We presentNetMesh, anew algorithmthat producesaconformingDelaunaymesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparison based algorithm runs in time O(nlogn+m), where n is the input size and m is the output size, and with constants depending only on the dimension and the desired element quality bounds. It can terminate early in O(nlogn) time returning a O(n) size Voronoi diagram of a superset of P with a relaxed quality bound, which again matches the known lower bounds. The previous best results in the comparison model depended on the log of the spread of the input, the ratio of the largest to smallest pairwise distance among input points. We reduce this dependence to O(logn) by using a sequence of ǫ-nets to determine input insertion order in an incremental Voronoi diagram. We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.

We propose algorithms to modify a mesh of a PSLG through out the interactive addition/deletion of elements to/from the PSLG, keeping the quality of the mesh all along the process. Our algorithms achieve quality by deleting, moving or inserting Steiner points. 1

Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, meshes are often used to discretely represent the value of a function over space. In addition to the quality requirement, the user often has input segments or polygons (generally, a piecewise linear complex) they would like see retained in the mesh; the mesh must respect these constraints. Finally, the mesh should be size-conforming: the size of mesh elements should be related to a particular sizing function based on the distance between input features. The static meshing problem is increasingly well-understood: one can download software with provable guarantees that on reasonable input, the meshes will have good quality, will respect the input, and will be size-conforming; more recently, these algorithms have started to come with optimal runtimes of O(n lg(L/s) + m), where L/s is the spread of the input. As a first result, I present experimental results of the first time-optimal code, available online at sparse-meshing.com. Increasingly, static meshing is insufficient: users want to modify the mesh over time. Throwing away the old mesh and remeshing from scratch is a common approach, but that suffers from slow runtime, and from reinterpolation error because the old and new meshes may be almost unrelated. Mesh stability analyzes the correspondence between meshes for two inputs. The main theoretical result of this thesis is an algorithm that has provable bounds on stability: upon inserting or removing a feature that in the final mesh is represented using k points, the mesh only modifies O(k lg(L/s)) mesh simplices. Finally, stability can be exploited to produce an efficient dynamic algorithm. Under the self-adjusting computation framework, with a small amount of additional effort, I show that my algorithm can be dynamized to run in O(k lg(L/s)) time per update, using O(n lg(L/s) + m) space.

Summary. In mesh refinement for scientific computing or graphics, the input is a description of an input geometry, and the problem is to produce a set of additional “Steiner ” points whose Delaunay triangulation respects the input geometry, and whose points are well-spaced. Ideally, we would like the minimum output size – as few Steiner points as possible. Ruppert showed how to come within a constant factor of optimal in two dimensions, and a number of techniques have been proposed to extend his work to various settings (higher dimension, curved features, fast runtime, parallelism, etc). Each extension needs a different choice of Steiner point, and thus its own tedious proof of the sizing guarantee. In this article, I show how to classify Steiner points as safe or potentially unsafe; any algorithm that chooses only safe points achieves the sizing guarantee. Furthermore, I show that it is easy to classify as safe the Steiner points chosen in most prior work. This work frees future meshing researchers to more easily consider varying the choice of Steiner points to achieve important new properties.

The mesh generation problem is to output a set of tetrahedra that discretize an input geometry. The input is given as a piecewise linear complex (PLC), a set of points, lines, and polygons to which the output tetrahedra must conform. Additionally, a mesh generation algorithm must make guarantees on the quality and number of output tetrahedra. Downstream applications in scientific computing and visualization necessitate these guarantees on the mesh. Recent advances have led to provably correct algorithms for a number of input classes. Particular difficulties arise when the input contains creases, regions where input segments or polygons meet at acute angles. When the input is without creases, the mesh generation problem is better understood. Algorithms for such inputs exist with near-optimal runtimes of O(n log ∆+m), where n and m are the size of the input and output, and ∆ is the ratio of largest-to-smallest distances in the input geometry. The principle result of this thesis is to extend this result to the general case of piecewise linear complexes with creases. Correct algorithms to handle inputs with creases involve explicitly constructing

We presentNetMesh, a newalgorithm that producesaconforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparison-based algorithm runs in O(nlogn + m) time, where n is the input size and m is the output size, and with constants depending only on the dimension and the desired element quality. It can terminate early in O(nlogn) time returning a O(n) size Voronoi diagram of a superset of P, which again matches the known lower bounds. The previous best results in the comparison model depended on the log of the spread of the input, the ratio of the largest to smallest pairwise distance. We reduce this dependence to O(logn) by using a sequence of ǫ-nets to determine input insertion order into a incremental Voronoi diagram. We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.