a linear analytical boundary element method 2d
TRANSCRIPT
-
7/27/2019 A linear analytical boundary element method 2D
1/14
Computers & Geosciences 28 (2002) 679692
A linear analytical boundary element method (BEM) for 2D
homogeneous potential problems$
J .urgen Friedrich*
Department of Geodesy & Photogrammetry, Karadeniz Technical University, 61080 Trabzon, Turkey
Received 24 August 2000; received in revised form 18 May 2001; accepted 31 May 2001
Abstract
The solution of potential problems is not only fundamental for geosciences, but also an essential part of related
subjects like electro- and fluid-mechanics. In all fields, solution algorithms are needed that should be as accurate as
possible, robust, simple to program, easy to use, fast and small in computer memory. An ideal technique to fulfill these
criteria is the boundary element method (BEM) which applies Greens identities to transform volume integrals into
boundary integrals. This work describes a linear analytical BEM for 2D homogeneous potential problems that is more
robust and precise than numerical methods because it avoids numerical schemes and coordinate transformations. After
deriving the solution algorithm, the introduced approach is tested against different benchmarks. Finally, the gained
method was incorporated into an existing software program described before in this journal by the same author.
r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: 2D homogeneous potential problems; Boundary element method; Linear analytical solution; Gravity potential; Geoid
determination
1. Introduction
Since its beginnings in the 1960s, the boundary
element method (BEM) has become a wellestablished
numerical technique which provides an efficient alter-
native to the finite difference and finite element method
for solving a variety of engineering problems (Brebbia,
1978; Banerjee, 1994). The classical BEM considered inthis work requires a fundamental solution to the
governing differential equation (here the Laplace equa-
tion) in order to obtain an equivalent boundary integral
equation. Regarding homogeneous potential problems,
BEMs have the following advantages (Brebbia and
Dominguez, 1989).
* The BEM is a boundary-only integral technique for
potential problems which does not require the ex/
interior of the problem volume to be discretized. The
Laplace equation can be solved just on the boundary,
then called boundary solution, which reduces theproblem dimension by one, thus requiring less
unknowns and therefore saving memory space and
CPU time.* The boundary solution is followed by the ex/internal
solution for ex/interior points where their location
and number can be chosen as wanted, again
minimizing computer space and time.* Both the boundary and ex/internal solution provide
exact results to the governing Laplace equation for
external and internal problems because the corre-
sponding BEM integral equation can be analytically
solved for linear elements applied in this work. The
$Code available from server at http://www.iamg.org/
CEEditor/index htm.
*Corresponding author. Tel.: 90-462-325-7977; fax: 90-462-
325-7977.
E-mail address:[email protected] (J. Friedrich).
0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 8 - 3 0 0 4 ( 0 1 ) 0 0 0 9 3 - 0
-
7/27/2019 A linear analytical boundary element method 2D
2/14
only BEM error for potential problems originates
from discretizing the boundary.
In conclusion, the BEM is the ideal technique for
solving homogeneous potential problems with precise
results while minimizing computer space and time in
combination with robustness, easy implementation, anduser-friendliness, fulfilling all criteria mentioned before.
This work concentrates on the precision and robustness
of the proposed algorithm in comparison to numerical
methods.
2. The boundary element method (BEM) for potential
problems
2.1. The boundary value problem
The governing 2D Laplace equation for a function
u ux;y; which will be considered from now on forsimplicity reasons (the 3D formulation can be derived in
an analogous way)
Duq
2u
qx2
q2u
qy20 inV; 1
is going to be solved for a volumeVbounded by a single
surface S with given boundary conditions (u is a
continuous function with continuous first partial deri-
vatives)
u %u on S1; q %u
qn %q onS2; a %ub %q onS3; 2
where n is the outward normal to the boundary SS1,S2,S3 (Fig. 1). The bars indicate known values of
potentials u and fluxes q; and real numbers a and b aregiven.
The starting point to solve Eqs. (1) and (2) is Greens
second identity (Heiskanen and Moritz, 1985)ZV
UDV VDU dv
ZS
UqV
qn V
qU
qn
ds; 3
where Uu and VG are chosen with G fulfilling the
following differential equation
DG dxx;ym: 4
G is the so called Greens function, and dxx;ymthe 2D Dirac delta function (Greenberg, 1971), defined
through the expressionZV
Fx;ydxx;ym dv Fx;m; 5
which picks out the value of the function Fat the pointx;m: Inserting Eqs. (1) and (4) into (3) gives
ux;y
ZS
Gqu
qn
qG
qn u
ds; 6
which is the basic BEM equation for potential problems
to compute both the boundary and the ex/internal
solution for potentials ux;y and fluxes qx;y; at apoint with coordinates x;y: In two dimensions, G isgiven by (Brebbia, 1978)
Gr 1
bln r; b 2p; r
ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffixx2 ym2
q :
7
The 3D Greens function can be analogously derived,
yielding (Banerjee, 1994)
Gr 1
br; b 4p; r
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffixx2 ym2 zz2
q :
8
Note that Eqs. (7) and (8) express the effect of a source
point of unit strength located at x;m; x on a field pointat position x;y; z: Inserting Eq. (7) into (6) yieldsGreens third identity for internal 2D Laplace problems
for a field point P inside the volume Vb 2p
b ux;y Z
s
lnr quqn
winir2
u
ds;
b
2p for P inside Volume;
p for P on Surface;
0 for P outside Volume;
8>:
qG
qn
wini
br2; wi xx;ymi; ni nx; nyi; 9
which can be rewritten as follows:
ux;y Z
SHu ds
ZS
Gq ds; Hq
Gqn : 10
For external problems, the sign of the right-hand side
of Eq. (9) is flipped and the value of b reversed,
if ni continues to be the outward normal which is
the case in this work, resulting in b 2p for P
outside V; and b 0 for P inside V (Heiskanen andMoritz, 1985).
2.2. The solution algorithm for the linear analytical BEM
For evaluating Eq. (10), the volumes surface is
discretized by boundary nodes connected by linearFig. 1. Definition sketch of single bounded volume.
J. Friedrich / Computers & Geosciences 28 (2002) 679692680
-
7/27/2019 A linear analytical boundary element method 2D
3/14
boundary elements of length ln as shown in Fig. 2
(Brebbia, 1978).
Thus, all boundary-related parameters, e.g. co-
ordinates, potentials and fluxes, can be linearly
interpolated by
xZ I1xn I2xn1; yZ I1yn I2Yn1;
uZ I1un I2un1; qZ I1qn I2qn1;11
where Z is a dimensionless parameter varying be-
tween 1 and +1, (xn;yn), (xn1;yn1) thenode coordinates of a linear boundary element
En; (un; qn), (un1; qn1) the end point potentialsand fluxes, and I1; I2 linear interpolation (shape)functions defined by
I1 1 Z=2; I2 1 Z=2; 1oZo1: 12
Applying Eqs. (11) and (12) to Eq. (10) and
approximating the boundary integrals by a summationover all linear boundary elements En; n f1; 2;y;Ng;results in
ux;y XNn1
H1n un H2n un1
XNn1
G1nqn G2nqn1;
13
where the terms G and H represent the following
integrals valid for internal problems that allow an
analytical integration, if the boundary is piecewise
planar which is the situation for the linear shape
functions used here to discretize the boundary (Camp,
1992; Klees, 1996).
G1n 1
b
ZEn
I1 lnr ds; G2n
1
b
ZEn
I2 ln r ds;
H1n 1
b
ZEn
I1wini
r2 ds; H2n
1
b
ZEn
I2wini
r2 ds:
14
By expressingwi;the distancer and the line element dsasfunctions ofZ using Eqs. (11) and (12)
r2Z aZ2 bZc; dsln
2 dZ;
ln ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi
xn1 xn
2
yn1 yn
2q ;
a ln
2
2b
xn1 xn
2 x
xn1xn
yn1 yn
2 m
yn1 yn;
c xn1 xn
2 x
2
yn1 yn
2 m
2;
d4acb2;
15
Eq. (14) can be analytically integrated without numer-
ical schemes from Z 1 to +1, yielding
G1n ln
4b 2 b
2a 1
2 ln rn1 3ln rn ;
2acb2
8a2
b
4a
ln r2n1 ln r
2n
b3 4abc
8a2
d
4a
fn
;
G2n ln
4b
2 b
2a
1
2
3ln rn1 ln rn
2acb2
8a2
b
4a
lnr2n1 lnr
2n
b3 4abc
8a2
d
4a
fn
H1n Sn
4b 1
b
2a
fn
1
2alnr2n1 lnr
2n
H2n sn
4b 1
b
2a
fn
1
2alnr2n1 lnr
2n
16
where
rn
ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffixn x
2 yn m 2;
qrn1
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffixn1 x
2yn1m2
qsn
xn1 xn
2 x
yn1 yn
yn1 yn
2
m xn1 xn; 17Fig. 2. Boundary discretization with linear boundary elementsnx yn1 yn=ln; ny xn1 xn=ln:
J. Friedrich / Computers & Geosciences 28 (2002) 679692 681
-
7/27/2019 A linear analytical boundary element method 2D
4/14
fn 2
2ab
2
2ab ford 0;
2
ffiffiffid
p atan
2ab
ffiffiffid
p a tan
2ab
ffiffiffid
p
! for d> 0;
1ffiffiffiffiffiffiffid
p ln2ab ffiffiffiffiffiffiffidp2ab
ffiffiffiffiffiffiffid
p
ln2ab
ffiffiffiffiffiffiffid
p2ab
ffiffiffiffiffiffiffid
p!
for do0:
Eq. (17) automatically handles singular integrals for
which the determinant d is equal to zero, when a source
point is identical to a boundary node, for example xn;yn;thus avoiding numerical schemes and coordinate trans-
formations which increases robustness and precision. In
this instance, Eq. (17) uses the values ln rn ln r2n 0
and rn1 ln so that the singular integrals become
(b p)
G1n ln
4b
3
2 ln ln
; G2n
ln
4b
1
2 ln ln
;
H1n H2n
1
4b atan
yn1 yn
xn1 xna tan
yn yn1
xn xn1
;
18
which are the normally used formulas (Brebbia and
Dominguez, 1989). Eq. (17) takes also care of ex/internal
problems with H1n ext 1 H1n
int and H2n ext
1 H2n int as the only modifications in Eq. (16).
Numerical integration, e.g. by Gaussian quadrature,
does not have these advantages and is less precise,
especially when approaching to a boundary as is shownin Section 3.
The boundary solution is based on Eq. (13) applied to
each boundary node so that every single equation can
incorporate one unknown (potential or flux) for every
node, corresponding to Greens third identity Eq. (9) for
a field point P on S; yielding a system ofN equations
XNk1
ukx;y XNn1
H1k;nunH2k;nun1
"
XN
n1
G1k;nqn G2k;nqn1
#; b p; 19
which can be rewritten in matrix form as
Hu Gq; 20
where the unknowns are on both sides. Reordering
Eq. (20) results in (Brebbia, 1978)
Ax y ) x A1y; 21
where A is a regular NN matrix, x a N 1 vector of
unknowns (potentials or fluxes), and y a given N1
vector. Eq. (21) can be solved by standard matrix
inversion formulas so that both potentials and fluxes
are known at every boundary node. This is the boundary
solution. Then the ex/internal solution for non-bound-
ary points can be directly computed from Eq. (13) which
is a discretized form of Greens third identity Eq. (9)
ux;y XNn1
G1nqn G2nqn1H
1n un H
2n un1;
b2p for P inside V
;0 for P outside V;
( 22
The fluxes of ex/internal points are given by
qx;y qux;y
qn
qux;y
@x
qx
qn
qux;y
qy
@y
@n
PN
n1
qG1nqx
qnqG2nqx
qn1
qH1nqx
un qH2nqx
un1
@x
@n
PNn1 qG1n
qy
qnqG2nqy
qn1qH1nqy
un qH2nqy
un1 @y
@n
; 23
where the analytically gained derivatives of the terms G
and H according to Eqs. (1517) can be found in the
appendix. Depending on the type of problem, internal or
external, Eqs. (22) and (23) produce a zero result for
external or internal points, respectively, allowing to
check the described solution algorithm apart from using
other benchmark tests.
3. Examples and benchmark tests
In this section, four examples and benchmark tests
(two internal, two external) will be described in order to
evaluate the proposed linear analytical BEM. For this,
the derived algorithm was incorporated in an update of
the CFDLab 1.1 for WindowsTM program (Friedrich,
1999), offering a visual interactive user interface that
allows simple and fast pre-processing, computations,
and post-processing to solve these type of problems.
3.1. Uniform potential flow through two parallel walls
The first example and benchmark test deals with
uniform potential flow through two parallel and
horizontal walls (no flux condition at top and bottom
boundary), and a potential difference of +1.0 between
the left and right boundary. Every side of the square in
Fig. 3 has a length of 1.0 (Prasuhn, 1980).
The proposed algorithm works with four boundary
elements Enn 4 and boundary nodes with x;ycoordinates P1x;y 0; 0; P2x;y 1; 0; P3x;y=(1,1), P4x;y 0; 1: After the boundary solution,which automatically delivers the correct solution on
the boundary, the internal solution can be obtained
J. Friedrich / Computers & Geosciences 28 (2002) 679692682
-
7/27/2019 A linear analytical boundary element method 2D
5/14
which gives exactly zero for potentials and fluxes at allpoints outside of the square (the problem volume)
according to Eqs. (22) and (23). The correct solution for
internal potential and flux values is given by
ux;y 1 x; qxx;y qux;y
qx 1;
qyx;y qux;y
qy 0: 24
The internal potentials linearly decrease from one to zero
when moving from the left to the right boundary (Fig. 4a),
whereas the internal fluxes are constant everywhere in the
volume, resulting in a uniform flow to the right as plotted
in Fig. 4b.
The results of CFDLab and the proposed algorithm
are labeled with the tag Linear Analytically BEM
(LABEM) and compared to the correct solution and
four other boundary element methods (Brebbia and
Dominguez, 1989):
* Constant Analytically BEM (CABEM),* Constant Numerically BEM (CNBEM) using 4-
point Gaussian quadrature formulas (GQF),* Linear Numerically BEM (LNBEM) with 6- point
GQF, and* Quadratic Numerically BEM (QNBEM) with 10-
point GQF, where four more boundary nodes are
placed in the center of each boundary element En in
Fig. 3.
The outcome is printed in Table 1. where the internal
points IPi; i f1; 2;y; 5g; are moved from the squarecenter along its diagonal to the left bottom corner in
Fig. 3.
As can be seen from this table, LABEM gives always
the correct results independent from the distance to a
boundary element or node, whereas all other methods
become unstable and inaccurate for a distance
rZ=lno
0:1 when a field point is moved closer to a
boundary. Thus, BEM techniques that approximate the
involved boundary integrals by numerical schemes like
Gaussian quadrature become unreliable when approach-
ing a boundary, whereby fluxes of field points are more
affected than their potentials because the flux singularity
is of order 1=rZ smaller than the singularity of
potentials according to Eqs. (22) and (23).
3.2. Warping function of an elliptical cross-section
The second example of an internal potential problem
considers the warping function ux;y as part of atorsion problem for an elliptical cross-section x2=a2
y2=b2 1 with a semimajor and -minor axis of a 2and b 1 (Brebbia and Dominguez, 1989). The
analytical solution is given by
ux;y b2 a2xy
a
2 b
2 25
and the normal derivatives by
qu
qn
a2 b2xyffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffia4y2 b4x2
p : 26Due its symmetry, only the first quarter of the problem
domain needs to be discretized, using Eq. (25) as
boundary condition on both the coordinate axis
(potential=u=0), and Eq. (26) as a non linear boundary
condition on the elliptical section with a total number of
12 boundary nodes (Fig. 5).
The described warping problem was again solved by
using CFDLab together with the proposed algorithm(Fig. 6a and b).
Repeating the comparison of different methods
as done in the last example, the following results
were gained (Table 2) when moving from an internal
point (x;y)=(0.5, 0.5) towards the boundary approach-ing the boundary node n=8 at (x;y)=(1.2400, 0.7846)(Fig. 5).
These results confirm the findings of the first example;
the LABEM provides more precise results compared to
the other methods except QNBEM for internal field
points not closer to the boundary than about
rZ=lno0:1; originating from a better approximationof curved boundaries and non-linear boundary condi-
tions by quadratic shape functions.
3.3. Dirichlets problem to determine a spherical earth
model
The first external problem to be considered is that
of Dirichlet or the first boundary value problem
of potential theory. If the boundary is a sphere of
radius ae; as is the situation for a spherical earthmodel, the analytical external solution in spherical
coordinates r; y; l in a geocentric, north-pole
Fig. 3. Potential flow in square with four boundary nodes and
elements (ln 1:0m).
J. Friedrich / Computers & Geosciences 28 (2002) 679692 683
-
7/27/2019 A linear analytical boundary element method 2D
6/14
oriented coordinate system is given by (Heiskanen and
Moritz, 1985)
ur; y; l kXNn0
ae
r
n1 Xnm0
Anm cos ml
Bnm sinml Pnmcos y; 27
where k is the gravity constant and Pnmcos y
the Legendres functions. The gradient (flux) of
Eq. (27) in Cartesian coordinates wi x;y; zi is
obtained by (Ilk, 1983)
qur; y; l
qwik
XNn0
an1e
Xnm0
AnmqCnm=r
n1
@wiBnm
qSnm=rn1
@wi
;
Cnm Pnmcos y cos ml; Snm Pnmcosy sin ml;28
Fig. 4. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of internal fluxes.
J. Friedrich / Computers & Geosciences 28 (2002) 679692684
-
7/27/2019 A linear analytical boundary element method 2D
7/14
yielding
where d ij is the Dirac delta tensor, and the argument of
Legendres functions is changed to Pnmsin y: Thedetermination of the constant coefficients Anm and Bnm
is based on the orthogonality relations of the spherical
harmonics (Heiskanen and Moritz, 1985):
An0 2n1
4p ZSFy; lPnmcos y ds;
Anm 2n1
2p
nm!
nm!
ZS
Fy; lPnmcos y cos ml ds
forma0;
Bnm 2n1
2p
nm!
nm!
ZS
Fy; lPnmcos y sin ml ds
forma0; 30
where Fy; l is a known function on the surface S;allowing to compute Anm and Bnm by integration of
Eq. (30). For a spherical earth model, the geoid is
approximated by a sphere of radiusaewith the following
potential as boundary condition (me and o are the
Earths mass and angular velocity)
Fy; l %ur ae; y; l 0 kme
ae
1
2o2a2e cos
2 y;31
qur; y; l=qw
1
2
3
264
375 kXN
n0
an1e2rn2
Xnm0
Anm
1 d0mnm2nm1Cn1;m1 1 d0mCn1;m1
1d0mnm2nm1Sn1;m1 1 d0mSn1;m1
2nm1Cn1;m
264
375
0B@
Bnm
1 d0mnm2nm1Sn1;m1 1 d0mSn1;m1
1 d0mnm2nm1Cn1;m1 1 d0mCn1;m1
2nm1Sn1;m
264
3751CA; 29
Table 1
Potentialsux;y;fluxesqx(x,y) andqyx;y for the potential flow problem in a square (all differences Correct x BEM are absolutevalues)
IPi Correct Correct - Correct - Correct - Correct - Correct -
xi yi solution LABEM CABEM CNBEM LNBEM QNBEM
Potentials ux;y (m2
s2
)0.5 0.5 0.50 00 0.00 00 0.00 00 0.00 07 0.00 00 0.00 00
0.1 0.1 0.90 00 0.00 00 0.09 18 0.09 66 0.01 41 0.00 27
0.01 0.01 0.99 00 0.00 00 0.15 17 0.59 12 0.30 56 0.32 01
0.001 0.001 0.99 90 0.00 00 0.16 18 0.72 03 0.72 33 0.66 96
0.0001 0.0001 0.99 99 0.00 00 0.16 32 0.73 06 0.74 96 0.25 03
Fluxes qxx;y (ms2)
0.5 0.5 1.00 00 0.00 00 0.09 44 0.09 66 0.00 00 0.00 00
0.1 0.1 1.00 00 0.00 00 0.12 64 1.26 86 0.85 79 0.10 50
0.01 0.01 1.00 00 0.00 00 3.68 76 9.91 76 30.86 51 7.36 91
0.001 0.001 1.00 00 0.00 00 39.49 56 7.50 79 15.74 17 46.60 71
0.0001 0.0001 1.00 00 0.00 00 397.59 41 7.28 38 14.41 91 36.89 69
Fluxes qyx;y (m s2)0.5 0.5 0.00 00 0.00 00 0.00 00 0.00 00 0.00 00 0.00 00
0.1 0.1 0.00 00 0.00 00 0.62 85 1.16 12 0.94 47 0.10 64
0.01 0.01 0.00 00 0.00 00 4.66 74 7.28 52 30.69 69 6.63 94
0.001 0.001 0.00 00 0.00 00 4.91 040 4.20 47 14.86 27 45.79 10
0.0001 0.0001 0.00 00 0.00 00 399.43 99 3.96 29 13.51 27 35.99 50
Fig. 5. Determination of a warping function in the first quarter
of an elliptic cross-section with 12 boundary nodes
(lmaxn 0:5m).
J. Friedrich / Computers & Geosciences 28 (2002) 679692 685
-
7/27/2019 A linear analytical boundary element method 2D
8/14
so that only zonal components are left in Eq. (30),
resulting in
A00 kme
ae
1
3o2a2e ; A20
1
3o2a2e : 32
All other coefficients are zero. Inserting Eq. (32) into
Eqs. (27) and (29) gives
ur; y kme
r
1
2
o2r2 cos2y; 33
qur; y
qwi kme
wir3
AijlojAlmnomwn; 34
where the first part is the central gravity potential and
acceleration, the second part the centrifugal potential
and acceleration, and eijk the epsilon tensor. For a 2D
problem (kme=r) and (kme i /r3) in Eqs. (33) and (34)
are replaced by (kmeln(r)) andkmewi=r2) according to
Eqs. (7) and (8). After discretizing the boundary (the
Greenwhich meridian l 0) with 36 boundary nodes
Fig. 6. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of internal fluxes.
J. Friedrich / Computers & Geosciences 28 (2002) 679692686
-
7/27/2019 A linear analytical boundary element method 2D
9/14
(Fig. 7) and using Eq. (33) as boundary condition, the
same procedure as in the previous paragraph is
analogously applied (y-components=0).
The described Dirichlet problem was solved with
CFDLab and the proposed algorithm (Fig. 8a and b),where z-components were treated as y-components
inside of CFDLab. For convenience, the problem
dimension was scaled to values between zero and 1000,
using the following parameters: ae=100.0 m, kme=
1000.0 m3 s2, o=0.01s1, ln 17:45m.The gained results for potentials and gradients were
compared with the correct values according to Eqs. (33)
and (34) and listed in Table 3 when moving from an ex-
ternal point EPi (i=1) with coordinates (r; y)=(150.0 m,30.01) towards the boundary approaching the boundary
node n=4 at (r; y)=(100.0 m, 30.01) (Fig. 7).
The figures in this table confirm the results of theother examples; LABEM provides more precise results
compared to the numerical methods which become
unsafe for external field points closer to the boundary
than about rZ=lno0:3 (E5/17), whereas the otheranalytical method CABEM offers the same precision as
LABEM, at least for the fluxes.
3.4. Third boundary value problem to determine geoidal
heights
The second external problem deals with the disturbing
potential Tx;y; z used to determine geoidal heights.
Tx;y; z; defined as the difference between the actualand the normal gravity potential (Heiskanen and
Moritz, 1985)
Tx;y; z Wx;y; z Ux;y; z; 35
satisfies Laplaces equation
DTq
2T
qx2
q2T
qy2
q2T
qz2 0; 36
and can therefore be determined by a third boundary
value problem of potential theory. In spherical approx-
Table 2
Potentialsux;y;fluxesqxx;yandqyx;yfor warping function in elliptic cross-section (all differences Correct - x BEM are absolutevalues)
IPi Correct - Correct - Correct - Correct - Correct - Correct -
xi yi solution LABEM CABEM CNBEM LNBEM QNBEM
Potentials ux;y (m2
s2
)0.5 0.5 0.30 00 0.00 20 0.00 48 0.00 48 0.00 20 0.00 02
1.2 0.7 0.50 40 0.00 76 0.02 16 0.02 14 0.00 76 0.00 30
1.23 0.77 0.56 83 0.00 70 0.02 86 0.06 27 0.01 55 0.17 13
1.239 0.783 0.58 21 0.00 63 0.03 03 0.28 53 0.23 93 0.28 81
1.2399 0.7845 0.58 36 0.00 62 0.03 24 0.31 69 0.30 73 0.30 21
Fluxes qxx;y (m s2)
0.5 0.5 0.30 00 0.00 48 0.00 91 0.00 91 0.00 48 0.00 03
1.2 0.7 0.42 00 0.00 84 0.00 23 0.04 96 0.01 56 0.11 79
1.23 0.77 0.46 20 0.00 06 0.28 20 5.37 33 1.52 85 3.02 43
1.239 0.783 0.46 98 0.01 24 3.46 26 6.94 04 15.23 14 3.13 60
1.2399 0.7845 0.47 07 0.03 42 39.02 30 6.93 84 14.87 96 3.14 84
Fluxes qyx;y (m s2)0.5 0.5 0.60 00 0.00 38 0.00 84 0.00 84 0.00 38 0.00 09
1.2 0.7 0.72 00 0.00 09 0.06 21 0.00 01 0.00 87 0.18 90
1.23 0.77 0.73 80 0.03 22 0.33 26 6.25 59 1.40 34 5.88 56
1.239 0.783 0.74 34 0.08 81 2.35 00 16.90 87 35.70 36 7.45 62
1.2399 0.7845 0.74 39 0.15 21 39.27 44 16.82 10 36.31 62 7.48 71
Fig. 7. Determination of spherical earth model on circle with
36 boundary nodes.
J. Friedrich / Computers & Geosciences 28 (2002) 679692 687
-
7/27/2019 A linear analytical boundary element method 2D
10/14
imation, the boundary values are given by
Dg qT
qn
2
aeT; ae 6:37110
6 m; 37
where Dg are the gravity anomalies assumed to be
known on the geoid and ae the radius of the sphere.
After obtaining the solution for Tx;y; z; the geoidalheights Nare computed via Bruns theorem
NT=g; g 9:798 m s2; 38
where g is the normal gravity on the sphere. A direct
formula to compute geoidal heights from gravity
anomalies is Stokes integral
N ae
4pg
ZS
DgSc ds; 39
where Sc is the Stokes function defined by
Sc 1 sin1c=2 6 sinc=2 5 cosc
3 cosc lnsinc=2 sin2c=2; 40
Fig. 8. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of gradients.
J. Friedrich / Computers & Geosciences 28 (2002) 679692688
-
7/27/2019 A linear analytical boundary element method 2D
11/14
and c is the spherical distance between a source and a
field point. Comparing Eqs. (37)(40) with (7)(10)
shows how Stokes integral is related to Greens
identities and the BEM formulation used in this work.
In other words, Stokes integral is an equivalent
expression in spherical polar coordinates for the external
BEM solution of potentials given by Eqs. (7)(10), or by
Eq. (22) after the boundary discretization for a 2D
problem. This becomes clearer when constant boundary
elements are used for which Eq. (22) simplifies to
Tx;y XNn1
GnqT
qn
n
HnTn
;
b2p forP insideV;
0 for P outsideV:
( 41
where the transformation equations
rn 2ae sinc=2; rn1 2ae sincn1=2; 42
are inserted into the following formulas for the termsGn
and Hn:
Gn 1
2G1n G
2n
ln
2b
"1 ln rn1 ln rn
b
4a
lnr2n1ln r2n
d
4a
fn#;Hn
1
2H1n H
2n
sn
2bfn: 43
In order to verify the proposed analytical algorithm with
this problem, the disturbing potential is developed in a
series of spherical harmonics using coordinates r; y; l;aswas done in the last paragraph (Heiskanen and Moritz,
1985)
Tr; y; l X
N
n0
ae
r
n1 Xnm0
%Anm cos ml
%Bnm sinmlPnmcos y: 44
The gradient of Eq. (44) is correspondingly given by
Table 3
Potentialsux;y; gradients qxx;y and qyx;y for spherical earth model (all differences Correctx BEM are absolute values)
EPi Correct - Correct - Correct - Correct - Correct - Correct -ri yi Solution LABEM CABEM CNBEM LNBEM QNBEM
Potentials ur; y (m 2 s2)150.0 30.01 5011.48 0.83 75 2.88 28 2.83 06 0.83 75 0.31 14110.0 30.01 4700.93 1.14 97 3.06 82 2.79 54 1.15 14 0.01 33105.0 30.01 4654.37 1.07 37 2.97 30 3.95 46 0.94 96 13.36 52101.0 30.0o 4615.50 0.56 86 2.45 12 112.01 61 70.53 13 1060.57 13100.1 30.01 4606.55 0.11 84 1.99 68 1834.51 60 1497.39 55 2158.32 72
Gradients qxr; y (m s2)
150.0 30.01 5.76 05 0.01 49 0.01 72 0.01 72 0.01 49 0.01 36110.0 30.01 7.86 34 0.01 91 0.02 24 0.14 18 0.01 74 0.07 73105.0 30.0o 8.23 88 0.05 41 0.05 80 3.21 93 0.29 60 12.90 04101.0 30.01 8.56 58 0.29 17 0.30 05 572.13 12 52.20 17 845. 32 85100.1 30.01 8.46 29 0.80 49 0.86 82 2845.31 80 5431.15 01 1197.53 97
Gradients qyr; y (ms2)
150.0 30.01 3.33 33 0.00 04 0.00 10 0.00 09 0.00 04 0.00 12
110.0 30.01 4.54 55 0.00 18 0.00 35 0.07 25 0.00 08 0.03 54105.0 30.01 4.76 19 0.02 17 0.02 29 1.86 93 0.16 13 7.43 84101.0 30.01 4.95 05 0.15 84 0.15 19 330.28 61 30.14 97 488.03 92100.1 30.01 4.99 50 0.45 43 0.35 48 1642.7490 3135.68 48 691.39 22
qur; y; l=qw
1
2
3
264
375 XN
n0
an1e2rn2
Xnm0
%Anm
1 d0mnm2nm1Cn1;m1 1 d0mCn1;m1
1d0mnm2nm1Sn1;m1 1 d0mSn1;m1
2nm1Cn1;m
264
375
0B@
%Bnm
1 d0mnm2nm1Sn1;m1 1 d0mSn1;m1
1 d0mnm2nm1Cn1;m1 1 d0mCn1;m1
2nm1Sn1;m
2
64
3
75
1
CA; 45
J. Friedrich / Computers & Geosciences 28 (2002) 679692 689
-
7/27/2019 A linear analytical boundary element method 2D
12/14
The problem is restricted to just zonal coefficients up to
an order ofn 2
%A00 10; %A20 %A21 %A22 300: 46
All other coefficients are zero. By applying Eqs. (44)
(46) as boundary conditions, the problem was solved
with CFDLab and the proposed algorithm (Fig. 9a and
b where y:
z), using the same boundary discretization
as in the earlier paragraph: 36 boundary nodes (Fig. 7)
with ae 6.371 106 m and lnE1.1 10
6 m.
The flux vectors in Fig. 9b are othogonal to the
geoidal surface defined by Eqs. (44)(46), displayed in
the right window of Fig. 9a. The gained results for
potentials and gradients were compared with the correct
values according to Eqs. (44) and 46) and listed in
Table 4 when moving from an external point
Fig. 9. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of gradients.
J. Friedrich / Computers & Geosciences 28 (2002) 679692690
-
7/27/2019 A linear analytical boundary element method 2D
13/14
(r; y)=(6.4 106 m, 30.01) towards the boundary ap-proaching the boundary node n=4 at
(r; y)=(6.371 106 m, 30.01) (Fig. 7).The numbers in this table affirm the results of the other
examples; the LABEM delivers more precise results
compared to the numerical methods which become
unreliable for external field points closer to the boundary
than aboutrZ=lno0:1:Here, all external field points arelocated within this limit. In this example, only the flux
precision of the other analytical method CABEM is as
good as or sometimes even better than the LABEM. This
may happen when the potential field as well as the
boundary geometry and conditions are better modeled
by constant than by linear analytical elements. Further,
relative flux errors of both analytical methods can exceed
100% in this example when approaching the boundary.
4. Conclusions
The objective of this work was to introduce a more
robust and precise algorithm for 2D homogeneous
potential problems in comparison to numerical meth-
ods. The proposed algorithm is based on analytically
integrated boundary elements with linear shape func-
tions which allow numerical schemes and coordinate
transformations to be avoided. But this method cannot
be extended to heterogeneous and non-linear problems
because in such cases, the governing equations contain
non-homogeneous parts which can only be incorporated
by means of domain integrals so that the BEM loses its
original attraction of a boundary-only method. The
dual reciprocity method (DRM) appears to be a solution
to this difficulty (Partridge et al., 1992), but it is less
accurate than analytical BEMs. The DRM uses a
fundamental solution to a simpler governing equation
and takes into account the remaining non-homogeneous
terms in the original equation by applying reciprocity
principles and certain approximating functions. This is
the reason why it is being used in the CFDLab program
for solving other type of problems like Poisson, diffusion
or convectiondiffusion ones. Due to the good results
gained so far, it is planned to extend the described
linear analytical BEM to three-dimensional potential
problems
Acknowledgements
I would like to thank the reviewers and Osman
B .orekci at Bosphorus University in Istanbul for their
helpful comments. My special thanks go to Hewlett-
Packard (HP) GmbH in Germany for their general
support of this work.
Appendix
The derivatives of the G- and H-terms according to
Eqs. (14)(17) to compute the fluxes of ex/internal
points are given by (b 2p)
Table 4
Disturbing potentials Tr; y; gradients Txr; y and Tyr; y (all differences Correctx BEM are absolute values)
EPi Correct - Correct - Correct - Correct - Correct - Correct -
ri (106 m) yi Solution LABEM CABEM CNBEM LNBEM QNBEM
Disturbing potentials Tr; y (m2 s2)
6.400 30.01 1023.27 9.11 41 361.88 89 545.49 16 26.47 09 269.98 036.380 30.01 1032.86 2.74 06 377.04 96 604.81 91 278.99 48 379.42 15
6.372 30.01 1036.73 0.28 04 383.19 18 629.79 50 464.97 26 424.97 54
6.37109 30.01 1037.17 0.02 27 383.89 33 632.67 08 488.08 74 430.18 21
6.371009 30.01 1037.21 0.00 20 383.95 57 632.92 71 490.15 48 430.64 57
Gradients Txr; y (104 m s2)
6.400 30.01 4.17 35 3.45 45 6.45 86 16.15 51 44.22 25 41.07 83
6.380 30.01 4.22 60 3.79 82 6.58 28 18.21 56 170.67 00 44.81 63
6.372 30.01 4.24 72 4.39 46 6.63 17 18.80 23 211.54 65 45.36 72
6.37109 30.01 4.24 96 5.03 05 6.63 73 18.85 87 213.62 42 45. 39 28
6.371009 30.01 4.24 98 5.71 46 6.63 78 18.86 36 213.77 75 45.39 47
Gradients Tyr; y (104 m s2)
6.400 30.0o 2.30 22 0.43 99 3.83 34 9.64 63 24.00 16 23.68 20
6.380 30.0o 2.33 12 0.29 15 3.89 86 10.83 21 96.29 91 25.81 24
6.372 30.0o 2.34 29 1.77 20 3.92 40 11.16 91 119.68 96 26.12 01
6.37109 30.01 2.34 42 3.43 08 3.92 69 11.20 15 120.87 42 26.13 37
6.371009 30.01 2.34 44 4.99 64 3.92 72 11.20 43 120.96 15 26.13 47
J. Friedrich / Computers & Geosciences 28 (2002) 679692 691
-
7/27/2019 A linear analytical boundary element method 2D
14/14
References
Banerjee, P.K., 1994. The Boundary Element Method in
Engineering. McGraw-Hill, New York, 452pp.
Brebbia, C.A., 1978. The Boundary Element Method for
Engineers. Pentech Press, London, 189pp.
Brebbia, C.A., Dominguez, J., 1989. Boundary Elements: AnIntroductory Course. McGraw-Hill, New York, 466pp.
Camp, C.C., 1992. Direct evaluation of singular boundary
integrals in two-dimensional biharmonic analysis. Interna-
tional Journal for Numerical Methods in Engineering 35,
20672078.
Friedrich, J., 1999. Object-oriented design and implementation
of CFDLab: a computer-assisted learning tool for fluid
dynamics using dual reciprocity boundary element metho-
dology. Computers & Geosciences 27 (7), 785800. The latest
version can be downloaded at http://bsttc.marhost.com.
Greenberg, M.D., 1971. Application of Greens Functions in
Science and Engineering. Prentice-Hall, Englewood Cliffs,
NJ, 236pp.
Heiskanen, W.A., Moritz, H., 1985. Physical Geodesy. Reprint
Institute of Physical Geodesy, Technical University of Graz,
Austria, 364pp.
Ilk, K.H., 1983. Ein Beitrag zur Dynamik ausgedehnter
K .orperFGravitationswechselwirkung. Deutsche Geod-
.atische Kommission Reihe C 288, M .unchen, 124pp.
Klees, R., 1996. Numerical calculation of weakly singular
surface integrals. Journal of Geodesy 70, 781797.
Partridge, P.W., Brebbia, C.A., Wrobel, L.C., 1992. The Dual
Reciprocity Boundary Element Method. Elsevier Science
Oxford, London, 280pp.
Prasuhn, A.L., 1980. Fundamentals of Fluid Mechanics.
Prentice Hall, Englewood Cliffs, NJ, 563pp.
qG1nqx
qG1nqy
qG2n
qxqG2nqy
8>>>>>>>>>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>=>>>>>>>>>>>>;
ln
4b
xn1 xn
2 x
xn x
xn1xn
2
yn1 yn
2 m
yn m
yn1yn
2
xn1 xn
2
x xn1 x xn1 xn
2
yn1 yn
2 m
yn1 m
yn1 yn
2
26666666664
37777777775
K0
K1
K2
2
64
3
75; A:1
K0fn; K1 1
2aln r2n1 ln r
2n
b
2afn; K2
2
a
b
2a2 ln r2n1 lnr
2n
b2 2ac
2a2 fn;
qH1nqx
qH1nqy
qH2nqx
@H2n@y
26666666666664
37777777777775
sn
2b
xn1 xn
2 x
xnx
xn1xn
2
yn1yn
2 m
yn m
yn1yn
2
xn1xn
2 x xn1x
xn1xn
2 yn1 yn
2 m
yn1m
yn1yn
2
26666666664
37777777775
M0
M1
M2
264
375
N0
N1
N2
N3
26664
37775; A:2
M0 2ab
dr2n1
2ab
dr2n
2a
d fn; M1
b 2c
dr2n1
b 2c
dr2n
b
d fn;
M2 b22acbc
adr2n1
b22acbc
adr2n
2c
d fn;
N0
N1
N2
N3
26664 37775 12b
yn1 yn
2
yn1yn2
xn1 xn
2 xn1 xn
2
yn1 yn
2
yn1 yn
2
xn1 xn
2
xn1xn2
2
6666666664
3
7777777775fn
12a
lnr2n1 lnr2n
b2a
fn8