Jonathan Richard Shewchuk

Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Rupperts Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

Delaunay refinement algorithms for triangular mesh generation

Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles. Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. A compromise is necessary. A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30^o or greater and no angle is smaller than arcsin[(3/2)sin(@f/2)]~(3/4)@f, where @f=<60^ois the smallest angle separating two segments of the input domain. New angles smaller than 30^o appear only near input angles smaller than 60^o. In practice, the algorithms performance is better than these bounds suggest. Another new result is that Rupperts analysis technique can be used to reanalyze one of Chews algorithms. Chew proved that his algorithm produces no angle smaller than 30^o (barring small input angles), but without any guarantees on grading or number of triangles. He conjectures that his algorithm offers such guarantees. His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 26.5^o, Chews algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal.

Tetrahedral mesh generation by Delaunay refinement

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.

Interpolating and approximating implicit surfaces from polygon soup

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers

This paper reports on the development of a parallel numerical methodology for simulating large-scale earthquake-induced ground motion in highly heterogeneous basins. We target large sedimentary basins with contrasts in wavelengths of over an order of magnitude. Regular grid methods prove intractable for such problems. We overcome the problem of multiple physical scales by using unstructured finite elements on locally-resolved Delaunay triangulations derived from octree-based grids. The extremely large mesh sizes require special mesh generation techniques. Despite the method’s multiresolution capability, large problem sizes necessitate the use of distributed memory parallel supercomputers to solve the elastic wave propagation problem. We have developed a system that helps automate the task of writing efficient portable unstrucmred mesh solvers for distributed memory parallel supercomputers. The numerical methodology and software system have been used to simulate the seismic response of the San Fernando Valley in Southern California to an aftershock of the 1994 Northridge Earthquake. We report on parallel performance on the Cray T3D for several models of the basin ranging in size from 35 000 to 77 million tetrahedra. The results indicate that, despite the highly irregular structure of the problem, excellent performance and scalability are achieved.

Isosurface stuffing: fast tetrahedral meshes with good dihedral angles

The isosurface stuffing algorithm fills an isosurface with a uniformly sized tetrahedral mesh whose dihedral angles are bounded between 10.7° and 164.8°, or (with a change in parameters) between 8.9° and 158.8°. The algorithm is whip fast, numerically robust, and easy to implement because, like Marching Cubes, it generates tetrahedra from a small set of precomputed stencils. A variant of the algorithm creates a mesh with internal grading: on the boundary, where high resolution is generally desired, the elements are fine and uniformly sized, and in the interior they may be coarser and vary in size. This combination of features makes isosurface stuffing a powerful tool for dynamic fluid simulation, large-deformation mechanics, and applications that require interactive remeshing or use objects defined by smooth implicit surfaces. It is the first algorithm that rigorously guarantees the suitability of tetrahedra for finite element methods in domains whose shapes are substantially more challenging than boxes. Our angle bounds are guaranteed by a computer-assisted proof. If the isosurface is a smooth 2-manifold with bounded curvature, and the tetrahedra are sufficiently small, then the boundary of the mesh is guaranteed to be a geometrically and topologically accurate approximation of the isosurface.

Spectral surface reconstruction from noisy point clouds

We introduce a noise-resistant algorithm for reconstructing a watertight surface from point cloud data. It forms a Delaunay tetrahedralization, then uses a variant of spectral graph partitioning to decide whether each tetrahedron is inside or outside the original object. The reconstructed surface triangulation is the set of triangular faces where inside and outside tetrahedra meet. Because the spectral partitioner makes local decisions based on a global view of the model, it can ignore outliers, patch holes and undersampled regions, and surmount ambiguity due to measurement errors. Our algorithm can optionally produce a manifold surface. We present empirical evidence that our implementation is substantially more robust than several closely related surface reconstruction programs.

Aggressive Tetrahedral Mesh Improvement

We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh clean-up.” Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poor-quality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques—smoothing, vertex insertion, and traditional transformations—are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31° and 149°, or (with a different objective function) between 23° and 136°

Streaming computation of Delaunay triangulations

We show how to greatly accelerate algorithms that compute Delaunay triangulations of huge, well-distributed point sets in 2D and 3D by exploiting the natural spatial coherence in a stream of points. We achieve large performance gains by introducing spatial finalization into point streams: we partition space into regions, and augment a stream of input points with finalization tags that indicate when a point is the last in its region. By extending an incremental algorithm for Delaunay triangulation to use finalization tags and produce streaming mesh output, we compute a billion-triangle terrain representation for the Neuse River system from 11.2 GB of LIDAR data in 48 minutes using only 70 MB of memory on a laptop with two hard drives. This is a factor of twelve faster than the previous fastest out-of-core Delaunay triangulation software.

Interactive simulation of surgical needle insertion and steering

We present algorithms for simulating and visualizing the insertion and steering of needles through deformable tissues for surgical training and planning. Needle insertion is an essential component of many clinical procedures such as biopsies, injections, neurosurgery, and brachytherapy cancer treatment. The success of these procedures depends on accurate guidance of the needle tip to a clinical target while avoiding vital tissues. Needle insertion deforms body tissues, making accurate placement difficult. Our interactive needle insertion simulator models the coupling between a steerable needle and deformable tissue. We introduce (1) a novel algorithm for local remeshing that quickly enforces the conformity of a tetrahedral mesh to a curvilinear needle path, enabling accurate computation of contact forces, (2) an efficient method for coupling a 3D finite element simulation with a 1D inextensible rod with stick-slip friction, and (3) optimizations that reduce the computation time for physically based simulations. We can realistically and interactively simulate needle insertion into a prostate mesh of 13,375 tetrahedra and 2,763 vertices at a 25 Hz frame rate on an 8-core 3.0 GHz Intel Xeon PC. The simulation models prostate brachytherapy with needles of varying stiffness, steering needles around obstacles, and supports motion planning for robotic needle insertion. We evaluate the accuracy of the simulation by comparing against real-world experiments in which flexible, steerable needles were inserted into gel tissue phantoms.