VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Conforming restricted Delaunay mesh generation for piecewise smooth complexes

A Frontal-Delaunay refinement algorithm for mesh generation in piecewise smooth domains is described. Built using a restricted Delaunay framework, this new algorithm combines a number of novel features, including: (i) an unweighted, conforming restricted Delaunay representation for domains specified as a (non-manifold) collection of piecewise smooth surface patches and curve segments, (ii) a protection strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of off-centre refinement rules designed to achieve high-quality point-placement along embedded curve features. Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from alpha-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an epsilon-sample on the bounding surface, with a weak sigma-sparsity condition, we work with the balls of radius delta times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of epsilon, sigma and delta, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples

Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries. The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems. Keywords: Anisotropic mesh generation; metric; Nash embedding theorem; isometric; geometry-conforming; restricted Voronoi diagram; constrained centroidal Voronoi tessellation; Steiner vertices; dimension-independen

Locally optimal Delaunay-refinement and optimisation-based mesh generation

The field of mesh generation concerns the development of efficient algorithmic techniques to construct high-quality tessellations of complex geometrical objects. In this thesis, I investigate the problem of unstructured simplicial mesh generation for problems in two- and three-dimensional spaces, in which meshes consist of collections of triangular and tetrahedral elements. I focus on the development of efficient algorithms and computer programs to produce high-quality meshes for planar, surface and volumetric objects of arbitrary complexity. I develop and implement a number of new algorithms for mesh construction based on the Frontal-Delaunay paradigm - a hybridisation of conventional Delaunay-refinement and advancing-front techniques. I show that the proposed algorithms are a significant improvement on existing approaches, typically outperforming the Delaunay-refinement technique in terms of both element shape- and size-quality, while offering significantly improved theoretical robustness compared to advancing-front techniques. I verify experimentally that the proposed methods achieve the same element shape- and size-guarantees that are typically associated with conventional Delaunay-refinement techniques. In addition to mesh construction, methods for mesh improvement are also investigated. I develop and implement a family of techniques designed to improve the element shape quality of existing simplicial meshes, using a combination of optimisation-based vertex smoothing, local topological transformation and vertex insertion techniques. These operations are interleaved according to a new priority-based schedule, and I show that the resulting algorithms are competitive with existing state-of-the-art approaches in terms of mesh quality, while offering significant improvements in computational efficiency. Optimised C++ implementations for the proposed mesh generation and mesh optimisation algorithms are provided in the JIGSAW and JITTERBUG software libraries

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains

Adaptive Sampling for Geometric Approximation

Geometric approximation of multi-dimensional data sets is an essential algorithmic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an epsilon-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1+epsilon) times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. Combined with a careful regularization of the distance minimizers, we obtain a generalized data structure that essentially matches state-of-the-art results specific to the Euclidean distance. Finally, we investigate the generation of Voronoi meshes, where a given domain is decomposed into Voronoi cells as desired for a number of important solvers in computational fluid dynamics. The challenge is to arrange the cells near the boundary to yield an accurate surface approximation without sacrificing quality. We propose a sampling algorithm for the placement of seeds to induce a boundary-conforming Voronoi mesh of the correct topology, with a careful treatment of sharp and non-manifold features. The proposed algorithm achieves significant quality improvements over state-of-the-art polyhedral meshing based on clipped Voronoi cells

Virtual Element based formulations for computational materials micro-mechanics and homogenization

In this thesis, a computational framework for microstructural modelling of transverse behaviour of heterogeneous materials is presented. The context of this research is part of the broad and active field of Computational Micromechanics, which has emerged as an effective tool both to understand the influence of complex microstructure on the macro-mechanical response of engineering materials and to tailor-design innovative materials for specific applications through a proper modification of their microstructure. While the classical continuum approximation does not account for microstructural details within the material, computational micromechanics allows detailed modelling of a heterogeneous material's internal structural arrangement by treating each constituent as a continuum. Such an approach requires modelling a certain material microstructure by considering most of the microstructure's morphological features. The most common numerical technique used in computational micromechanics analysis is the Finite Element Method (FEM). Its use has been driven by the development of mesh generation programs, which lead to the quasi-automatic discretisation of the artificial microstructure domain and the possibility of implementing appropriate constitutive equations for the different phases and their interfaces. In FEM's applications to computational micromechanics, the phase arrangements are discretised using continuum elements. The mesh is created so that element boundaries and, wherever required, special interface elements are located at all interfaces between material's constituents. This approach can be effective in modelling many microstructures, and it is readily available in commercial codes. However, the need to accurately resolve the kinematic and stress fields related to complex material behaviours may lead to very large models that may need prohibitive processing time despite the increasing modern computers' performance. When rather complex microstructure's morphologies are considered, the quasi-automatic discretisation process stated before might fail to generate high-quality meshes. Time-consuming mesh regularisation techniques, both automatic and operator-driven, may be needed to obtain accurate numeric results. Indeed, the preparation of high-quality meshes is today one of the steps requiring more attention, and time, from the analyst. In this respect, the development of computational techniques to deal with complex and evolving geometries and meshes with accuracy, effectiveness, and robustness attracts relevant interest. The computational framework presented in this thesis is based on the Virtual Element Method (VEM), a recently developed numerical technique that has proven to provide robust numerical results even with highly-distorted mesh. These peculiar features have been exploited to analyse two-dimensional representations of heterogeneous materials' microstructures. Ad-hoc polygonal multi-domain meshing strategies have been developed and tested to exploit the discretisation freedom that VEM allows. To further simplify the preprocessing stage of the analysis and reduce the total computational cost, a novel hybrid formulation for analysing multi-domain problems has been developed by combining the Virtual Element Method with the well-known Boundary Element Method (BEM). The hybrid approach has been used to study both composite material's transverse behaviour in the presence of inclusions with complex geometries and damage and crack propagation in the matrix phase. Numerical results are presented that demonstrate the potential of the developed framework

Addressing some current issues in linear and high-order meshing

The thesis explores the generation of anisotropic and boundary conforming Voronoi regions and Delaunay triangulations, high-order mesh quality and the development of mesh enhancement techniques which incorporate quality measures to preserve mesh validity longer. In the first part an analogy with crystal growth is proposed to handle mesh anisotropy and boundary conformity in Voronoi diagrams and Delaunay mesh generation. A Voronoi partition of a domain corresponds to the steady-state configuration of many crystals growing from their seed points. Mesh anisotropy is incorporated and the shape of the boundary of an isolated crystal is guided by re-interpreting a user-defined Riemann metric in terms of the velocity of the crystal boundary. A straightforward implementation of conformity to boundaries is achieved by treating the boundary of the computational domain as the boundary of a stationary crystal. The second part attempts to answer the question: what is a good highorder element? A review of a priori mesh quality measures suitable for high-order elements is presented. A systematic analysis of the quality measures for interior and boundary elements is then carried out utilising a number of test cases that consist of a set of carefully selected elements with various degrees of distortion. Their ability to identify severe geometrical distortion is discussed. The effect of boundary curvature on the performance of quality measures is also investigated. The last part proposes improvements to a conventional mesh deformation method based on the equations of elasticity to maintain highorder mesh validity and enhance mesh quality. This is accomplished by incorporating additional terms, that can be interpreted as body forces and thermal stresses in the elastic analogy. Different test cases are designed to prolong mesh validity, and their performance is reported. A proposal of how to formulate these terms to incorporate anisotropy is also presented.Open Acces

7th International Meshing Roundtable '98

The goal of the 7th International Meshing Roundtable is to bring together researchers and developers from industry, academia, and government labs in a stimulating, open environment for the exchange of technical information related to the meshing process. In the past, the Roundtable has enjoyed significant participation from each of these groups from a wide variety of countries