6 elements produced by the quadtree and octree

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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 00 11 00 11 00 Boundary quadrant Interior quadrant Exterior quadrant 1 0 1 0 Finite element 1111 0000 1111 0000 11 00 11 00 111111 000000 111111 000000 111111 000000 11 00 11 00 11 00 1 0 11 11 00 00 1 0 11 11 00 00 1 0 11 1 00 0 11 00 11 1 00 0 11 00 11 1 00 0 11 00 11 1 1 00 0 0 11 1 1 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. One-level (top) and three-level (bottom) tree structures are shown. The mesh of triangular elements associated with the three-level 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 re-positioned 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 quadrilateral-element 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 three-dimensional domains may be discretized by quadtree and octree decomposition to produce unstructured grids. Further solution-based 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.

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