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Unformatted text preview: s may have poor geometric
shapes near boundaries. A nal \smoothing" of the mesh improves element shapes and 5.2. Mesh Generation 7 11
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00 0 0 Figure 5.2.5: Finite quadtree mesh generation for a domain consisting of a rectangle and
a region within a curved arc. Onelevel (top) and threelevel (bottom) tree structures
are shown. The mesh of triangular elements associated with the threelevel quadtree is
shown superimposed.
further reduces mesh gradation near @ . Element vertices on @ are moved along the
boundary to provide a better approximation to it. Pairs of boundary vertices that are too
close to each other may be collapsed to a single vertex. Interior vertices are smoothed by a
Laplacian operation that places each vertex at the \centroid" of its neighboring vertices.
To be speci c, let i be the index of a node to be repositioned xi be its coordinates Pi
be the set of indices of all vertices that are connected to Node i by an element edge and
Qi contain the indices of vertices that are in the same quadrant as Node i but are not 8 Mesh Generation and Assembly Figure 5.2.6: Quadtree structure and mixed triangular and quadrilateralelement mesh
generated from it.
connected to it by an edge. Then P P 2 j2P xj + j2 x
xi = 2 dim(iP ) + dim(Qi )j
Qi
i (5.2.4) where dim(S ) is the number of element vertices in set S . Additional details appear in 5.3. Data Structures 9 Baehmann et al. 2].
Arbitrarily complex two and threedimensional domains may be discretized by quadtree
and octree decomposition to produce unstructured grids. Further solutionbased mesh
re nement may be done by subdividing appropriate terminal quadrants or octants and
generating a new mesh locally. This unites mesh generation and adaptive mesh re nement by a common...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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