Motivated by Lagrangian simulation of elastic deformation, we propose a new tetrahedral mesh generation algorithm that produces both high quality elements and a mesh that is well conditioned for subsequent large deformations. We use a signed distance function defined on a Cartesian grid in order to represent the object geometry. After tiling space with a uniform lattice based on crystallography, we use the signed distance function or other user defined criteria to guide a red green mesh subdivision algorithm that results in a candidate mesh with the appropriate level of detail. Then, we carefully select the final topology so that the connectivity is suitable for large deformation and the mesh approximates the desired shape. Finally, we compress the mesh to tightly fit the object boundary using either masses and springs, the finite element method or an optimization approach to relax the positions of the nodes. The resulting mesh is well suited for simulation since it is highly structured, has robust topological connectivity in the face of large deformations, and is readily refined if deemed necessary during subsequent simulation.

We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in IR 3 with spread A has complexity il(min{A 3 , uA, u2}) and O (min{A 4, u2}). For the case A = D(v/), our lower bound construction consists of a grid-like sample of a right circular cylinder with constant height and radius. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.

Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra called slivers that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic point sets, but not with boundaries. Recently a randomized point-placement strategy has been proposed to remove slivers while conforming to a given boundary. In this paper we present a deterministic algorithm for generating a weighted Delaunay mesh which respects the input boundary and has no poor quality tetrahedron including slivers. This success is achieved by combining the weight pumping method for sliver exudation and the Delaunay refinement method for boundary conformation. We show that an incremental weight pumping can be mixed seamlessly with vertex insertions in our weighted Delaunay refinement paradigm. 1

We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteed-quality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionality---most notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20 # , as measured from the skewed perspective of any point in the triangle.

Summary. We present a method to decompose an arbitrary 3D piecewise linear complex (PLC) into a constrained Delaunay tetrahedralization (CDT). It successfully resolves the problem of non-existence of a CDT by updating the input PLC into another PLC which is topologically and geometrically equivalent to the original one and does have a CDT. Based on a strong CDT existence condition, the redefinition is done by a segment splitting and vertex perturbation. Once the CDT exists, a practically fast cavity retetrahedralization algorithm recovers the missing facets. This method has been implemented and tested through various examples. In practice, it behaves rather robust and efficient for relatively complicated 3D domains. 1

We present a procedural approach to authoring layered, solid models. Using a simple scripting language, we define the internal structure of a volume from one or more input meshes. Sculpting and simulation operators are applied within the context of the language to shape and modify the model. Our framework treats simulation as a modeling operator rather than simply as a tool for animation, thereby suggesting a new paradigm for modeling as well as a new level of abstraction for interacting with simulation environments.

We present an algorithm to compute a Delaunay mesh conforming to a polyhedron possibly with small input angles. The radius-edge ratio of most output tetrahedra are bounded by a constant, except possibly those that are provably close to small angles. Furthermore, the mesh is not unnecessarily dense in the sense that the edge lengths are at least a constant fraction of the local feature sizes at the edge endpoints. This algorithm is simple to implement as it eliminates most of the computation of local feature sizes and explicit protective zones. Our experimental results validate that few skinny tetrahedra remain and they lie close to small acute input angles. 1

Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearest-neighbor searching, clustering, finite-element mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since three-dimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worst-case running time \Omega (n2). However, this behavior is almost never observed in practice except for highly-contrived inputs. For all practical purposes, three-dimensional Delaunay triangulations appear to have linear complexity. This frustrating

a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in output-sensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement. 1

In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is O(log²(L/s)), where L is the diameter of the domain, and s is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasi-uniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to O(log(L/s)). To the best of our knowledge, these are the first provably polylog(L/s) parallel time Delaunay meshing algorithms that generate well-shaped meshes of size optimal to within a constant.