Rening quadrilateral and hexahedral element meshes R. Schneiders RWTH Aachen Lehrstuhl fur Angewandte Mathematik, insb. Informatik Ahornstr. 55, 5056 Aachen, F.R. Germany (robert@feanor.informatik.rwth-aachen.de) Abstract We consider the problem of rening quadrilateral and hexahedral element meshes. For the two-dimensional case we present algorithms that solve the problem. A renement algorithm for hexahedral element meshes is given which in some cases produces an unnecesseraly large number of elements. Possible solutions of the problem are indicated, and we conclude with a discussion of the renement problem for structured meshes. Introduction It is a well-known fact that the generation of solution-adaptive meshes is crucial in order to get a guaranteed-quality solution at minimal computational cost. This is usually achieved by a two-step-procedure: an initial solution is computed for a coarse mesh which is then rened in regions where the solution error is too large. Therefore one needs { besides an error estimator { an algorithm that renes a given mesh in a way that the size of an element does not exceed a prespecied bound. Algorithms for various types of mesh renement can be found in literature. [] discusses the renement problem for triangular, [] for tetrahedral meshes. [3] gives solutions for meshes of tetrahedra, pyramids and prisms, and [4] gives an algorithm for the renement of structured quadrilateral element meshes. In the paper we present algorithms for the renement of quadrilateral and hexahedral element meshes. We start with a formal statement of the problem. Then we give a
general algorithm that can be used for dierent types of renement, solutions for the D case and partial solutions for the 3D case. Formal statement of the problem Let M be a mesh of quadrilateral (hexahedral) elements, constituted by the set V of its nodes and the set H of its elements. Other entities of interest are the set E of mesh edges and the set F of faces (for the three-dimensional case). Def. A quadrilateral (hexahedral) element mesh is conforming if two distinct, nondisjoint elements intersect at common nodes or edges (or faces) only. Def. A quadrilateral (hexahedral) element mesh is structured if each interior node v belongs to 4 quadrilateral elements (8 hexahedral elements); otherwise it is called unstructured. In the following we consider the mesh renement problem for unstructured conforming meshes. We start with a problem specication for the D case: given a mesh M; the elements and edges of M are assigned subdivision levels S that indicate that { f H is split up into 4 S(f) quadrilaterals { e E is split up into S(e) edges Figure : Subdivision level assignement a) b) Subdivision level transition elements (g. a shows an example). The renement levels can easily be obtained by estimating the numerical error: if an element f of size h has to be replaced by elements of size h 0 h < h, f is assigned the subdivision level dlog h e. Rening edges makes sense since 0 one can enforce a better approximation of curved boundaries in that way. The resulting mesh is not conforming; therefore one must rene the neighbor elements appropriately (g. a). Due to the choice of the basis b = this technique is called -renement. In the same way we can dene 3-renement for the basis b = 3 (g. b): { f H is split up into 9 S(f) quadrilaterals { e E is split up into 3 S(e) edges
Figure : Rened meshes a) b) Figure 3: Unstable/stable transition a) v b) 0 0 In principle -renement is favorable since the resulting element size is likely more closed to the desired size. Nevertheless, 3-renement is considered since it can be achieved with more elegant algorithms. The algorithmic diculty in the mesh renement problem is to connect the ne to the coarse part of the mesh in a conformity-preserving way (g. b!g. a). Fig. 3a shows a possible \solution" which is not useful since the smallest angle min at node v decreases with increasing subdivision level S(f) (the number of elements adjacent to v increases). Def. 3 A renement is stable if the minimum angle of the rened mesh does not depend on the subdivision level S: min > 0 S! Stability is a necessary condition for the convergence of the numerical solutions, and the rened meshes must full this property. Fig. 3b shows a transition which is stable because of the use of an additional layer of elements. It remains to give a formal denition of the mesh renement problem for hexahedral element meshes. In the 3D-case subdivision levels S can be assigned to the elements, edges and faces of the coarse mesh so that { h H is split up into 8 S(H) (7 S(H) ) elements { f F is split up into 4 S(f) (9 S(f) ) quadrilateral faces { e E is split up into S(e) (3 S(e) ) edges
Figure 4: 3-Renement of a hexahedral element mesh An example of 3-renement is shown in g. 4. 3-Renement As already pointed out, the diculty in mesh renement is to nd a good transition from the coarse to the rened zone of the mesh (conforming closure). The way an element is rened depends, besides its own subdivision level, on the subdivision levels of the adjacent elements and edges. The idea is to propagate them to the mesh nodes (g. 5): Figure 5: Dening nodal subdivision levels Def. 4 The subdivision level of a node v V is dened as the maximum subdivision level of its n adjacent elements and edges: A node v with S(v) > 0 is called marked. S(v) = maxfmax i=;n S(e i); max i=;n S(f i)g It turns out that the nodal subdivision levels uniquely dene how to rene an element, and that element renement can be done in a purely local manner. Consider an element f M and its marked nodes (g. 7): if one only considers whether a node is marked or not, there are six possible combinations (g. 6).
Figure 6: Templates for element renement 0: : a: b: 3: 4: The element is then split up by selecting the appropriate template from g. 6. The nodal subdivision levels are decreased by, and appropriate renement levels of the newly introduced nodes are dened (g. 7a). Figure 7: Example of element renement a) b) The new subdivision levels S 0 are determined with the help of auxiliary subdivision levels of the element f = (v; v; v3; v4) and its edges e i ; i = ; 4: S 0 (v i ) = maxfv i? ; 0g; i = ; 4 S 0 (e i ) = maxfs 0 (v a ); S 0 (v b )g; e = (v a ; v b ) S 0 (f) = max i=;4 S0 (v i ) The subdivision levels of the newly introduced nodes are then dened as follows: { S 0 (v) = S 0 (e) if v is inserted on the edge e { S 0 (v) = S 0 (f) if v lies in the interior of f. In the example of g. 7a the subdivision levels of the interior nodes are set to zero while the nodes on the basic edge are assigned the level. The procedure is then applied recursively to the newly generated elements (g. 7b). Formally the mesh renement algorithm can be stated as follows: Alg. procedure refine ( element ) Select appropriate template; Refine element; Update nodal subdivision levels; forall newly generated elements elenew refine ( elenew );
procedure mesh_refinement Define nodal subdivision levels; forall elements refine ( element ); It remains to prove that the rened meshes are conforming and that the renement is stable. Note that { an edge with marked nodes is split into 3 edges { an edge with marked node is split into { an edge with no edge marked is not split. These rules, together with the rules for updating the nodal subdivision levels, uniquely dene how to split up an edge. The number of new points that are inserted on an edge is determined by its nodal subdivision levels only, and thus the renement process results in a conforming mesh. Stability is ensured by the fact that the newly dened elements in g. 6 that have at least one node marked are self-similar to their \parent" element. Fig. 8 shows how the algorithm proceeds for the example mesh. Figure 8: 3-Renement: example The algorithm can, in principle, be used for the 3D-case: nodal subdivision levels are dened as the maximum subdivision levels of the adjacent elements, faces and edges: S(v) = maxfmaxs(e i ); maxs(f i ); max S(h i )g The problem in the 3D-case is to nd an appropriate set of templates. Fig. 9 shows 5 out of possible congurations of marked nodes (in practice this subset is sucient for \convex" renements, g. 4). Note that the faces of the hexahedra are rened according to the templates of g. 6; this ensures conformity. Stability is also ensured as in the D-case by taking care that element which might be rened recursively are self-similar to their \parent" elements (as an example see the template 4 in g. 0, for the other templates in g. 9 see [5]). Unfortunately one cannot nd a tesselation for every conguration of marked nodes (since the tesselation of the faces of a hexahedron is determined by the D-templates, in some cases one gets an odd number of quadrilaterals on the surface so that a hex mesh
Figure 9: Selected 3D-templates 3 4 5 Figure 0: Template for conguration 4 that matches the surface does not exists; the problem of nding a hex mesh for a given surface is treated in [6]). Fig. shows an example of a subdivision level assignement with non-admissible templates. Two solutions exist: one is to propagate the renement levels so that there are only admissible combinations (conforming closure, g. a); the resulting mesh has an unnecessarily large number of elements. The second one is to insert a new layer of elements in the coarse mesh so that only template 4 must be used. A combination of both methods might be the best solution. Figure : Problematic case a) b) The 3-renement algorithm has been used for the simulation of micromechanical processes [7] and for the octree-based generation of hexahedral element meshes [8]. -Renement -Renement (g. a) is better than 3-renement since the resulting element sizes are
likely more closed to the maximum sizes allowed. Usually the rened meshes can be expected to have fewer elements. It is, however, more dicult to give an algorithm for -renement, especially in the 3D-case. The simplicity of the 3-renement algorithm is due to the fact that the transition could be done locally. Unfortunately this does not hold for -renement where one must consider pairs of adjacent elements (g. ). Figure : -Renement -Renement can be achieved with a modied version of algorithm. The mesh nodes are assigned nodal subdivision levels as usual. In order to determine the element pairs in the transition zone, one has to consider the sequence fv; v; : : : ; vk?; vkg on the boundary of the submesh of M that must be rened (note that the number of boundary nodes of a quadrilateral element mesh is even). The nodes vi; i k, are deactivated, and the remaining active notes determine the way an element is split up: if an element has one active marked node, template (g. 3) is inserted, if it has more than one active node, template is used. All boundary nodes of M are set active while the newly inserted nodes remain inactive. This procedure is recursively applied to the newly generated elements. Figure 3: Templates for -renement of structured quadrilateral element meshes : : Because of the choice of the templates and the activation strategy one can easily show that the renement process is stable and results in a conforming mesh. However, sometimes elements with sidelength diering by a factor 4 are generated in the transition zone (g. ). One can avoid this problem if one uses two layers of M for transitioning (g. 4). One can easily modify the -renement algorithm for unstructured quadrilateral element meshes. This is, however, not possible in the 3D-case. Fig. 5 shows a subdivision level assignement for a structured hexahedral element mesh (all marked nodes are assigned the subdivision level ).
Figure 4: -Renement (two transition layers) Figure 5: -Renement of structured hexahedral element meshes a) b) In the 3D-case one must nd templates for the transition at faces and edges of the submesh to be rened (g. 5a). At faces the transition can only be achieved by considering four elements that are adjacent to a node P (g. 6a). Fig. 6b shows how each of the four elements is split up. The transition at an edge is done with the help of the template shown in g. 6c. Surprisingly, the quality of the elements generated by the latter template is worse since in this case a node of degree 4 is inserted in the mesh (the degree of a node v is dened as the number of elements adjacent to v). Figure 6: Templates for -renement of structured hexahedral element meshes a) b) c) P P Mesh renement with renement levels S > can only be realized with the help of two transition layers. The algorithm is a straightforward-generalization of the D-algorithm, g. 7 shows an example of how it works. In principle the proposed algorithm for -renement can handle the same renement specications as the 3-renement algorithm. In particular, the \concavity" problem remains and can be solved in the same way. This problem can be avoided if one accepts
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