Lecture on Mesh Generation PDF
Preview Lecture on Mesh Generation
Mathematical Foundations of Mesh Generation Alper Üngör University of Florida International Meshing Roundtable, 2008, Pittshburgh Meshing Problem ✦ Compute quickly the smallest size (simplicial, cubical) mesh of a given domain such that all the elements are of good quality. ✦ Applications: Engineering, Graphics, Visualization, GIS, CAD, Medical Imaging, Computational Biology... Meshing Problem ✦ Compute quickly the smallest size (simplicial, cubical) mesh of a given domain such that all the elements are of good quality. ✦ Applications: Engineering, Graphics, Visualization, GIS, CAD, Medical Imaging, Computational Biology... Meshing Problem ✦ Compute quickly the smallest size (simplicial, cubical) mesh of a given domain such that all the elements are of good quality. ✦ Applications: Engineering, Graphics, Visualization, GIS, CAD, Medical Imaging, Computational Biology... Meshing Problem ✦ Compute quickly the smallest size (simplicial, cubical) mesh of a given domain such that all the elements are of good quality. ✦ Applications: Engineering, Graphics, Visualization, GIS, CAD, Medical Imaging, Computational Biology... Meshing Problem ✦ Compute quickly the smallest size (simplicial, cubical) mesh of a given domain such that all the elements are of good quality. ✦ Applications: Engineering, Graphics, Visualization, GIS, CAD, Medical Imaging, Computational Biology... Input: PSLG A collection Ω of vertices and segments is called a planar straight line graph (PSLG) if both endpoints of any segment in Ω are vertices of Ω; • two segments in Ω intersect only at their endpoints. • Local Feature Size Definition. The local feature size at point x of a domain Ω, lfs (x), Ω is the radius of the smallest ball at x that intersects two non-incident features of Ω. lfs(p) lfs(q) + pq (Lipschitz property) ≤ | | INPUT: PLC A collection Ω of vertices, segments, and facet is called a pievewise linear complex (PLC) [MillerTTW96] if boundary of an element in Ω also belong to Ω; • Definitions 87 two elements intersect only at the elements of Ω. • (a) (b) Figure 4.3: (a) Any facet of a PLC may contain holes, slits, and vertices. (b) When a PLC is tetrahedralized, each facet of the PLC is partitioned into triangular subfacets, which respect the holes, slits, and vertices. S F Figure 4.4: The orthogonal projections of points and sets of points onto facets and segments. 4.1 Definitions Tetrahedral mesh generation necessarily divides each facet of a PLC, like that depicted in Figure 4.3(a), into triangular faces, as illustrated in Figure 4.3(b). Just as the triangulation edges that comprise a segment are called subsegments, the triangular faces that comprise a facet are called subfacets. The bold edges in Figure 4.3(b) are subsegments; other edges are not. All of the triangular faces visible in Figure 4.3(b) are subfacets, but most of the faces in the interior of the tetrahedralization are not. Frequently in this chapter, I use the notion of the orthogonal projection of a geometric entity onto a line or plane. Given a facet or subfacet and a point , the orthogonal projection proj of onto is the point that is coplanar with and lies in the line that passes through orthogonally to , as illustrated in Figure 4.4. The projection exists whether or not it falls in . Similarly, the orthogonal projection proj of onto a segment or subsegment is the point that is collinear with and lies in the plane through orthogonal to . Sets of points, as well as points, may be projected. If and are facets, then proj is the set proj . 4.2 A Three-Dimensional Delaunay Refinement Algorithm In this section, I describe a three-dimensional Delaunay refinement algorithm that produces well-graded tetrahedral meshes satisfying any circumradius-to-shortest edge ratio bound greater than two. Miller, Tal- Quality Constraint ✦ Bad quality elements cause interpolation, conditioning, discretization errors BabuskaA76, Knupp98, Shewchuk02 ✦ Various quality criteria: small angles undesired large angles undesired obtuse or non-acute angles undesired elements stretched in certain directions are desired
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