28 to keep angles bounded away from zero as the mesh

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Unformatted text preview: ay from zero as the mesh is re ned. Figure 8.2.7: Uniform bisection of a triangular element into four and the division of neighboring elements in two (shown dashed). Rivara 34, 33] developed a mesh re nement algorithm based on bisecting the longest edge of an element. Rivara's procedure avoids irregular nodes by additional re nement as described in the algorithm of Figure 8.2.9. In this procedure, we suppose that elements 8 Adaptive Finite Element Techniques Figure 8.2.8: Uniform re nement of green triangles of the mesh shown in Figure 8.2.7 to avoid the second bisection of vertex angles. New re nements are shown as dashed lines. of a sub-mesh of mesh h are scheduled for re nement. All elements of are bisected by their longest edges to create a mesh 1 , which may contain irregular nodes. Those h 1 that contain irregular nodes are placed in the set 1 . Elements of 1 are elements e of h bisected by their longest edge to create two triangles. This bisection may create another node Q that is di erent from the original irregular node P of element e. If so, P and Q are joined to produce another element (Figure 8.2.10). The process is continued until all irregular nodes are removed. procedure rivara( h, ) Obtain 1 by bisecting all triangles of by their longest edges h Let 1 contain those elements of 1 having irregular nodes h i := 1 while i is not do Let e 2 i have an irregular node P and bisect e by its longest edge Let Q be the intersection point of this bisection if P 6= Q then Join P and Q end if Let ih+1 be the mesh created by this process Let i+1 be the set of elements in ih+1 with irregular nodes i := i + 1 end while return i h Figure 8.2.9: Rivara's mesh bisection algorithm. Rivara's 33] algorithm has been proven to terminate with a regular mesh in a nite number of steps. It also keep angles bounded away from 0 and . In fact, if is the 8.2. h-Re nement 9 P P e Q Figure 8.2.10: Elimination of an irregular node P (left) as part of Rivara's algorithm shown in Figure 8.2.9 by dividing the longest edge of Element e and connecting vertices as indicated. smallest angle of any triangle in the original mesh, the smallest angle in the mesh obtained after an arbitrary number of applications of the algorithm of Figure 8.2.10 is no smaller than =2 35]. Similar procedures were developed by Sewell 37] and used by Mitchell 28] by dividing the newest vertex of a triangle. Tree structures can be used to represent the data associated with Bank's 10] and Rivara's 33] procedures. As with structured-mesh computation, elements introduced by re nement are regarded as o spring of coarser parent elements. The actual data representations vary somewhat from the tree described earlier (Figure 8.2.4) and readers seeking more detail should consult Bank 10] or Rivara 34, 33]. With tree structures, any coarsening may be done by pruning \leaf" elements from the tree. Thus, those elements nested within a coarser parent are removed and the parent is restored as the element. As mentioned earlier, coarsening bey...
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