] the applicability of the grid-generation system proposedby Tsay et al [12 14 16] is still limited due to the above-mentioned problem of inaccurate boundary partial deriva-tivesThis problem not only precludes the derivatives and thetransformation Jacobian determinants from being evaluatedexactly on the boundaries but also decreases the accuracy ofthe finite difference computation after the transformation

In this paper by applying the rectangular properties ofthe transformed region and the Cauchy-Riemann equationsthe derivatives and the Jacobian determinants of the transfor-mation can be evaluated on the boundaries This evaluationsignificantly improves the transformation accuracy

2 Mathematical Description ofthe Coordinate Transformation

Based on the Cauchy-Riemann equations the forward con-formal transformation from the physical domain (119909-119910 coor-dinates) to the computational domain (120585-120578 coordinates) canbe reduced to a problem governed by two Laplace equations

1205972

120585

1205971199092+

1205972

120585

1205971199102= 0 (1a)

1205972

120578

1205971199092+

1205972

120578

1205971199102= 0 (1b)

with the boundary conditions

n sdot nabla120578 = 0 on 120585 = 0 120585 = 1205850 (2a)

n sdot nabla120585 = 0 on 120578 = 0 120578 = 1205780 (2b)

where n is the outward normal vector of the boundariesand nabla is the gradient operator Here the range of either 120585

or 120578 can be chosen arbitrarily while the range of the othercoordinate is restricted by the satisfaction of the Cauchy-Riemann equations

120597120585

120597119909=

120597120578

120597119910 (3a)

120597120585

120597119910= minus

120597120578

120597119909 (3b)

That is when satisfying the Cauchy-Riemann equationsthere is always a specific value of 120578

0that corresponds to a

given 1205850 and vice versa To find this specific value Seidl and

Klose [15] proposed an iterative scheme Besides Wu et al[17] showed that the transformation is not conformal but stillorthogonal if one simply chooses both values of 120585

0and 120578

0

arbitrarily In Section 4 we will propose a numerical integralmethod to find the specific ratio of 120585

0to 1205780which satisfies the

Cauchy-Riemann equationsBy applying the divergence theorem (1a) and (1b) can be

represented as an integral equation [13]

120572Φ (119875) = intΓ

[Φ (119876)120597119903

120597119899minus ln 119903

120597

120597119899Φ (119876)] 119889120591 (4)

where Φ represents 120585 or 120578 Γ is the bounding contour of theproblem domain 119863 119899 is the out-normal direction of Γ 119875

represents a source point 119876 represents a boundary point119903 is the distance from 119875 to 119876 120572 = 2120587 when 119875 is locatedinside the domain and 120572 is 1205872 or 120587 when 119875 is locatedon the corners or the straight boundaries respectively asshown in Figure 1 An elementwise local coordinate system(119904-120582) is adopted to discretize the geometry as linear elementsas shown in Figure 2 where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element and thesubscript 119894 represents the 119894th source point After discretizingthe boundary and choosing the source points as the nodalpoints of the elements the unknown function values Φand their normal derivatives 120597Φ120597119899 can be formulated assimultaneous equation system and then be solved [13 16] Toease the explanation the above solution for the unknowns ofboundary elements is named boundary-unknown solving stepin this paper

The inverse conformal transformation is also governed byLaplace equations whose function values are 119909 and 119910 withindependent variables of 120585 and 120578

1205972

119909

1205971205852+

1205972

119909

1205971205782= 0 (5a)

1205972

119910

1205971205852+

1205972

119910

1205971205782= 0 (5b)

and the following Cauchy-Riemann equations should besatisfied

120597119909

120597120578= minus

120597119910

120597120585 (6a)

120597119910

120597120578=

120597119909

120597120585 (6b)

Since the values of 119909 and 119910 on the boundary are alreadyknown the boundary condition of the inverse transformationis theDirichlet typeThe formulation of integral equations for(5a) and (5b) is the same as (4) but Φ represents 119909 or 119910 Asketch of the transformation is illustrated in Figure 3

After Φ and 120597Φ120597119899 on the boundary are solved Φ

inside the domain can be obtained by using (4) explicitlywhere 119875 is now located either inside the domain or onthe boundary In this research a hyperrectangular region istransformed into a rectangular region It is easy to generate

Mathematical Problems in Engineering 3

Γ

Q

r

DP

120572

(a)Γ

P

r

Q

D

120572

(b)

Figure 1 Configuration for 119875 119876 119903 and 120572 associated with the problem domain 119863 with the bounding contour Γ (a) 119875 is enclosed by Γ (b) 119875

is located on Γ

y

x

120579

s

Pi

ri

sej

120582ei

n120595

sej+1

Q(s 120582)

eth element

Figure 2 Definition of the elementwise local coordinates s-120582 coordinates The 119909-119910 coordinates are global coordinates The 119890th element119904119895

minus 119904119895+1

is a local element with outward normal vector n and an included angle 120595 with the global 119909-119910 coordinates The local coordinate 120582 ispositive if 120582 and n are in the same direction 119903

119894is the distance from the 119894th source point 119875

119894to a boundary point 119876(119904 120582)

1205972x

1205971205852+1205972x

1205971205782= 0

1205972y

1205971205852+1205972y

1205971205782= 0

1205972120585

120597x2+

1205972120585

120597y2= 0

1205972120578

120597x2+

1205972120578

120597y2= 0

120585 = 0

120597120578

120597n= 0

120597120585

120597n= 0

120578 = 1205780

120585 = 1205850

A

B C

D

A998400

B998400 C998400

D998400

120578

120578 = 0 120585

120597120578

120597n= 0

120597120585

120597n= 0x

y

Figure 3 A sketch of the orthogonal transformation from the physical domain (119909-119910 coordinates) to the computational domain (120585-120578coordinates)

4 Mathematical Problems in Engineering

120597x

120597120585(directly solved)

120597y

120597120585(directly solved)

120597x

120597120578= minus

120597y

120597120585

120597y

120597120585= minus

120597x

120597120578

120597y

120597120578=

120597x

120597120585

120597x

120597120585=

120597y

120597120578

120597x

120597120578(directly solved)

120597y

120597120578(directly solved)120585

120578

Figure 4 An illustration for the solution of the partial derivatives on the boundaries The values of 120597119909120597120585 and 120597119910120597120585 on the top and bottomboundaries can be solved for directly using the BIEMwith the improved formulation and then the values of 120597119909120597120578 and 120597119910120597120578 can be obtainedusing theCauchy-Riemann equations Similarly the values of 120597119909120597120578 and 120597119910120597120578 on the right and left boundaries can be solved for directly usingthe BIEM with the improved formulation and then the values of 120597119909120597120585 and 120597119910120597120585 can be obtained using the Cauchy-Riemann equations

an orthogonal grid in the rectangular region by using (4)where Φ represents the coordinate of the rectangular regionThen each grid node is mapped onto the hyperrectangularregion by using (4) again where Φ represents the coordinateof the hyperrectangular region now The above evaluationof grid nodes is named grid-generation step in this paperThe regular orthogonal grid in the rectangular region is veryconvenient for applying FDM To perform FDM on the gridin the rectangular region the governing equation usually hasto transform into a formulation with Jacobian determinants[18ndash20] For example the Biharmonic equation nabla4120601

(119909119910)= 0

has to transform into nabla2

(119869(119909119910)

nabla2

120601(120585120578)

) = 0 where 119909 and 119910

are coordinates of the hyperrectangular region 120585 and 120578 arecoordinates of the rectangular region 119869

(119909119910)is the Jacobian

determinant for the forward transformationThe Jacobian determinants for the forward transforma-

tion and inverse transformation are defined respectively asbelow

119869(119909119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597120585

120597119909

120597120578

120597119909

120597120585

120597119910

120597120578

120597119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597120585

120597119909

120597120578

120597119910minus

120597120585

120597119910

120597120578

120597119909 (7a)

119869(120585120578)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597119909

120597120585

120597119910

120597120585

120597119909

120597120578

120597119910

120597120578

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597119909

120597120585

120597119910

120597120578minus

120597119909

120597120578

120597119910

120597120585 (7b)

With the help of Chain rule and these Jacobian determinantsthe relationships between the partial derivatives of 119909-119910 and120585-120578 can be built as follows

120597119909

120597120585=

1

119869(119909119910)

120597120578

120597119910 (8a)

120597119909

120597120578= minus

1

119869(119909119910)

120597120585

120597119910 (8b)

120597119910

120597120585= minus

1

119869(119909119910)

120597120578

120597119909 (8c)

120597119910

120597120578=

1

119869(119909119910)

120597120585

120597119909 (8d)

120597120585

120597119909=

1

119869(120585120578)

120597119910

120597120578 (8e)

120597120585

120597119910= minus

1

119869(120585120578)

120597119909

120597120578 (8f)

120597120578

120597119909= minus

1

119869(120585120578)

120597119910

120597120585 (8g)

120597120578

120597119910=

1

119869(120585120578)

120597119909

120597120585 (8h)

After substituting (8a)ndash(8d) into (7b) or substituting (8e)ndash(8h) into (7a) the relation between 119869

(119909119910)and 119869

(120585120578)can be

derived as follows

119869(119909119910)

119869(120585120578)

= 1 (9)

3 The Derivatives and Jacobian Determinantson the Boundaries

The Jacobian determinant is a combination of partial deriva-tives Since these derivatives are evaluated by the BIEMsingularities occur when the unknown derivatives are locatedon the boundaries A previous research has suggested thatthese boundary derivatives can be approximated by evalu-ating the inner points that are very close to the boundaries[16] In this study the derivative formulations of the BIEMare carefully investigated By applying the Cauchy-Riemannequations a method of evaluating the boundary derivativesis proposed Unlike conventional methods that just calculate

Mathematical Problems in Engineering 5

the derivative close to the boundary as succedaneums thismethod can produce accurate numerical results exactly onthe boundaries

The partial derivatives of Φ can be found by differenti-ating (4) and then the following formulations are obtained[13]

120572120597

1205971199091

Φ (119875) = intΓ

[Φ (119876)120597

1205971199091

(1

119903

120597119903

120597119899)

minus120597

1205971199091

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10a)

120572120597

1205971199092

Φ (119875) = intΓ

[Φ (119876)120597

1205971199092

(1

119903

120597119903

120597119899)

minus120597

1205971199092

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10b)

The function value and its normal derivative on the boundaryΦ(119876) and (120597120597119899)Φ(119876) are already known after the evaluationof (4) 119909

1and 119909

2represent 119909 and 119910 respectively when the

function value Φ is 120585 or 120578 Conversely 1199091and 119909

2represent 120585

and 120578 respectively when the function value Φ is 119909 or 119910 Afterthe geometry discretization the following derivative termscan be represented with the local coordinate system (119904-120582)

120597119903

120597119899=

120582

119903 (11a)

120597120582

1205971199091

= sin120595 (11b)

120597119903

1205971199091

= minus119904 cos120595 minus 120582 sin120595

119903 (11c)

120597120582

1205971199092

= minus cos120595 (11d)

120597119903

1205971199092

= minus120582 cos120595 + 119904 sin120595

119903 (11e)

By applying (11a)ndash(11e) (10a) and (10b) can be representedwith the following formulations respectively

120597

1205971199091

Φ (119875) =1

120572intΓ

[Φ sin120595

1199032

+2Φ120582 (119904 cos120595 minus 120582 sin120595)

1199034

+119904 cos120595 minus 120582 sin120595

1199032

120597Φ

120597119899] 119889120591

(12a)

120597

1205971199092

Φ (119875) =1

120572intΓ

[minusΦ cos120595

1199032

+2Φ120582 (120582 cos120595 + 119904 sin120595)

1199034

+120582 cos120595 + 119904 sin120595

1199032

120597Φ

120597119899] 119889120591

(12b)

Liggett and Liu [13] have indicated that these cannot beevaluated on the boundaries because singularities will occurHowever they are actually applicable on the boundariesin the conformal mapping process through the followinginvestigation

The discretization of these boundary integral equations isperformed by linear approximations and the potential and itsnormal derivative are subsequently approximated by

Φ =

[(Φ119895+1

minus Φ119895) 119904 + (119904

119895+1Φ119895

minus 119904119895Φ119895+1

)]

(119904119895+1

minus 119904119895)

120597Φ

120597119899=

[(120597Φ120597119899)119895+1

minus (120597Φ120597119899)119895] 119904 + [119904

119895+1(120597Φ120597119899)

119895minus 119904119895(120597Φ120597119899)

119895+1]

(119904119895+1

minus 119904119895)

(13)

for 119904119895

le 119904 le 119904119895+1

where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element Aftersubstituting (13) into (12a) and integrating each term in thelocal coordinate system (119904-120582) the discretized formulationof the integral equation for the derivative in the horizontaldirection can then be represented as

120597

1205971199091

Φ (119875)

=1

120572

119899

sum

119890=1

[119860 119861] [

Φ119895

Φ119895+1

] + [119862 119863][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(14)

where 119890 represents the 119890th discretized element on the bound-ary 119899 is the number of the elements and

119860 = minus1198681199091

11minus 1198691199091

11+ 1198701199091

11+ 119904119895+1

(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15a)

119861 = 1198681199091

11+ 1198691199091

11minus 1198701199091

11minus 119904119895(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15b)

119862 = minus1198681199091

21+ 1198691199091

21+ 119904119895+1

(1198681199091

22minus 1198691199091

22) (15c)

119863 = 1198681199091

21minus 1198691199091

21minus 119904119895(1198681199091

22minus 1198691199091

22) (15d)

in which

1198681199091

11=

sin120595

119904119895+1

minus 119904119895

times1

2ln

119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16a)

6 Mathematical Problems in Engineering

1198681199091

12=

sin120595

119904119895+1

minus 119904119895

times1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16b)

1198691199091

11=

cos120595

119904119895+1

+ 119904119895

[

120582119894119904119895

119904119895

2 + 120582119894

2minus

120582119894119904119895+1

119904119895+1

2 + 120582119894

2

+ (tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16c)

1198691199091

12=

120582119894cos120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16d)

1198701199091

11=

120582119894

2 sin120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16e)

1198701199091

12=

sin120595

119904119895+1

minus 119904119895

[minus

119904119895

119904119895

2 + 120582119894

2+

119904119895+1

119904119895+1

2 + 120582119894

2

+1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16f)

1198681199091

21=

cos120595

119904119895+1

minus 119904119895

[119904119895+1

minus 119904119895

minus 120582119894(tanminus1

119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16g)

1198681199091

22=

cos120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16h)

1198691199091

21=

120582119894sin120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16i)

1198691199091

22=

sin120595

(119904119895+1

minus 119904119895)

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16j)

During the grid-generation step when 119875 is located onthe top or bottom boundary of the transformed rectangularregion where 120582

119894= 0 singularity will occur in (16b) and

(16f) However at the same time the value of 120595 is either 0or 120587 which makes sin120595 = 0 This zero value of sin120595 makesthe singularities of (16b) and (16f) vanish Besides it shouldbe emphasized that the value of the term (tanminus1(119904

119895+1120582119894) minus

tanminus1(119904119895120582119894)) is also zero when 120582

119894= 0 [14] After substituting

120582119894= 0 sin120595 = 0 and (tanminus1(119904

119895+1120582119894) minus tanminus1(119904

119895120582119894)) = 0 into

(16a)ndash(16j) the original discretized equations are reduced tothe following formulations

1198681199091

11= 1198681199091

12= 1198691199091

11= 1198691199091

12= 1198701199091

11= 1198701199091

12= 1198691199091

21= 1198691199091

22= 0 (17a)

1198681199091

21=

1 120595 = 0

minus1 120595 = 120587

(17b)

1198681199091

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 0

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 120587

(17c)

This result not only is relatively simple but can also bedetermined except at the source point (119904

119895= 0 or 119904

119895+1=

0) Similarly (12b) the derivative formulation in the verticaldirection can be discretized to the following equation

120597

1205971199092

Φ (119875)

=1

120572

119899

sum

119890

[119864 119865] [

Φ119895

Φ119895+1

] + [119866 119867][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(18)

where

119864 = 1198681199092

11minus 1198691199092

11minus 1198701199092

11+ 119904119895+1

(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19a)

119865 = minus1198681199092

11+ 1198691199092

11+ 1198701199092

11minus 119904119895(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19b)

119866 = minus1198681199092

21minus 1198691199092

21+ 119904119895+1

(1198681199092

22+ 1198691199092

22) (19c)

119867 = 1198681199092

21+ 1198691199092

21minus 119904119895(1198691199092

22+ 1198681199092

22) (19d)

Notice that the formulation of 1198681199092

11 1198681199092

12 1198691199092

11 1198691199092

12 1198701199092

11 1198701199092

12

1198681199092

21 1198681199092

22 1198691199092

21 and 119869

1199092

22can be obtained by changing sin120595 to

cos120595 and cos120595 to sin120595 in (16a)ndash(16j) and similar reducedformulations can be derived when 119875 is located on the left orright boundary of the transformed rectangular region

1198681199092

11= 1198681199092

12= 1198691199092

11= 1198691199092

12= 1198701199092

11= 1198701199092

12= 1198691199092

21= 1198691199092

22= 0 (20a)

1198681199092

21=

1 120595 =1

2120587

minus1 120595 =3

2120587

(20b)

1198681199092

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

1

2120587

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

3

2120587

(20c)

The Jacobian determinant of the inverse transformationfrom the rectangular region to the irregular region defined as(7b) is a combination of partial derivatives in the rectangularregion On the top and bottom boundaries of the rectangularregion the horizontal partial derivatives 120597119909120597120585 and 120597119910120597120585

can be evaluated using (14)ndash(17c) Instead of using the BIEMdirectly high-accuracy vertical derivatives 120597119909120597120578 and 120597119910120597120578

can be evaluated by applying the Cauchy-Riemann equationsfor inverse transformation (6a) and (6b) Similarly on theright and left boundaries the vertical derivatives 120597119909120597120578 and120597119910120597120578 can be derived using (18)ndash(20c) and the horizontalpartial derivatives 120597119909120597120585 and 120597119910120597120585 can be subsequently

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Γ

Q

r

DP

120572

(a)Γ

P

r

Q

D

120572

(b)

Figure 1 Configuration for 119875 119876 119903 and 120572 associated with the problem domain 119863 with the bounding contour Γ (a) 119875 is enclosed by Γ (b) 119875

is located on Γ

y

x

120579

s

Pi

ri

sej

120582ei

n120595

sej+1

Q(s 120582)

eth element

Figure 2 Definition of the elementwise local coordinates s-120582 coordinates The 119909-119910 coordinates are global coordinates The 119890th element119904119895

minus 119904119895+1

is a local element with outward normal vector n and an included angle 120595 with the global 119909-119910 coordinates The local coordinate 120582 ispositive if 120582 and n are in the same direction 119903

119894is the distance from the 119894th source point 119875

119894to a boundary point 119876(119904 120582)

1205972x

1205971205852+1205972x

1205971205782= 0

1205972y

1205971205852+1205972y

1205971205782= 0

1205972120585

120597x2+

1205972120585

120597y2= 0

1205972120578

120597x2+

1205972120578

120597y2= 0

120585 = 0

120597120578

120597n= 0

120597120585

120597n= 0

120578 = 1205780

120585 = 1205850

A

B C

D

A998400

B998400 C998400

D998400

120578

120578 = 0 120585

120597120578

120597n= 0

120597120585

120597n= 0x

y

Figure 3 A sketch of the orthogonal transformation from the physical domain (119909-119910 coordinates) to the computational domain (120585-120578coordinates)

4 Mathematical Problems in Engineering

120597x

120597120585(directly solved)

120597y

120597120585(directly solved)

120597x

120597120578= minus

120597y

120597120585

120597y

120597120585= minus

120597x

120597120578

120597y

120597120578=

120597x

120597120585

120597x

120597120585=

120597y

120597120578

120597x

120597120578(directly solved)

120597y

120597120578(directly solved)120585

120578

Figure 4 An illustration for the solution of the partial derivatives on the boundaries The values of 120597119909120597120585 and 120597119910120597120585 on the top and bottomboundaries can be solved for directly using the BIEMwith the improved formulation and then the values of 120597119909120597120578 and 120597119910120597120578 can be obtainedusing theCauchy-Riemann equations Similarly the values of 120597119909120597120578 and 120597119910120597120578 on the right and left boundaries can be solved for directly usingthe BIEM with the improved formulation and then the values of 120597119909120597120585 and 120597119910120597120585 can be obtained using the Cauchy-Riemann equations

an orthogonal grid in the rectangular region by using (4)where Φ represents the coordinate of the rectangular regionThen each grid node is mapped onto the hyperrectangularregion by using (4) again where Φ represents the coordinateof the hyperrectangular region now The above evaluationof grid nodes is named grid-generation step in this paperThe regular orthogonal grid in the rectangular region is veryconvenient for applying FDM To perform FDM on the gridin the rectangular region the governing equation usually hasto transform into a formulation with Jacobian determinants[18ndash20] For example the Biharmonic equation nabla4120601

(119909119910)= 0

has to transform into nabla2

(119869(119909119910)

nabla2

120601(120585120578)

) = 0 where 119909 and 119910

are coordinates of the hyperrectangular region 120585 and 120578 arecoordinates of the rectangular region 119869

(119909119910)is the Jacobian

determinant for the forward transformationThe Jacobian determinants for the forward transforma-

tion and inverse transformation are defined respectively asbelow

119869(119909119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597120585

120597119909

120597120578

120597119909

120597120585

120597119910

120597120578

120597119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597120585

120597119909

120597120578

120597119910minus

120597120585

120597119910

120597120578

120597119909 (7a)

119869(120585120578)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597119909

120597120585

120597119910

120597120585

120597119909

120597120578

120597119910

120597120578

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597119909

120597120585

120597119910

120597120578minus

120597119909

120597120578

120597119910

120597120585 (7b)

With the help of Chain rule and these Jacobian determinantsthe relationships between the partial derivatives of 119909-119910 and120585-120578 can be built as follows

120597119909

120597120585=

1

119869(119909119910)

120597120578

120597119910 (8a)

120597119909

120597120578= minus

1

119869(119909119910)

120597120585

120597119910 (8b)

120597119910

120597120585= minus

1

119869(119909119910)

120597120578

120597119909 (8c)

120597119910

120597120578=

1

119869(119909119910)

120597120585

120597119909 (8d)

120597120585

120597119909=

1

119869(120585120578)

120597119910

120597120578 (8e)

120597120585

120597119910= minus

1

119869(120585120578)

120597119909

120597120578 (8f)

120597120578

120597119909= minus

1

119869(120585120578)

120597119910

120597120585 (8g)

120597120578

120597119910=

1

119869(120585120578)

120597119909

120597120585 (8h)

After substituting (8a)ndash(8d) into (7b) or substituting (8e)ndash(8h) into (7a) the relation between 119869

(119909119910)and 119869

(120585120578)can be

derived as follows

119869(119909119910)

119869(120585120578)

= 1 (9)

3 The Derivatives and Jacobian Determinantson the Boundaries

The Jacobian determinant is a combination of partial deriva-tives Since these derivatives are evaluated by the BIEMsingularities occur when the unknown derivatives are locatedon the boundaries A previous research has suggested thatthese boundary derivatives can be approximated by evalu-ating the inner points that are very close to the boundaries[16] In this study the derivative formulations of the BIEMare carefully investigated By applying the Cauchy-Riemannequations a method of evaluating the boundary derivativesis proposed Unlike conventional methods that just calculate

Mathematical Problems in Engineering 5

the derivative close to the boundary as succedaneums thismethod can produce accurate numerical results exactly onthe boundaries

The partial derivatives of Φ can be found by differenti-ating (4) and then the following formulations are obtained[13]

120572120597

1205971199091

Φ (119875) = intΓ

[Φ (119876)120597

1205971199091

(1

119903

120597119903

120597119899)

minus120597

1205971199091

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10a)

120572120597

1205971199092

Φ (119875) = intΓ

[Φ (119876)120597

1205971199092

(1

119903

120597119903

120597119899)

minus120597

1205971199092

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10b)

The function value and its normal derivative on the boundaryΦ(119876) and (120597120597119899)Φ(119876) are already known after the evaluationof (4) 119909

1and 119909

2represent 119909 and 119910 respectively when the

function value Φ is 120585 or 120578 Conversely 1199091and 119909

2represent 120585

and 120578 respectively when the function value Φ is 119909 or 119910 Afterthe geometry discretization the following derivative termscan be represented with the local coordinate system (119904-120582)

120597119903

120597119899=

120582

119903 (11a)

120597120582

1205971199091

= sin120595 (11b)

120597119903

1205971199091

= minus119904 cos120595 minus 120582 sin120595

119903 (11c)

120597120582

1205971199092

= minus cos120595 (11d)

120597119903

1205971199092

= minus120582 cos120595 + 119904 sin120595

119903 (11e)

By applying (11a)ndash(11e) (10a) and (10b) can be representedwith the following formulations respectively

120597

1205971199091

Φ (119875) =1

120572intΓ

[Φ sin120595

1199032

+2Φ120582 (119904 cos120595 minus 120582 sin120595)

1199034

+119904 cos120595 minus 120582 sin120595

1199032

120597Φ

120597119899] 119889120591

(12a)

120597

1205971199092

Φ (119875) =1

120572intΓ

[minusΦ cos120595

1199032

+2Φ120582 (120582 cos120595 + 119904 sin120595)

1199034

+120582 cos120595 + 119904 sin120595

1199032

120597Φ

120597119899] 119889120591

(12b)

Liggett and Liu [13] have indicated that these cannot beevaluated on the boundaries because singularities will occurHowever they are actually applicable on the boundariesin the conformal mapping process through the followinginvestigation

The discretization of these boundary integral equations isperformed by linear approximations and the potential and itsnormal derivative are subsequently approximated by

Φ =

[(Φ119895+1

minus Φ119895) 119904 + (119904

119895+1Φ119895

minus 119904119895Φ119895+1

)]

(119904119895+1

minus 119904119895)

120597Φ

120597119899=

[(120597Φ120597119899)119895+1

minus (120597Φ120597119899)119895] 119904 + [119904

119895+1(120597Φ120597119899)

119895minus 119904119895(120597Φ120597119899)

119895+1]

(119904119895+1

minus 119904119895)

(13)

for 119904119895

le 119904 le 119904119895+1

where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element Aftersubstituting (13) into (12a) and integrating each term in thelocal coordinate system (119904-120582) the discretized formulationof the integral equation for the derivative in the horizontaldirection can then be represented as

120597

1205971199091

Φ (119875)

=1

120572

119899

sum

119890=1

[119860 119861] [

Φ119895

Φ119895+1

] + [119862 119863][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(14)

where 119890 represents the 119890th discretized element on the bound-ary 119899 is the number of the elements and

119860 = minus1198681199091

11minus 1198691199091

11+ 1198701199091

11+ 119904119895+1

(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15a)

119861 = 1198681199091

11+ 1198691199091

11minus 1198701199091

11minus 119904119895(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15b)

119862 = minus1198681199091

21+ 1198691199091

21+ 119904119895+1

(1198681199091

22minus 1198691199091

22) (15c)

119863 = 1198681199091

21minus 1198691199091

21minus 119904119895(1198681199091

22minus 1198691199091

22) (15d)

in which

1198681199091

11=

sin120595

119904119895+1

minus 119904119895

times1

2ln

119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16a)

6 Mathematical Problems in Engineering

1198681199091

12=

sin120595

119904119895+1

minus 119904119895

times1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16b)

1198691199091

11=

cos120595

119904119895+1

+ 119904119895

[

120582119894119904119895

119904119895

2 + 120582119894

2minus

120582119894119904119895+1

119904119895+1

2 + 120582119894

2

+ (tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16c)

1198691199091

12=

120582119894cos120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16d)

1198701199091

11=

120582119894

2 sin120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16e)

1198701199091

12=

sin120595

119904119895+1

minus 119904119895

[minus

119904119895

119904119895

2 + 120582119894

2+

119904119895+1

119904119895+1

2 + 120582119894

2

+1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16f)

1198681199091

21=

cos120595

119904119895+1

minus 119904119895

[119904119895+1

minus 119904119895

minus 120582119894(tanminus1

119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16g)

1198681199091

22=

cos120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16h)

1198691199091

21=

120582119894sin120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16i)

1198691199091

22=

sin120595

(119904119895+1

minus 119904119895)

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16j)

During the grid-generation step when 119875 is located onthe top or bottom boundary of the transformed rectangularregion where 120582

119894= 0 singularity will occur in (16b) and

(16f) However at the same time the value of 120595 is either 0or 120587 which makes sin120595 = 0 This zero value of sin120595 makesthe singularities of (16b) and (16f) vanish Besides it shouldbe emphasized that the value of the term (tanminus1(119904

119895+1120582119894) minus

tanminus1(119904119895120582119894)) is also zero when 120582

119894= 0 [14] After substituting

120582119894= 0 sin120595 = 0 and (tanminus1(119904

119895+1120582119894) minus tanminus1(119904

119895120582119894)) = 0 into

(16a)ndash(16j) the original discretized equations are reduced tothe following formulations

1198681199091

11= 1198681199091

12= 1198691199091

11= 1198691199091

12= 1198701199091

11= 1198701199091

12= 1198691199091

21= 1198691199091

22= 0 (17a)

1198681199091

21=

1 120595 = 0

minus1 120595 = 120587

(17b)

1198681199091

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 0

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 120587

(17c)

This result not only is relatively simple but can also bedetermined except at the source point (119904

119895= 0 or 119904

119895+1=

0) Similarly (12b) the derivative formulation in the verticaldirection can be discretized to the following equation

120597

1205971199092

Φ (119875)

=1

120572

119899

sum

119890

[119864 119865] [

Φ119895

Φ119895+1

] + [119866 119867][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(18)

where

119864 = 1198681199092

11minus 1198691199092

11minus 1198701199092

11+ 119904119895+1

(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19a)

119865 = minus1198681199092

11+ 1198691199092

11+ 1198701199092

11minus 119904119895(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19b)

119866 = minus1198681199092

21minus 1198691199092

21+ 119904119895+1

(1198681199092

22+ 1198691199092

22) (19c)

119867 = 1198681199092

21+ 1198691199092

21minus 119904119895(1198691199092

22+ 1198681199092

22) (19d)

Notice that the formulation of 1198681199092

11 1198681199092

12 1198691199092

11 1198691199092

12 1198701199092

11 1198701199092

12

1198681199092

21 1198681199092

22 1198691199092

21 and 119869

1199092

22can be obtained by changing sin120595 to

cos120595 and cos120595 to sin120595 in (16a)ndash(16j) and similar reducedformulations can be derived when 119875 is located on the left orright boundary of the transformed rectangular region

1198681199092

11= 1198681199092

12= 1198691199092

11= 1198691199092

12= 1198701199092

11= 1198701199092

12= 1198691199092

21= 1198691199092

22= 0 (20a)

1198681199092

21=

1 120595 =1

2120587

minus1 120595 =3

2120587

(20b)

1198681199092

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

1

2120587

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

3

2120587

(20c)

The Jacobian determinant of the inverse transformationfrom the rectangular region to the irregular region defined as(7b) is a combination of partial derivatives in the rectangularregion On the top and bottom boundaries of the rectangularregion the horizontal partial derivatives 120597119909120597120585 and 120597119910120597120585

can be evaluated using (14)ndash(17c) Instead of using the BIEMdirectly high-accuracy vertical derivatives 120597119909120597120578 and 120597119910120597120578

can be evaluated by applying the Cauchy-Riemann equationsfor inverse transformation (6a) and (6b) Similarly on theright and left boundaries the vertical derivatives 120597119909120597120578 and120597119910120597120578 can be derived using (18)ndash(20c) and the horizontalpartial derivatives 120597119909120597120585 and 120597119910120597120585 can be subsequently

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

120597x

120597120585(directly solved)

120597y

120597120585(directly solved)

120597x

120597120578= minus

120597y

120597120585

120597y

120597120585= minus

120597x

120597120578

120597y

120597120578=

120597x

120597120585

120597x

120597120585=

120597y

120597120578

120597x

120597120578(directly solved)

120597y

120597120578(directly solved)120585

120578

Figure 4 An illustration for the solution of the partial derivatives on the boundaries The values of 120597119909120597120585 and 120597119910120597120585 on the top and bottomboundaries can be solved for directly using the BIEMwith the improved formulation and then the values of 120597119909120597120578 and 120597119910120597120578 can be obtainedusing theCauchy-Riemann equations Similarly the values of 120597119909120597120578 and 120597119910120597120578 on the right and left boundaries can be solved for directly usingthe BIEM with the improved formulation and then the values of 120597119909120597120585 and 120597119910120597120585 can be obtained using the Cauchy-Riemann equations

an orthogonal grid in the rectangular region by using (4)where Φ represents the coordinate of the rectangular regionThen each grid node is mapped onto the hyperrectangularregion by using (4) again where Φ represents the coordinateof the hyperrectangular region now The above evaluationof grid nodes is named grid-generation step in this paperThe regular orthogonal grid in the rectangular region is veryconvenient for applying FDM To perform FDM on the gridin the rectangular region the governing equation usually hasto transform into a formulation with Jacobian determinants[18ndash20] For example the Biharmonic equation nabla4120601

(119909119910)= 0

has to transform into nabla2

(119869(119909119910)

nabla2

120601(120585120578)

) = 0 where 119909 and 119910

are coordinates of the hyperrectangular region 120585 and 120578 arecoordinates of the rectangular region 119869

(119909119910)is the Jacobian

determinant for the forward transformationThe Jacobian determinants for the forward transforma-

tion and inverse transformation are defined respectively asbelow

119869(119909119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597120585

120597119909

120597120578

120597119909

120597120585

120597119910

120597120578

120597119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597120585

120597119909

120597120578

120597119910minus

120597120585

120597119910

120597120578

120597119909 (7a)

119869(120585120578)

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597119909

120597120585

120597119910

120597120585

120597119909

120597120578

120597119910

120597120578

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=120597119909

120597120585

120597119910

120597120578minus

120597119909

120597120578

120597119910

120597120585 (7b)

With the help of Chain rule and these Jacobian determinantsthe relationships between the partial derivatives of 119909-119910 and120585-120578 can be built as follows

120597119909

120597120585=

1

119869(119909119910)

120597120578

120597119910 (8a)

120597119909

120597120578= minus

1

119869(119909119910)

120597120585

120597119910 (8b)

120597119910

120597120585= minus

1

119869(119909119910)

120597120578

120597119909 (8c)

120597119910

120597120578=

1

119869(119909119910)

120597120585

120597119909 (8d)

120597120585

120597119909=

1

119869(120585120578)

120597119910

120597120578 (8e)

120597120585

120597119910= minus

1

119869(120585120578)

120597119909

120597120578 (8f)

120597120578

120597119909= minus

1

119869(120585120578)

120597119910

120597120585 (8g)

120597120578

120597119910=

1

119869(120585120578)

120597119909

120597120585 (8h)

After substituting (8a)ndash(8d) into (7b) or substituting (8e)ndash(8h) into (7a) the relation between 119869

(119909119910)and 119869

(120585120578)can be

derived as follows

119869(119909119910)

119869(120585120578)

= 1 (9)

3 The Derivatives and Jacobian Determinantson the Boundaries

The Jacobian determinant is a combination of partial deriva-tives Since these derivatives are evaluated by the BIEMsingularities occur when the unknown derivatives are locatedon the boundaries A previous research has suggested thatthese boundary derivatives can be approximated by evalu-ating the inner points that are very close to the boundaries[16] In this study the derivative formulations of the BIEMare carefully investigated By applying the Cauchy-Riemannequations a method of evaluating the boundary derivativesis proposed Unlike conventional methods that just calculate

Mathematical Problems in Engineering 5

the derivative close to the boundary as succedaneums thismethod can produce accurate numerical results exactly onthe boundaries

The partial derivatives of Φ can be found by differenti-ating (4) and then the following formulations are obtained[13]

120572120597

1205971199091

Φ (119875) = intΓ

[Φ (119876)120597

1205971199091

(1

119903

120597119903

120597119899)

minus120597

1205971199091

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10a)

120572120597

1205971199092

Φ (119875) = intΓ

[Φ (119876)120597

1205971199092

(1

119903

120597119903

120597119899)

minus120597

1205971199092

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10b)

The function value and its normal derivative on the boundaryΦ(119876) and (120597120597119899)Φ(119876) are already known after the evaluationof (4) 119909

1and 119909

2represent 119909 and 119910 respectively when the

function value Φ is 120585 or 120578 Conversely 1199091and 119909

2represent 120585

and 120578 respectively when the function value Φ is 119909 or 119910 Afterthe geometry discretization the following derivative termscan be represented with the local coordinate system (119904-120582)

120597119903

120597119899=

120582

119903 (11a)

120597120582

1205971199091

= sin120595 (11b)

120597119903

1205971199091

= minus119904 cos120595 minus 120582 sin120595

119903 (11c)

120597120582

1205971199092

= minus cos120595 (11d)

120597119903

1205971199092

= minus120582 cos120595 + 119904 sin120595

119903 (11e)

By applying (11a)ndash(11e) (10a) and (10b) can be representedwith the following formulations respectively

120597

1205971199091

Φ (119875) =1

120572intΓ

[Φ sin120595

1199032

+2Φ120582 (119904 cos120595 minus 120582 sin120595)

1199034

+119904 cos120595 minus 120582 sin120595

1199032

120597Φ

120597119899] 119889120591

(12a)

120597

1205971199092

Φ (119875) =1

120572intΓ

[minusΦ cos120595

1199032

+2Φ120582 (120582 cos120595 + 119904 sin120595)

1199034

+120582 cos120595 + 119904 sin120595

1199032

120597Φ

120597119899] 119889120591

(12b)

Liggett and Liu [13] have indicated that these cannot beevaluated on the boundaries because singularities will occurHowever they are actually applicable on the boundariesin the conformal mapping process through the followinginvestigation

The discretization of these boundary integral equations isperformed by linear approximations and the potential and itsnormal derivative are subsequently approximated by

Φ =

[(Φ119895+1

minus Φ119895) 119904 + (119904

119895+1Φ119895

minus 119904119895Φ119895+1

)]

(119904119895+1

minus 119904119895)

120597Φ

120597119899=

[(120597Φ120597119899)119895+1

minus (120597Φ120597119899)119895] 119904 + [119904

119895+1(120597Φ120597119899)

119895minus 119904119895(120597Φ120597119899)

119895+1]

(119904119895+1

minus 119904119895)

(13)

for 119904119895

le 119904 le 119904119895+1

where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element Aftersubstituting (13) into (12a) and integrating each term in thelocal coordinate system (119904-120582) the discretized formulationof the integral equation for the derivative in the horizontaldirection can then be represented as

120597

1205971199091

Φ (119875)

=1

120572

119899

sum

119890=1

[119860 119861] [

Φ119895

Φ119895+1

] + [119862 119863][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(14)

where 119890 represents the 119890th discretized element on the bound-ary 119899 is the number of the elements and

119860 = minus1198681199091

11minus 1198691199091

11+ 1198701199091

11+ 119904119895+1

(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15a)

119861 = 1198681199091

11+ 1198691199091

11minus 1198701199091

11minus 119904119895(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15b)

119862 = minus1198681199091

21+ 1198691199091

21+ 119904119895+1

(1198681199091

22minus 1198691199091

22) (15c)

119863 = 1198681199091

21minus 1198691199091

21minus 119904119895(1198681199091

22minus 1198691199091

22) (15d)

in which

1198681199091

11=

sin120595

119904119895+1

minus 119904119895

times1

2ln

119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16a)

6 Mathematical Problems in Engineering

1198681199091

12=

sin120595

119904119895+1

minus 119904119895

times1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16b)

1198691199091

11=

cos120595

119904119895+1

+ 119904119895

[

120582119894119904119895

119904119895

2 + 120582119894

2minus

120582119894119904119895+1

119904119895+1

2 + 120582119894

2

+ (tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16c)

1198691199091

12=

120582119894cos120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16d)

1198701199091

11=

120582119894

2 sin120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16e)

1198701199091

12=

sin120595

119904119895+1

minus 119904119895

[minus

119904119895

119904119895

2 + 120582119894

2+

119904119895+1

119904119895+1

2 + 120582119894

2

+1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16f)

1198681199091

21=

cos120595

119904119895+1

minus 119904119895

[119904119895+1

minus 119904119895

minus 120582119894(tanminus1

119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16g)

1198681199091

22=

cos120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16h)

1198691199091

21=

120582119894sin120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16i)

1198691199091

22=

sin120595

(119904119895+1

minus 119904119895)

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16j)

During the grid-generation step when 119875 is located onthe top or bottom boundary of the transformed rectangularregion where 120582

119894= 0 singularity will occur in (16b) and

(16f) However at the same time the value of 120595 is either 0or 120587 which makes sin120595 = 0 This zero value of sin120595 makesthe singularities of (16b) and (16f) vanish Besides it shouldbe emphasized that the value of the term (tanminus1(119904

119895+1120582119894) minus

tanminus1(119904119895120582119894)) is also zero when 120582

119894= 0 [14] After substituting

120582119894= 0 sin120595 = 0 and (tanminus1(119904

119895+1120582119894) minus tanminus1(119904

119895120582119894)) = 0 into

(16a)ndash(16j) the original discretized equations are reduced tothe following formulations

1198681199091

11= 1198681199091

12= 1198691199091

11= 1198691199091

12= 1198701199091

11= 1198701199091

12= 1198691199091

21= 1198691199091

22= 0 (17a)

1198681199091

21=

1 120595 = 0

minus1 120595 = 120587

(17b)

1198681199091

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 0

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 120587

(17c)

This result not only is relatively simple but can also bedetermined except at the source point (119904

119895= 0 or 119904

119895+1=

0) Similarly (12b) the derivative formulation in the verticaldirection can be discretized to the following equation

120597

1205971199092

Φ (119875)

=1

120572

119899

sum

119890

[119864 119865] [

Φ119895

Φ119895+1

] + [119866 119867][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(18)

where

119864 = 1198681199092

11minus 1198691199092

11minus 1198701199092

11+ 119904119895+1

(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19a)

119865 = minus1198681199092

11+ 1198691199092

11+ 1198701199092

11minus 119904119895(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19b)

119866 = minus1198681199092

21minus 1198691199092

21+ 119904119895+1

(1198681199092

22+ 1198691199092

22) (19c)

119867 = 1198681199092

21+ 1198691199092

21minus 119904119895(1198691199092

22+ 1198681199092

22) (19d)

Notice that the formulation of 1198681199092

11 1198681199092

12 1198691199092

11 1198691199092

12 1198701199092

11 1198701199092

12

1198681199092

21 1198681199092

22 1198691199092

21 and 119869

1199092

22can be obtained by changing sin120595 to

cos120595 and cos120595 to sin120595 in (16a)ndash(16j) and similar reducedformulations can be derived when 119875 is located on the left orright boundary of the transformed rectangular region

1198681199092

11= 1198681199092

12= 1198691199092

11= 1198691199092

12= 1198701199092

11= 1198701199092

12= 1198691199092

21= 1198691199092

22= 0 (20a)

1198681199092

21=

1 120595 =1

2120587

minus1 120595 =3

2120587

(20b)

1198681199092

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

1

2120587

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

3

2120587

(20c)

The Jacobian determinant of the inverse transformationfrom the rectangular region to the irregular region defined as(7b) is a combination of partial derivatives in the rectangularregion On the top and bottom boundaries of the rectangularregion the horizontal partial derivatives 120597119909120597120585 and 120597119910120597120585

can be evaluated using (14)ndash(17c) Instead of using the BIEMdirectly high-accuracy vertical derivatives 120597119909120597120578 and 120597119910120597120578

can be evaluated by applying the Cauchy-Riemann equationsfor inverse transformation (6a) and (6b) Similarly on theright and left boundaries the vertical derivatives 120597119909120597120578 and120597119910120597120578 can be derived using (18)ndash(20c) and the horizontalpartial derivatives 120597119909120597120585 and 120597119910120597120585 can be subsequently

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

the derivative close to the boundary as succedaneums thismethod can produce accurate numerical results exactly onthe boundaries

The partial derivatives of Φ can be found by differenti-ating (4) and then the following formulations are obtained[13]

120572120597

1205971199091

Φ (119875) = intΓ

[Φ (119876)120597

1205971199091

(1

119903

120597119903

120597119899)

minus120597

1205971199091

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10a)

120572120597

1205971199092

Φ (119875) = intΓ

[Φ (119876)120597

1205971199092

(1

119903

120597119903

120597119899)

minus120597

1205971199092

(ln 119903)120597

120597119899Φ (119876)] 119889120591

(10b)

The function value and its normal derivative on the boundaryΦ(119876) and (120597120597119899)Φ(119876) are already known after the evaluationof (4) 119909

1and 119909

2represent 119909 and 119910 respectively when the

function value Φ is 120585 or 120578 Conversely 1199091and 119909

2represent 120585

and 120578 respectively when the function value Φ is 119909 or 119910 Afterthe geometry discretization the following derivative termscan be represented with the local coordinate system (119904-120582)

120597119903

120597119899=

120582

119903 (11a)

120597120582

1205971199091

= sin120595 (11b)

120597119903

1205971199091

= minus119904 cos120595 minus 120582 sin120595

119903 (11c)

120597120582

1205971199092

= minus cos120595 (11d)

120597119903

1205971199092

= minus120582 cos120595 + 119904 sin120595

119903 (11e)

By applying (11a)ndash(11e) (10a) and (10b) can be representedwith the following formulations respectively

120597

1205971199091

Φ (119875) =1

120572intΓ

[Φ sin120595

1199032

+2Φ120582 (119904 cos120595 minus 120582 sin120595)

1199034

+119904 cos120595 minus 120582 sin120595

1199032

120597Φ

120597119899] 119889120591

(12a)

120597

1205971199092

Φ (119875) =1

120572intΓ

[minusΦ cos120595

1199032

+2Φ120582 (120582 cos120595 + 119904 sin120595)

1199034

+120582 cos120595 + 119904 sin120595

1199032

120597Φ

120597119899] 119889120591

(12b)

Liggett and Liu [13] have indicated that these cannot beevaluated on the boundaries because singularities will occurHowever they are actually applicable on the boundariesin the conformal mapping process through the followinginvestigation

The discretization of these boundary integral equations isperformed by linear approximations and the potential and itsnormal derivative are subsequently approximated by

Φ =

[(Φ119895+1

minus Φ119895) 119904 + (119904

119895+1Φ119895

minus 119904119895Φ119895+1

)]

(119904119895+1

minus 119904119895)

120597Φ

120597119899=

[(120597Φ120597119899)119895+1

minus (120597Φ120597119899)119895] 119904 + [119904

119895+1(120597Φ120597119899)

119895minus 119904119895(120597Φ120597119899)

119895+1]

(119904119895+1

minus 119904119895)

(13)

for 119904119895

le 119904 le 119904119895+1

where the subscripts 119895 and 119895 + 1

represent the starting and ending nodes of an element Aftersubstituting (13) into (12a) and integrating each term in thelocal coordinate system (119904-120582) the discretized formulationof the integral equation for the derivative in the horizontaldirection can then be represented as

120597

1205971199091

Φ (119875)

=1

120572

119899

sum

119890=1

[119860 119861] [

Φ119895

Φ119895+1

] + [119862 119863][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(14)

where 119890 represents the 119890th discretized element on the bound-ary 119899 is the number of the elements and

119860 = minus1198681199091

11minus 1198691199091

11+ 1198701199091

11+ 119904119895+1

(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15a)

119861 = 1198681199091

11+ 1198691199091

11minus 1198701199091

11minus 119904119895(1198681199091

12+ 1198691199091

12minus 1198701199091

12) (15b)

119862 = minus1198681199091

21+ 1198691199091

21+ 119904119895+1

(1198681199091

22minus 1198691199091

22) (15c)

119863 = 1198681199091

21minus 1198691199091

21minus 119904119895(1198681199091

22minus 1198691199091

22) (15d)

in which

1198681199091

11=

sin120595

119904119895+1

minus 119904119895

times1

2ln

119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16a)

6 Mathematical Problems in Engineering

1198681199091

12=

sin120595

119904119895+1

minus 119904119895

times1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16b)

1198691199091

11=

cos120595

119904119895+1

+ 119904119895

[

120582119894119904119895

119904119895

2 + 120582119894

2minus

120582119894119904119895+1

119904119895+1

2 + 120582119894

2

+ (tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16c)

1198691199091

12=

120582119894cos120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16d)

1198701199091

11=

120582119894

2 sin120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16e)

1198701199091

12=

sin120595

119904119895+1

minus 119904119895

[minus

119904119895

119904119895

2 + 120582119894

2+

119904119895+1

119904119895+1

2 + 120582119894

2

+1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16f)

1198681199091

21=

cos120595

119904119895+1

minus 119904119895

[119904119895+1

minus 119904119895

minus 120582119894(tanminus1

119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16g)

1198681199091

22=

cos120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16h)

1198691199091

21=

120582119894sin120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16i)

1198691199091

22=

sin120595

(119904119895+1

minus 119904119895)

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16j)

During the grid-generation step when 119875 is located onthe top or bottom boundary of the transformed rectangularregion where 120582

119894= 0 singularity will occur in (16b) and

(16f) However at the same time the value of 120595 is either 0or 120587 which makes sin120595 = 0 This zero value of sin120595 makesthe singularities of (16b) and (16f) vanish Besides it shouldbe emphasized that the value of the term (tanminus1(119904

119895+1120582119894) minus

tanminus1(119904119895120582119894)) is also zero when 120582

119894= 0 [14] After substituting

120582119894= 0 sin120595 = 0 and (tanminus1(119904

119895+1120582119894) minus tanminus1(119904

119895120582119894)) = 0 into

(16a)ndash(16j) the original discretized equations are reduced tothe following formulations

1198681199091

11= 1198681199091

12= 1198691199091

11= 1198691199091

12= 1198701199091

11= 1198701199091

12= 1198691199091

21= 1198691199091

22= 0 (17a)

1198681199091

21=

1 120595 = 0

minus1 120595 = 120587

(17b)

1198681199091

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 0

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 120587

(17c)

This result not only is relatively simple but can also bedetermined except at the source point (119904

119895= 0 or 119904

119895+1=

0) Similarly (12b) the derivative formulation in the verticaldirection can be discretized to the following equation

120597

1205971199092

Φ (119875)

=1

120572

119899

sum

119890

[119864 119865] [

Φ119895

Φ119895+1

] + [119866 119867][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(18)

where

119864 = 1198681199092

11minus 1198691199092

11minus 1198701199092

11+ 119904119895+1

(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19a)

119865 = minus1198681199092

11+ 1198691199092

11+ 1198701199092

11minus 119904119895(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19b)

119866 = minus1198681199092

21minus 1198691199092

21+ 119904119895+1

(1198681199092

22+ 1198691199092

22) (19c)

119867 = 1198681199092

21+ 1198691199092

21minus 119904119895(1198691199092

22+ 1198681199092

22) (19d)

Notice that the formulation of 1198681199092

11 1198681199092

12 1198691199092

11 1198691199092

12 1198701199092

11 1198701199092

12

1198681199092

21 1198681199092

22 1198691199092

21 and 119869

1199092

22can be obtained by changing sin120595 to

cos120595 and cos120595 to sin120595 in (16a)ndash(16j) and similar reducedformulations can be derived when 119875 is located on the left orright boundary of the transformed rectangular region

1198681199092

11= 1198681199092

12= 1198691199092

11= 1198691199092

12= 1198701199092

11= 1198701199092

12= 1198691199092

21= 1198691199092

22= 0 (20a)

1198681199092

21=

1 120595 =1

2120587

minus1 120595 =3

2120587

(20b)

1198681199092

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

1

2120587

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

3

2120587

(20c)

The Jacobian determinant of the inverse transformationfrom the rectangular region to the irregular region defined as(7b) is a combination of partial derivatives in the rectangularregion On the top and bottom boundaries of the rectangularregion the horizontal partial derivatives 120597119909120597120585 and 120597119910120597120585

can be evaluated using (14)ndash(17c) Instead of using the BIEMdirectly high-accuracy vertical derivatives 120597119909120597120578 and 120597119910120597120578

can be evaluated by applying the Cauchy-Riemann equationsfor inverse transformation (6a) and (6b) Similarly on theright and left boundaries the vertical derivatives 120597119909120597120578 and120597119910120597120578 can be derived using (18)ndash(20c) and the horizontalpartial derivatives 120597119909120597120585 and 120597119910120597120585 can be subsequently

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

1198681199091

12=

sin120595

119904119895+1

minus 119904119895

times1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16b)

1198691199091

11=

cos120595

119904119895+1

+ 119904119895

[

120582119894119904119895

119904119895

2 + 120582119894

2minus

120582119894119904119895+1

119904119895+1

2 + 120582119894

2

+ (tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16c)

1198691199091

12=

120582119894cos120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16d)

1198701199091

11=

120582119894

2 sin120595

119904119895+1

minus 119904119895

[1

119904119895

2 + 120582119894

2minus

1

119904119895+1

2 + 120582119894

2] (16e)

1198701199091

12=

sin120595

119904119895+1

minus 119904119895

[minus

119904119895

119904119895

2 + 120582119894

2+

119904119895+1

119904119895+1

2 + 120582119894

2

+1

120582119894

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16f)

1198681199091

21=

cos120595

119904119895+1

minus 119904119895

[119904119895+1

minus 119904119895

minus 120582119894(tanminus1

119904119895+1

120582119894

minus tanminus1119904119895

120582119894

)]

(16g)

1198681199091

22=

cos120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16h)

1198691199091

21=

120582119894sin120595

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

+ 120582119894

2

119904119895

2 + 120582119894

2 (16i)

1198691199091

22=

sin120595

(119904119895+1

minus 119904119895)

(tanminus1119904119895+1

120582119894

minus tanminus1119904119895

120582119894

) (16j)

During the grid-generation step when 119875 is located onthe top or bottom boundary of the transformed rectangularregion where 120582

119894= 0 singularity will occur in (16b) and

(16f) However at the same time the value of 120595 is either 0or 120587 which makes sin120595 = 0 This zero value of sin120595 makesthe singularities of (16b) and (16f) vanish Besides it shouldbe emphasized that the value of the term (tanminus1(119904

119895+1120582119894) minus

tanminus1(119904119895120582119894)) is also zero when 120582

119894= 0 [14] After substituting

120582119894= 0 sin120595 = 0 and (tanminus1(119904

119895+1120582119894) minus tanminus1(119904

119895120582119894)) = 0 into

(16a)ndash(16j) the original discretized equations are reduced tothe following formulations

1198681199091

11= 1198681199091

12= 1198691199091

11= 1198691199091

12= 1198701199091

11= 1198701199091

12= 1198691199091

21= 1198691199091

22= 0 (17a)

1198681199091

21=

1 120595 = 0

minus1 120595 = 120587

(17b)

1198681199091

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 0

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 = 120587

(17c)

This result not only is relatively simple but can also bedetermined except at the source point (119904

119895= 0 or 119904

119895+1=

0) Similarly (12b) the derivative formulation in the verticaldirection can be discretized to the following equation

120597

1205971199092

Φ (119875)

=1

120572

119899

sum

119890

[119864 119865] [

Φ119895

Φ119895+1

] + [119866 119867][[[

[

(120597Φ

120597119899)

119895

(120597Φ

120597119899)

119895+1

]]]

]

(18)

where

119864 = 1198681199092

11minus 1198691199092

11minus 1198701199092

11+ 119904119895+1

(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19a)

119865 = minus1198681199092

11+ 1198691199092

11+ 1198701199092

11minus 119904119895(minus1198681199092

12+ 1198691199092

12+ 1198701199092

12) (19b)

119866 = minus1198681199092

21minus 1198691199092

21+ 119904119895+1

(1198681199092

22+ 1198691199092

22) (19c)

119867 = 1198681199092

21+ 1198691199092

21minus 119904119895(1198691199092

22+ 1198681199092

22) (19d)

Notice that the formulation of 1198681199092

11 1198681199092

12 1198691199092

11 1198691199092

12 1198701199092

11 1198701199092

12

1198681199092

21 1198681199092

22 1198691199092

21 and 119869

1199092

22can be obtained by changing sin120595 to

cos120595 and cos120595 to sin120595 in (16a)ndash(16j) and similar reducedformulations can be derived when 119875 is located on the left orright boundary of the transformed rectangular region

1198681199092

11= 1198681199092

12= 1198691199092

11= 1198691199092

12= 1198701199092

11= 1198701199092

12= 1198691199092

21= 1198691199092

22= 0 (20a)

1198681199092

21=

1 120595 =1

2120587

minus1 120595 =3

2120587

(20b)

1198681199092

22=

1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

1

2120587

minus1

2 (119904119895+1

minus 119904119895)

ln119904119895+1

2

119904119895

2 120595 =

3

2120587

(20c)

The Jacobian determinant of the inverse transformationfrom the rectangular region to the irregular region defined as(7b) is a combination of partial derivatives in the rectangularregion On the top and bottom boundaries of the rectangularregion the horizontal partial derivatives 120597119909120597120585 and 120597119910120597120585

can be evaluated using (14)ndash(17c) Instead of using the BIEMdirectly high-accuracy vertical derivatives 120597119909120597120578 and 120597119910120597120578

can be evaluated by applying the Cauchy-Riemann equationsfor inverse transformation (6a) and (6b) Similarly on theright and left boundaries the vertical derivatives 120597119909120597120578 and120597119910120597120578 can be derived using (18)ndash(20c) and the horizontalpartial derivatives 120597119909120597120585 and 120597119910120597120585 can be subsequently

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

x

y

A1

A2

A3

O1

O2

O3

Figure 5 The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from ahyperrectangular region onto a rectangular region

Present approach pathApproach path by Tsay and Hsu [16]

= sin(minus05)cosh(y) minus cos(minus05)sinh(y)

120601(x 05) = cosh(05)cos(x) + sinh(05)sin(x)

= minussin(05)cosh(y) + cos(05)sinh(y)

120601(x minus05) = cosh(minus05)cos(x) + sinh(minus05)sin(x)

(00 05)

x

y

120597120601

120597n (05y)=

120597120601

120597x (05y)||

120597120601

120597n (minus05y)= minus

120597120601

120597x (minus05y)||

Figure 6 The unit square with complicated boundary conditions In order to evaluate the derivative on a source point this researchapproached the point along the boundary while Tsay and Hsu [16] approached the point from inside the boundary

evaluated by applying the Cauchy-Riemann equations Anillustration of this solution is provided in Figure 4Once thesepartial derivatives on the boundaries are evaluated using thereduced formulations high-accuracy Jacobian determinantsof the inverse transformation 119869

(120585120578) can be constructed

directly on the boundaries using (7b) After constructing119869(120585120578)

the derivatives of the forward transformation 120597120585120597119909120597120585120597119910 120597120578120597119909 and 120597120578120597119910 can be obtained using (8e)ndash(8h)Finally the Jacobian determinant of the forward transforma-tion 119869

(119909119910) can be constructed by using (7a)

Taking advantage of the transformed rectangular geome-trymost singular terms in the conventional derivative formu-lations vanish and the derivatives can then be evaluated onthe boundaries Furthermore most terms in the reduced for-mulations are equal to zero which implies far fewer computa-tional errors than using conventional formulations Although

the present reduced formulations still cannot evaluate thederivatives at the source points adopting approximationswith the present formulations can yield accurate results Anillustrated example is provided in Section 5

4 Length-to-Width Ratio ofConformal Modulus

To perform conformal mapping from a hyperrectangularregion (119909-119910) to a rectangular region (120585-120578) a specific ratio oflength to width have to be fit on the rectangular region Aniterative method has been presented by Seidl and Klose [15]In this section we propose a numerical integral method tofind this ratio as follows

Let the width of the rectangular region (1205780) be equal to a

suitable value Then 120597120578120597119909 120597120578120597119910 and 119869(119909119910)

can be obtained

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Boundary derivative approximation of a source pointTsay and Hsu [16]Present

Erro

r (

)1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1Eminus006

1Eminus009

1Eminus012

1Eminus005

1Eminus010

1Eminus011

1Eminus007

1Eminus008

Approach distance

Figure 7 The percentage errors of the boundary derivatives near the source point (00 05) The errors were significantly reduced by usingthe present scheme

120582 of Ej

Ei

C998400

Ej

C

Figure 8 An illustration of the corner singularity While evaluating the derivative of C1015840 the small distance 120582 to 119864119895will become a singularity

source

by solving nabla2

120578 = 0 and (10a) and (10b) The correspondinglength (120585

0) satisfying the Cauchy-Riemann equations can be

evaluated by the following integral equation

1205850

= int 119889120585 = int120597120585

120597119909119889119909 + int

120597120585

120597119910119889119910 (21)

where 120597120585120597119909 = 120597120578120597119910 and 120597120585120597119910 = minus120597120578120597119909 By usingdifferencemethod (21) was discretized to the following equa-tion

1205850

=

119873

sum

119896=1

(1205851003816100381610038161003816119896+1 minus 120585

1003816100381610038161003816119896)

=

119873

sum

119896=1

[120597120578

120597119910

10038161003816100381610038161003816100381610038161003816119896lowast(119909|119896+1

minus 119909|119896)]

+

119873

sum

119896=1

[minus120597120578

120597119909

10038161003816100381610038161003816100381610038161003816119896lowast(119910

1003816100381610038161003816119896+1 minus 1199101003816100381610038161003816119896)]

(22)

where the subscript 119896 represents the 119896th node on the integralpath the subscript 119896

lowast means a middle location between 119896

and 119896 + 1 119873 is the number of the nodes The numerical

integration is performed from the side mapped onto 120585 =

0 to the other side mapped onto 120585 = 1205850 Theoretically

the integration is independent of integral path Since thenumerical error is unavoidable several integral paths arechosen in the hyperrectangular region (119909-119910) to reduce thenumerical error for example 997888997888997888997888O

1A1 997888997888997888997888O2A2 and 997888997888997888997888O

3A3 in

Figure 5 Then perform integration using (22) respectivelyAfter the integrations are completed the average of 120585

0s from

these integral paths 1205850 is used as the length of the rectangular

region that fitted Cauchy-Riemann equations

5 Results and Discussion

This research proposes a numerical integral method tocalculate the length-to-with ratio of a rectangular regionwhen a conformalmapping is performed from a hyperrectan-gular region to the rectangular region Besides an improvedscheme to directly solve for the partial derivatives on theboundaries when conformal mapping is performed usingthe BIEM is also proposed in this research By applying theproperties of rectangular geometry the reduced formulationsof BIEM discretization (17a)ndash(17c) and (20a)ndash(20c) signifi-cantly improved the accuracy of the calculation results

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Rout

Rin

Rpath

120579

(a)

120578

1205780

120585

1205850A998400

B998400

C998400

D998400

(b)

Figure 9 Conformal mapping from (a) an arc region to (b) a rectangular region where 120579 is from 1205874 to 31205874 119877out = 2 119877in = 1 1205780

= 1 and1205850

= 226618 119877path is the integral path equal to 13 14 15 16 and 17 in this case

120585 0

226640

226636

226632

226628

226624

226620

226616

Numerical integral methodAnalytical solution

100 200 300 400 5000N

(a)

9297E minus 006

9298E minus 006

9299E minus 006

120590 9300E minus 006

9301E minus 006

9302E minus 006

9303E minus 006

100 200 300 400 5000N

(b)

Figure 10 (a) The average length of rectangular region 1205850 obtained by using the numerical integral method proposed in this research (b)

The standard deviation of 1205850s from different integral paths

In order to validate the improved boundary-derivative-solving scheme three examples are provided in this sectionthe first is a transformation from a unit-square region intoanother unit-square region the second is a transformationfrom an arc region into a rectangular region the third is atransformation from a wave-block region (one side is a wavecurve and other sides are straight lines) into a rectangularregion

51 Unit-Square Transformation The case of unit-squaretransformation and the error estimates at a source pointare introduced first When the calculation point is locatedat a source point neither the conventional nor the present

numerical scheme can provide the derivative However sincethe governing equation is the Laplace equation the analyticalsolution should be continuous and smooth These propertiesmake it reasonable to approach the derivative at the sourcepoint using a numerical result from a nearby point Tsay andHsu [16] suggested that an approach distance of 10minus6 from theinner domain be used to perform the derivative calculationwhile in this research the approximation approach along theboundaries is used To assess which approach can provideresults that are more accurate a unit-square region withtwo complicated Dirichlet and two Neumann boundaryconditions is considered The Dirichlet boundaries were seton the top and bottom and the Neumann boundaries were

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Erro

r (

)

1E minus 005

1E minus 004

1E minus 003

1E minus 002

100 200 300 400 5000N

Figure 11The percentage errors of the average length of rectangularregion 120585

0 As the discretized elements of the integral path are equal

to 500 the numerical error is negligible

set on the right and left as shown in Figure 6 The analyticalsolution of the potential value and its partial derivative in the119909-direction are

120601 = cos (119909) cosh (119910) + sin (119909) sinh (119910)

120597120601

120597119909= minus sin (119909) cosh (119910) + cos (119909) sinh (119910)

(23)

In order to capture the complicated analytical solution100 elements were applied on each side of the computationalregion and then a series of computations in the derivativewith respect to 119909 near the source point (00 05) wereperformed Figure 7 provides the percentage errors betweenthe analytical solutions at the source point (00 05) andthe numerical results near the source point Figure 7 showsthat the percentage errors from the approximations made byTsay and Hsu and in the present study both decrease as thecalculation point approaches the source point However thepresent study generates significantly fewer errors In additionwhen the approach distance is shorter than 10minus12 Tsay andHsursquos numerical result becomes singular whereas the presentscheme still provides an accurate result

Unfortunately a numerical result cannot be achieved atthe corners using this boundary approach The reason forthis is as follows Either (17a)ndash(17c) or (20a)ndash(20c) can beapplied to overcome the singularity that occurswhen the localcoordinate 120582 is equal to 0 for an element on one side close to acorner However the local coordinate 120582 of the element on theadjacent side is very small while the original formulations in(16a)ndash(16j) are applied This substitution of a small 120582 in theoriginal formulations results in a singularity

For example in order to calculate the derivative value oncorner C in Figure 8 the calculation should be performed asmall distance away along the boundary at C1015840 because thecorner is a source point When the derivative calculation is

performed using (18) and (19a)ndash(19d) the reduced formu-lations in (20a)ndash(20c) are applied to solve the 120582 singularityon the boundary elements on the left-hand side 119864

119894 Original

formulations in (16a)ndash(16j) are applied on the boundaryelements at the bottom 119864

119895 The small distance 120582 of 119864

119895

results in another numerical singularity To overcome thisproblem the conventional forward and backward differenceschemes are suggested although they cannot provide thesame accuracy as the present scheme does at other points

52 Arc-Region Transformation The second example is atransformation from an arc region into a rectangular regionas shown in Figures 9(a) and 9(b) where 120579 is from 1205874 to31205874 119877out = 2 119877in = 1 and 120578

0= 1 The remaining

unknown 1205850can be determined by the numerical integral

method proposed in this research using 5 integral paths with119877path = 13 14 15 16 and 17 respectively as the dash linesshown in Figure 9(a)The development of 120585

0with the number

of discretized elements of the integral path 119873 is shown inFigure 10(a) As 119873 is increasing 120585

0approaches the analytical

solution in (24a)ndash(24h) The standard deviation of these 1205850s

from different integral paths is observed to be independentof 119873 when 119873 is greater than 200 as shown in Figure 10(b)The error evaluations shown in Figure 11 illustrate that thepercentage errors decrease with 119873 When 119873 is equal to500 the percentage error is only 155 times 10

minus5() Sincethe analytical solution is available in this case conformalmapping was performed using the analytical 120585

0= 120587 ln 4 =

226618 which derived from (24g) To perform the conformalmapping with BIEM either inner or outer curve boundaryof the arc region is discretized into 200 elements and eachstraight boundary is discretized into 100 elements By usingthe governing equations and boundary conditions for theforward transformations shown in Figures 12(a) and 12(b)AB BC CD and DA sections in the arc region are mappedonto AB1015840 BC1015840 CD1015840 and DA1015840 sections in the rectangularregion respectively Then the inverse transformation isperformed with the governing equations nabla2119909 = 0 and nabla

2

119910 =

0 and all the Dirichlet type boundary conditions After theconformal mapping is completed a 13 times 5 regular grid isbuilt in the rectangular region and then mapped onto the arcregion as shown in Figures 13(a) and 13(b)

The numerical boundary derivatives and Jacobian deter-minants were examined with their analytical values Theanalytical solutions of the forward transformation and therelated derivatives are listed below

120585 = minus119880 (tanminus1119910

119909minus

3

4120587) (24a)

120578 = 119880 lnradic1199092 + 1199102 + 119881 (24b)

120597120585

120597119909= 119880

119910

1199092 + 1199102 (24c)

120597120585

120597119910= minus119880

119909

1199092 + 1199102 (24d)

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

A

C

D

B

120585 = 0 120585 = 1205850

nabla2120585 = 0

120597120585

120597n= 0

120597120585

120597n= 0

(a)A

B C

D

nabla2120578 = 0

120578 = 0

120578 = 1205780

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 12 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

1

15

Y 2

25

3

minus1 minus05 05 10X

(a)

0

05

120578 1

15

2

05 1 15 20120585

(b)

Figure 13 Orthogonal 13 times 5 grids generated by using BIEM (a) arc region (b) rectangular region

120597120578

120597119909= 119880

119909

1199092 + 1199102 (24e)

120597120578

120597119910= 119880

119910

1199092 + 1199102 (24f)

where

119880 =2

1205871205850

=1205780

ln (119877out119877in) (24g)

119881 = minus1205780

ln (119877out119877in)ln119877in (24h)

Then the analytical solutions of the inverse transformationand related derivatives are as follows

119909 = 119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25a)

119910 = 119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25b)

120597119909

120597120585=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25c)

120597119909

120597120578=

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25d)

120597119910

120597120585= minus

1

119880119890(120578minus119881)119880 cos(minus

120585

119880+

3

4120587) (25e)

120597119910

120597120578=

1

119880119890(120578minus119881)119880 sin(minus

120585

119880+

3

4120587) (25f)

Finally the analytical Jacobian determinants can be derivedusing above equations and shown as below

119869(120585120578)

=1

11988021198902((120578minus119881)119880)

=1199092

+ 1199102

1198802=

1

119869(119909119910)

(26)

which agrees with (9) Figures 14(a)ndash14(d) respectivelycompare the errors of 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 on theboundaries evaluated using the scheme suggested byTsay andHsu [16] and the scheme provided in the present researchThe horizontal axis represents the node indices clockwise

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(c)

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(d)

Figure 14 Comparison of the percentage errors of (a) 120597119909120597120585 (b) 120597119909120597120578 (c) 120597119910120597120585 and (d) 120597119910120597120578 between the present results and those fromTsay and Hsu [16] After adopting the present numerical scheme the derivatives on the boundary are much more accurate

along the boundary excluding the corners The vertical axisrepresents the relative errors 120576 defined as

120576 () =

1003816100381610038161003816120594 minus 1205741003816100381610038161003816

120574times 100 (27)

where 120574 represents analytical solutions and 120594 representsnumerical solutions Although 120576 from Tsay and Hsursquos schemeis small it is observed to be further smaller than in the presentscheme The errors of 119869

(120585120578)and 119869(119909119910)

on the boundaries by

using the two methods are compared in Figures 15(a) and15(b) Since 119869

(119909119910)has a direct relation with 119869

(120585120578) as shown in

(9) and (26) the trends of error distribution of 119869(120585120578)

and 119869(119909119910)

are similar As can be seen the present numerical schemeprovides more accurate results To quantify the improvedaccuracy an average error was defined as 120576 = sum

119873

119894120576119894119873 where

120576119894is the percentage error () at each boundary node and 119873

is the number of the boundary nodes excluding the cornersAlthough the scheme of Tsay and Hsu [16] provides accurate

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

120576(

)

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(a)120576

()

1E minus 006

1E minus 005

1E minus 004

1E minus 003

1E minus 002

1E minus 001

1E + 000

1E + 001

1E + 002

10 20 30 400Node index

Tsay and Hsu [16]Present

(b)

Figure 15 Comparison of the percentage errors of (a) 119869(120585120578)

and (b) 119869(119909119910)

between the present results and those from Tsay and Hsu [16] Afteradopting the present numerical scheme the transformed Jacobian determinants on the boundary are much more accurate

A

B C

D

3

2

2

1

1

0

0x

y

minus2 minus1

(a)

A D

B C

120578

1205780

120585

1205850

(b)

Figure 16 Conformal mapping from (a) a wave-block region to (b) a rectangular region where 119877 = 1 1205780

= 3 1205850

= 4966196 The dash linein (a) is the numerical integral path for evaluating 120585

0

results with 120576 = 0074 in both 119869(120585120578)

and 119869(119909119910)

the averageerror farther sharply decreases to 120576 = 000068 after usingthe present scheme

53Wave-Block Region Transformation The third example isa transformation from a wave-block region into a rectangularregion as shown in Figures 16(a) and 16(b) where 119877 = 1 and1205780

= 3 1205850was obtained by the numerical integral method

since the analytical 1205850does not exist To perform the numer-

ical integral method 6 straight lines are chosen as integralpaths as the dash lines in Figure 16(a) Figures 17(a) and17(b) respectively show the convergence of 120585

0and its stand-

ard deviation 120590 with the discrete elements using the numer-ical integral method As the number of the discretized

elements is equal to 25600 1205850was convergent to 4966196

used in the conformal mapping process 120590 was observed todecrease with the number of the discretized elements Ortho-gonality of the grid is a criterion to examine whether 120585

0

approached the analytical 1205850 The following relative errors

quantify the grid orthogonality [17]

AREo = radicsum119898

119894=1(nabla120585 sdot nabla120578)

2

119894

sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

MREo = radicmax (nabla120585 sdot nabla120578)

2

119894 119894 = 1 sim 119898

((1119898) sum119898

119894=1(1003816100381610038161003816nabla120585

1003816100381610038161003816

2 1003816100381610038161003816nabla1205781003816100381610038161003816

2

)119894

)

(28)

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

120585 0

10000 20000 300000N

4952

4956

4960

4964

4968

(a)

0E + 000

120590

2E minus 004

4E minus 004

6E minus 004

8E minus 004

10000 20000 300000N

(b)

Figure 17 (a) The average length of the rectangular region transformed from the wave-block region (b) The standard deviation of 1205850s from

different integral paths

4952 4956 496 4964 4968

1205850

0E + 000

1E minus 004

2E minus 004

3E minus 004

4E minus 004

5E minus 004

ARE

o

AREo = minus003936 lowast 1205850 + 019546

(a)

4952 4956 496 4964 4968

1205850

0E + 000

4E minus 003

8E minus 003

1E minus 002

2E minus 002

2E minus 002

MRE

o

MREo = minus156293 lowast 1205850 + 776214

(b)

Figure 18 The relative errors of orthogonality (a) AREo and (b) MREo The two errors are observed to linearly decrease with 1205850

The smaller the values of AREo andMREo are the higher theorthogonality of the grid will be A linear relation is observedbetween not only AREo and 120585

0 but also MREo and 120585

0 as

shown in Figure 18 When 1205850is equal to 4966196 AREo and

MREo are equal to 814 times 10minus6 and 323 times 10minus4 respectivelyThe small values of AREo and MREo indicate that 120585

0is very

close to the analytical valueTo perform the conformal mapping with BIEM the

boundary is discretized into 1400 elements where AB andCD sections are discretized into 300 elements and AD andBC sections are discretized into 400 elements AB BC CDand DA sections of the wave-block region are mapped onto

A1015840B1015840 B1015840C1015840 C1015840D1015840 andD1015840A1015840 sections of the rectangular regionrespectively In the forward transformation the governingequations and boundary conditions are set as in Figure 19In the backward transformation the governing equations arenabla2

119909 = 0 and nabla2

119910 = 0 with all the Dirichlet type boundaryconditions After the conformal mapping was completed a50 times 30 regular grid is built in the rectangular region andthenmapped onto thewave-block region as shown in Figures20(a) and 20(b)

The boundary derivatives of the conformal mappingare evaluated using the improved scheme proposed in thisresearch Since there is no analytical solution in this case aconventional numerical method FDM is used to examine

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

A

B C

D

3

2

2

1

1

0

0

y

x

120585 = 0 120585 = 1205850nabla2120585 = 0

minus2 minus1

120597120585

120597n= 0

120597120585

120597n= 0

(a)

A

B C

D

3

2

1

0

ynabla2120578 = 0

120578 = 1205780

120578 = 0

210

x

minus2 minus1

120597120578

120597n= 0

120597120578

120597n= 0

(b)

Figure 19 The governing equations and boundary conditions of (a) 120585 transformation and (b) 120578 transformation

0

1

2Y

3

minus2 minus1 1 20X

(a)

321 40120585

0

1

2120578

3

4

(b)

Figure 20 Orthogonal 50 times 30 grids generated by using BIEM (a) wave-block region (b) rectangular region

the correctness of the proposed scheme It is applied in therectangular region (120585-120578 coordinates) to obtain the boundaryderivatives 120597119909120597120585 120597119909120597120578 120597119910120597120585 and 120597119910120597120578 Then (8e)ndash(8h)are used to obtain the boundary derivatives 120597120585120597119909 120597120585120597119910120597120578120597119909 and 120597120578120597119910 To increase the accuracy the second-orderformulations of FDM are used as shown in the following

Φ1015840

119894=

minus3Φ0

119886+ 4Φ+Δ

119886minus Φ+2Δ

119886

2Δ+ 1198742

(forward difference formulation)

Φ1015840

119894=

3Φ0

119886minus 4ΦminusΔ

119886+ Φminus2Δ

119886

2Δ+ 1198742

(backward difference formulation)

Φ1015840

119894=

Φ+05Δ

119886minus Φminus05Δ

119886

Δ+ 1198742

(central difference formulation)

(29)

where Δ = 001 in this case the subscript 119886 representsthe 119886th point to evaluate its derivative Φ

plusmn119887Δ

119894represents

the function value (119909 or 119910) from the 119886th point plus orminus 119887Δ distance Since the boundary-unknown solving stephas been finished any function value can be obtained byusing (4) explicitly without constraint on the grid nodesThese difference formulations are applied either along orperpendicular to the boundaries The positions of Φ are allset within the rectangular region

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Mathematical Problems in Engineering

PresentFDM

minus1E minus 003

minus5E minus 004

120597y120597120585

0E + 000

5E minus 004

1E minus 003

1 2 30120578

(a)

PresentFDM

minus12

minus08

minus04

0

120597y120597120585

04

08

12

1 2 3 4 50120585

(b)

PresentFDM

1 2 30120578

minus1E minus 003

minus5E minus 004

0E + 000

120597y120597120585

5E minus 004

1E minus 003

(c)

PresentFDM

1 2 3 4 50120585

minus4E minus 007

minus2E minus 007

0120597y120597120585

2E minus 007

4E minus 007

6E minus 007

(d)

Figure 21The derivative 120597120578120597119909 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

The derivatives of the conformal mapping 120597120578120597119909 and120597120578120597119910 on the boundaries of the wave-block region evaluatedby the two numerical methods are shown in Figures 21(a)ndash21(d) and 22(a)ndash22(d) respectively The results of 120597120578120597119910 areobserved to be consistent on each section of the boundarieswhile 120597120578120597119909 are observed with some difference on AB CDand DA sections Because the boundary conditions enforce120597120578120597119909 = 0 on AB CD and DA sections the nonzero valueson these sections are numerical errors Although the numer-ical errors of present scheme are observed to be higher than

FDM on DA section their absolute values are constrainedbelow a small value of about 3 times 10

minus7 On the other hand theabsolute values of numerical errors of FDM are observed tobe about 5 times 10

minus4 on AB and CD sections much higher thanthe errors of the present scheme Similar results take place on120597120585120597119909 and 120597120585120597119910 which are not shown in this paper for ele-gance Despite the tiny numerical errors the boundary deri-vatives are observed to be consistent by using the pro-posed scheme and FDM These consistent results demon-strate that the proposed scheme can be applied to irregular

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 17

1 2 30

PresentFDM

120578

08

1

12

120597y120597120578

14

16

18

2

(a)

4 51 2 30120585

PresentFDM

04

08

12

120597y120597120578

16

2

24

(b)

PresentFDM

08

1

12

14

120597y120597120578

16

18

2

1 2 30120578

(c)

PresentFDM

078

079

08

120597y120597120578

081

082

083

1 2 3 4 50120585

(d)

Figure 22The derivative 120597120578120597119910 on (a) AB (b) BC (c) CD and (d) DA sections of the wave-block boundariesThe results obtained by usingthe proposed scheme and FDM are consistent on each section

regions that have no analytical solutions for the conformalmapping Since the improved boundary derivative formula-tions proposed in this research are coupled with the Cauchy-Riemann equations their accuracy are affected by the correct-ness of the length-to-width ratio of the rectangular regionThe consistent results of the boundary derivatives validate notonly the proposed boundary derivative formulations but alsothe integral method for the length-to-width ratio

6 Conclusions

A numerical integral method to find the length-to-widthratio of the rectangular region that fitted Cauchy-Riemann

equations is proposed in this research An arc exampledemonstrates that the numerical results can approach theanalytical solution with negligible error (155 times 10

minus5)A wave-block example shows that by using the length-to-width ratio produced by the proposed integral method theconformal mapping can perform successfully

By applying the geometric property of the transformedrectangular region and the Cauchy-Riemann equations thederivatives and Jacobian determinant of the numerical con-formal mapping method developed by Tsay and Hsu [16] canbe evaluated on the boundaries and more accurate resultsare obtained The approach for calculating the Jacobian

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

18 Mathematical Problems in Engineering

determinant at the source points adopted in the presentnumerical scheme can produce numerical results that aremore accurate than the conventional approach Although thederivatives and the Jacobian determinant at the four cornersof the transformed rectangular region cannot be improvedby the present scheme this research was able to improvethe accuracy of the numerical conformal mapping on theboundaries and makes the mapping method more reliablewhen solving problems that are sensitive to accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chuin-Shan Chen would like to acknowledge the grantsupport from the Ministry of Science and Technology inTaiwan

References

[1] D B Spalding ldquoA novel finite difference formulation for differ-ential expressions involving both first and second derivativesrdquoInternational Journal for Numerical Methods in Engineering vol4 no 4 pp 551ndash559 1972

[2] A Nicholls and B Honig ldquoA rapid finite difference algo-rithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equationrdquo Journal of Computational Chemistry vol12 no 4 pp 435ndash445 1991

[3] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[4] S G Rubin and P K Khosla ldquoPolynomial interpolation meth-ods for viscous flow calculationsrdquo Journal of ComputationalPhysics vol 24 no 3 pp 217ndash244 1977

[5] A R Mitchell and D F Griffiths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[6] J FThompson Z UWarsi and CWMastin ldquoBoundary-fittedcoordinate systems for numerical solution of partial differentialequationsmdasha reviewrdquo Journal of Computational Physics vol 47no 1 pp 1ndash108 1982

[7] P R Eiseman ldquoGrid generation for fluid mechanics computa-tionsrdquo Annual Review of Fluid Mechanics vol 17 pp 487ndash5221985

[8] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[9] AKhamayseh andCWMastin ldquoSurface grid generation basedon elliptic PDE modelsrdquo Applied Mathematics and Computa-tion vol 65 no 1ndash3 pp 253ndash264 1994

[10] Y Zhang Y Jia and S S Y Wang ldquo2D nearly orthogonalmesh generationrdquo International Journal for Numerical Methodsin Fluids vol 46 no 7 pp 685ndash707 2004

[11] J F Thompson Z U A Warsi and C W Mastin NumericalGrid Generation Foundation and Applications North-HollandAmsterdam The Netherlands 1985

[12] T-K Tsay B A Ebersole and P L F Liu ldquoNumericalmodellingof wave propagation using parabolic approximation with aboundary-fitted co-ordinate systemrdquo International Journal forNumerical Methods in Engineering vol 27 no 1 pp 37ndash55 1989

[13] J A Liggett and P L-F Liu The Boundary Integral ElementMethod for Porous Media Flow Allen amp Unwin London UK1983

[14] J Wang and T-K Tsay ldquoAnalytical evaluation and applicationof the singularities in boundary element methodrdquo EngineeringAnalysis with Boundary Elements vol 29 no 3 pp 241ndash2562005

[15] A Seidl and H Klose ldquoNumerical conformal mapping of atowel-shaped region onto a rectanglerdquo SIAM Journal on Scienti-fic and Statistical Computing vol 6 no 4 pp 833ndash842 1985

[16] T-K Tsay and F-S Hsu ldquoNumerical grid generation of anirregular regionrdquo International Journal for Numerical Methodsin Engineering vol 40 no 2 pp 343ndash356 1997

[17] N-J Wu T-K Tsay T-C Yang and H-Y Chang ldquoOrthogonalgrid generation of an irregular region using a local polynomialcollocationmethodrdquo Journal of Computational Physics vol 243pp 58ndash73 2013

[18] S T Grilli J Skourup and I A Svendsen ldquoAn efficient bound-ary element method for nonlinear water wavesrdquo EngineeringAnalysis with Boundary Elements vol 6 no 2 pp 97ndash107 1989

[19] R P Shaw and G D Manolis ldquoGeneralized Helmholtz equa-tion fundamental solution using a conformal mapping anddependent variable transformationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 177ndash188 2000

[20] G D Manolis and R P Shaw ldquoFundamental solutions for vari-able density two-dimensional elastodynamic problemsrdquo Engi-neering Analysis with Boundary Elements vol 24 no 10 pp739ndash750 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of