Delaunay Triangulations and Control-Volume Meshing Michael Murphy

Motivation: Unstructured mesh generation

Outline Algorithms to obtain boundary conformity in Delaunay triangulations Algorithms for computing and manipulating Delaunay triangulations Algorithms for quality triangular and tetrahedral meshes

Computing a Delaunay Triangulation Given a set of points S, find all triangles with vertices from S whose circumcircles are empty. Assume general position: No four points are cocircular.

A triangulation is Delaunay iff the circumcircles of the two triangles incident upon each interior edge are locally empty. Local Delaunay Characterization Locally Delaunay Always Locally Delaunay Not

Lawson 1977 Given a triangulation T, flip edges which are not locally Delaunay. Runs in O(n ) time. 2 Lawson’s Algorithm

22 (x, y, x + y ) (From Jonathan Shewchuck) Brown circa 1977

Flipping in Lifted Space

Inserting Points into a Triangulation 1-3 flip Incremental DT algorithm: Create a “big triangle” For each x in S 1-3 flip x into the triangulation 2-2 flip until Delaunay Remove the big triangle Runs in O(n log n) expected time when insertion order is randomized. Guibas, Knuth, Sharir 1992

Deleting a Point from a Delaunay Triangulation Flip edges incident upon q to create triangles on the link of q whose circumcircles contain q and no other point. Perform a 1-3 flip to remove q. q Runs in O(k ) time. It is not asymptotically optimal but perhaps the fastest in practice. 2

Moving a Point in a Delaunay Triangulation To change the coordinates of a point from (x,y) to (x’,y’), one can imagine moving the point continuously along the line segment (x,y)-(x’,y’). A priority queue can be used to manage the points on this segment at which flips must occur to maintain the Delaunay triangulation.

Any two triangulations of the 2-sphere can be related by a sequence of 2-2 topological flips Proof (sketch): Find a Tutte embedding of both triangulations in the plane so that the coordinates of the outer faces coincide. Flip both to the Delaunay triangulation. Move the points of one triangulation so that the coordinates correspond to the vertices of the other while maintaining the Delaunay triangulation. Wagner’s Theorem Wagner 1936 Topologically unflippable Because of this edge

Flip-Graph Connectivity Flip graph G=(V,E): V: A set of triangulations E: (u,v) such that v can be obtained from u with 1 flip. When is the flip graph connected? Triangulations of point sets in the plane. (Lawsons algorithm) n-vertex triangulations of the 2-sphere (Wagner’s theorem) (De Lorea 95)

3-d Bistellar Flips a b c d e abcd abce a b c d e abde bcde acde For any collection of 5 points in convex position, one of these two triangulations is “Locally Delaunay”

3-d Bistellar Flips (Continued)

Flip-based algorithms that work in 3-d Inserting a point into a “regular” tetrahedralization (Incremental Insertion) Edelsbrunner,Shaw 1996 Deleting a point from a regular tetrahedralizationDevillers 1999 Flipping amongst regular tetrahedralizations Edelsbrunner et al Moving a point in a regular tetrahedralization All manipulate one point at a time

Does Lawson’s algorithm generalize? No! Joe 89 2 problems occur Geometrically unflippable faces Topologically unflippable edges No 2-3 flip No 3-2 flip

Is the flip graph connected for 3-d point sets? Open There is a triangulation of a 5-dimensional point set which has no flips. However, the point set is degenerate. De Lorea,Santos,Urrutia 1999 Tetrahedralizations of convex 3-polytopes with no interior vertices have flips. Santos flips There are flip graphs of 3-d point sets which have “low connectivity”De Lorea 1995 The flip graph for such tetrahedralizations is believed to be connected. Bespamyatnik 2000

What about performing topological flips? “Warning: This algorithm may transiently create ‘inverted’ or ‘degenerate’ tetrahedra even if they do not appear in the final tetrahedralization.” Shewchuk 2002 Not every maximal tetrahedralization of a point set is combinatorially equivalent to a regular tetrahedralization. (There is no 3-d Steinitz’s theorem). Grunbaum 1968

A topologically unflippable complex The 1-skeleton is the complete graph. Every edge has 4 or more incident tetrahedra. We have a pure tetrahedral complex, U, with the following two properties: 16 vertices, 120 edges, 208 triangles, 104 tetrahedra Faber, Dougherty, Murphy 2001

Orbits of tuples Let n be the number of vertices in the target complex. Let (a,b,c,d) be a 4-tuple with a,b,c,d < n. The orbit of (a,b,c,d) is the set of tuples: (a+k mod n, b+k mod n, c+k mod n, d+k mod n) for k=0,1,…,n e.g, when n=5 we have {(0,1,2,3), (1,2,3,4)} For n=16, there are 116 distinct orbits.

Tree search Method: Take a union of orbits and ask if it forms an interesting simplicial complex. Prune the search tree by not following branches that will only yield invalid manifolds or uninteresting complexes. Examples: A face appears more than twice. An edge has 3 incident tetrahedra and no more can be added.

Pure tetrahedral complexes satisfying V-E+F-T=0 Homology spheres Homotopy spheres 3-spheres 3-diagrams Boundary complexes of simplicial 4-polytopes

The d-sphere recognition problem Given a triangulation without boundaries, decide if it is topologically equivalent to a sphere. 2-sphere recognition: Linear-time Hopcraft, Tarjan sphere recognition: At most exponential time. Thompson 1994 Mijatovic 1999 Believed to be NP-hard Folklore 4-sphere recognition: Undecidable Markov 1950

Specialized Pachner’s theorem: All tetrahedralizations of the 3-sphere can be related by a sequence of 2-3, 3-2, 1-4, and 4-1 flips. In general, bistellar flips are topology preserving. Pachner 1991

A flip-based 3-manifold recognition heuristic 4-1 flips are always taken 3-2 flips are preferred over 2-3 flips A “heating” and “cooling” schedule is used to kick the complex out of local minima Lutz 1999 Wagner’s theorem does not generalize to 3 dimensions.

U is not straight-line embeddable in R 3 Every neighborly 3-diagram has a topological flip.

Non-Neighborly Unflippable Complexes Every edge has 4 or more incident tetrahedra. For each face (a,b,c) in C shared by the tetrahedra (a,b,c,d) and (a,b,c,e), the edge (d,e) is in C. A tetrahedral complex C is topologically unflippable if: We found a complex with 20 vertices, 170 edges, 300 triangles, and 150 tetrahedra but it is not a 3-sphere.

Does there exist a non-neighborly unflippable 3-sphere? Can it be a 3-diagram? Is there an unflippable 3-sphere smaller than 16 vertices? Questions:

Outline Algorithms to obtain boundary conformity in Delaunay triangulations Algorithms for computing and manipulating Delaunay triangulations Algorithms for “quality” triangular and tetrahedral meshes

Given an irregular domain, taking the Delaunay triangulation of its vertices does not solve most mesh generation problems

2-d piecewise-linear domain representation Polygons with holesPlanar straight-line graphs (PSLG) Can be used to represent simple domains (e.g., single materials) Can be used to represent arbitarily complicated multi-material domains

3-d piecewise-linear domain representation