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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
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
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
kE k : (8.2.5a) Thus, using (8.2.4c), an enriched mesh of
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
where kE ke is the error indicator of Element e o...
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- Spring '14