A Consistent Parallel Isotropic Unstructured Mesh Generation Method based on Multi-phase SPH

In this paper, we propose a consistent parallel unstructured mesh generator based on a multi-phase SPH method. A set of physics-motivated modeling equations are developed to achieve the targets of domain decomposition, communication volume optimization and high-quality unstructured mesh generation simultaneously. A unified density field is defined as the target function for both partitioning the geometry and distributing the mesh-vertexes. A multi-phase Smoothing Particle Hydrodynamics (SPH) method is employed to solve the governing equations. All the optimization targets are achieved implicitly and consistently by the particle relaxation procedure without constructing triangulation/tetrahedralization explicitly. The target of communication reduction is achieved by introducing a surface tension model between distinct partitioning sub-domains, which are characterized by colored SPH particles. The resulting partitioning diagram features physically localized sub-domains and optimized interface communication. The target of optimizing the mesh quality is achieved by introducing a tailored equation-of-state (EOS) and a smooth isotropic kernel function. The mesh quality near the interface of neighboring sub-domains is improved by gradually removing the surface-tension force once a steady state is achieved. The proposed method is developed basing on a new parallel environment for multi-resolution SPH to exploit both coarse- and fine-grained parallelism. A set of benchmarks are conducted to verify that all the optimization targets are achieved consistently within the current framework.Comment: 47 pages 13 figure

Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric

International audienceThis article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We show here how to solve the inverse problem, i.e., given an input metric, compute a smooth intersection-free high-d embedding of the input such that the pullback metric of the embedding matches the input metric. Our numerical solution mechanism matches the deformation gradient of the 3D → higher-d mapping with the given Riemannian metric. We demonstrate applications of the method, by being used to construct anisotropic Restricted Voronoi Diagram (RVD) and anisotropic meshing, that are otherwise extremely difficult to compute. In the SIFHDE-space constructed by our algorithm, difficult 3D anisotropic computations are replaced with simple Euclidean computations, resulting in an isotropic RVD and its dual mesh on this high-d embedding. The results are compared with the state-ofthe-art in anisotropic surface and volume meshings using several examples and evaluation metrics

Variational Volumetric Meshing

Domain discretization, also referred to as mesh generation, is one of the fundamental steps of many computation based applications. Although mesh generation techniques have evolved rapidly over the years, some volumetric meshing problems like sliver suppressing in tetrahedral meshing, field-aligned tetrahedral meshing, and hexahedral meshing are still not fully resolved. In this dissertation, we bring some insights to those problems. This dissertation discusses variational-based methods to tackle mesh generation problems, i.e., we model these problems in the energy optimization framework. An energy which inhibits small heights is proposed to suppress almost all the badly-shaped elements in tetrahedral meshing. By iteratively optimizing vertex positions and mesh connectivity, slivers are harshly suppressed even in anisotropic tetrahedral meshing. Besides that, a particle-based field alignment framework is introduced. Specifically, a Gaussian Hole Kernel is constructed associated with each particle to constrain the formation of the desired one ring structure aligned with the frame field. The minimization of the sum of Gaussian hole kernels induces an inter-particle potential energy whose minimization encourages particles to have the desired layout. A cubic one ring structure leads to high quality hexahedral-dominant meshing. The one ring structures of the Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) lattice leads to high quality field-aligned tetrahedral meshing. This is the first time both Riemannian distance alignment and direction alignment problems have been considered in tetrahedral meshing. Also, field-aligned tetrahedral meshing better preserves the rotation geometry and also creates better anisotropic meshes