An algorithm for the optimization of directionally stretched triangulations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS I N ENGINEERING, VOL. 37, 1481- 1497 (1 994)

AN ALGORITHM FOR THE OPTIMIZATION OF DIRECTIONALLY STRETCHED TRIANGULATIONS

HOU ZHANG AND JEAN-YVES TREPANIER

Department ol Mechanical Engineering, €Cole Polyrechnique de Montreal, Montreal, Quebec, Canada, H3C 3A7

SUMMARY A procedure for the optimization of stretched triangular grids is described. The method is based on the construction and minimization of a function that represents a generalized version for stretched grids of a non-linear spring system. The function is minimized using a gradient method based on the steepest descent. Examples are provided to show the applicability of the method to computational fluid dynamics problems.

1. INTRODUCTION

Triangulations are now widely used for the discretization of complex planar domains needed in the solution of engineering problems. For example, finite element and finite volume analyses in Computational Fluid Dynamics (CFD) have benefited from the high flexibility of such meshes, especially when grid adaptation is used.'-3 Adaptive methods can be classified in two main families. The first allows modification in mesh connectivity and comprises various discrete operators such as refinement (node addition), coarsening (node deletion) and/or diagonal swap- ping. The second family allows only for a continuous operator related to nodal movement or relocation, keeping the initial connectivity of the triangulation. These families are complementary and both are needed in order to produce a grid that satisfies arbitrary requirements for triangle size and stretching distributions.

A general relocation method can be useful in many situations. It may be combined with a method from the first family to smooth out the large variation in grid characteristics generated by the discrete behaviour of these operators. It may be useful in smoothing grids generated by other types of grid generators, such as the advancing front method4 or a Delaunay-based method.' Finally, it may be used alone as an adaptation strategy in cases where the increased book keeping associated with connectivity changes must be avoided (as in the case of structured grids6).

Laplacian smoothing, the simplest and most frequently used method, moves each interior point to the centroid of its neighbours in an iterative p r o ~ e s s . ~ For convex geometries, this algorithm is satisfactory, but the method may not work for some non-convex domains because some triangles can overlap. To solve this problem, Kennon and Anderson' have developed a non-linear spring scheme. A different approach for the smoothing was presented by Parthasarathy and Kodilayamg in which a constrained optimization method was used to minimize a cost function defined from aspect ratios of the triangles.

These procedures assume that the objective of the relocation method is to produce triangles that are as close as possible to equilateral. This is geometrically correct, but for fluid-flow computations, grid stretching must also be considered by any grid optimization method. Cabello

CCC 0029-5981/94/091481- 17%9.00 0 1994 by John Wiley & Sons, Ltd.

Received I September 1992 Revised 4 October 1993

1482 H. ZHANG AND J.-Y. TREPANIER

et a/." first studied this problem from an optimization point of view based on the smoothing of the triangulation in a transformed plane. In this paper, we introduce a variational method for smoothing an arbitrarily stretched triangulation. The method is derived from the optimization of a function which represents a non-linear spring system in the transformed plane, but the iterative procedure is directly carried out in the physical plane. The paper will concentrate on describing the formulation of the system of equations to be minimized and a simple method will be implemented to realize this optimization.

2. SMOOTHING REGULAR TRIANGULATIONS

2.1. Basic geometric relations

Let us assume that we have a list of nodes N, and assume that we also have an element list T giving the connectivity of the triangulation. A triangular element is defined by three points r l , r, and r3 in a counterclockwise direction, while its side vectors are defined by Arl , Ar2 and Ar3 (see Figure 1). Each point r is defined by its two components x and y in Cartesian co-ordinates. Some basic requirements must be fulfilled for the triangulation be valid for domain discretization: the elements must cover the whole domain to be discretized (except for discretization errors at the boundaries) and no triangle overlapping is allowed. The quality of the individual triangles may be related to the following two quantities:

A = J?Ar, x Ar,

A is proportional to the Jacobian of the triangle (twice its area), while B is the so-called potential energy of the triangle.* It can be shown that the modulus of A is always less than or equal to the value of B. They are equal if and only if the triangle is equilateral. Thus, the dimensionless quantity

A Q=, which varies from zero to one, can be used as a measure of the equilarity of the triangle.

f-1

Figure 1. Definition of triangle points and side vectors

DIRECTIONALLY STRETCHED TRIANGULATIONS

2.2. A minimization problem

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A spring smoothing scheme is derived by minimizing the function that represents the total potential energy of the triangulation given by

O = C K B T

(3)

where K is a penalty for the spring system. The Laplacian smoothing scheme is obtained by using K = 1. However, the penalty can be accomplished differently for better control of the grid. Such a penalty was introduced by Kennon and Anderson’ to treat the case of non-convex domains, and K = 1/A was used.

This example exhibits an important fact that should be considered: which is most important, size or quality? If it is quality then the coefficients of the nodal points are expected to be a function of the quality Q only. This requirement can be accomplished by choosing

K = 14(Q) (4)

where 4(Q) is a function that depends only on the quality of the triangle and ,? is a scalar weighting to be defined later. Since equation (3) is continuous with respect to the position of the nodes r, a minimum of the function will be found when the gradient of equation (3) with respect to r is zero.

2.3. Gradient of @

Consider Figure 2. We can observe that the movement of a nodal point r affects only equation (3) through its neighbouring triangles. For a given node in N, therefore, the two components of the gradient of O can be written as (see Appendix I for details)

Ti+, Ti+ ...

Figure 2. Effect of node movement on @ in its neighbourhood


where i = 1,2,. . . , Nb(rNb+ = r l ) refers to the neighbouring triangles and nodes of point r and

& = Q l l - 4 aA ar 5 = - = f iD(r i -

and

D = [ - 1 0 '1 (7)

3. EXTENSION T O STRETCHED TRIANGULATIONS

Cabello et a/." were among the first to propose an optimization procedure for stretched triangulations. Their algorithm is composed of three steps: they first transform stretched triangles into unstretched ones defined in an image plane; then, the transformed triangles are optimized by comparing them will equilateral ones; and finally, they are transformed back to the physical space. In the last step, since a single physical point may not be obtained, an averaging procedure is applied. In this section, we will show that it is possible to work directly on the physical plane without invoking the last transformation. The crux of the method is the definition of the quality of a stretched triangle. To introduce this concept, the following definition of triangle stretching is needed.

3.1. Dejnition of stretching

Definitions of stretching modulus can be obtained, for example, by measuring the ratio of the longest to the shortest side, the ratio of circumscribed-circle radius to inscribed-circle radius, or other similar scalar procedures. A vector alternative, which allows the computation of stretching modulus and orientation, consists in considering the transformation of a given triangle into an equilateral one, as illustrated in Figure 3. The triangle is first described in the reference frame given by the new axis (k, 5 ) and then stretched by a factor 1/E in the direction 1. Such a transformation can be used to measure the stretching of any triangle." This means that, for a given triangle, a couple (E, 0) can always be found such that when the transformation is applied, the triangle becomes equilateral. This couple (E, 0) can thus be used as the definition and the measure of triangle stretching.

Let the direction of stretching be specified equivalently by a vector (a, b) and the stretching modulus by E = Jm. Then, let the transformed quantities be denoted by an overline; the transformation may be expressed in the following matrix form:

(8) where the subscript 0 denotes the centroid of the triangle, and the matrix M is given by (see Appendix I1 for details)

i = M(r - ro)

DIRECTIONALLY STRETCHED TRIANGULATIONS 1485

Figure 3. Transformation of a triangle into an equilateral one

3.2. Quality of stretched triangles

The quality of a stretched triangle can now be computed. First, an objective must be defined which, in the isotropic case, is implicitly an equilateral triangle. A spatial distribution of stretching amplitude and orientation is needed which is considered as data from the mesh optimization point of view. For any given triangle, suppose that a local objective is given by a couple ( E o , Oo). If the actual triangle, after the application of the transformation with couple ( E o , O0), maps into an equilateral triangle, then the triangle meets the objective and its quality is good. On the other hand, if, after transformation, the triangle maps into a highly stretched triangle, its quality is poor.

The quality of stretched grids can now be measured: we first apply the transformation with the couple ( E o , 0,) to a triangle and then compute the quality of the resulting triangle in the transformed plane:

- A

B Q = -

3.3. Optimization of stretched triangles

system given by equations (3) and (4) can be obtained by the minimization of the function Following the previous discussions and definitions, a generalized form of the non-linear spring

Since equation (8) provides the relationship between the physical and transformed triangles, equation (10) depends directly on the nodal point co-ordinates in the physical plane. On

1486

observing that

H. ZHANG AND J.-Y. TREPANIER

‘M 2r - ri - T aB

ar M - = M (

M - = f i M T D M ( r i - r i + l ) = - ( r i - r i + l ) a2 J S D ar E

the generalized version of equation (5) for stretched triangulations can be written as

where ij = P(2r - ri - r i + l ) - r = Q q - f

z - 5 5 = ,

and a2/E4 + b2/E2 ab(l/E4 - 1/E2)]

P = M T M = [ab(l/E4 - 1/E2) a 2 / E 2 + b2/E4 (14)

4. TREATMENT OF BOUNDARY NODES

Unlike internal points, which may move in any direction, boundary points are constrained to move along the curves that define the boundaries. A boundary is usually defined as a curve that may be expressed as a function of its arc length s as

r = rb(s) (15)

Consequently, the functional a will be the function of internal points r and the arc lengths of the boundary points. For a boundary point r = rb(s), the s component of Q, is given by

This requires at least r’(s) to be continuous. In this study, it is further required that the second derivative be continuous. However, if r’(s) or r”(s) are not continuous, they must be approximated by a smooth function around that point. A simple treatment is to let r”(s) vary continuously and linearly from its left value to the right value within a given interval around the point and r’(s) is evaluated by integrating r”(s) in that interval. Also, corner points are usually fixed during the smoothing procedure.

5. MINIMIZATION METHODOLOGY

The solution of the previously defined optimization problem requires a numerical algorithm that is simple and non-oscillating, because, in the context of adapting a mesh, the optimization will


seldom be pursued until full convergence. In this work, the steepest descent method has been used, mainly because of its simplicity. It is well known that this simple algorithm can converge very slowly.12 However, it is sufficient for testing the formulation of the problem, leaving the implementation of some of the more sophisticated optimization strategies to a future work.

5.1. Direction of steepest descent

the internal and boundary points may be calculated, respectively, by The steepest descent is given by the negative of equations (1 2) and (16). So, the displacements of

6so = - --.rk(s) [: Consequently,

r = ro + v6ro

will provide new positions for the internal points and

will give the position of the new boundary points. The superscript 0 represents points from the previous iteration, and v is a scaling factor. In order to find the best value for the scaling factor v, we again minimize the function along the descent direction, which is now a function of v given only by

F(v) = @(r; rEN) (20)

The minimization requires that F’ = 0 and F“ > 0. Their explicit expressions are (see Appendix 111 for details)

F’(v) = - 6r(v) - 6r0

F”(v) = - x 6 r ‘ ( v ) . 6 r 0 N

N

where 6r(v) = - aO/ar and 6r’(v) = - (d/dv)a@/dr is given by

where

- 1 - Q‘= -=([-6r0 + &.dry + &+l-6r!+,) €3

= @J

ti‘ = &jo

Similar formulas can be obtained for the nodes on the boundaries.


5.2. Minimum along the descent direction

In order to solve F'(v) = 0, a Newton iteration has been used. Thus, the value of v is updated following the iterative procedure

It is well known that the difficulty in obtaining a solution via equation (24) is to specify a proper initial value for v . Obviously, there is no general method for specifying this value for diverse meshes. However, we can obtain estimates for the minimum and maximum values of v. The minimum value vmin may be set to zero, while the maximum value v,,, is computed as the minimum value of the equation A = 0 for all triangles in the list T, since triangles are not allowed to overlap.

To complete search strategy in the descent direction can now be defined: the initial value for the Newton iteration is set to be the average of the minimum and the maximum values of v. Once the computed F' and F" satisfy F' < 0 and F" > 0, v is improved by the application of equation (24). Otherwise, v,,, is updated to the previous value of v and, consequently, v is replaced by the new averaged value. This procedure is repeated until I F'(v)l is less than a given tolerance.

6. EXAMPLES OF MESH OPTIMIZATION

6.1. Non-convex domain

The first example has been selected to show the performance of the method in the case of a non-convex domain. The initial grid is reproduced in Figure 4. This grid has been produced by cutting an initially structured grid. The application of the Laplacian smoothing operator (linear spring system) to this grid results in overlapping triangles, as illustrated in Figure 5.

The current algorithm has then been used to smooth this grid. For better comparability with the Laplacian operator, the stretching objective for all triangles was set to (E, 0) = (1 ,O) (no

Figure 4. initial grid in a non-convex domain


Figure 5. Grid in a non-convex domain after Laplacian smoothing

Figure 6. Grid in a non-convex domain after optimization

stretching) and the boundary nodes were not allowed to move. For this case, as well as for the following tests, the function $(Q) was taken as

The resulting smoothed grid is reproduced in Figure 6. The optimized grid resembles the grid of Figure 5 away from the concave boundary, but the effect of the function $(Q) allows for better control of the grid in this region.


6.2. CFD-like problems

In this section, we will present two examples to illustrate the applicability of the method to computational fluid dynamics problems. These examples will be computed using the mesh of Figure 7 as the initial grid.

6.2.1. Shock-like optimization. In cases where a shock wave appears on the suction side of an aerofoil, it is essential that the grid be stretched in the direction of the shock, from the boundary of the aerofoil to some distance in the flow field. These flow demands are simulated by imposing

Figure 7. Initial grid used in the CFD-like examples

Figure 8. Stretching requirements for simulated shock-like optimization

Figure 9. Optimized grid for simulated shock


stretching requirements on the initial grid, as illustrated in Figure 8, where these requirements are associated with the triangles directly. The optimized grid is illustrated in Figure 9.

6.2.2. Shock-boundary-layer interaction-like optimization. In the case of a shock-boundary- layer interaction, the grid requirements for stretching can be simulated as illustrated in Figure 10. In the shock and boundary-layer regions, the required stretching modulus is high, while in the

Figure 10. Stretching requirements for simulated shock-boundary-layer interaction

Figure 11. Optimized grid for simulated shock-boundary-layer interaction

p = 1.0 u = 2.9 u = 0.0

E = 5.991

u = 2.6185 u = -0.5063 E = 9.8702

Reflected shock B I 1 1.0

Figure 12. Computational domain for the shock reflection problem


interaction region, an isotropic grid was requested. The resulting optimized grid is reproduced in Figure 11.

For the CFD-like problems described above, the triangular elements in the central region of the shock and the boundary layer have almost reached their objective for stretching and the transition region allows for a smooth gradation from stretched to regular triangles.

6.3. Inviscid shock reflection

As a final example, the optimization algorithm is used in an adaptive strategy for the computation of an inviscid shock reflection. Figure 12 illustrates the computational domain and the boundary conditions used. This is a standard test case for the inviscid Euler equations.

The initial grid and solution are reproduced in Figure 13. The shock is diffused over two or three cells, which is typical of the solver used, which is an unstructured grid finite volume implementation3 of the approximate Riemann solver of Roe."

Starting from this solution, a simple strategy has been implemented as a coupling between the flow solver and the optimization algorithm. The gradient of the density field has been estimated on each triangle based on neighbouring information and the stretching requirements have been set to

E = 1 + klVpl

0

- - """-"" "-"---"--" - -"--" "-"-"--

Figure 13. Initial grid and solution for the shock reflection problem: (a) grid, (b) isolines of Mach number


By doing so, the stretching direction I ) is set orthogonally to the gradient of the density field, while the stretching modulus is set proportional to the modulus of this gradient. The constant k is adjusted to control the maximum stretching imposed in the computational domain.

Figure 14 illustrates the grid and solution for a case where the maximum stretching was set to three. It can be appreciated that even a small grid distortion in the direction of the shock can lead

Figure 14. Optimized grid and solution lor the shock reflection problem: (a) grid, (b) isolines 6f Mach number

x Y .A u1 c al Q

3 . 5 0

3.00

2.50

2.00

l.SO

1.00

I

Analytical - Initial grid ---- Optimized grid ----.

1 0.00 0.50 1.00 1.50 2.00 2.50 3.00

Position

Figure IS. Density profile for the shock reflection problem


to an important improvement on the solution accuracy. This is because Roe's approximate Riemann solver introduces more diffusion when the grid lines are not aligned with the main waves in the solution. Grid optimization is a simple way to perform this alignment. The improvement of the solution accuracy can be best appreciated in Figure 15, where the density profile at y = 0.3 is compared with the analytical solution.

7. CONCLUSION

A methodology has been designed and implemented for the optimization of arbitrarily stretched triangulations, The construction of a function representing a non-linear spring system and its minimization has produced very good results in non-convex domains, as well as for two CFD-like grid requirements. The application of the optimization algorithm in an adaptive strategy for the computation of an inviscid shock reflection problem has shown the great potential of the approach.

The rate of convergence of the algorithm can be improved by using a more sophisticated optimization strategy. Also, the method can produce triangles with obtuse angles, which can cause some problems in the stability of the discretized eq~at ions . '~ . l 5 A different measure of quality is being developed to improve this situation.

ACKNOWLEDGEMENTS

The authors would like to thank the NSERC of Canada for their financial support. Also, our special thanks go to Dr. Dominique Pelletier, who gave us useful comments on the first draft of this paper.

APPENDIX I: GRADIENT OF 0

Consider the function representing a non-linear spring system as

~ = C K B T

where

K = WQ) (26)

For a given node in N with Nb neighbouring triangles (Figure 2), the two components of the gradient of @ can be written as

The evaluation of the gradient of A and B is obtained as follows. Consider a neighbouring triangle, say (r, ri, r i+ (see Figure 2); we have

A = f i [ ( x i - x ) ( y i + 1 - Y ) - (yi - y ) (x i+ 1 - X)I


and the two components of its gradient are

1 - ( Y i t 1 - Y) + (Yi - Y) - (Xi - x) + ( X i + 1 - x)

The potential energy for this triangle is

B = f [ ( x - x i ) ’ + (X - X i t 1 ) ’ + ( - x i - x ~ + I ) ’ + ( y - Yi)’ + ( Y - ~ i t l ) ’ + ( Y i - Yi+1)’]

and its gradient is

aB 1 - 2(Xi - x) + 2(x - x i + I )

- 2(yi - Y ) + 2(y - ~ i + l )

Finally, equation (27) may be written in a compact form,

where

and

aB ar

q = - = 2r - ri - r i f l

C = Q t l - 5

aA 4 = - = fiD(ri - r i t l ) ar

D = [ - 1 0 ’1 APPENDIX 11: MATRIX OF TRANSFORMATION

The transformation that maps a given triangle (rl, r2, r3) into an equilateral one comprises three steps. The reference frame is first translated into the centroid of the triangle, giving

where f = r - r o

ro = +(r, + r2 + r3)

Then the triangle is described in the reference frame given by the rotated axes (2, j ) , yielding

1496 H. ZHANG AND J.-Y. T R ~ P A N I E R

Finally, a scaling by 1/E is applied in the ;direction, giving - - 2

E j j = y

x = -

- (33)

Combining equations (3 1)-(33), the complete transformation can thus be written as

f = M ( r - ro) (34)

where the matrix M is given by

APPENDIX 111: DERIVATIVE ALONG THE DESCENT DIRECTION

Once the descent direction has been chosen, the optimization problem is now a function of v only, given by

F(v) = #(r; r E N)

In order to find the best value for the scaling factor v, we minimize equation (36), and this requires that F' = 0 and F" > 0. On observing that r is a linear function of v, a r p v = bro (from equation (1 8)), we have

(36)

F"(v) = - 1 br'(v) bro N

where, using equation (12),

Noting that

a2 - a B - - 6 r - Q - - . S r o

ar

(37)


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An algorithm for the optimization of directionally stretched triangulations

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