Of element h2ae h 2e 3e v4 v4 v5 1e v5 v0 v2 v6 v3 v2

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Unformatted text preview: V2 V6 V3 V2 V3 V6 V7 V7 V1 V1 Figure 8.2.13: A situation where the collapse of Vertex V0 (left) creates an invalid mesh (right). Ω1 E Ω2 Ω1 Ω2 E Figure 8.2.14: Swapping an edge of a pair of elements (left) to improve element shape (right). The shape of elements containing small or large angles that were created during re nement or coarsening may be improved by edge swapping. This procedure operates on pairs of triangles 1 and 2 that share a common edge E . If Q = 1 2 , edge swapping occurs deleting Edge E and re-triangulating Q by connecting the vertices opposite to Edge E (Figure 8.2.14). Swapping can be regarded as a re nement of Edge E followed by a collapsing of this new vertex onto a vertex not on Edge E . As such, we recognize that swapping will have to be checked for mesh validity and topological compatibility. Of course, it will also have to provide an improved mesh quality. 8.2.3 Re nement Criteria Following the introductory discussion of error estimates in Section 8.1, we assume the existence of a set of re nement indicators e , e = 1 2 : : : N , which are large where re nement is desired and small where coarsening is appropriate. As noted, these might 12 Adaptive Finite Element Techniques be the restriction of a global error estimate to Element e 2 e = kE k2 e (8.2.2) or an ad hoc re nement indicator such as the magnitude of the solution gradient on the element. In either case, how do we use this error information to re ne the mesh. Perhaps the simplest approach is to re ne a xed percentage of elements having the largest error indicators, i.e., re ne all elements e satisfying e 1 max jN j : (8.2.3) A typical choice of the parameter 2 0 1] is 0.8. We can be more precise when an error estimate of the form (8.1.1) with indicators given by (8.2.2) is available. Suppose that we have an a priori error estimate of the form kek C hp: (8.2.4a) After obtaining an a posteriori error estimate kE k on a mesh with spacing h, we could compute an estimate of the error constant C as C kE k hp : (8.2.4b) The mesh spacing parameter h may be taken as, e.g., the average element size r A h= N (8.2.4c) where A is the area of . Suppose that adaptivity is to be terminated when kE k where is a prescribed tolerance. Using (8.2.4a), we would like to construct an enriched mesh with a spacing ~ parameter h such that ~ C hp : Using the estimate of C computed by (8.2.4b), we have ~ h h 1=p kE k : (8.2.5a) Thus, using (8.2.4c), an enriched mesh of ~2 ~ =h NA h2 A 2=p kE k (8.2.5b) 8.2. h-Re nement 13 elements will reduce kE k to approximately . ~ Having selected an estimate of the number of elements N to be in the enriched mesh, we have to decide how to re ne the current mesh in order to attain the prescribed tolerence. We may do this by equidistributing the error over the mesh. Thus, we would like each element of the enriched mesh to have approximately the same error. Using (8.1.1), this implies that 2 ~e kE k2 ~ N ~ where kE ke is the error indicator of Element e o...
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