Tetrahedral mesh generation by Delaunay refinement

Abstract

Given a complex of vertices, constraining segments, and planar straight-line constraining facets in E3, with no input angle less than 90'. an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradius-to-shortest edge ratios are no greater than two. The sizes of the tetrahedra can provably gr<ade from small to large over a relatively short distance. An implementation demonstrates that the algorithm generates excellent meshes, generally surpassing the theoretical bounds, and is effective in eliminating tetrahedra with small or large dihedral angles, although they are not all covered by the theoretical guarantee. I IntroductionMeshes of triangles or tetrahedra have many applications, including interpolation. rendering, and numerical methods such as the finite element method. Most such applications demand more than just a triangulation of the object or domain being rendered or simulated. To ensure accurate results, the triangles or tetrahedra must be "wellshaped," having small aspect ratios or bounds on their smallest and largest angles.Mesh generation algorithms based on Deluunq refinement are effective both in theory and in practice. Delaunay refinement algorithms operate by maintaining a Ddaunay or constrained Delaunay triangulation, which is refined by inserting carefully placed vertices until the mesh meets constraints on triangle quality and size.It is difficult to trace who first used Delaunay triangulations for finite element meshing, and equally difficult to tell where the suggestion arose to use the triangulation to guide vertes creation. These ideas have been intensively studied in the engineering community since the mid-1980s; for instance, Frey [7] eliminates poorly shaped triangles from a triangulation by inserting new vertices at their circumcenters (defined in Section 2), whereas Weatherill [17] inserts new vertices at their centroids.These ideas bore vital theoretical fruit when the problem of mesh generation began to attract interest from the computational $

Study of the high speed blanking proccess with the C-NEM

“…Some techniques for constructing 3D constrained Delaunay tessellations are available by adding Steiner points [6,7]. These points are always set on triangles belonging to the boundary of the domain.…”

3D finite element meshing from imaging data

“…Shewchuk [53] solved the problem of enforcing boundary conformity by constrained Delaunay triangulation (CDT). Delaunay refinement [52], edge removal and multi-face removal optimization algorithm [54] were used to improve the tetrahedral quality. Shewchuk [55] provided some valuable conclusions on quality measures for finite element method.…”

Particle Systems for Adaptive, Isotropic Meshing of CAD Models

“…Technologies to construct simplicial meshes vary greatly. Some of the most important paradigms include advancing-front [19,20,26], Delaunay refinement [10,25,29], and particle systems [22,33]. However, to build meshes that are adaptive, many variants of these techniques require an input oracle that evaluates a sizing field over the domain [4,8,23,27,32].…”