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8Harmonic Mappings

8.1 Introduction 8.2 Nondegenerate Planar Grids

Two-Dimensional Regular Grids • Discrete Analog of the Jacobian Positiveness • Irregular Two-Dimensional Meshes

8.3 Planar Harmonic Grid GenerationProblem Formulation • Variational Method for Irregular Planar Mesh Smoothing

8.4 Harmonic Maps Between Surfaces. Derivation of Governing EquationsIntroductory Remarks • Theory of Harmonic Maps • Derivation of Governing Equations

8.5 Two-Dimensional Adaptive-Harmonic Structured GridsDerivation of Equations • Numerical Implementation

8.6 Two-Dimensional Adaptive-Harmonic Irregular MeshesProblem Formulation • Approximation of the Functional •Minimization of the Functional • Derivation of Computational Formulas

8.7 Adaptive-Harmonic Structured Surface Grid GenerationDerivation of Equations • Numerical Implementation

8.8 Irregular Surface MeshesProblem Formulation • Approximation of the Functional • Minimization of the Functional • Derivation of Computational Formulas

8.9 Three-Dimensional Regular GridsDerivation of Equations • Numerical Implementation

8.10 Three-Dimensional Irregular MeshesDiscrete Analog of the Jacobian Positiveness • Problem Formulation • Approximation of the Functional • Minimization of the Functional • Derivation of Computational Formulas

8.11 Results of Test ComputationsComparison Between the Winslow Method and the Variational Approach • Comparison Between the Finite-Difference Method for Two-DimensionalAdaptive-Harmonic Meshes and the Variational Approach • Comparison Between the Finite-Difference Method for Adaptive-HarmonicGrid Generation on Surfaces and the Variational Approach • Comparison Between the Finite-Difference Method for Adaptive-Harmonic Three-Dimensional Meshes and the Variational Approach

8.12 ConclusionsSergey A. Ivanenko

8.1 Introduction


Methods of grid generation based on the theory of harmonic maps are presented in this chapter.Algorithms for structured and unstructured adaptive grids in two-dimensional and three-dimensionalcases as well as for grids on surfaces are described in detail. All methods are based on grid nodes movement(r-refinement).

Two fundamental problems in grid generation are considered in the present chapter.The first problem is to find conditions for discrete mappings to the nondegenerate. The condition of

convexity of all the grid cells in two dimensions is assumed as a discrete analog of the Jacobian positiveness.It guarantees the grid to be nondegenerate. Indeed, if all grid cells are convex, then all grid nodes do notleave a domain, and such a grid does not contain self-intersecting cells. In the three-dimensional case, amore complicated analog of Jacobian positiveness is presented.

The second problem is to develop a suitable theoretical framework for grid generation. The theory ofharmonic maps has been chosen as a basis for this purpose. The problem of constructing harmoniccoordinates on the surface of the graph of control functions is formulated. Harmonic coordinates areconstructed from harmonic mapping of the surface onto a parametric square (or cube in the three-dimensional case). The projection of these coordinates onto a physical region produces an adaptive–harmonicgrid [Liseikin, 1991, 1993; Ivanenko, 1993, 1995]. The application of such monitoring surfaces was alsoconsidered by Dwyer, et al. [1982], Eiseman [1987], and Spekreijse, et al. [1996].

Two methods are used for numerical solution. The first one is based on the finite-difference approx-imation of Euler equations. The second method is based on a direct minimization of the discrete analogof the harmonic functional.

The variational approach has been extended to the case of irregular meshes [Ivanenko, 1995b]. Themain principle can be formulated as follows. Recall that harmonic coordinates are generated by the globalharmonic mapping of the physical domain or the surface of control function onto a parametric square.The result will be a regular grid. Irregular (unstructured) grids can be considered as a set of localcoordinates, different for each cell or element. Hence, each cell, for example a quadrilateral, can beharmonically mapped onto the same auxiliary unit square. The total irregular grid with fixed connectionscan be computed by minimizing the sum of harmonic functionals, written for each grid cell. This willbe a smoothing and adaption stage in the method of irregular grid generation. For triangular grids, eachtriangle should be mapped harmonically onto an equilateral triangle and so on.

A very important property of variational approaches is that the functionals are approximated in sucha way that all their discrete analogues have infinite barrier on the boundary of the set of nondegenerategrids. The resulting algorithms assure generation of nondegenerate grids according to developed discreteconditions of the Jacobian positiveness. Consequently, the theory of harmonic maps, applied to gridgeneration, can be assumed as a general framework for the development of fully automated algorithms.Moreover, as on the continuous level, the theory of harmonic maps provides construction of nondegen-erate curvilinear coordinates; on the discrete level, the developed application of this theory guaranteesgeneration of nondegenerate grids in arbitrary domains.

8.2 Nondegenerate Planar Grids

Two types of grids/meshes are used in computations: regular (structured) and irregular (unstructured).Regular grids contain only regular nodes, or nodes whose neighbors are known only from the indexation.A typical example is a curvilinear grid constructed by a mapping of a parametric square onto a physicaldomain. Grid nodes are enumerated with double indices in the two-dimensional and by triple indicesin the three-dimensional case. This is not the case of irregular meshes. For such a mesh, neighbors ofnodes must be specified. In spite of the fact that the set of regular grids is a reduction of the set ofirregular meshes, we will start with the consideration of regular grids. The condition of the Jacobianpositiveness is considered as the condition for a regular grid to be nondegenerate. An irregular mesh can


be assumed as a set of local coordinates, so the condition of the Jacobian positiveness can be used alsoto define discrete conditions for an irregular mesh to be nondegenerate.

8.2.1 Two-Dimensional Regular Grids

The problem of grid generation in two dimensions will be considered in the following formulation. Ina simply connected domain Ω on the plane x, y a grid

(8.1)

must be constructed with given coordinates of boundary nodes

(8.2)

The problem can be treated as a discrete analog of the problem of finding functions x(ξ, η) and y(ξ, η),ensuring one-to-one mapping of the parametric square

(8.3)

onto a domain Ω (see Figure 8.1) with a given transformation of the square boundary onto the boundaryof Ω , associated with the boundary conditions Eq. 8.2, i.e., on each side of the parametric square thefollowing eight functions are specified:

Instead of the parametric square Eq. 8.3 on the plane ξ, η the parametric rectangle is often introducedto simplify the computational formulas

FIGURE 8.1 Correspondence of nodes numbers for a mapping of the square cell 2+1/2, 2+1/2 in the plane ξ, ηonto a corresponding quadrilateral cell in the plane x, y.

( , ) ,...., ,...,,x y i i j ji j = =∗ ∗1 1

( . ) ( , ) ( , ) ( , ),

x y x y x y x yi i j j i j1 1 ∗ ∗

0 1 0 1< < < <ξ η

x x x x x x x x

y y y y y y y y

down up left right

down up left right

ξ ξ ξ ξ η η η η

ξ ξ ξ ξ η η η η

, , , ,

, , , ,

0 1 0 1

0 1 0 1

( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( )( ) = ( ) ( ) ( ) ( ) = ( ) ( ) = ( )

=

(8.4)

1 1< < < <∗ ∗ξ ηi j


associated with the square grid (ξi, ηj) on the plane ξ, η such that

In the paper by Bobilev, Ivanenko, and Ismailov [1996], the following theorem has been proven:THEOREM 1. If a smooth mapping of one domain onto another with a one-to-one mapping between

boundaries possesses a positive Jacobian not only inside a domain but also on its boundary, then sucha mapping will be one-to-one.

Hence, the curvilinear coordinate system constructed in a domain Ω will be nondegenerate if theJacobian of the mapping x(ξ, η), y(ξ, η) is positive:

(8.5)

Thus, the problem of constructing curvilinear coordinates in a domain Ω can be formulated as theproblem of finding of smooth mapping of a parametric square onto a domain Ω that satisfies the conditionof the Jacobian positiveness Eq. 8.5. The mapping between boundaries must be one-to-one, which canbe easily provided from the condition of monotonic variations of ξ and η along the appropriate partsof the boundary of a domain Ω .

Consequently, in the discrete case for the grid (Eq. 8.1) a discrete analog of the Jacobian positivenessmust be also applied.

8.2.2 Discrete Analog of the Jacobian Positiveness

The condition of grid cell convexity was introduced by Ivanenko and Charakhch’yan [1988] as a discreteanalog of the Jacobian positiveness. The mapping x(ξ, η), y(ξ, η) was approximated by quadrilateralfinite elements.

Let the coordinates (x, y)ij of grid nodes be given. To construct the mapping xh(ξ, η), yh(ξ, η) of theparametric rectangle Eq. 8.4 onto the domain Ω such that xh(i, j) = xi,j and yh(i, j) = yij we use quadrilateralisoparametric finite elements [Strang and Fix, 1973]. The square cell numbered i + 1/2, j + 1/2 on theplane ξ, η is mapped onto the quadrilateral cell on the plane x, y, formed by nodes with coordinates

The cell vertices are numbered from 1 to 4 in the clockwise direction as is shown in Figure 8.1. The node(i, j) corresponds to the vertex 1, node (i, j + 1) to vertex 2 and so on. Each vertex is associated with atriangle: vertex 1 with ∆412, vertex 2 with ∆123 and so on. The doubled area Jk, k = 1, 2, 3, 4, of thesetriangles is introduced as follows:

In the first expression the vertex indices are used and in the second the corresponding node indices areused. Functions xh, yh for i ≤ ξ ≤ i + 1, j ≤ η ≤ j + 1 are represented in the form

(8.6)

ξ ηi ji j i i j j= = = ∗ ∗ = 1,...,1,...,

J x y x y= − > ≤ ≤ ≤ ≤η ξ η ξ ξ η0 0 1 0 1

x y x y x y x yi j i j i j i j

, , , ,, , , ,( ) ( ) ( ) ( )+ + + +

1 1 1 1

J x x y y y y x x

x x y y y y x xi j i j i j i j i j i j i j i j

1 4 1 2 1 4 1 2 1

1 1 1 1

= −( ) −( ) − −( ) −( ) =

−( ) −( ) − −( ) −( )+ + + +, , , , , , , ,

x x x x i x x j x x x x i j

y y y y i y y j y y y y i

h

h

ξ η ξ η ξ η

ξ η ξ η ξ η

,

,

( ) = + −( ) −( ) + −( ) −( ) + − − +( ) −( ) −( )( ) = + −( ) −( ) + −( ) −( ) + − − +( ) −( ) −

1 4 1 2 1 3 4 2 1

1 4 1 2 1 3 4 2 1 jj( )

Each side of the square is linearly transformed onto the appropriate side of the quadrilateral. Conse-quently, the global transformation

x

h

,

y

h

is continuous on the cell boundaries. To check the one-to-one


property of the transformation Eq. 8.6, we write out the expression for the Jacobian

where A = x3 – x4 – x2 + x1, B = y3 – y4 – y2 + y1. Jacobian is linear, not bilinear, since the coefficientbefore ξη in this determinant is equal to zero. Consequently, if J h > 0 in all corners of the square, it doesnot vanish inside this square. In the corner 1 (ξ = i, η = j) of the cell i + 1/2, j + 1/2 the Jacobian

i.e., Jh(i, j) = J1 is the doubled area of triangle ∆412, introduced above.From this it follows that the condition of the Jacobian positiveness for the mapping xh(ξ, η), yh(ξ, η)

is equivalent to the system of inequalities

(8.7)

where Jk = (xk–1 – xk)(yk+1 – yk) – (yk–1 – yk)(xk+1 – xk), and in expressions for Jk one should put k – 1 = 4if k = 1, and k + 1 = 1 if k = 4.

If conditions Eq. 8.7 are satisfied, then all grid cells are convex quadrilaterals. Hence, if the mappingx(ξ, η), y(ξ, η) is approximated by piecewise-bilinear functions, then the one-to-one condition is equiv-alent to the condition of convexity of all grid cells Eq. 8.7. Such grids were called convex grids [Ivanenkoand Charakhch’yan, 1988], and only convex grids can be used in the finite element method with con-forming quadrilateral elements.

The set of grids satisfying inequalities Eq. 8.7 is called a convex grid set and denoted by D. This setbelongs to the Euclidean space RN, where N = 2(i* – 2)(j* – 2) is the total number of degrees of freedomof the grid equal to double the number of its internal nodes. In this space D is an open bounded set. Itsboundary ∂D is the set if grids for which at least one of the inequalities Eq. 8.7 becomes an equality.

8.2.3 Irregular Two-Dimensional Meshes

In the employment of irregular meshes we must define the correspondence between local (for eachelement) and global nodes numeration. In Figure 8.2 the simplest example of an irregular mesh is shown.Element numbers are shown in circles. The local numeration is shown only for the element 1. The globalnumeration is shown with a bold font.

The function of COR(N, k) is introduced to define a correspondence between local and global nodenumbers:

where n is a global node number, Nn is a total number of mesh nodes, N is an element number, Ne is anumber of elements, k is a local node number in the element. This function is implemented in thecomputer program as a function for a regular grid and as an array for an irregular mesh. For example,

J x y x yx x A j x x A i

y y B j y y Bh h h h h= − =

− + −( ) − + −−( )

ξ η η ξ

η ξη ξ

det( )4 1 2 1

24 1 1- + - + ( - i)

J i j x x y y y y x xh ,( ) = −( ) −( ) − −( ) −( )4 1 2 1 4 1 2 1

x y x y i jh h h hξ η η ξ ξ η− > ≤ ≤ ≤ ≤∗ ∗0 1 1

J k i i j jk i j[ ] > = = − = −+ +

∗ ∗1 2 1 2

0 1 2 3 4 1 1 1 1,

, , , ,..., ,...,

COR N k n n N N N kn e, ,..., ,... , , ,( ) = = = = 1 1 1 2 3 4


for the irregular mesh shown in Figure 8.2 the correspondence between local and global numerations isdefined as follows:

For irregular meshes the array COR is filled up during the mesh construction, for example, by a frontmethod. It is often necessary to use other correspondence functions, for example, when we must definenumbers of two elements from the number of their common edge or to define the neighbor numbersfor a given node. The choice of these functions depends on the type of elements used and on the solverpeculiarities. We will consider below only the simplest data structure, defined by COR(N, k), which isenough for our purposes.

For regular grids we can use the function with the same name instead of the array COR. It is convenientto use one-dimensional numeration instead of double indices. For node numbers of a regular grid,introduced above in Eq. 8.1, we have

where n(i, j) corresponds to the node i, j, and N(i, j) corresponds to the cell number i + 1/2, j + 1/2.Then the correspondence function is defined as follows:

Now we consider conditions for the mesh node coordinates to assure a mesh to be nondegenerate.Note, that in the case of a regular grid instead of the mapping x(ξ, η), y(ξ, η) of the parametric rectangleEq. 8.4 onto a domain Ω , a bilinear mapping of the same unit square onto each quadrilateral cell can be

FIGURE 8.2 Correspondence of nodes numbers for a mapping of the unit square in the plane ξ, η onto thequadrilateral cell 1 of irregular mesh in the plane x, y.

COR COR COR COR1 1 1 1 2 3 1 3 4 1 4 2, , , ,( ) = ( ) = ( ) = ( ) =

N i j i j i i i j j

n i j i j i i i j j

, ,..., ,...,

, ,...,

( ) = + −( ) −( ) = − = −

( ) = + −( ) =

∗ ∗ ∗

∗ ∗ ∗

1 1 1 1 1 1

1 1

= 1,...,

COR N i j n i j COR N i j n i j

COR N i j n i j COR N i j n i j

, , , , , ,

, , , , , ,

( )( ) = ( ) ( )( ) = +( )( )( ) = + +( ) ( )( ) = +( )

1 2 1

3 1 1 4 1

considered. All argumentation in Section 8.2.1 will be true in this case, since the Jacobian of the mappingxh(ξ, η), yh(ξ, η) is not changed if the square cell is shifted in the plane ξ, η. Hence, for each cell of


irregular mesh a bilinear mapping of the unit square on the plane ξ, η onto this cell can be introduced(see Figure 8.2). The condition of the Jacobian positiveness can be written as follows:

(8.8)

where Jk = (xk–1 – xk)(yk+1 – yk) – (yk–1 – yk)(xk+1 – xk) is the area of the triangle, written in local numeration.Consequently, all the mesh cells satisfying inequalities Eq. 8.8 will be convex quadrilaterals.

As in the case of regular grids, irregular meshes, satisfying inequalities Eq. 8.8 will be called convex meshes.As in the previous subsection the set of meshes, satisfying inequalities Eq. 8.8 is called a convex mesh

set and denoted by D. This set belongs to the Euclidean space RNin, where Nin is the total number ofdegrees of freedom of the mesh equal to double the number of its internal nodes. In this space D is anopen bounded set. Its boundary ∂D is the set of meshes for which at least one of the inequalities Eq. 8.8becomes an equality.

8.3 Planar Harmonic Grid Generation

Experience has shown the efficiency and the reliability of the method based on harmonic mapping,proposed by Winslow [1966]. This is consistent with the theoretical foundation of the method, since thetheory guarantees that the generated curvilinear coordinate system is nondegenerate. This propertyfollows from the general result on existence and uniqueness of the one-to-one harmonic mapping of anarbitrary domain onto a parametric square.

Development of the method suggested by Godunov and Prokopov [1972] is based on the use of suchadditional parameters that there was no loss of the one-to-one property. This approach was introducedto control the grid spacing (adaption). Further developments of this approach were presented by Thomp-son, et al. [1985].

The system of two Laplace equations is used for constructing harmonic mapping. The natural way toextend this method is to use more common elliptic equations with right-hand sides. However, in thegeneral case it is not clear how to obtain conditions on control parameters under which the generationof a nondegenerate curvilinear coordinate system (regular grid) is guaranteed.

8.3.1 Problem Formulation

The simplest and the most investigated elliptic equation is Laplace equation. That is why the system

or its direct extensions may be considered for grid generation.However, these equations cannot guarantee the generation of a nondegenerate grid. A simple example

was constructed by Prokopov [1993]. Let us consider the transformation

defined on the unit square 0 < ξ < 1, 0 < η < 1.Obviously, this transformation satisfies Laplace equations and the Jacobian

J k N Nk N e[ ] > = =0 1 2 3 4 1 , , , ,...,

x x y yξξ ηη ξξ ηη+ = + =0 0

x yξ η ξ η ξ ξ η ξη ξ η, , ,( ) = −( ) − ( ) = + −12

23

12

13

2 2

J x y x yξ η ξ ξ η ηξ η η ξ,( ) = − = −

−

+ +

23

13

12

Since J(ξ, 0) = (ξ – 2/3)(ξ – 1/3) < 0 on the interval η = 0, 1/3 < ξ < 2/3, the transformation is folded nearthe image of the lower part of the square boundary. The example is interesting because the image of the


square has a very simple form so the transform degeneration and the grid folding seems absolutelyunexpected.

The method of grid generation guaranteeing the one-to-one mapping on the continuous level wasproposed by Winslow [1966]. Two families of grid lines are constructed as contours of functions ξ(x, y),η(x, y) satisfying two Laplace equations

(8.9)

with Dirichlet boundary conditions associated with the one-to-one mapping of the boundary of para-metric square Eq. 8.4 onto the boundary of domain.

After transforming to independent variables ξ, η, these equations take the form

(8.10)

where α = x2η + y2

η, β = xξxη + yξyη, γ = x2ξ + y2

ξ.The standard approximation of Eq. 8.10 with centered differences for the first-order derivatives was

used by Winslow [1966] and Godunov and Prokopov [1972]. Computational formulas for the extensionof the method to the case of adaptive planar grids will be described in detail in the next section.

8.3.2 Variational Method for Irregular Planar Mesh Smoothing

The process of irregular mesh generation usually contains two stages. The meshes produced at the firststage by automated techniques often exhibit large variations of mesh cells. The smoothing techniquesare used then to form better shaped cells and yield more accurate analyses. Various approaches have beendeveloped, but the most promising is, in our opinion, an approach based on harmonic mappings. Forregular grids such algorithms were proposed by Yanenko, et al. [1977], Brackbill and Saltzman [1982],and Ivanenko and Charakhch’yan [1988]. In this section we will consider extension of the methodpresented in papers by Ivanenko and Charakch’yan [1988] and Ivanenko [1988], guaranteeing the con-vexity of all the grid cells to the case of irregular meshes.

The Dirichlet (harmonic) functional was considered by Brackbill and Saltzman [1982]:

(8.11)

The minimum of this functional is attained on the harmonic mapping of a domain Ω onto a parametricsquare. This functional and its generalizations have been used in many papers for regular grid generation.

The problem of irregular mesh smoothing or relaxation is formulated as follows. Let the coordinatesof irregular mesh be given:

(8.12)

The mesh is formed by quadrilateral elements, i.e., the array COR(N, k) is also defined. The problem isto find new coordinates of the mesh nodes, minimizing the sum of the functional Eq. 8.11 values,computed for a mapping of the unit square onto an each cell of a mesh.

It is clear that for a regular grid, this formulation reduces to a discrete analog of the problem toconstruct harmonic coordinates ξ and η in a domain Ω . Now we will consider the approximation of thefunctional Eq. 8.11.

ξ ξ η ηxx yy xx yy+ = + =0 0

α β γ α β γξξ ξη ηη ξξ ξη ηηx x x y y y− + = − + =2 0 2 0

Ix y x y

Jd d=

+ + +∫ ξ ξ η η ξ η

2 2 2 2

x y n Nn n, ,...,( ) = 1

The present algorithm is based on a particular approximation of the functional Eq. 8.11 whereby theminimum ensures all mesh cells to be convex quadrilaterals and guarantees no folding for the mesh. In


its implementation the peculiarity of vanishing the Jacobian when the one-to-one property is lost canbe used explicitly.

The mapping x(ξ, η), y(ξ, η) is approximated by functions xh(ξ, η), yh(ξ, η) introduced above.Substituting these expressions into Eq. 8.11 and replacing integrals over the square cell by the quadratureformulas with nodes coinciding with the square corners on the plane ξ, η, the following discrete analogcan be obtained:

(8.13)

where Fk is the integrand evaluated in the kth grid node

and Jk is the doubled area of triangle introduced above.Note that the approximation Eq. 8. 13 of the functional Eq. 8.11 can be obtained as follows. The square

cell on the plane ξ, η is divided into two triangles first by the diagonal 13, and then by 24. The mappingof the square onto a quadrilateral cell in the plane x, y is approximated by a function which is linear ineach triangle. Denote this function as before xh(ξ, η), yh(ξ, η). All derivatives in the integrand of Eq. 8.11are easy to compute, for example, for one of two triangles obtained by splitting the quadrilateral cellwith the diagonal 13 we have

The integral Eq. 8.11 over the quadrilateral cell in the plane ξ, η is approximated by half of the sum ofvalues of this integral, computed for piecewise-linear approximations on triangles, obtained for the firstand the second splittings. The result is the approximation Eq. 8.13.

The function Ih has the following property, which can be formulated as a theorem:THEOREM 2. The function Ih has an infinite barrier at the boundary of the set of convex meshes, i.e.,

if at least one of the quantities Jk tends to zero for some cell while remaining positive, then .Proof. In fact, suppose that in Eq. 8.13 for some cell, but Ih does not tend to +∞. Then the

numerator in Eq. 8.13 must also tend to zero, i.e., the lengths of two sides of the cell tend to zero. Consequently,the areas of all triangles that contain these sides must also tend to zero. Repeating the argument as manytimes as necessary, we conclude that the lengths of the sides of all grid cells, including those at the boundaryof the domain, must tend to zero, i.e., the mesh compresses into a point, which is impossible.

Thus, if the set D is not empty, the system of algebraic equations

has at least one solution that is a convex mesh. To find it, one must first find a certain initial convexmesh, and then use a method of unconstrained minimization. Since the function Eq. 8.13 has a infinitebarrier on the boundary of the set of convex meshes, each step of the method can be chosen so that themesh always remains convex.

I Fh

kN

N

k N

e

= [ ]==

∑∑ 141

4

1

F x x x x y y y y Jk k k k k k k k k k= −( ) + −( ) + −( ) + −( )[ ]+ − + −−

1

2

1

2

1

2

1

2 1

x x x y y y x x x y y y

J x x y y y y x x

h h h h

h

ξ ξ η η= − = − = − = −

+ −( ) −( ) − −( ) −( )3 2 3 2 2 1 2 1

1 2 3 2 1 2 3 2

I h +∞→Jk 0→

RI

xR

I

yx

h

ny

h

n

= = = =∂∂

∂∂

0 0

We first consider a method of minimizing the function assuming that the initial convex mesh has beenfound. Suppose the mesh at the lth step of the iterations is determined. We use the quasi-Newtonian


procedure when the (l + 1)-th step is accomplished by solving two linear equations for each interior node:

(8.14)

From this follows

(8.15)

where τ is the iteration parameter, that is chosen so that the mesh remains convex. For this purpose,after each step the conditions Eq. 8.8 are checked and if they are not satisfied, this parameter is multipliedby 0.5. Note that Eq. 8.15 is not the Newton–Raphson iteration process, because not all the secondderivatives are taken into account. The rate of convergence is low by comparison. At the same time, theNewton–Raphson method gives a much more complex system of linear equations.

Each of the derivatives in Eq. 8.15 is the sum of a proper number of terms, in accordance with thenumber of triangles containing the given node as a vertex. For example, for the irregular mesh shownin Figure 8.2, the number of such triangles for the node 3 is equal to 9. Rather than write out suchcumbersome expressions, we consider the first and second derivatives of the terms in Eq. 8.15. Arraysstoring the derivatives are first cleared, and then all mesh triangles are scanned and the appropriatederivatives are added to the relevant elements of the arrays. The use of formulas Eq. 8.15 for the boundarynode (if its position on the boundary is not fixed) should be completed by the projection of this nodeonto the boundary.

If the initial mesh is not convex, the computational formulas should be modified so that the initialgrid need not belong to the set of convex meshes [Ivanenko, 1988]. To achieve this, the quantities Jk

appearing in the expressions for Rx, Ry and in their derivatives are replaced with new quantities~Jk:

where ε > 0 is some sufficiently small quantity.It is important to choose an optimal value of ε so that the convex mesh is constructed as fast as possible.

The method used for specifying the value of ε is based on the computation of the absolute value of theaverage area of triangles with negative areas:

where Sneg is double the absolute value of the total area of triangles with negative areas, and Nneg is thenumber of these triangles. The quantity ε1 > 0 sets a lower bound on ε to avoid very large values appearingin computations. The coefficient α is chosen experimentally and is in the range 0.3 ≤ α ≤ 0.7.

τ ∂∂

∂∂

τ∂∂

∂∂

RR

xx x

R

yy y

RR

xx x

R

yy y

xx

nnl

nl x

nnl

nl

yy

nnl

nl y

nnl

nl

+ −( ) + −( ) =

+ −( ) + −( ) =

+ +

+ +

1 1

1 1

0

0

x x RR

yR

R

y

R

x

R

y

R

x

R

y

y y RR

xR

R

x

R

x

R

y

R

nl

nl

xy

ny

x

n

x

n

y

n

y

n

x

n

nl

nl

yx

nx

y

n

x

n

y

n

y

+−

+

= − −

−

= − −

−

1

1

1

τ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

τ ∂∂

∂∂

∂∂

∂∂

∂∂∂

∂∂x

R

yn

x

n

−1

JJ if J

if Jkk k

k

=>≤

εε ε

ε α ε= +( )[ ]max . ,S Nneg neg 0 01 1

Computational formulas for the direct extension of the method to the case of adaptive planar gridswill be described in detail below.


8.4 Harmonic Maps Between Surfaces. Derivation of Governing Equations

8.4.1 Introductory Remarks

Recall that for grid generation in a domain Ω the auxiliary problem of constructing a harmonic mappingof this domain onto the parametric square is involved. A mapping of the domain boundary onto thesquare boundary is given. Laplace equations for unknown functions ξ and η are “inverted” into theequations for the functions x and y, Eq. 8.10, which are then solved numerically, as described in Section8.3.1. On the other hand, the problem can be stated as a variational minimization of the functionalEq. 8.12 dependent on the unknown functions x(ξ, η) and y(ξ, η). The variational approach is convenientfor the method extension to the case of surfaces. To achieve this, the problem of finding the harmonicmapping of the surface onto the parametric square is formulated. The one-to-one mapping betweenboundaries should be specified.

In the following subsection a more common problem of constructing harmonic maps between man-ifolds is considered. The emphasis is placed on the formulation of the conditions, providing the one-to-one mapping.

8.4.2 Theory of Harmonic Maps

First we present some common definitions from the survey by Eells and Lemaire [1988]. Let M and Nbe two n-dimensional manifolds (surfaces) with metrics g and h, defined in local coordinates ui and ξα,i, α = 1, …, n. The energy density of a map ξ(u): (M, g) → (N, h) is called the function e(ξ): M → R(≥ 0),defined in local coordinates as follows:

(8.16)

where the standard summation convention is assumed, gij and hij are the elements of metric tensors Gand H manifolds M and N, and gij is the inverse metric:

This means if gij are the elements of matrix G, then gij are the elements of the inverse matrix G–1.The generalization of Dirichlet functional for the mapping ξ(u) is called the energy of the mapping

and is defined as follows:

(8.17)

A smooth map ξ(u): (M, g) → (N, h) is called harmonic if it is an external of the energy functional E.The Euler equations, whose solution minimizes the energy [Eells and Lemaire, 1988] contain Christ-

offel symbols. The simplified solution form of these equations will be presented below.The fundamental result on the sufficient conditions of existence and uniqueness of harmonic maps,

proved by Hamilton [1975] and Shoen and Yau [1978], can be formulated as the theorem.

e u g uu

u

u

uh uij

i jξ ∂ξ∂

∂ξ∂

ξα β

αβ( )( ) = ( ) ( ) ( ) ( )( )

g gif i k

if i kij

jk ki= =

≠

δ1

0

=

E e u du dudM dM G n

M

ξ ξ( ) = ( )( ) ( )=∫ , det ... where 1

THEOREM 3. Let the smooth one-to-one map φ : M → N exist that is also one-to-one betweenboundaries ∂M and ∂N. The curvature of the manifold N is nonpositive, and its boundary ∂N is convex.


Then there exists a unique harmonic map ϕ : M → N, such that ϕ is homotopy equivalent to φ andϕ(∂M) = φ(∂M).

Here we consider the case when N has a simple shape, for example, it is a unit cube in the Euclideanspace. The conditions of the theorem (nonpositive curvature and convex boundary) are obviously satisfiedin this case. Consequently, the theory of harmonic maps includes the theoretic foundation of the method,proposed by Winslow [1966].

So, consider when M is a n-dimensional manifold, N is a unit cube in Rn: 0 < ξ i < 1, i = 1, …, n. TheEuclidean metric in Rn is hαβ = δαβ . If the local coordinates ui and ξα are the same, then Eq. 8.16 can besimplified to give

Hence, the energy functional Eq. 8.17 will be

(8.18)

The Euler equations for the functional Eq. 8.18 can be also simplified, and we can avoid the appearanceof Christoffel symbols in these equations. Now we will derive these equations following Liseikin [1991].

8.4.3 Derivation of Governing Equations

We denote by Srn a n-dimensional in Rn+k with a local coordinate system

The surface is defined by a nondegenerate transform

(8.19)

The new parameterization of the surface Srn is defined by a mapping of a unit cube Qn : 0 < ξ i < 1,i = 1, …, n in Rn onto a surface Srn:

(8.20)

which is the composition of r(u) and some nondegenerate transform

(8.21)

The problem of finding a new parameterization of the surface is stated as the problem of constructionat this transformation u(ξ). The mapping of r(u(ξ)) defines on a surface Srn a new coordinate system(ξ1 … ξn) = ξ, which generates a local metric tensor

e g g g Tr Giji j

ijij

iiξ ∂ξ∂ξ

∂ξ∂ξ

δ δα β

αβ( ) = = = = ( )−1

E g dM Tr G G d dii

M

nξ ξ ξ ξ( ) = ( ) = ( ) ( )−∫∫∫ ... det ...1

0

1

0

11

u u u S Ri n n n,...,( ) = ∈ ⊂

r u S S r r rn rn n k( ) → = ( )+: ,..., 1

r u Q S Qn rn n nξ ξ ξ ξ( )( ) → = ( ) ∈: ,..., 1

u Q Sn nξ( ) →:

G g i j nrijrξ ξ= = , , ...,1 2

whose elements are scalar products of the vectors ri = ∂r/∂ξ i and rj = ∂r/∂ξ j:


The elements of the metric tensor defined by the transformation r(u) are given by

These elements are the scalar products of the vectors ∂r/∂ui and ∂r/∂uj:

Consider the contravariant metric tensors whose elements form the symmetric matrices Gξr and Gur,inverse to the matrices Grξ and Gru:

where djiξr and dji

ur are the determinants of cofactors of the elements grξij and gru

ij in the matrices Grξ and Gru

correspondingly.Let us prove the following relation:

(8.22)

Indeed, substituting in the following identity the right-hand side of Eq. 8.22 instead of glpξr we obtain

The summation is performed on repeated indices, here α = 1, 2, …, n + k; i, j, l, p, t, h, m = 1, …, n. Nowtaking Eq. 8.22 into account, the functional Eq. 8.17 takes the form

(8.23)

In the derivation of the Euler equations the integration domain in Eq. 8.23 will be replaced by Srn, andthe surface element is transformed as follows:

g rrr r

ijr

i j

m

im

n k m

jξ ∂

∂ξ∂∂ξ

= ==

+

∑1

G g i j nruijru= = , , ...,1 2

gr

u

r

uijru

m

im

n k m

j==

+

∑ ∂∂

∂∂1

g d G g d Grij i j

rji r

urij i j

urji ru

ξ ξξ= −( ) ( ) = −( ) ( )+ +1 1det det

g gu ur

ijurml

i

m

j

lm l

n

ξ∂ξ∂

∂ξ∂

= ∑,

δ ∂∂ξ

∂∂ξ

∂∂

∂∂

∂∂ξ

∂∂ξ

∂∂ξ

∂∂ξ

∂ξ∂

∂ξ∂

δ ∂∂ξ

∂ξ∂

ξξ

α α

ξ

α α

ξip

ilr

rlp

i l rlp

t h

t

i

h

l rlp

thru

urmj

t

i

h

l

l

m

p

j

thru

urmj

mh

t

i

p

j thru

g gr r

gr

u

r

u

u ug g g

u u

u u

g gu

ug

= = = = =

= ggu

u

u

uurhj

t

i

p

j tj

t

i

p

j ip∂

∂ξ∂ξ∂

δ ∂∂ξ

∂ξ∂

δ= =

I g dS gu u

dSrii

S

rnurml

i

mi m l

n

S

i

lrn

rn rn

= =∫ ∑∫ξ∂ξ∂

∂ξ∂, ,

dS G dS G du durn ru n ru n= ( ) = ( )det det ...1

Consequently, the functional Eq. 8.23 can be written as


(8.24)

The quantities and gmlur in the functional Eq. 8.24 are independent on the functions ξ i(u) and

their derivatives, and hence remain unchanged when ξ(u) is varied. Therefore the Euler equations forthe functions ξ i(u), minimizing Eq. 8.24 are of the form

(8.25)

The equations which each component ui(ξ) of the function u(ξ) satisfies can be derived from Eq. 8.25.To achieve this, ith equation of the system Eq. 8.25 is multiplied by ∂uj/∂ξ i and summed over i. As aresult, we have

Here j = 1, …, n.Now, multiplying each equation on 1/ and taking into account Eq. 8.22 and the relation

we finally obtain

(8.26)

This is a quasilinear system of elliptic equations that is a direct extension of the system Eq. 8.10. It willbe the basis of the algorithms for structured two-dimensional adaptive grids, grids on surfaces and three-dimensional grids. For derivation of governing equations in all these cases, we need only to express thecontravariant components gij

ξr and gijur as functions on the covariant components gξr

ij and gurij and substitute

the associate expressions into Eq. 8.26 for n = 2 and n = 3.

8.5 Two-Dimensional Adaptive-Harmonic Structured Grids

8.5.1 Derivation of Equations

Let Ω be a two-dimensional domain in R2, and let in a Euclidean space R3, the surface Sr2 is given as z =f(x, y). We introduce new notations

I G gu u

du duruurml

i

m

i

li m l

n

S

n

n

=

∑∫ det( ) ...

, ,

∂ξ∂

∂ξ∂

1

det Gru( )

Lu

G gu

i nim

ruurml

i

ll

n

m

n

ξ ∂∂

∂ξ∂( ) =

= ===∑∑ det( ) ,...,

11

0 1

Lu

uG g

u

u

uG g

u

uG g

ij

i mi m

n

i

nru

urmp

i

pp

n j

i

mm

nru

urmp

i

p

j

ii p

nru

urmp

i

( ) det

det det

,

,

ξ ∂∂ξ

∂∂

∂ξ∂

∂∂ξ

∂∂

∂ξ∂

∂∂ξ

∂ξ

= ( )

=

( )

− ( )

== =

= =

∑∑ ∑

∑ ∑

11 1

1 1 ∂∂∂ξ∂

∂∂ξ ∂ξu u

up

t

mi m p t

n j

i t, , , =∑ =

1

2

0

det Gru( )

∂ξ∂

∂∂ξ

δi

p

j

i pj

i

n

u

u ==∑

1

gu

G uG j nr

itj

i t ru mru

urmj

m

n

ξ∂

∂ξ ∂ξ∂

∂

2

1

11=

( ) ( )( ) ==

∑det

det ,...,g


The problem formulation is the following. Suppose we are given a simply connected domain Ω witha smooth boundary in the x, y plane. Consider the surface z = f(x, y) of the graph of the function f ∈C2(Ω). It is required to find a mapping of the parametric square Q2 onto the domain Ω under a givenmapping between boundaries such that the mapping of the surface onto the parametric square beharmonic (see Figure 8.3). Thus, the problem is to minimize the Dirichlet functional, written for a surface

(8.27)

where g11ξr g12

ξr g22ξr are the elements of the contravariant metric tensor Gξr dependent on the elements of

the covariant metric tensor Grξ as follows:

where

(8.28)

FIGURE 8.3 Harmonic coordinates on the surface of the graph of the graph of a function z = f(x, y).

r r r r x y z x y f x y S R u u u x y R

Q R r x y z r x y z

r= ( ) = ( ) = ( )( ) = ( ) = ( )= ( ) = ( ) = ( ) = ( )

∈ ⊂ ∈ ⊂

∈ ⊂

1 2 3 2 3 1 2 2

1 2 2 2

, , , , , , , , ,

, , , , , , ,

Ω

ξ ξ ξ ξ η ξ ξ ξ ξ η η η η

I g g dSr rr= +( )∫ ξ ξ

11 22 2

g g G g g G g g g Grr r

rr r

r rr r

ξξ ξ

ξξ ξ

ξ ξξ ξ11

2222

1112 21

12= = det det det( ) ( ) = = − ( )

g r x y z g g r r x x y y z z g r x y z

G g g g z f x f y z

r r r r

r r r rx y

112 2 2 2

12 21 222 2 2 2

11 22 12

2

ξξ ξ ξ ξ

ξ ξξ η ξ η ξ η ξ η

ξη η η η

ξ ξ ξ ξξ ξ ξ η

= = + + = = ⋅( ) = + + = = + +

( ) = − ( ) = +

det , == +f x f yx yη η

Inverting dependent and independent variables in Eq. 8.27 and taking in account


we obtain

(8.29)

Euler equation for the functional Eq. 8.29 follow from Eq. 8.26 for n = 2, k = 1. We need only tocompute the elements of the covariant metric tensor Grξ and contravariant metric tensor Gξr of thetransform r(u) = r(x, y) : Ω → Sr2:

Substituting these expressions into Eq. 8.26, we obtain equations, written in a form convenient forpractical use:

(8.30)

where

8.5.2 Numerical Implementation

Eq. 8.30 are approximated on the square grid with the unit size Eq. 8.4, introduced above with the simplestdifference relations

dS g g g d dr r r r211 22 12

2= − ( )ξ ξ ξ ξ η

Ig g

g g gd d

r r

r r r= +

− ( )∫ 11 22

11 22 12

2

ξ ξ

ξ ξ ξξ η

r x y f x y r f r f

g r f g g r r f f g r f

G g g g f f

x x y y

rux x

ru rux y x y

ruy y

ru ru ru rux y

= ( )( ) = ( ) = ( )= = + = = ⋅ = = = +

( ) = − ( ) = + +

, , , , , , ,

det

1 0 0 1

1 1

1

112 2

12 21 222 2

11 22 122 2 22 2

1122

2 2 2

1221

2 2

22 2 2

1 1

1

1

det det

det

det

G G x y x y

g g G f f f

g g G f f f f

g g f f g

r ru

urru ru

y x y

urru ru

x y x y

ur yru

x y

ξξ η η ξ( ) = ( ) −( )

= ( ) = +( ) + +( )= − ( ) = − + +( )

= + +( ) ururrTg22 =

L x x x x J Dx

f

D y

f f

D

L y y y y J Dx

f f

D y

f

D

y x y

x y x

( ) = − + −+

−

=

( ) = − + −−

+ +

=

α β γ ∂∂

∂∂

α β γ ∂∂

∂∂

ξξ ξη ηη

ξξ ξη ηη

21

0

21

0

22

22

D f f J x y x y f x x y y f f x y fx y= + + = = + + = + + = + +1 2 2 2 2 2 2 2 2, , , , . η ξ η η η ξ η ξ η ξ η ξ ξ ξα β γ

x x x x x x x xi j i j ij i j i jξ ξ η η≈ [ ] = −( ) ≈ [ ] = −( )+ − + −0 5 0 51 1 1 1. ., , , ,


(8.31)

Substitute these expressions into Eq. 8.30 and denote the difference approximations of L(x) and L(y) as[L(x)]ij and [L(y)]ij correspondingly. Suppose that the coordinates of grid nodes (x, y)ij at the lth step ofiterations are determined. Then the (l + 1)-th step is accomplished as follows:

(8.32)

The expressions in square brackets denote the corresponding approximations of expressions in the gridnode (i, j) at the lth iteration step. The value of iteration parameter τ is chosen in limits 0 < τ < 1, usuallyτ = 0.5.

Derivatives [fx]ij and [fy]ij in the ijth grid node are evaluated with the centered differences

These formulas must be modified for the boundary nodes. Indices, “leaving” the computational domainmust be replaced by the nearest boundary indices. For example, if j = 1, then (i, j – 1) must be replacedby (i, j).

Note that if [fξ]ij = 0 and [fη]ij = 0, then [fx]ij = 0 and [fy]ij = 0 and the method Eq. 8.32 reduces to theWinslow method, described briefly in Section 8.3.1.

The adaptive-harmonic grid generation algorithm is formulated as follows:

1. Compute the values of the control function at each grid node. The result is fij.2. Evaluate derivatives (fx)ij and (fy)ij and other expressions in Eq. 8.32 using the above formulas.3. Make one iteration step and compute new values of xij and yij.4. Repeat, starting with Step 1 to convergency.

y y y y y y y y

ij

iji j i j ij i j i jξ ξ η η≈ [ ] = −( ) ≈ [ ] = −( )+ − + −0 5 0 51 1 1 1. ., , , ,

f f f f f f f f

x x x x x

x x x

iji j i j ij i j i j

iji j ij i j

i ji j

ξ ξ η η

ξξ ξξ

ξη ξη

≈ [ ] = −( ) ≈ [ ] = −( )≈ [ ] = − +

≈ [ ] =

+ − + −

+ −

+

0 5 0 5

2

0 25

1 1 1 1

1 1

1

. .

.

, , , ,

, ,

,, ++ + − − + − −

+ − + −

+ +

− − +( )≈ [ ] = − + ≈ [ ] = − +

≈ [ ] =

1 1 1 1 1 1 1

1 1 1 1

1

2 2

0 25

x x x

x x x x x y y y y y

y y y

i j i j i j

ij i j i j i jij

i j ij i j

iji j

, , ,

, , , , ,

,.

ηη ηη ξξ ξξ

ξη ξη 11 1 1 1 1 1 1

1 1

2 2 2

2

− − +( )≈ [ ] = − +

≈ [ ] + [ ] + [ ] ≈ [ ] [ ] + [ ] [ ] + [ ]

+ − − + − −

+ −

y y y

y y y y y

x y f x x y y f f

i j i j i j

ij i j ij i j

ij ij ij ij ij ij ij ij

, , ,

, ,

ηη ηη

η η η ξ η ξ η ξ ηα β [[ ] ≈ [ ] + [ ] + [ ]ij ij ij ijx y f γ ξ ξ ξ

2 2 2

x xL x

y yL y

ijl

ijl ij

ij ij

ijl

ijl ij

ij ij

+ += +( )[ ]

[ ] + [ ]( )[ ]

[ ] + [ ]1 1

2 2 2 2τ

α γτ

α γ = +

ff f y y f f y y

x x y y x x yx ij

i j i j i j i j i j i j i j i j

i j i j i j i j i j i j i

[ ] =−( ) −( ) − −( ) −( )−( ) −( ) − −( )

+ − + − + − + −

+ − + − + − +

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

, , , , , , , ,

, , , , , , ,, ,

, , , , , , , ,

, , , , ,

j i j

y ij


i j i j i j i j i j

y

ff f x x f f x x

x x y y x

−( )

[ ] = −−( ) −( ) − −( ) −( )−( ) −( ) −

−

+ − + − + − + −

+ − + − +

1

1 1 1 1 1 1 1 1

1 1 1 1 1 −−( ) −( )− + −x y yi j i j i j, , ,1 1 1

The resulting algorithm can be used in the numerical solution of the partial differential equations. Inthis case, at the first step of the algorithm the values fij in each grid node are taken from the finite-


difference or finite element solution of the host equations.Note that for control of the number of grid nodes in the layers of high gradients, it is convenient to

use Cf instead of f(x, y). The larger the coefficient C, the greater the number of nodes in the layer of highgradients of the function f.

8.6 Two-Dimensional Adaptive-Harmonic Irregular Meshes


In notations of Section 8.5.1 the problem is formulated as follows. Suppose we are given a simplyconnected domain Ω with a smooth boundary in the x, y plane. Consider the surface z = f(x, y) of thegraph of the function f ∈ C1(Ω). It is required to find a mapping of the parametric square Q2 onto adomain Ω under a given mapping between boundaries such that the mapping of the surface onto theparametric square be harmonic (see Figure 8.3). Thus, the problem is to minimize the harmonic func-tional Eq. 8.27.

Substituting expressions Eq. 8.28 for zξ and zη into Eq. 8.29, we obtain the functional from the paperby Ivanenko [1993] to define adaptive-harmonic grid, clustered in regions of high gradients of thefunction f(x, y):

(8.33)

The problem of irregular mesh smoothing and adaption is formulated as follows. Let the coordinates ofirregular mesh be given. The mesh is formed by quadrilateral elements, i.e., the array COR(N, k) is alsodefined. The problem is to find new coordinates of the mesh nodes, minimizing the sum of the functionalEq. 8.33 values, computed for a mapping of the unit square onto each cell of a mesh (see Figure 8.3).

8.6.2 Approximation of the Functional

The functional Eq. 8.33 possesses the same properties as the functional Eq. 8.11, and it can be alsoapproximated in such a way that its minimum is attained on a grid/mesh of convex quadrilaterals:

(8.34)

where

Ix x f y y f f f x y x y

x y x y f fd d

x y x y

x y

=+( ) +( ) + +( ) +( ) + +( )

−( ) + +∫ ξ η ξ η ξ ξ η η

ξ η η ξ

ξ η2 2 2 2 2 2

2 2

1 1 2

1

I Fhk N

kN

Ne

= [ ]==

∑∑ 141

4

1

FD f D f D f f

J f f

D x x x x D y y y y

D x

k

x k y k x k y k

k x k y k

k k k k k k k k

=+ ( )[ ] + + ( )[ ] + ( ) ( )

+ ( ) + ( )[ ]= −( ) + −( ) = −( ) + −( )

=− + − +

1

2

2

2

3

2 2 1 2

1 1

2

1

2

2 1

2

1

2

3

1 1 2

1

kk k k k k k k k

k k k k k k k k k

x y y x x y y

J x x y y x x y y

− − + +

− + + −

−( ) −( ) + −( ) −( )= −( ) −( ) − −( ) −( )

1 1 1 1

1 1 1 1

Here (fx)k and (fy)k are the values of derivatives, computed in the node number k of the cell number N.If the set of convex meshes D is not empty, the system of algebraic equations


has at least one solution which is a convex mesh. To find it, one must first find a certain initial convexmesh, and then use some method of unconstrained minimization of the function Ih. Since this functionhas an infinite barrier on the boundary of the set D, each step of the method can be chosen so that themesh always remains convex.

8.6.3 Minimization of the Functional

Suppose the mesh at the lth step of the iterations is determined. We use the quasi-Newtonian procedurewhen the (l + 1)-th step is accomplished as follows:

(8.35)

where τ is the iteration parameter, which is chosen so that the mesh remains convex. For this purposeafter each step conditions Eq. 8.8 are checked and if they are not satisfied, this parameter is multipliedby 0.5. Then conditions Eq. 8.8 are checked for the grid, computed with a new value of τ and if they arenot satisfied, this parameter is multiplied by 0.25 and so on.

The adaptive-harmonic algorithm for r–refinement is formulated as follows:

1. Generate an initial mesh by the use of a marching method.2. Compute the values of the control function fn at each mesh node.3. Evaluate derivatives (fx)n and (fy)n and other expressions in Eq. 8.35.4. Make one iteration step and compute new values of xn and yn.5. Repeat starting with Step 2 to convergency.

Computational formulas for [fx]n and [fy]n will be presented below.

8.6.4 Derivation of Computational Formulas

Note that the approximation Eq. 8.34 of the functional Eq. 8.33 can be obtained as it was done for thefunctional Eq. 8.11 in Section 8.3.2. The square cell on the plane ξ, η is divided into two triangles firstby the diagonal 13, and then by 24. The mapping of the square onto a quadrilateral cell in the plane x,y is approximated by two functions which are linear in each triangle. Denote this functions as beforexh(ξ, η), yh(ξ, η). All derivatives in the integrand of Eq. 8.33 is easy to compute, as it was done in Section8.3.2. Then the integral Eq. 8.33 over the square cell in the plane ξ, η is approximated by a half of thesum of values of this integral, computed for piecewise linear approximations on triangles, obtained forthe first and the second splittings. The result is the approximation Eq. 8.34.

Four triangles, introduced above are considered for the quadrilateral cell number N. Each of thesetriangles corresponds to a corner with the number k and gives a proper contribution to the functionaland also to the values of its derivatives. Since the integrand in Eq. 8.33 does not depend on the rotationof the coordinate system ξ, η, then all the computational formulas will be the same for all triangles. Weenumerate nodes of triangle corresponding to the corner with the local number k from 1 to 3 as follows:

RI

xR

I

yx

h

ny

h

n

= = = =∂∂

∂∂

0 0

x x RR

yR

R

y

R

x

R

y

R

x

R

y

y y RR

xR

R

x

R

x

R

y

R

nl

nl

xy

ny

x

n

x

n

y

n

y

n

x

n

nl

nl

yx

nx

y

n

x

n

y

n

y

+−

+

= − −

−

= − −

−

1

1

1

τ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

τ ∂∂

∂∂

∂∂

∂∂

∂∂∂

∂∂x

R

yn

x

n

−1

node 1 corresponds to the local node number k – 1 of the cell N,node 2 corresponds to the local node number k of the cell N,


node 3 corresponds to the local node number k + 1 of the cell N.

Then in the new numeration the expression for Fk will be

(8.36)

where

We introduce notations

We use formulas for the derivatives of the relation of two functions. Differentiating, we obtain

(8.37)

For the triangle vertex with the number 1, we should substitute appropriate expressions instead of U andV, Ux and Vx and so on into Eq. 8.37 and replace x and y by x1 and y1.

For the vertex 1 we have

FD f D f D f f

J f f

x k y k x k y k

x k y k

=+[ ] + +[ ] +

+ +[ ]1

22

23

22 2 1 2

1 1 2

1

( ) ( ) ( ) ( )

( ) ( )

D x x x x D y y y y

D x x y y x x y y

J x x y y x x y y

1 1 2

2

3 2

2

2 1 2

2

3 2

2

3 1 2 1 2 3 2 3 2

2 1 2 3 2 3 2 1 2

= −( ) + −( ) = −( ) + −( )= −( ) −( ) + −( ) −( )= −( ) −( ) − −( ) −( )

UD f D f D f f

f f

V x x y y x x y y

x k y k x k y k

x k y k

=+ ( )[ ] + + ( )[ ] + ( ) ( )

+ ( ) + ( )[ ]= −( ) −( ) − −( ) −( )

1

2

2

2

3

2 2 1 2

1 2 3 2 3 2 1 2

1 1 2

1

FU

V

FU FV

VF

U FV

VF

U F V FV

V

FU F V FV

VF F

U F V F V FV

V

xx x

yy y

xxxx x x xx

yyyy y y yy

xy yxxy x y y x xy

=

= − =−

= − −

=− −

= =− − −

2

2

V y y V x x

V V V

Uf x x f f y y

f fU

f y

x y

xx xy yy

x

x k x k y k

x k y k

y

y k

= − = −

= = =

=+ ( )[ ] −( ) + ( ) ( ) −( )

+ ( ) + ( )[ ]=

+ ( )[ ]

3 2 2 3

2

1 2 1 2

2 2 1 2

2

1

0 0 0

21

12

1

,

, ,

−−( ) + ( ) ( ) −( )+ ( ) + ( )[ ]

=+ ( )

+ ( ) + ( )[ ]=

( ) ( )+ ( ) + ( )[ ]

=+

y f f x x

f f

Uf

f fU

f f

f fU

f

x k y k

x k y k

xxx k

x k y k

xy

x k y k

x k y k

yy

y

2 1 2

2 2 1 2

2

2 2 1 22 2 1 2

1

21

12

12

1

(( )+ ( ) + ( )[ ]

k

x k y kf f

2

2 2 1 2

1




Computations are performed as follows. Let F and its derivatives on x1 and y1 be computed with the useof formulas Eq. 8.37 for the cell number N and triangle number k. Then the computed values are addedto the appropriate array elements

where n = COR(N, k – 1).Similarly for the vertex 2, the correspondence between local and global number is n = COR(N, k).

Similarly for the vertex 3, the correspondence between local and global number is n = COR(N, k + 1).Derivatives [fx]n and [fy]n are computed as follows. All triangles of the mesh are scanned and for the

triangle number k of the cell number N the following values are computed:

V y y V x x

V V V

Uf x x x f f y y y

f f

Uf

x y

xx xy yy

x

x k x k y k

x k y k

y

= − = −

= = =

=+ ( )[ ] − −( ) + ( ) ( ) − −( )

+ ( ) + ( )[ ]=

+

1 3 3 1

2

2 1 3 2 1 3

2 2 1 2

0 0 0

21 2 2

1

21 yy k x k y k

x k y k

xxx k

x k y k

xy

x k y k

x k y

y y y f f x x x

f f

Uf

f fU

f f

f f

( )[ ] − −( ) + ( ) ( ) − −( )+ ( ) + ( )[ ]

=+ ( )

+ ( ) + ( )[ ]=

( ) ( )+ ( ) +

2

2 1 3 2 1 3

2 2 1 2

2

2 2 1 22

2 2

1

41

14

1

(( )[ ]=

+ ( )+ ( ) + ( )[ ]k

yy

y k

x k y k

Uf

f f2 1 2

2

2 2 1 221

1

V y y V x x

V V V

Uf x x f f y y

f f

Uf y

x y

xx xy yy

x

x k x k y k

x k y k

y

y k

= − = −

= = =

=+ ( )[ ] −( ) + ( ) ( ) −( )

+ ( ) + ( )[ ]=

+ ( )[ ]

2 1 1 2

2

3 2 3 2

2 2 1 2

2

0 0 0

21

1

21 33 2 3 2

2 2 1 2

2

2 2 1 22 2 1 2

1

21

12

12

1

−( ) + ( ) ( ) −( )+ ( ) + ( )[ ]

=+ ( )

+ ( ) + ( )[ ]=

( ) ( )+ ( ) + ( )[ ]

=+

y f f x x

f f

Uf

f fU

f f

f fU

f

x k y k

x k y k

xxx k

x k y k

xy

x k y k

x k y k

yy yy k

x k y kf f

( )+ ( ) + ( )[ ]

2

2 2 1 2

1

I F R F R F

R F R F R F

hx n x y n y

xx n xx xy n xy yy n yy

+ = [ ] + = [ ] + =

[ ] + = [ ] + = [ ] + =

f f f y y f f y yx = −( ) −( ) − −( ) −( )1 2 3 2 3 2 1 2


where f1, f2, and f3 are values of the function f at vertices of the triangle, numbered 1, 2, and 3, corre-sponding to local numbers of corners of a quadrilateral cell k – 1, k and k + 1. Computed values areadded to corresponding array elements (which were first cleared):

New values of derivatives are computed as follows:

Here, according to C-language notations, a+ = b means that the new value of a becomes equal to a + b,and a/ = b means that the new value of a becomes equal to a/b.

So, the iteration method for irregular mesh relaxation and adaption is described in detail.

8.7 Adaptive-Harmonic Structured Surface Grid Generation


Introduce the following notations:

Thus, consider a two-dimensional surface in a four-dimensional space, defined as x = x(u,v), y = y(u, v),z = z(u,v), f = f(u,v). Let functions ξ = ξ(u,v), η = η(u,v) are used to define a new parameterization ofa surface.

The problem of construction the adaptive-harmonic grid on a surface is stated as the problem offinding the new parameterization u = u(ξ,η), v = v(ξ,η), minimizing the functional Eq. 8.24, specifiedfor this surface.

The result of minimization will be a new parameterization u = u(ξ,η), v(ξ,η), defining the adaptive-harmonic grid on a surface. Difficulties encountered in this problem are concerned with nonuniquesolutions of its discrete analog, in spite of the result from the harmonic map theory that the continuousproblem has a unique solution [Steinberg and Roache, 1990].

Metric tensor elements g ruij are defined

We write out the Euler equations in the case of adaption. These equations follow from Eq. 8.26 if n = 2, k = 2:

f x x f f x x f f

J x x y y x x y y

y = −( ) −( ) − −( ) −( )= −( ) −( ) − −( ) −( )

1 2 3 2 3 2 1 2

2 1 2 3 2 3 2 1 2

f f f f J J n COR N kx n x y n y n[ ] + = [ ] + = [ ] + = = ( ) 2 ,

f J f Jx n n y n n[ ] = [ ] [ ] = [ ]

r r r r r x y z f S R

u u u u v Q R Q R

r x y z f r x y z f r

r

u

= ( ) = ( )= ( ) = ( ) = ( ) = ( )

= ( ) = ( ) =

∈ ⊂

∈ ⊂ ∈ ⊂

1 2 3 4 2 4

1 2 2 2 1 2 2 2

, , , , , ,

, , , ,

, , , , , ,

ξ ξ ξ ξ η

ξ ξ ξ ξ ξ η η η η η xx y z f r x y z fu u u u v v v v v, , , , , ,( ) = ( )

g x y z f g x x y y z z f f g x y z fruu u u u

ruu v u v u v u v

ruv v v v11

2 2 2 212 22

2 2 2 2= + + + = + + + = + + +

L u u u u J Du

g

D v

g

D

ru ru

( ) = − + − −

=α β γ ∂

∂∂∂ξξ ξη ηη2 02 22 12


(8.38)

where


The method Eq. 8.32 is used for the numerical solution of Eq. 8.38, where x and y are replaced by u andv and [L(u)]ij and [L(v)]ij are the approximations of Eq. 8.38 at the grid node ij. All derivatives on u andv are computed with the use of formulas similar to formulas from Section 8.5.2:

The adaptive-harmonic surface grid generation algorithm is formulated as follows:

1. Generate a quasi-uniform harmonic surface grid using the same algorithm as for adaption, but f = 0.2. Compute the values of the control function at each grid node. The result is fij.3. Evaluate derivatives (fu)ij and (fv)ij and other expressions in Eq. 8.38 using the above formulas.4. Make one iteration step and compute new values of uij and vij.5. Repeat starting with Step 2 to convergency.

The resulting algorithm is simple in implementation but can demand a special procedure for the choiceof the parameter τ to achieve the numerical stability.

8.8 Irregular Surface Meshes


In notations of the previous section, consider a two-dimensional surface in a four-dimensional space,defined as x = x(u, v), y = y(u, v), z = z(u, v), f = f(u, v). Let functions ξ = ξ(u, v), η = η(u, v) are usedto define a new parameterization of a surface.

The problem of construction of the adaptive-harmonic grid on a surface is stated as the problem offinding the new parameterization u = u(ξ, η), v = v(ξ, η) minimizing the functional

(8.39)

L v v v v J Du

g

D v

g

D

ru ru

( )

= − + − − +

=α β γ ∂∂

∂∂ξξ ξη ηη2 02 12 11

D g g g J u v u v g D J x y z f

g D J x x y y z z f f g D J x y z f

ru ru ru r

r r

= − ( ) = − = = + + +

= = + + + = = + + +

11 22 12

2

222 2 2 2 2 2

122 2

112 2 2 2 2 2

ξ η η ξξ

η η η η

ξξ η ξ η ξ η ξ η

ξξ ξ ξ ξ

α

β γ

ff f v v f f v v

u u v v u u vu ij


i j i j i j i j i j i j i

[ ] =−( ) −( ) − −( ) −( )−( ) −( ) − −( )

+ − + − + − + −

+ − + − + − +

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

, , , , , , , ,

, , , , , , ,, ,

, , , , , , , ,

, , , , ,

j i j

v ij


i j i j i j i j i j

v

ff f u u f f u u

u u v v u

−( )

[ ] = −−( ) −( ) − −( ) −( )−( ) −( ) −

−

+ − + − + − + −

+ − + − +

1

1 1 1 1 1 1 1 1

1 1 1 1 1 −−( ) −( )− + −u v vi j i j i j, , ,1 1 1

Ig u u g u v u v g v v

g g g u v u vd d

ru ru ru

ru ru ru=

+( ) + +( ) + +( )− ( ) −( )∫ 11

2 212 22

2 2

11 22 12

2

2ξ η ξ ξ η η ξ η

ξ η η ξ

ξ η

where


(8.40)

The result of minimization will be a new parameterization u = u(ξ, η), v = v(ξ, η).Now we can formulate the problem of irregular surface mesh smoothing and adaption. Let coordinates

of an irregular mesh in the plane u, v be given:

The mesh is formed by quadrilateral elements, i.e., the array COR(N, k) is also defined. Functions x =x(u, v), y = y(u, v), z = z(u, v) and f = f(u, v) are assumed to be specified, for example, can be computedby analytic formulas.

The problem is to find new coordinates of the mesh nodes, minimizing the sum of the functionalEq. 8.39 values, computed for a mapping of the unit square in the plane ξ, η onto each cell of a meshin the plane x, y.


Note that if in the functional Eq. 8.33 we replace expressions for 1 + ( fx)2 by gru11, fx fy by gru

12, and 1 +( fy)2 by gru

22, we obtain the functional Eq. 8.39. Hence, the last one possesses all the properties of thefunctional Eq. 8.33 and also can be approximated in such a way that the minimum of its discrete analogis attained on a nondegenerate grid of convex quadrilaterals on the plane u, v. The algorithm from theSection 8.5 can be used for its approximation and minimization:

(8.41)

where

Here the values g ruij are computed at the node number k of the cell number N.

If the set D of convex meshes on the plane u, v is not empty, the system of algebraic equations

has at least one solution that is a convex mesh. To find it, one must first find a certain initial convexmesh, and then use some method of unconstrained minimization of the function Ih. Since this functionhas an infinite barrier on the boundary of the set of convex meshes, each step of the method can bechosen so that the mesh always remains convex.

g x y z f g x x y y z z f f g x y z fruu u u u

ruu v u v u v u v

ruv v v v11

2 2 2 212 22

2 2 2 2= + + + = + + + = + + +

u v n Nn n, ,...,( ) = 1

I Fhk N

kN

Ne

= [ ]==

∑∑ 141

4

1

FD g D g D g

J g g g

D u u u u D v v v v

D u u v v u

k

ru ru ru

kru ru ru

k k k k k k k k

k k k k k

= + +

− ( )= −( ) + −( ) = −( ) + −( )

= −( ) −( ) + −− + − +

− − +

1 11 2 22 3 12

11 22 12

2

1 1

2

1

2

2 1

2

1

2

3 1 1 1

2

uu v v

J u u v v u u v v

k k k

k k k k k k k k k

( ) −( )= −( ) −( ) − −( ) −( )

+

− + + −

1

1 1 1 1

RI

uR

I

vu

h

nv

h

n

= = = =∂∂

∂∂

0 0


Suppose the mesh at the lth step of the iterations is determined. We use the quasi-Newtonian procedure


when the (l+1)-th step is accomplished by solving two linear equations for each interior node:

(8.42)

where τ is the iteration parameter, which is chosen so that the mesh remains convex. For this purposeafter each step the conditions of grid convexity on the plane u, v are checked and if they are not satisfied,this parameter is multiplied by 0.5.

The adaptive-harmonic algorithm for the mesh smoothing and adaption on a surface is formulatedas follows:

1. Generate an initial mesh with the use of a marching method.2. Compute new values xn, yn, zn and fn at each mesh node.3. Evaluate derivatives [xu]n and [xv]n, [yu]n and [yv]n, [zu]n and [zv]n, [ fu]n and [ fv]n used in Eq. 8.42.4. Make an iteration step and compute new values of un and vn.5. Repeat starting with Step 2 to convergency.

Computational formulas for [fu]n and [ fv]n can be obtained as described in Section 8.6.4.


Recall that if in the functional Eq. 8.33 we replace expressions for 1 + ( fx)2 by g ru11, fx fy by g ru

12, and 1 +(fy)2 by g ru

22, we obtain the functional Eq. 8.39. From this follows that for derivation of computationalformulas for surface meshes we need only to perform these replacements in computational formulas foradaptive planar meshes, described in Section 8.6.4.

8.9 Three-Dimensional Regular Grids


We will derive equations at once for the case of adaptation. Introduce notations

The functional Eq. 8.24 in the three-dimensional case has the form

(8.43)

where dSr3 is the element of the surface Sr3.

τ ∂∂

∂∂

τ ∂∂

∂∂

RR

uu u

R

vv v

RR

uu u

R

vv v

uu

nnl

nl u

nnl

nl

vv

nnl

nl v

nnl

nl

+ −( ) + −( ) =

+ −( ) + −( ) =

+ +

+ +

1 1

1 1

0

0

r r r r r x y z f S R

u u u u x y z R Q R

r x y z f r x y z

r= ( ) = ( )= ( ) = ( ) = ( ) = ( )

= ( ) =

∈ ⊂

∈ ⊂ ∈ ⊂

1 2 3 4 3 4

1 2 3 3 1 2 3 3 3

, , , , , ,

, , , , , , , ,

, , , , , ,

Ω ξ ξ ξ ξ ξ η µ

ξ ξ ξ ξ ξ η η η η ff r x y z f

r f r f r fx x y y z z

η µ µ µ µ µ( ) = ( )= ( ) = ( ) = ( )

, , , ,

, , , , , , , , , .1 0 0 0 1 0 0 0 1

I g g g dSr r rr= + +( )∫ ξ ξ ξ

11 22 33 3

The functional Eq. 8.43 can be used for constructing harmonic coordinates on the surface of the graphof control function dependent on three variables. Projection of these coordinates onto a physical domain


gives an adaptive-harmonic grid, clustered in regions of high gradients of adapted function f(x, y, z).The Euler equations of the functional Eq. 8.43 follow from Eq. 8.26 for n = 3, k = 1. We need only to

compute the elements of the covariant metric tensor Gru and contravariant tensor Gur of the transformr(u) = r(x, y, z) : Ω → Sr3:

Substituting these expressions into Eq. 8.26, we obtain equations convenient for practical use:

(8.44)

where

g r f g r f g r f

g g r r f f g g r r f f g g r r f f

G g g g

rux x

ruy y

ruz z

ru rux y x y

ru rux z x z

ru ruy z y z

ru ru ru ru

112 2

222 2

332 2

12 21 13 31 23 32

11 22 33

1 1 1= = + = = + = = +

= = ⋅ = = = ⋅ = = = ⋅ =

( ) =

,

det −− ( )[ ] − −( ) + −( ) =

+( ) + +( ) − − = + + +

( ) =

g g g g g g g g g g g

f f f f f f f f f f

G g g g

ru ru ru ru ru ru ru ru ru ru ru

x y z x y x z x y z

r r r r

23

2

12 12 33 13 23 13 12 23 22 13

2 2 2 2 2 2 2 2 2 2

11 22 33

1 1 1

det ξ ξ ξ ξ −− ( )[ ] − −( ) + −( ) =

( ) −( ) − −( ) + −( )[ ]=

g g g g g g g g g g g

G x y z y z y x z x z z x y x y

g g

r r r r r r r r r r r

ru

r

23

2

12 12 33 13 23 13 12 23 22 13

2

11

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

ξ η µ µ η ξ η µ µ η ξ ξ η η ξ

ξ

det

2222 33 23

2 1212 33 13 23

1312 23 13 22

2211 33 12

2

r r r rr

r r r r r

rr r r r r

rr r r r

g g G g g g g g G

g g g g g G g g g g G

ξ ξ ξ ξξ

ξ ξ ξ ξ ξ

ξξ ξ ξ ξ ξ

ξξ ξ ξ ξ

− ( )[ ] ( ) = − −( ) ( )= −[ ] ( ) = − ( )[ ] (

det det

det det

))= − −[ ] ( ) = − ( )[ ] ( )g g g g g G g g g g Gr

r r r r rr

r r r rξ

ξ ξ ξ ξ ξξ

ξ ξ ξ ξ2311 23 13 12

3311 22 12

2det det

g f f f f f g f f f f f

g f f f f f g f f f f f

g f

ur y z x y z ur x y x y z

ur x z x y z ur x z x y z

ur

11 2 2 2 2 2 12 2 2 2

13 2 2 2 22 2 2 2 2 2

23

1 1 1

1 1 1

= + +( ) + + +( ) = − + + +( )= − + + +( ) = + +( ) + + +( )= −

yy z x y z ur x y x y zf f f f g f f f f f1 1 12 2 2 33 2 2 2 2 2+ + +( ) = + +( ) + + +( )

L x g x g x g x g x g x g x

D x

f f

D y

f f

D z

f f

D

L y g y g

r r r r r r

y z x y x z

r

( ) = + + + + + −

+ ++

−+ −

=

= +

ξ ξξ ξ ξη ξ ξµ ξ ηη ξ ηµ ξ µµ

ξ ξξ ξ

∂∂

∂∂

∂∂

11 12 13 22 23 33

2 2

11

2 2 2

1 10

2

( ) rr r r r r

x y x z y z

r r r

y g y g y g y g y

D x

f f

D y

f f

D z

f f

D

L z g z g z g z

12 13 22 23 33

2 2

11 12 13

2 2

1 10

2 2

ξη ξ ξµ ξ ηη ξ ηµ ξ µµ

ξ ξξ ξ ξη ξ ξµ

∂∂

∂∂

∂∂

+ + + + −

−+ + + +

−

=

= + + +

( ) gg z g z g z

D x

f f

D y

f f

D z

f f

D

r r r

x z y z x y

ξ ηη ξ ηµ ξ µµ

∂∂

∂∂

∂∂

22 23 33

2 2

2

1 10

+ + −

− +−

++ +

=

D f f fx y z= + + +1 2 2 2


The problem of grid generation in three dimensions will be considered in the following formulation. In


a simply connected domain Ω in the space x, y, z a grid

must be constructed with given coordinates of boundary nodes

Instead of the parametric cube the following parametric domain can be introduced to simplify thecomputational formulas:

associated with the cube grid (ξi, ηj, µm) such that

Eq. 8.44 are approximated on this grid with the use of simplest finite-difference relations for derivativeson ξ, η, µ. For example, derivatives of f(ξ, η, µ) are approximated as

The method similar to Eq. 8.32 is used for the numerical solution of the resulting finite-differenceequations:

(8.45)

x y z i i j j m mijm

, , ,..., ,..., ,...,( ) = = =∗ ∗ ∗ 1 1 1

x y z x y z x y z x y z x y z x y zij ijm i m ij m jm i jm

, , , , , , , , , , , ,* * *( ) ( ) ( ) ( ) ( ) ( )1

1 1

1 1 1< < < < < <ξ η µi j m* * *

ξ η µi j mi j m i i j j m m= = = = = =, ,..., * ,..., ,..., * * 1 1 1

f f f f f f f f

f f f f f f f f

ijm i j m i j m ijm i j m i j m

ijm i j m i j m ijm i j m ijm

ξ ξ η η

µ µ ξξ ξξ

≈ [ ] = −( ) ≈ [ ] = −( )≈ [ ] = −( ) ≈ [ ] = −

+ − + −

+ − +

12

12

12

2

1 1 1 1

1 1 1

, , , , , , , ,

, , , , , , ++

≈ [ ] = − − +( )≈ [ ] = − − +

−

+ + − + + − − −

+ + + − − +

f

f f f f f f

f f f f f

i j m

ijm i j m i j m i j m i j m

ijm i j m i j m i j m

1

1 1 1 1 1 1 1 1

1 1 1 1 1 1

14

14

, ,

, , , , , , , ,

, , , , , ,

ξη ξη

ξµ ξµ ff

f f f f f

f f f f f f

f f

i j m

ijm i j m ijm i j m

ijm i j m i j m i j m i j m

− −

+ −

+ + + − − + − −

( )≈ [ ] = − +

≈ [ ] = − − +( )≈

1 1

1 1

1 1 1 1 1 1 1 1

2

14

, ,

, , , ,

, , , , , , , ,

ηη ηη

ηµ ηµ

µµ µµ[[ ] = − ++ −ijm i j m ijm i j mf f f, , , ,1 12

x xL x

g g g

y yL y

g g g

z zL z

g

ijml

ijml ijm

r ijm r ijm r ijm

ijml

ijml ijm

r ijm r ijm r ijm

ijml

ijml ijm

r ijm

+

+

+

= +( )[ ]

[ ] + [ ] + [ ]= +

( )[ ][ ] + [ ] + [ ]

= +( )[ ]

[ ] +

1

11 22 33

1

11 22 33

1

11

2 2 2

2 2 2

2

τ

τ

τ

ξ ξ ξ

ξ ξ ξ

ξ 22 222 33g gr ijm r ijmξ ξ[ ] + [ ]

Consider formulas for the transformation of derivatives in the three-dimensional case:


From this follows

where J = xξ(yηzµ – yµzη) – xη(yξzµ – yµzξ) + xµ(yξzη – yηzξ).Approximating all derivatives in these expressions with the use of the above formulas, we obtain the

approximation of derivatives [fx]ijm, [fy]ijm, and [fz]ijm, used in Eq.8.45.The adaptive-harmonic grid generation algorithm is formulated as follows:

1. Generate a quasi-uniform grid using the same algorithm as for adaption, but f = 0.2. Compute the values of the control function fijm at each grid node.3. Evaluate derivatives [fx]ijm, [fy]ijm, and [fz]ijm and substitute them into Eq. 8.45.4. Make on iteration step and compute new values of xijm, yijm, and zijm.5. Repeat starting with Step 2 to convergency.

The resulting algorithm is simple in implementation and can be used for meshing the three-dimensionaldomains until the increased complexity of domain or boundary layers produce the appearance of self-intersecting cells. Then the special algorithm should be employed, based on a variational formulationand guaranteeing nondegenerate grid generation.

8.10 Three-Dimensional Irregular Meshes

8.10.1 Discrete Analog of the Jacobian Positiveness

The three-dimensional case is much more complicated than the two-dimensional case, because simpleconditions of the Jacobian positiveness cannot be obtained for the trilinear mapping of the unit cubeonto a hexahedral cell. The notation of convexity also cannot be used, since faces of a hexahedron arenot plane. This is why the approach developed for two-dimensional meshes in Section 8.2 cannot bedirectly extended to the three-dimensional case.

Nevertheless, the discrete analog of the Jacobian positiveness for the mapping of the unit cube ontoa hexahedral cell can be obtained. We use the decomposition of the parametric cube to tetrahedra, whichare mapped onto the corresponding tetrahedra of the decomposed hexahedral cell. The mapping of eachtetrahedra is one-to-one. This approach is analogous to the approach used in 2D case for approximationof the functional Eq. 8.11 in such a way that it has an infinite barrier at the boundary of the set ofnondengenerate meshes. Recall that in Section 8.3.2 the quadrilateral cell is decomposed to two trianglesfirst by the one diagonal and then by the other. In the first and second decompositions the mapping isapproximated by the functions which are linear in each triangle. All the conditions of the Jacobianpositiveness for each of such mappings coincide with the condition for all the mesh cells to be convexquadrilaterals.

Consider a unit cube in the three-dimensional space ξ, η, µ, shown in Figure 8.4. We divide it intotwo prisms by the plane 1584. Then we devide the prism shown in Figure 8.4 into three tetrahedradrawing the diagonals 14, 25, 58, 45, and 46. Obtained tetrahedra denote as Tξ

5124, Tξ5684 and Tξ

5624. Note

x f y f z f f x f y f z f f x f y f z f fx y z x y z x y zξ ξ ξ ξ η η η η µ µ µ µ+ + = + + = + + =

f f y z y z J f y z y z J f y z y z J

f f x z x z J f x z x z J f x z x z J

f f x y x y J f x

x

y

z

= −( ) − −( ) + −( )= − −( ) + −( ) − −( )= −( ) −

ξ η µ µ η η ξ µ µ ξ µ ξ η η ξ


ξ η µ µ η η ξ yy x y J f x y x y Jµ µ ξ µ ξ η η ξ−( ) + −( )


that all these tetrahedra are equal to each other (with rotation and reflection taken into account) andone of the edges of the cube corresponds to each of them. For example, tetrahedron Tξ

5124 can be referredto the edge 12. Only one extra tetrahedrons is referred to this edge, namely Tξ

3126. What is the differencebetween tetrahedra Tξ

5124 and Tξ3126? The answer is that each of them corresponds to a proper type of

coordinate system, right-hand or left-hand. It is easy to compute the total number of such tetrahedra. Itis equal to double the number of the cube edges, i.e., 24. For the unit cube the volume of one tetrahedronis equal to 1/6, and the total volume of all such tetrahedra is equal to 4.

FIGURE 8.4 Vertex numeration and decomposition of the cube to tetrahedrons.

FIGURE 8.5 Vertex numeration for the base tetrahedron.

Consider the base tetrahedron shown in Figure 8.5. Vertices are enumerated from 1 to 4 as shown inFigure 8.5. Each vertex corresponds to a radius-vector r1, r2, r3, or r4 in the space x, y, z. All these vectors


define tetrahedron in the space x, y, z. We introduce the base vectors

Note that the coordinate system e1, e2, e3 is a right-hand system, which is easy to see from the orientationof the base tetrahedron in Figure 8.5. Hence, the volume of the “right” tetrahedron is equal to

At the same time, the volume of the “left” tetrahedron is equal to

Now, in analogy with the two-dimensional case, the condition for the mesh to be nondegenerate forthe three-dimensional hexahedral mesh can be expressed as follows:

(8.46)

where (JT left)m is a volume of the tetrahedron corresponding to the edge number m and defining the left-hand coordinate system, (JT right)m is a volume of the tetrahedron corresponding to the edge number mand defining the right-hand coordinate system (each cube has 12 edges), N is the cell number, Ne is thetotal number of cells. Conditions Eq. 8.46 define the discrete analog of the Jacobian positiveness in thethree-dimensional case. Meshes satisfying inequalities Eq. 8.46 we will call nondegenerate hexahedralmeshes.

As in the two-dimensional case, we should introduce the function COR(N,k) to define a correspon-dence between local and global node numbers:

where n is a global node number, Nn is a total number of mesh nodes, N is an element number, Ne is anumber of elements, k is a local node number in the element. This function is implemented in thecomputer program as a function for a regular grid and as an array for an irregular mesh.


Let adapted function f(x, y, z) define a three-dimensional surface in the four-dimensional space. Innotations of the previous section, the functional Eq. 8.24 can be written as follows:

(8.47)

where

e r r e r r e r r1 2 1 2 3 2 3 4 3= − = − = −, , .

J e e eT right = ×( ) ⋅1 2 3

J e e eT left = − ×( ) ⋅1 2 3

J J m N NT left m NT right m N

e ( )[ ] > ( )[ ] > = =0 0 1 12 1,..., ; ,...,

n COR N k n N N N Kn e= ( ) = =, ,..., ,..., ,..., = 1 1 1 8

Ig g g g g g g g g

g g g g g g g g g g g g g

r r r r r r r r r

r r r r r r r r r r r r r

=− ( ) + − ( ) + − ( )

− ( )[ ] − −( ) + −

11 22 12

2

11 33 13

2

22 33 23

2

11 22 33 23

2

12 12 33 13 23 13 12 23 22

ξ ξ ξ ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξggd d d

r13

ξξ η µ

( )∫

g r g r g r g g r r g g r r g g r rr r r r r r r r r11

222

233

212 21 13 31 23 32

ξξ

ξη

ξµ

ξ ξξ η

ξ ξξ µ

ξ ξη µ= = = = = ⋅( ) = = ⋅( ) = = ⋅( )

here


The functional Eq. 8.47 can be used for constructing harmonic coordinates on the surface of the graphof control function dependent on three variables. Projection of these coordinates onto a physical domaingives an adaptive-harmonic grid, clustered in regions of high gradients of adapted function f(x, y, z).

The problem of irregular three-dimensional mesh smoothing and adaption is formulated as follows.Let the coordinates of irregular mesh be given:

(8.48)

The mesh is formed by hexahedral elements, i.e., the array COR(N, k) is also defined. The problem is tofind new coordinates of the mesh nodes, minimizing the sum of the functional Eq. 8.47 values, computedfor a mapping of the unit cube onto each cell of a mesh.


First consider the case, where f(x, y, z) = 0. The functional Eq. 8.47 in this case can be written in a moresimple form:

(8.49)

where × is a vector product, and ⋅ is a scalar product,

Let the linear transform xh(ξ, η, µ), yh(ξ, η, µ), zh(ξ, η, µ) map the base tetrahedron Tξ1234 in the space

ξ, η, µ onto a tetrahedron T1234 in the space x, y, z. The value of the functional with the linear functionsxh(ξ, η, µ), yh(ξ, η, µ) and zh(ξ, η, µ) can be computed precisely. Consequently, the approximation ofthis functional can be written as

(8.50)

where

Consider one term in Eq. 8.50, for example, (Fm)left, and suppose that the Jacobian (Jm)left tends to zero,remaining positive. For Ih not to tend to infinity in this situation it is necessary that the numerator in(Fm)left must also tend to zero. From the form of the numerator it follows that vectors e1 = r2 – r1, e2 = r3

f f x f y f z f f x f y f z f f x f y f zx y z x y z x y zξ ξ ξ ξ η η η η µ µ µ µ= + + = + + = + +

x y z n Nn n, , ,...,( ) = 1

Ir r r r r r

r r rd d d=

×( ) + ×( ) + ×( )×( ) ⋅∫ ξ η ξ µ η µ

ξ η µ

ξ η µ2 2 2

r x y z r x y z r x y zξ ξ ξ ξ η η η η µ µ µ µ= ( ) = ( ) = ( ), , , , , ,

I F Fhm left m right

mN

N

N

e

= ( ) + ( )[ ]==

∑∑ 1241

12

1

Fr r r r r r

JF

r r r r r r

J

J r r r J

m left

h h h h h h

m left

m right

h h h h h h

m right

m left

h h hm right

( ) =×( ) + ×( ) + ×( )

( ) ( ) =×( ) + ×( ) + ×( )

( )( ) = − ×( ) ⋅ ( )

ξ η ξ µ η µ ξ η ξ µ η µ

ξ η µ

2 2 2 2 2 2

== ×( ) ⋅r r rh h hξ η µ

– r2 and e3 = r4 – r3 are parallel, hence all points r1, r2, r3, and r4 lie on a straight line. Consequently, thevolumes of all tetrahedra that contain corresponding faces must also tend to zero, including the tetra-


hedron defined by the edge 34 and containing the edge 23. Repeating the argument as many times asnecessary, we conclude that all mesh nodes, including those at the boundary of the domain, must lie ona straight line, which is impossible.

From this follows that the function Ih has an infinite barrier at the boundary of nondegenerate three-dimensional hexahedral meshes, satisfying inequalities Eq. 8.46. Hence, if this set is not empty, the systemof algebraic equations

has at least one solution which is a nondegenerate mesh. To find it, one must first find a certain initialnondegenerate mesh, and then use some method of unconstrained minimization of the function Ih. Sincethis function has an infinite barrier on the boundary of the set of nondegenerate meshes, each step ofthe method can be chosen so that the mesh always satisfies inequalities (Eq. 8.46).

For adaptive mesh generation with the employment of the functional Eq. 8.47, we use the sameapproach: consider T tetrahedra, described above. Then the mapping of the base tetrahedron onto eachof these tetrahedra is approximated by linear functions, with assumption that f is also approximated bya linear function defined by its values in tetrahedron vertices. Then the integrand in Eq. 8.47 will beequal to constant. Note that the integrand in Eq. 8.47 differs from Eq. 8.49: the first is an invariant forthe orthogonal transformations of the base tetrahedron. This means that we do not need to use twoterms in the approximation of Eq. 8.47 corresponding to right-hand and left-hand coordinate systems.The value of this functional depends only on the numeration of nodes of the base tetrahedron, not onits type.


Suppose the mesh at the lth step of the iterations is determined. We use the quasi-Newtonian procedurewhen the (l+1)-th step is accomplished by solving two linear equations for each interior node:

(8.51)

where τ is the iteration parameter, which is chosen so that the mesh remains nondegenerate. For thispurpose after each step the conditions Eq. 8.46 are checked and if they are not satisfied, this parameteris multiplied by 0.5.

The adaptive-harmonic algorithm for the three-dimensional mesh is formulated as follows:

1. Generate initial mesh with the use of a marching method.2. Compute new values fn at each mesh node.3. Make one iteration step Eq. 8.51 and compute new values of xn, yn, and zn.4. Repeat starting with Step 2 to convergency.

Note, that the algorithm contains computational formulas for [fx]n, [fy]n and [fz]n which will bepresented below.

RI

xR

I

yR

I

zx

h

ny

h

nz

h

n

= = = = = =∂∂

∂∂

∂∂

0 0 0

τ ∂∂

∂∂

∂∂

τ∂∂

∂∂

∂∂

RR

xx x

R

yy y

R

zz z

RR

xx x

R

yy y

R

zz z

xx

nnl

nl x

nnl

nl x

nnl

nl

yy

nnl

nl y

nnl

nl y

nnl

nl

+ −( ) + −( ) + −( ) =

+ −( ) + −( ) + −( ) =

+ + +

+ + +

1 1 1

1 1 1

0

00

01 1 1τ ∂∂

∂∂

∂∂

RR

xx x

R

yy y

R

zz zz

z

nnl

nl z

nnl

nl z

nnl

nl+ −( ) + −( ) + −( ) =+ + +


We will obtain computational formulas in the case of adaption, i.e., we will approximate the functional


Eq. 8.47. The used approach is similar to the method of approximation to the functional described inSection 8.3.

Consider the linear transform xh(ξ, η, µ), y h(ξ, η, µ), zh(ξ, η, µ) of the base tetrahedron shown inFigure 8.5 onto one of tetrahedra of the cell decomposition. Function f will be approximated by the linearfunction fh(ξ, η, µ). Derivatives of these functions can be easily computed taking into account thenumeration of the vertices of the base tetrahedron:

From this follows

i.e.,

(8.52)

Substituting these expressions into the integrand of Eq. 8.47 we obtain

where

(8.53)

(8.54)

We use formulas for differentiating the relation of two functions. After differentiating we obtain

(8.55)

r x y z f r r x x y y z z f f

r x y z f r r x x y y z z f f

r x y

h h h h h

h h h h h

h h h

ξ ξ ξ ξ ξ

η η η η η

µ µ µ

= ( ) = − = − − − −( )= ( ) = − = − − − −( )=

, , , , , ,

, , , , , ,

,

2 1 2 1 2 1 2 1 2 1

3 2 3 2 3 2 3 2 3 2

,, , , , ,z f r r x x y y z z f fh hµ µ( ) = − = − − − −( )4 3 4 3 4 3 4 3 4 3

g r r r rij i i j j= −( ) ⋅ −( )+ +1 1

g r r g r r g r r

g g r r r r

g g r r r r

g g r r r r

11 2 1

2

22 3 2

2

33 4 3

2

12 21 3 2 2 1

13 31 4 3 2 1

23 32 4 3 3 2

= −( ) = −( ) = −( )= = −( ) ⋅ −( )( )= = −( ) ⋅ −( )( )= = −( ) ⋅ −( )( )

FU

V=

U g g g g g g g g g= − ( ) + − ( ) + − ( )11 22 12

2

11 33 13

2

22 33 23

2

V g g g g g g g g g g g g g g= − ( )[ ] − −( ) + −( )11 22 33 23

2

12 12 33 13 23 13 12 23 22 13

FU FV

VF

U FV

VF

U FV

V

FU F V FV

VF

U F V FV

VF

U F V FV

V

F FU F V F V FV

VF F

U F V

xx x

yy y

zz z

xxxx x x xx

yyyy y y yy

zzzz z z zz

xy yxxy x y y x xy

xz zxxz x

= − =−

= −

= − − =− −

= − −

= =− − −

= = −

2 2 2

zz z x xz

yz zyyz z y y z yz

F V FV

V

F FU F V F V FV

V

− −

= =− − −

For the vertex 1 of the tetrahedron we should substitute the expressions Eq. 8.52, Eq. 8.53, and Eq. 8.54into Eq. 8.55, and also replace x, y and z by x1, y1 and z1 in the resulting formulas.


For the vertex 2 x, y, and z in Eq. 8.55 are replaced by x2, y2, and z2.For the vertex 3 x, y, and z in Eq. 8.55 are replaced by x3, y3, and z3.For the vertex 4 x, y, and z in Eq. 8.55 are replaced by x4, y4, and z4.

In computing the derivatives of fi on xj, yj, and zj, i = 1, …, 4, j = 1, …, 4, we use the formulas for thetransformation of derivatives in the three-dimensional space:

From this follows

(8.56)

where

Note that the derivatives on x, y, and z are independent on which system of coordinates, right-hand orleft-hand is used in Eq. 8.56. Substituting the expressions for the derivatives of xh, yh and zh on ξ, η, µinto Eq. 8.56, we obtain formulas for the derivatives f h

x, f hy, and f h

z. We use the following formulas incomputations:

Computations are performed as follows. Let F and its derivatives on x1, y1 and z1 in the numerationof the base tetrahedron be computed with the use of formulas Eq. 8.55 for the cell number N and thelocal node number k. Then the computed values are added to the appropriate array elements (whichwere first cleared):

(8.57)

where n = COR(N, k1). Here, a+ = b means that the new value of a becomes equal to a + b. Similarly for the vertex 2, the correspondence between local and global number is n = COR

(N, k2).Similarly for the vertex 3, the correspondence between local and global number is n = COR

(N, k3).

x f y f z f f x f y f z f f x f y f z f fx y z x y z x y zξ ξ ξ ξ η η η η µ µ µ µ+ + = + + = + + =

f f y z y z J f y z y z J f y z y z J

f f x z x z J f x z x z J f x z x z J

f f x y x y J f x

x

y

z

= −( ) − −( ) + −( )= − −( ) + −( ) − −( )= −( ) −



ξ η µ µ η η ξ yy x y J f x y x y Jµ µ ξ µ ξ η η ξ−( ) + −( )

J x y z y z x y z y z x y z y z= −( ) − −( ) + −( )ξ η µ µ η η ξ µ µ ξ µ ξ η η ξ

∂∂

∂∂

∂∂

f

x

f if i j

if i j

f

y

f if i j

if i j

f

z

f if i j

if i ji

j

xh

i

j

yh

i

j

zh

==≠

==

≠

==≠

0 0 0

I F R F R F R F

R F R F R F

R F R F R F

hx n x y n y z n z

xx n xx yy n yy zz n zz

xy n xy xz n xz yz n yz

+ = [ ] + = [ ] + = [ ] + =

[ ] + = [ ] + = [ ] + =

[ ] + = [ ] + = [ ] + =

Similarly for the vertex 4, the correspondence between local and global number is n = COR(N, k4).


So, the iteration method for irregular three-dimensional mesh relaxation and adaption is describedin detail.

8.11 Results of Test Computations

8.11.1 Comparison Between the Winslow Method and the Variational Approach

Comparison between the variational algorithm described in Section 8.3 and the Winslow method waspresented in the paper by Ivanenko and Charakhch’yan [1988]. We will describe here results of compu-tations shown in Figure 8.6. In Figure 8.6 the regular grids 10 × 10, 19 × 19 and 37 × 37, generated forbackward facing step by the Winslow method (Figures 8.6a, 8.6c, 8.6e) and by the variational barriermethods (Figs. 8.6b, 8.6d, 8.6f) are shown. The choice of this example is concerned with the discussionabout the applicability of the Winslow method. There is an opinion that this method can generate quitesatisfactory grids if the number of grid nodes is sufficiently large, despite the fact that in many cases thismethod generates grids with self-intersecting cells. Indeed, if the number of grid nodes tends to infinity,the limit will be a continuous mapping which is one-to-one. Such a mapping can be used then for thereplacement of independent variables (Jacobian is positive inside a domain). This is not the case of adiscrete mapping (a grid). If the Jacobian is negative on the boundary, then the Winslow method mightgenerate grids with degenerate cells near the boundary for any number of grid nodes. As shown in thepresented example, the form of degenerate cell near the internal corner is worse with increasing thenumber of nodes (the Winslow method, Figures 8.6a, 8.6c, 8.6e). At the same time, the variational methodgenerates satisfactory (convex) grids for any number of grid nodes (Figures 8.6b, 8.6d, 8.6f).

The geometric sense of the smoothing procedure defined by harmonic functional is that the shape ofeach cell tends to be a square. From this follows constraints on the application of the variational methodfor irregular meshes. In fact, satisfactory mesh with square cells might not exist for the given meshstructure. It is clear that if the square cell is used as initial, the variational method will not change it (theWinslow finite-difference method will not change it, too). If the initial mesh has the form shown inFigure 8.7a we obtain the irregular smoothed mesh shown in Figure 8.7b after 700 iterations.

The grid quality was estimated with the following parameters: Jmin is the minimum of the areas of alltriangles, scaled by the maximum area, Aspect is the maximum ratio of edge lengths in quadrilateral, andSkew is the minimum cells angle in degrees. For meshes in Figure 8.7 the minimum area decreases from0.13 to 0.0002, the maximum ratio increases from 10 to 10.3 and minimum angle decreases from 13.9to 11.7. But the mesh in Figure 8.7b looks more smooth than the mesh in Figure 8.7a. This means thatall these quality parameters do not estimate the mesh quality properly. Note that the mesh after smoothinglooks like several cobwebs and is extremely nonuniform. This example shows that in some cases thevariational method can be unsatisfactory for smoothing of irregular meshes, for example, if refinementis used for several blocks with regular grid structure in each as shown in Figure 8.7a.

8.11.2 Comparison Between the Finite-Difference Method for Two-Dimensional Adaptive-Harmonic Meshes and the Variational Approach

Methods for adaptive mesh generation are illustrated by the following example of control function[Ivanenko, 1993]. The square domain 0 < x < 1, 0 < y < 1 is considered. The cubic curve

y x x x x0 25 0 5 0 75 0 25 0 5( ) = −( ) −( ) −( ) +. . . .


determines the form of a layer of high gradients. For a given point x, y the function f(x, y) is calculatedas follows:

Here

FIGURE 8.6 Regular grid 10 × 10, 19 × 19, and 37 × 37 generated by the Winslow method (a,c,e), and by thevariational barrier method (b, d, f).

f

if y y

y y if y y y

if y y

=

≥ +

− +( ) + ≥ ≥ −

≤ −

1

0 5

0

0

0 0 0

0

δ

δ δ δ δ

δ

.

δ δ ∂∂

= +

0

0

2 1 2

1y

x


The value of δ is chosen so that the width of the layer will be about 2δ0 everywhere along the curve. Inall test computation this value was chosen to be δ0 = 0.02.

An additional control parameter C is introduced to control the number of mesh nodes inside theboundary or internal layers. The function Cf(x, y) is used in computational formulas instead of f(x, y).Increasing the value C, more mesh nodes will be in the layer of high gradients. This value is chosen inthe range from 0.1 to 0.5. A number of points in a layer is approximately C/(C + 1), i.e., if C = 0.5 onethird part of points will be in a layer of high gradients.

The grid, generated by the finite-difference method with C = 0.2 slightly differs from the gridgenerated by variational method with the same value of parameter C. But with the value of parameterC = 0.5, the satisfactory grid cannot be generated by the finite-difference method (Figure 8.8a). Thegrid generated for this value of parameter by the variational method is shown in Figure 8.8b. All gridcells are convex.

FIGURE 8.7 Smoothing of irregular mesh; (a) initial mesh, (b) smoothed mesh.


8.11.3 Comparison Between the Finite-Difference Method for Adaptive-Harmonic Grid Generation on Surfaces and the Variational Approach

The comparison of the finite-difference method for grid generation on surfaces with the variationalmethod was performed on an example of a surface defined parametrically:

“Monkeys saddle”

Methods for adaptive mesh generation on surfaces are illustrated on the example of control function, definedin previous subsection with u and v replaced by x and y. An additional control parameter C is also introducedto control the number of mesh nodes inside the boundary or internal layer. If C < 0.4, the finite-differencemethod generates quite satisfactory grids on the surface. But if C = 0.5, the finite-difference method generates

FIGURE 8.8 Adaptive-harmonic grids; (a) generated by the finite-difference method, (b) generated by the varia-tional method.

x u y v z v u v u v= = = −( ) − −( ) −( ) < < < <, . . . 8 0 3 24 0 5 0 5 0 1 0 13 2


degenerate grid shown in Figure 8.9, i.e., triangles with negative areas appear in the parametric space u, v, asshown in Figure 8.9a. There is also a problem with convergency of iterative process. Such meshes are oftenunsuitable for computations. At the same time, variational method gives us a satisfactory mesh, shown inFigure 8.10. The grid generated in the parametric space u, v is shown in Figure 8.10a.

8.11.4 Comparison Between the Finite-Difference Method for Adaptive-Harmonic Three-Dimensional Meshes and the Variational Approach

The comparison between variational and finite-difference methods was performed with the grid qualityestimated by the following parameters: Jmin is the minimum of the tetrahedra volumes, scaled by themaximum volume, Aspect is the maximum ratio of lengths of adjacent edges, and Skew is the minimalangle between edges in degrees.

FIGURE 8.9 Adaptive-harmonic grid on the surface generated by the finite-difference method; (a) the grid in theparametric space u, v, (b) the grid on the surface.


Methods for adaptive mesh generation are illustrated using the same example of the control functiondependent only on two variables x and y. An additional control parameter C is introduced to control thenumber of mesh nodes inside the boundary or internal layer.

The domain is a cube with a pedestal in the middle of the down face.An adaptive grid generated in the domain by the finite-difference method with C = 0.2 is shown in

Figure 8.11. Values of quality parameters are shown in the figure. The projection of the mesh surface µ= 3 onto the plane z = 0 is shown in Figure 8.11a. The section of the mesh in Figure 8.11c shows thepresence of degenerate cells (Jmin = – 0.3). At the same time, the mesh shown in Figure 8.12 generatedfor the same domain with the same parameter C by the variational method does not contain degeneratecells (Jmin = 0.02).

Note that the control function is two-dimensional, but the generated adaptive grids are substantiallythree-dimensional. Moreover, variational method generates are more fitted to control function mesh.The same results can be obtained for irregular mesh smoothing and adaption.

FIGURE 8.10 Adaptive-harmonic grid on the surface generated by the variational barrier method. (a) The grid inthe parametric space u, v, (b) the grid on the surface.


8.12 Conclusions

Algorithms for adaptive regular and irregular mesh generation in two and three dimensions as well asfor surfaces are considered in the present chapter. The approach is based on the theory of harmonicmaps. Formulated algorithms can be used for grid/mesh generation with strong clustering of mesh nodesand assure generation of nondegenerate meshes. The main conclusion is the following. The meshesproduced by irregular mesh smoothing and adaption are better for more regular meshes.

The variational algorithm for three-dimensional meshes appear to be cumbersome. At the same timeit is approximately 10 times more expensive than the finite-difference method for regular grids.

These investigations have been stimulated by the need in fully automatic numerical solvers for thecomplex problems of mathematical physics. This means that the human intervention into the solutionprocess, especially into adaptive grid generation, should be minimized. Modern methods do not alwayssatisfy these conditions, so the development of new fully automatic grid generation algorithms is of greatimportance today.

FIGURE 8.11 Adaptive-harmonic three-dimensional grid 19 × 19 × 7 generated by the finite-difference method;(a) projection of the coordinate surface µ = 3 onto the x, y plane, (b) coordinate surfaces µ = 1 and η = 19, (c)coordinate surfaces µ = 2 and η = 11, (d) coordinate surfaces µ = 4 and η = 11.


References

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FIGURE 8.12 Adaptive-harmonic three-dimensional grid 19 × 19 × 7 generated by the variational barrier method;(a) projection of the coordinate surface µ = 3 onto the x, y plane, (b) coordinate surfaces µ = 1 and η = 19, (c)coordinate surfaces µ = 2 and η = 11, (d) coordinate surfaces µ = 4 and η = 11.

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NJ, 1973.24. Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation, North-Holland, NY, 1985. 25. Winslow, A. M., Numerical solution of quasilinear Poisson equation in nonuniform triangle mesh,

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(in Russian) pp. 157–163, 1977.

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