c 1999 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J. SCI. COMPUT. Vol., No. 6, pp c 999 Societ for Industrial and Applied Mathematics A STUDY OF MONITOR FUNCTIONS FOR TWO-DIMENSIONAL ADAPTIVE MESH GENERATION WEIMING CAO, WEIZHANG HUANG, AND ROBERT D. RUSSELL Abstract. In this paper we stud the problem of two-dimensional adaptive mesh generation using a variational approach and, specificall, the effect that the monitor function has on the resulting mesh behavior. The basic theoretical tools emploed are Green s function for elliptic problems and the eigendecomposition of smmetric positive definite matrices. Based upon this stud, a general strateg is suggested for how to choose the monitor function, and numerical results are presented for illustrative purposes. The three-dimensional case is also briefl discussed. It is noted that the strateg used here can be applied to other elliptic mesh generation techniques as well. Ke words. variational mesh generation, mesh adaptation, monitor function, weight function, Green s function AMS subject classifications. 65M5, 65M, 35K5 PII. S Introduction. One of the major tasks in solving partial differential equations (PDEs) is the adaptive generation of the mesh, or grid. In two (or higher) spatial dimensions, this mesh generation and adaptation is commonl done using the variational approach, specificall, b minimizing a functional of the coordinate mapping ξ = ξ(, ), η = η(, ) between the phsical domain Ω p with variables, and the computational domain Ω c with variables ξ, η. The functional is chosen so that the minimum is suitabl influenced b the desired properties of the solution of the PDE itself. In most applications, this involves balancing several critical properties; e.g., mesh concentration in areas needing high resolution of the phsical solution, mesh alignment to some prescribed vector fields, and preservation of the smoothness and the orthogonalit of the mesh lines; e.g., see [3, 4, ]. Obviousl, different meshes are generated depending upon how the functional is formulated and how the minimization problem is solved. Due to the compleit of the mesh generation process, and also the fact that the shapes of the phsical and computational domains themselves can have a strong effect on the behavior of the coordinate mapping, it is usuall ver hard to predict the overall resulting mesh behavior from the functional itself. This has led to a situation where a preponderance of mesh generation approaches have been developed in the past, e.g., see [3, 4, 8, 4, 9,, ], et the understanding of these approaches is relativel limited. The functionals used in eisting variational approaches for mesh generation and adaptation can usuall be epressed in the form [3,, 4, 9, ] Received b the editors September 9, 997; accepted for publication (in revised form) June, 998; published electronicall Ma 6, 999. This research was supported in part b NSERC (Canada) grant OGP-878 and NSF (USA) grant DMS The computations were done in part on the machines of the Scientific Computing and Visualization Laborator (supported b NSF (USA) grant DMS-96866) at the Department of Mathematics, the Universit of Kansas. Department of Mathematics and Statistics, Simon Fraser Universit, Burnab, BC V5A S6, Canada (wcao@cs.sfu.ca). Department of Mathematics, the Universit of Kansas, Lawrence, KS 6645 (huang@ math.ukans.edu). Department of Mathematics and Statistics, Simon Fraser Universit, Burnab, BC V5A S6, Canada (rdr@cs.sfu.ca). 978

2 (.) STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 979 I[ξ,η] = dd ( ξ T G ξ + ηt G η), Ω p where G = G (, ) and G = G (, ) are smmetric positive definite matrices, referred to as the monitor functions, and =(, )T. For simplicit, we consider in this paper the simpler et still fairl general class of functionals of the form I[ξ,η] = (.) dd ( ξ T G ξ + η T G η ), Ω p where G = G(, ) is a smmetric positive definite matri. As will be clear later, our strateg used for understanding (.) can be straightforwardl applied to (.). With (.), the coordinate transformation and the mesh are determined from the Euler Lagrange equation (.3) (G ξ ) =, (G η ) = or its transformed form (via interchanging the roles of the dependent and independent variables) [ ( ) ξ gj T η G η ( )] η gj T ξ G η ξ (.4) [ + ( ) ξ gj T η G ξ + ( )] η gj T ξ G ξ η =, supplemented with suitable boundar conditions. Here, =(, ) T is the phsical coordinate, g = det(g), and J = ξ η η ξ is the Jacobian of the coordinate transformation. Functional (.) includes as particular eamples Winslow s well-known method and the method based upon harmonic maps. The functional for Winslow s method is I[ξ,η] = ( (.5) dd ξ + η ), Ω p w(, ) where w(, ) is a weight function depending on the phsical solution to be adapted, and this corresponds to (.) with the monitor function (.6) G = w(, )I. Brackbill and Saltzman [4] generalize Winslow s method to produce satisfactor mesh concentration while maintaining relativel good smoothness and orthogonalit. Theirs has become one of the most popular methods used for mesh generation and adaptation. The method based upon harmonic maps uses the functional I[ξ,η] = dd ( (.7) ξ T M ξ + η T M η ), Ω p m where m = det(m(, )) and M(, ) is a smmetric positive definite matri. It can be written in the general form (.) with (.8) G = M m.

3 98 W. CAO, W. HUANG, AND R. D. RUSSELL A nice feature of this tpe of method is that for two-dimensional problems, eistence, uniqueness, and nonsingularit for the continuous map can be guaranteed from the theor of harmonic maps (e.g., see [6, 9], and see [5, 6] for discussion on singularit of three-dimensional harmonic maps), and such theoretical guarantees are rare in the field of mesh generation. The success of a mesh adaptation strateg with functional (.) (and all other methods using a variational approach or an elliptic equation sstem) hinges on choosing an appropriate monitor function G, so understanding how the monitor function influences the resulting mesh properties is clearl crucial. There has been onl limited stud of this aspect of the adaptive mesh generation problem, e.g., see [,, 7, 4, 8], although practical eperience has undoubtedl led to insight into a suitable choice for the monitor function in some cases such as in [3, 4, ]. Our objective in this paper is to more sstematicall analze the adaptivit effects for the monitor function G in (.). We find that, in general, the eigenvectors of G determine the directions of the mesh concentration, while the eigenvalues determine the strength of mesh compression or epansion. This insight motivates how G can be chosen in practice. For instance, it can be fortuitous for problems in fluid dnamics to choose an eigenvector to be the streamline direction or the characteristic direction, and for general problems with steep wave fronts to choose an eigenvector in the gradient direction. The eigenvalues are in turn chosen to var the strength of the mesh concentration. An outline of the paper is as follows. In section, the Dirichlet problem for a general elliptic PDE is considered. The influence of the source term on the solution behavior is discussed both analticall and geometricall using Green s function. Then the eigendecomposition of the monitor function is used in section 3 to investigate the effect of the monitor function on the behavior of the mesh corresponding to the functional form (.). The above-mentioned general guide for the choice of the monitor function is also developed in this section. After numerical eamples for common choices of the monitor function are presented in section 4, the case of adaptive mesh generation for a disk is studied in section 5 in order to illustrate some of the complicated two-dimensional effects for mesh adaptation along the eigendirections of the monitor functions. A brief discussion of the three-dimensional case is given in section 6. Finall, section 7 contains the conclusions.. Green s function. Our basic tools for providing insight into the mesh behavior are Green s function for elliptic PDEs and the eigendecomposition of smmetric matrices. Using Green s function, we show in this section how the source function for an elliptic PDE influences the behavior of the solution. These results are crucial to the analsis of the influence of monitor functions on the mesh behavior developed in the net section. Detailed mathematical discussion of Green s function and its applications appears in man places, e.g., in [5, 7], but here the main purpose is to eamine certain geometrical effects. Consider the differential operator (.) L n a ij ( ) + i j i,j= n i= b i ( ) i, where the coefficients a ij = a ji and b i are continuous functions of =(,..., n ) T in a bounded domain Ω in n-dimensional space. The quadratic form n i,j= a ijξ i ξ j is assumed to be positive definite in the parameter ξ =(ξ,...,ξ n ) T for all Ω.

4 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 98 If the boundar Ω is sufficientl smooth, then Green s function for the differential operator L, denoted b G = G(, ), eists in Ω. Furthermore, G is positive in Ω and zero on Ω and has the asmptotic behavior that as, G(, ) n for n>orlog for n =, where is the distance from to. For the Dirichlet problem { L[u] =f in Ω, (.) u = h on Ω, the solution can be epressed b (.3) u( ) = G(, )f( )d where G n Ω Ω h( ) G (, )ds, n = n i,j= a ijcos( n, e i ) j, n is the outward normal to the boundar Ω, and e i is the unit vector in the i ais. To consider the effect the function f has on the behavior of the solution u, define v as the solution of the homogeneous problem { L[v] = inω, (.4) v = h on Ω. Clearl, (.5) v( ) = Ω h( ) G (, )ds, n and it follows from (.3), (.5), and the positivit of G(, ) that (.6) f inω = u v in Ω. This is of course the well-known conclusion of the comparison theorem [5]. For geometric meaning of the inequalit u v, Fig.. gives a sketch of u- and v-contour lines (surfaces when n>) on which u and v are constant. Since u( ) v( ) for all Ω, for an constant c the line u = c lies in the region where v c. Therefore, the u-contour lines are shifted awa from the uniform reference v-contour lines in the direction toward which u increases. Generall, a large positive value of f causes u-contour lines to move in the direction of increasing u, and a large negative value causes them to move in the direction of decreasing u. The larger the magnitude of f, the greater the shift. When the function f( ) changes sign over the domain, the situation becomes more complicated. To see the influence of such a source function, we consider the etreme (point charge) case (.7) f( ) =δ( ) δ( ), where and are two distinct points in the domain and δ is the Dirac delta function. (We note that this function fails to give a solution u in the strong sense, but it can be smoothed and have sufficientl small compact support that the following discussion can be made rigorous.) In this case, the solution to (.) is (.8) u( ) = G(, )+G(, )+v( ).

5 98 W. CAO, W. HUANG, AND R. D. RUSSELL v=c 3 v=c v=c u=c 3 u=c u increases u=c Fig... A sketch of u- and v-contour lines, where u<vin Ω and C <C <C 3 are three arbitrar constants. Since G(, ) + as, (.8) implies that (u v)( ) as and (u v)( ) + as. In other words, the influence of the function f on the behavior of u comes predominantl from the value f( ) near and from the value f( ) near. Geometricall, this means that in a neighborhood of, f causes u-contour lines to shift (relative to the v-contour lines) in the direction of increasing u, and in a neighborhood of in the direction of decreasing u. As a result, the u-contour lines in the region between and are compressed when u increases in the direction from to and are epanded when u decreases. An illustration is given in Fig... This analsis of the point charge source function can be etended to a general function f. Because of the singular behavior of Green s function G(, ) at =, the main contribution to the first integral on the right-hand side of (.3) comes from the values of f near. Thus, (u v) becomes negative in the region where f is sufficientl large and positive. Geometricall, u-contour lines will shift from the reference v- contour lines in the increasing u direction in the region. A similar interpretation can of course be made if (u v) becomes positive in the region where f is sufficientl large and negative. Consequentl, u-contour lines will be compressed or epanded about the region where f changes sign. We have thus far assumed that f is a function of onl. It is not difficult to see that the analsis in this section holds for the more general case where f depends also upon the unknown function u; i.e., f = f(, u). In this case, the sign of f at a point is regarded as the sign of f(, u( )), where u = u( ) is the solution of (.). 3. Monitor function in two dimensions. We now use the geometrical interpretations in the previous section to analze the behavior of the function minimizing the functional (.), or the solution of the Euler Lagrange equation (.3). Being smmetric and positive definite, the monitor function G and its inverse G in (.) have the eigendecompositions (3.) (3.) G = λ v v T + λ v v T, G = λ v v T + λ v v T,

6 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 983 Shift direction v=const. (f( )<) (f( )<) Shift direction v=const. u=const. u increases (f( )>) Shift direction u=const. u increases Shift direction (f( )>) (a) Compression (b) Epansion Fig... Compression and epansion of u-contour lines caused b the sources f( ) and f( ). (u- and v-contour lines are solid and dotted lines, respectivel.) where λ and λ are the two positive eigenvalues of G and v and v are corresponding normalized orthogonal eigenvectors. Using this decomposition and recalling that the directional derivative in the v direction is defined as v = v, we can rewrite the functional (.) as I[ξ,η] = [ ( ξ dd Ω p λ v + η ) ( v + ξ λ v + η )] (3.3). v The Euler Lagrange equation (.3) is (3.4) which after some manipulation becomes ξ λ v + ξ λ v ( ) ( ) v ξ v ξ + =, λ v λ v ( ) ( ) v η v η + =, λ v λ v = ( ) λ ξ v + ( ) λ ξ v, λ λ v v λ λ v v (3.5) η λ v + η λ v = ( ) λ η v + ( ) λ η v. λ λ v v λ λ v v (3.6)

7 984 W. CAO, W. HUANG, AND R. D. RUSSELL Consider now the effect a change in λ has on the behavior of ξ: Rewriting (3.5) in the form (.), with (3.7) ( λ ) L[ξ] ξ ξ v + λ v +( v ) ξ + v f(, ) λ ξ, λ v v ( λ λ ) ( v ) ξ v ( λ λ )( ) λ ξ, λ v v we easil show that the operator L has a form as in (.) and is elliptic. Thus, the analsis in the previous section applies. Moreover, since f is proportional to λ λ v, a rapid change in λ in the v direction will result in a significant change in f in the same direction. One can then conclude from the analsis in section that if λ changes rapidl in the v direction, ξ-coordinate lines can be epected to compress or epand in this direction compared with the reference coordinate lines corresponding to not having the source term λ ξ λ v v. Specificall, one can easil see from the special form of f that compression of ξ-coordinate lines occurs in the v direction if λ first increases and then decreases along this direction, while ξ-coordinate lines will be epanded if the λ change is the reverse. We emphasize that this compression and/or epansion of ξ-coordinate lines caused b the change in λ is relative motion compared with the reference coordinate lines. To determine the location of the actual ξ-coordinate lines, the impact of the other terms in the operator L in (3.7) on the mesh adaptation in the v direction must be taken into account. These include the change in v (the third term), the change in v (the fourth term), the relative change in λ in the v direction (the fifth term), and the ellipticit of the underling equation (the second term). The latter, the effect of the ellipticit, is well understood ellipticit tends to space coordinate lines more equall in the absence of boundar curvature. However, the effects from changes in v, v, and λ are complicated and difficult to analze in general. We do not perform a detailed theoretical analsis of these effects here; rather, the are illustrated for several eamples in sections 4 and 5. We refer to all of them as two-dimensional effects on the mesh adaptation in the v direction. It is worth pointing out that a decrease of the ratio λ λ reduces all of the two-dimensional effects on the mesh adaptation in the v direction ecept that resulting from the change in v (see (3.7)). A similar analsis holds for the ξ-coordinate line adaptation in the v direction and the η-coordinate lines in the v and v directions. It is useful to note that if, for instance, λ first increases and then decreases rapidl in the v direction, then both the ξ- and η-coordinate lines will be compressed in the same direction and the resulting mesh can become ver skewed. The analsis also etends to the case where G G in (.), in which ξ- and η-coordinate lines can have different compression and/or epansion directions. The above shows how the monitor function can be naturall defined directl in terms of suitabl chosen v, v, λ, and λ using (3.). Since v and v are orthogonal, the monitor function can be epressed as (3.8) G = λ v v T + λ v v T. If one wishes the mesh concentration or epansion to occur mainl along a direction, e.g., the streamline direction in the convection diffusion problem or the gradient

8 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 985 direction of the numerical solution, then one ma choose this direction as the eigenvector v, and choose the corresponding eigenvalue λ increasing and then decreasing (or vice versa) in certain regions to give mesh concentration (or epansion) along the v direction in the regions. Similarl, λ can be chosen to give mesh concentration or epansion along the v direction. Since the resulting mesh is affected b both the changes of λ in v and the changes of λ in v, which are inevitabl competing (two-dimensional effects), the magnitude of λ and λ should be chosen to reflect the relative preference of mesh adaptation along one direction over the other. Generall, smaller λ /λ produces smaller two-dimensional effects and thus better adaptation along just the v direction. It is instructive to consider (3.8) for the monitor functions used in Winslow s method and for the method based upon harmonic maps. For Winslow s method, (3.8) becomes (.6) with (3.9) λ = λ = w(, ). In this case, v can be chosen as an unit vector. Coordinate lines are compressed or epanded in the v direction if w changes rapidl in this direction and otherwise are relativel equall spaced. Thus, the direction in which w = w(, ) changes most rapidl is the one in which mesh lines are also compressed or epanded most. For the method based upon harmonic maps, we have { λ = α /α, λ = α /α, (3.) v = the normalized eigenvector associated with α, where α and α are the eigenvalues of the matri M (see (.7)). Unlike for Winslow s map, the eigendecomposition (3.) is unique for the harmonic map unless α = α. Thus, the mesh adaptation can be epected to occur mainl in the directions of v and its orthogonal complement. One class of mesh adaptation methods uses the arclength-like monitor function [,, 9]. That is, for a phsical solution u = u(, ), G is defined through (.8) with M = I + u u T. In this case, the eigenvalues are α =+ u and α =, and the corresponding eigenvectors are u/ u and its orthogonal complement. Motivated b this arclength-like monitor function, we can construct a class of monitor functions with (3.) v = u/ u, λ = + u, λ = a function of λ to perform mesh adaptation in the gradient direction of u. The choices for λ which correspond to Winslow s method and the method based upon harmonic maps are, respectivel, λ = λ and λ = λ. For problems in which u has steep fronts or even discontinuities, λ and λ change much faster in the gradient direction than in the tangential direction. It can be epected that with the choice (3.), coordinate line compression and epansion will mainl occur in the gradient direction. Furthermore, since the ratio of λ to λ is smaller for Winslow s method than for the one based upon harmonic maps (i.e., λ λ = versus λ λ = λ ), the two-dimensional effects on the mesh adaptation in v will generall be less significant for the former method than

9 986 W. CAO, W. HUANG, AND R. D. RUSSELL the latter. In other words, the mesh adaptation in this direction is more like onedimensional adaptation for Winslow s method than for the one based upon harmonic maps. On the other hand, adaptation with the method based upon harmonic maps has a stronger two-dimensional coupling, and our eperience has shown that in this case the control of skewness of the mesh can be easier. It is important to realize that this analsis is a local one. Other factors, such as the shapes of the phsical and computational domains and the grid point distributions on boundar, can also strongl influence the mesh adaptation in two dimensions. The influence of the boundar on the elliptic mesh generation is discussed in [9]. To conclude this section, we observe that the above analsis can also be applied to interpret the widel used Poisson mesh generation sstem (3.) ξ = P, η = Q, where P and Q are control functions, as well as other elliptic mesh generation sstems. For (3.) we easil see that a positive (or negative) value of the Laplacian of one of the curvilinear coordinates implies that the contour lines for that coordinate will shift in the increasing (or decreasing) direction of that coordinate. This result is also obtained in [9] b generalizing their analsis of the mesh behavior near boundaries. 4. Numerical eperiments. Numerical results obtained using the mesh equation (.4) together with Dirichlet boundar conditions (i.e., mesh points fied along the boundar Ω p ) are presented in this section. Three eamples are chosen to demonstrate the analsis of the previous section. The monitor function is defined using (3.8) with v and λ to be chosen and with λ being a function of λ. For the first two eamples, v and λ are artificiall designed to illustrate our mesh adaptation analsis. The third is somewhat more realistic, with a phsical solution prescribed and the comple effects on grid adaptation more in displa. For simplicit, both the phsical and computational domains are chosen as the unit square. The mesh equation (.4) is converted to a so-called moving mesh PDE (MMPDE) [, ], a time-dependent PDE with the desired solution of (.4) as its stead state solution. This MMPDE is discretized in space using central finite differences on a uniform mesh in the computational domain. The resulting sstem of ordinar differential equations is solved using an ADI-like scheme spatial eigenvalue approimate factorization (SEAF), integrating it until there is little mesh movement, viz., until the L norm of the difference between two consecutive solutions is less than 3. Also, a low pass filter is applied twice to the monitor function in order to give a smoother mesh and improve the convergence of the scheme. Since our purpose here is to show how the monitor function affects the behavior of the generated mesh and not to stud the details of the numerical solution process itself, we refer the interested reader to [, ] for the details of the finite difference discretization of (.4) and the SEAF scheme. The computations were performed on Silicon Graphics workstations, an Indigo and an On with double precision algorithms. Eample 4.. For the first eample, we choose (4.) v = [ ] [, v = λ = + sech(5( + ) ). ],

10 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION Fig. 4.. The function λ used in Eample 4.. Note that the desired direction of mesh adaptation v is constant. The function λ changes fastest in this direction and is constant in the perpendicular direction v. The function λ is shown in Figure 4.. Along the line =, for instance, the function λ increases until the center point (.5,.5) and then decreases. If mesh adaptation along this line (or direction) is one-dimensional, then from one-dimensional equidistribution arguments (e.g., see []) we know that the mesh densit increases until this center point and then decreases, so the mesh is densest near the center. On the other hand, from the analsis in the previous section, the change in λ will cause coordinate lines to compress in the v direction, and this will compete with the two-dimensional effect resulting from the ellipticit which tends to space coordinate lines equall. Note that in this eample the other two-dimensional effects do not take effect because both v and v are constant and ( λ )/( v ) =. Moreover, we cannot epect there to be significant mesh adaptation in the v direction because, once again, ( λ )/( v )=. Figure 4. shows the 3 3 meshes generated using four different values of λ : (a) λ =/λ (the method based upon harmonic maps), (b) λ =.λ, (c) λ = λ (Winslow s method), and (d) λ =λ. For the four cases, the compression of mesh lines in the v direction and basic absence of compression and epansion in the v direction can be clearl seen. Recall that the two-dimensional effect on the mesh adaptation in the v direction is stronger for the harmonic map than for Winslow s map. For all ecept the harmonic map case, the mesh is concentrating along the v direction as one would epect from one-dimensional mesh adaptation. However, from Figure 4.a we see that the mesh adaptation, sa along the line =, is obviousl not one dimensional. This is caused b the two-dimensional effect which tends to space mesh lines more equall. This two-dimensional effect can also be seen b comparing Figure 4.b and Figure 4.d, corresponding to λ /λ = and., respectivel. From our analsis, the two-dimensional effect on the mesh adaptation in the v direction is stronger in the former case than in the latter. As a result, the mesh line compression in this direction is weaker in Figure 4.b than in Figure 4.d. Finall, it is worth pointing out that for all the cases, mesh lines tend to be uniform near the boundar, particularl near the top-left and bottom-right corners, due to the influence of the uniform boundar point distribution.

11 988 W. CAO, W. HUANG, AND R. D. RUSSELL a: lambda_ = /lambda_ b: lambda_ =.*lambda_ c: lambda_ = lambda_ d: lambda_ = *lambda_ Fig. 4.. The meshes obtained for Eample 4. with different λ : (a) λ =/λ (method based upon harmonic maps), (b) λ =.λ, (c) λ = λ (Winslow s method), (d) λ =λ. Eample 4.. Our second eample is ver similar to the first eample ecept the function λ is now chosen such that its fastest variation direction changes: [ ] [ ] v =, v =, (4.) λ = + sech(5((.5) +(.5).3 )). Figure 4.3 shows the meshes generated using four different choices of λ. For the harmonic map shown in Figure 4.3a, mesh lines are compressed in the v direction in the area around the circle (.5) +(.5) =.3, as one would epect from the analsis in the previous section. In the perpendicular v direction, since λ ( /λ ) first decreases and then increases around the circle, mesh lines are epanded. From the analsis we also know that the two-dimensional effect is stronger in the v direction than the v direction, although it is hard to detect from this figure precisel what the effect is. For Winslow s map, there is no preferred direction of adaptation. Since λ changes in ever direction, mesh lines are compressed uniforml around the

12 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 989 a: lambda_ = /lambda_ b: lambda_ =.*lambda_ c: lambda_ = lambda_ d: lambda_ = *lambda_ Fig The meshes obtained for Eample 4. with different λ : (a) λ =/λ (method based upon harmonic maps), (b) λ =.λ, (c) λ = λ (Winslow s method), (d) λ =λ. circle (see Figure 4.3c). For the other two cases, λ is chosen to be.λ and λ, respectivel. As predicted, mesh lines are compressed in both the v and v direction. However, since the ratio λ /λ is in the first case, the two-dimensional effect is stronger in the v direction than the v direction. Thus, the mesh line compression is less in the v direction than the v direction (see Figure 4.3b). Conversel, for the case in Figure 4.3d, the ratio λ /λ is., and, therefore, the mesh line compression is stronger in the v direction than the v direction. Eample 4.3. The third eample is more realistic than the previous two. Here, we assume that the phsical solution u is given and the desired mesh adaptation direction is taken as the gradient direction of the solution. According to (3.), we define G with u(, ) = sech(5((.5) +(.5).3 )), (4.3) v = u/ u, λ = + u.

13 99 W. CAO, W. HUANG, AND R. D. RUSSELL a: lambda_ = /lambda_ b: lambda_ =.*lambda_ c: lambda_ = lambda_ d: lambda_ = *lambda_ Fig The meshes obtained for Eample 4.3 with different λ : (a) λ =/λ (method based upon harmonic maps), (b) λ =.λ, (c) λ = λ (Winslow s method), (d) λ =λ. Note that the function λ behaves similarl to that in Eample 4., changing most significantl across the circle (.5) +(.5) =.3. The meshes obtained with four different choices of λ are shown in Figure 4.4. As epected from the theoretical analsis, mesh line compression occurs mainl in the radial direction oriented from the center (.5,.5) in all the cases. It is eas to show that ( λ )/( v ) = since λ is a function of λ. Thus the two-dimensional effects onl come from the ellipticit and the change of v. However, in this case the are much more complicated than in the previous eamples. The will be analzed in the net section. 5. Mesh adaptation on the unit disk. In the previous two sections we have investigated theoreticall and numericall how the qualitative behavior of the mesh is related to the change in the eigensstems of the monitor function. The twodimensional effect is etremel complicated to analze for the general case. To get insight into this effect, we consider the special case where Ω p and Ω c are unit disks, and the monitor function is given b (3.8) with

14 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 99 (5.) { v = (cosθ, sinθ) T, v =( sinθ, cosθ) T, λ = λ (r), λ = λ (r) both positive and bounded above. Here (r, θ) are polar coordinates in Ω p, v is the unit radial vector, and v is the unit angular vector. Under these assumptions, the mesh adaptation problem has angular smmetr, so the mesh equations can be reduced to ordinar differential equations. In fact, using the identities = v r + r v θ, (5.) v = r, the mesh equations (3.5) and (3.6) become v = r θ, (5.3) ξ λ r + ξ λ r θ = λ λ η r + λ r η θ = λ ( dλ λ dr ) ξ r r + ξ ( ) λ r θ, ( dλ λ dr ) η r r + η ( ) λ r θ. The two-dimensional effects on the mesh adaptation in the radial direction result onl from the change in v (( v) =/r) and the ellipticit (from the second term of each of the equations in (5.3)). Using the polar coordinates (R, Θ) in Ω c,so (5.4) ξ = R cos Θ, η = R sin Θ, we have b smmetr that Θ = θ and R = R(r). Then (5.3) leads to (5.5) d R dr = ( dλ λ dr ) dr dr ( r ) dr dr + ( λ λ ) R r. It is eas to show from this equation and the boundar conditions R() = and R() = that (5.6) dr R = R(r) (, ), (r) > for r (, ). dr The mesh point densit function in the radial direction is dr dr, and the first, second, and third terms on the right-hand side of (5.5) represent the effects of the change in λ, the change in v, and the ellipticit of the underling sstem. We can readil conclude that as r increases, the two-dimensional effect from a change of v corresponds to a decrease in the densit function and the ellipticit term causes an increase in the densit function. In particular, mesh point densit near the origin will be increased b change in v and decreased b the ellipticit. Of course, these two-dimensional effects compete with one another and with the effect caused b the relative change in λ. The degree of the effect from ellipticit is controlled b the ratio λ /λ. To see this, we consider the case where λ and λ are constants, for which an analtical solution of (5.5) is available, viz. R(r) =r λ/λ. Note that for λ λ =4,, and 4, dr dr =r,,and r /, respectivel. The smaller the ratio λ λ, the stronger the effect of the change in v, and therefore the denser the mesh will be near the origin. For Eample 4.3, the phsical domain is not a disk, but because the boundar point distribution does not strongl affect the adaptation inside the domain and

15 99 W. CAO, W. HUANG, AND R. D. RUSSELL because λ and v behave similarl to those in this section, one can epect that the above qualitative arguments should remain valid. For the harmonic map shown in Figure 4.4a, since λ changes significantl in r (r is the distance to the point (.5,.5)) and the ratio λ /λ is large (i.e., λ ) near r =.3, the adaptation and the two-dimensional effect from ellipticit dominate the mesh line placement. Thus, we can see that the mesh points are pulled awa from point (.5,.5) (the ellipticit effect) and are denser near r =.3 (the adaptation effect). On the other hand, for the case shown in Figure 4.4d, the ratio λ /λ is. (small), and the adaptation and the two-dimensional effect from the change in v dominate the mesh line placement. Thus, the mesh points are concentrated near the center point (.5,.5) (the effect of the change in v) and in the area where r.3 (the effect of adaptation). The case shown in Figure 4.4b is similar to that in Figure 4.4a. For the case shown in Figure 4.4c, the two-dimensional effects from the change in v and the ellipticit appear to be more or less balanced. 6. The three-dimensional case. We briefl discuss the three-dimensional case. Denote the phsical and computational coordinates b =(,, 3 ) T and ξ = (ξ,ξ,ξ 3 ) T, respectivel. The general functional corresponding to (.) is I[ ξ]= d ( ξ i ) T (6.) G ξ i, Ω p i where G is a smmetric positive definite matri (monitor function). The Euler Lagrange equation is (6.) (G ξ i) =. If (λ i, v i ), i =,, 3 are the three pairs of eigenvalues and eigenvectors of G, it has the eigendecomposition (6.3) G = i λ i v i v T i. The general functional (6.) can then be rewritten as I[ ξ]= (6.4) d Ω p i,j and the Euler Lagrange equation becomes ξ j v i, λ i (6.5) j ξ i λ j v j = j λ j ( ) λ j ξ i v j. λ j v j v j An etension of the analsis in sections and 3 enables us to similarl conclude that (i) coordinate lines can be epected to compress and/or epand in the v i direction if the function λ i changes significantl in this direction; (ii) the compression and/or epansion (i.e., adaptation) of coordinate lines in the v i direction have to compete with the three-dimensional effects resulting from the changes in v, v, and v 3, the relative change in λ j along the direction of v j (j =,, 3 and j i), and the ellipticit of the underling sstem; (iii) among the three-dimensional effects, ellipticit tends to give more even placement of mesh lines. All of the three-dimensional effects on the mesh adaptation in v i ecept that from the change in v i can be controlled b

16 STUDY OF MONITOR FUNCTIONS FOR MESH ADAPTATION 993 controlling the ratios λ i /λ j (j =,, 3 and j i). More specificall, the smaller the ratios λ i /λ j (j =,, 3 and j i), the weaker these three-dimensional effects and the more dominant the effects of changes in λ i and v i are on the adaptation along v i direction. Motivated b this analsis, it is natural to define the monitor function using (6.3) with v = u/ u, v, v 3 are orthogonal complements of v, (6.6) λ = + u, λ and λ 3 are functions of λ, where u = u(,, 3 ) is the solution of the phsical PDE. One of course has considerable fleibilit in these choices. For instance, curvature information for u could also be included in the definition of λ. What is important is that the eigenvectors and eigenvalues of the monitor function determine, respectivel, the directions of adaptation and strength of compression/epansion in these directions, and this is used to tailor G to meet the needs of a particular application area. 7. Conclusions. In the previous sections, we have studied the effect of the monitor function on the behavior of the mesh generated with a variational approach. The stud has been carried out using Green s function for elliptic problems. Numerical results have also been presented for illustrative purposes. The main conclusion of this stud is that a significant change (first increasing and then decreasing, or vice versa) in an eigenvalue λ of the monitor function G along the corresponding eigendirection v will result in adaptation of coordinate lines along this direction. Meanwhile, this adaptation will compete with far more complicated two-dimensional effects, including the effect of the ellipticit and those from changes in eigenvectors and another eigenvalue λ of G. Fortunatel, most of these twodimensional effects can be controlled b the ratio λ /λ. That is, the smaller this ratio, the weaker the two-dimensional effects and the more dominant the effect that change in λ has on the adaptation along the v direction. The same analsis holds along the v direction. Based upon this analsis, general guidelines for defining monitor functions to achieve the desired mesh adaptation in practical computation have been given. In fact, a monitor function can often be naturall defined in terms of suitabl chosen λ, λ, v, and v through formula (3.) or (3.8). Specificall, when one wishes mesh concentration or epansion to occur along a given direction, e.g., the streamline direction in a convection diffusion problem or the gradient direction for a general problem, then one ma choose this direction as the eigenvector v and choose λ increasing and then decreasing (or vice versa) in certain regions to give mesh concentration (or epansion) along the v direction. There are a variet of natural was in which λ can be chosen as a function of λ. Last, the class of monitor functions given in (3.) or (6.6) is useful when one wants the coordinate lines to adapt along the gradient direction of a phsical variable u. We would like to point out that the results obtained in this stud are onl qualitative. Besides the competing effects of the mesh adaptation along the two eigendirections, mesh adaptation is also greatl affected b the shapes of the phsical and computational domains and the grid point distribution on the boundaries. These factors make the mesh adaptation procedure ver complicated. Man difficulties still

17 994 W. CAO, W. HUANG, AND R. D. RUSSELL remain in predicting the precise behavior of adaptive meshes. Nevertheless, it is our hope that this stud will provide a better understanding of monitor functions thereb providing some guidelines for one to choose them more easil and will be a useful complement to related recent work in grid generation such as that in [3, 3]. REFERENCES [] D. A. Anderson, Equidistribution schemes, Poisson generators, and adaptive grids, Appl. Math. Comput., 4 (987), pp. 7. [] D. A. Anderson, Grid cell volume control with an adaptive grid generator, Appl. Math. Comput., 35 (99), pp [3] J. U. Brackbill, An adaptive grid with direction control, J. Comput. Phs., 8 (993), pp [4] J. U. Brackbill and J. S. Saltzman, Adaptive zoning for singular problems in two dimensions, J. Comput. Phs., 46 (98), pp [5] R. Courant and D. Hilbert, Methods of Mathematical Phsics, Vol. II, John Wile and Sons, New York, 96. [6] A. S. Dvinsk, Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phs., 95 (99), pp [7] P. R. Eiseman, Adaptive grid generation, Comput. Methods Appl. Mech. Engrg., 64 (987), pp [8] R. Hagmeijer, Grid adaption based on modified anisotropic diffusion equations formulated in the parametric domain, J. Comput. Phs., 5 (994), pp [9] R. Hamilton, Harmonic Maps of Manifolds with Boundar, Lecture Notes in Mathematics 47, Springer-Verlag, Berlin, New York, 975. [] W. Huang and R. D. Russell, A high dimensional moving mesh strateg, Appl. Numer. Math., 6 (998), pp [] W. Huang and R. D. Russell, Moving mesh strateg based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput., to appear. [] W. Huang, Y. Ren, and R. D. Russell, Moving mesh partial differential equations (MM- PDEs) based upon the equidistribution principle, SIAM J. Numer Anal., 3 (994), pp [3] P. Knupp, Mesh generation using vector-fields, J. Comput. Phs., 9 (995), pp [4] P. Knupp and S. Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Raton, FL, 994. [5] G. Liao, A stud of regularit problem of harmonic maps, Pacific J. Math., 3 (988), pp [6] G. Liao and N. Smale, Harmonic maps with nontrivial higher-dimensional singularities, Lecture Notes in Pure and Appl. Math. 44, Dekker, New York, (993), pp [7] M. H. Protter and H. F. Weinberger, Maimum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 967. [8] Y. Tu and J. F. Thompson, Three-dimensional solution-adaptive grid generation on composite configurations, AIAA J., 9 (99), pp [9] J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, Numerical Grid Generation, Foundation and Applications, North-Holland, New York, 985. [] Z. U. A. Warsi and J. F. Thompson, Application of variational methods in the fied and adaptive grid generation, Comput. Math. Appl., 9 (99), pp [] A. Winslow, Numerical solution of the quasi-linear Poisson equation in a nonuniform triangle mesh, J. Comput. Phs., (967), pp