# This structure is depicted in figure 824 coarsening

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Unformatted text preview: picted in Figure 8.2.4. Coarsening can be done by \pruning" re ned quadrants. It's customary, but not essential, to assume that elements cannot be removed (by coarsening) from the original mesh 3]. Irregular nodes can be avoided by using transition elements as shown in Figure 8.2.5. The strategy on the right uses triangular elements as a transition between the coarse and ne elements. If triangular elements are not desirable, the transition element on the left uses rectangles but only adds a mid-edge shape functions at Node 3. There is no node at the midpoint of Edge 4-5. The shape functions on the transition element are N11 = ( h + x )( y ; h=2 ) h h=2 2 N21 = ( h + x )( h=h=; y ) h 2 6 Adaptive Finite Element Techniques 11 00 11 00 11 11 11 11 11 11 00 00 00 00 00 00 11 11 11 11 11 11 00 00 00 00 00 00 11 11 11 11 11 11 00 00 00 00 00 00 11 11 11 11 11 11 00 00 00 00 00 00 11 11 11 11 00 00 00 00 11 11 11 11 00 00 00 00 11 11 11 11 00 00 00 00 11 11 11 11 00 00 00 00 Figure 8.2.4: Original structured mesh and the bisection of two elements (left). The tree structure used to represent this mesh (right). 40 1 1 0 1 0 1 0 1 0 50 1 1 1 0y 1 0 1 0 1 0 1 10 1 0 1 0 1 0 2 30 1 1 0 1 0 3 1 0 1 0 1 20 1 0 1 0 1 0 1111 1 0000 0 1 0 1 0 x 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 Figure 8.2.5: Transition elements between coarse and ne elements using rectangles (left) and triangles (right). (y h + x ) ( h=2 ) if 0 y h=2 N31 = ( h ( h;2y ) if h=2 y h h= y N41 = ( ;x )( h ) N51 = ( ;x )( h ; y ): h h h Again, the origin of the coordinate system is at Node 2. Those shape functions associated with nodes on the right edge are piecewise-bilinear on Element 1, whereas those associated with nodes on the left edge are linear. Berger and Oliger 12] considered structured meshes with structured mesh re nement, but allowed elements of ner meshes to overlap those of coarser ones (Figure 8.2.6). This method has principally used with adaptive nite di erence computation, but it has had some use with nite element methods 29]. 8.2.2 Unstructured Meshes Computation with triangular-element meshes has been done since the beginning of adaptive methods. Bank 9, 11] developed the rst software system PLTMG, which solves 8.2. h-Re nement 7 Figure 8.2.6: Composite grid construction where ner grids overlap elements of coarser ones. our model problem with a piecewise-linear polynomial basis. It uses a multigrid iterative procedure to solve the resulting linear algebraic system on the sequence of adaptive meshes. Bank uses uniform bisection of a triangular element into four smaller elements. Irregular nodes are eliminated by dividing adjacent triangles sharing a bisected edge in two (Figure 8.2.7). Triangles divided to eliminate irregular nodes are called \green triangles" 10]. Bank imposes one-irregular and three-neighbor rules relative to green triangles. Thus, e.g., an intended second bisection of a vertex angle of a green triangle would not be done. Instead, the green triangle would be uniformly re ned (Figure 8.2.8) to keep angles bounded aw...
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