Variational Volumetric Meshing
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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.