Lecture on Mesh Generation

154 Pages·2008·7.96 MB·English

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