Towards High Quality Hexahedral Meshes: Generation, Optimization, and Evaluation

Hexahedral meshes are a preferred volumetric representation in a wide range of scientific and engineering applications that require solving partial differential equations (PDEs) and fitting tensor product/trivariate splines, such as mechanical analysis, kinematic and dynamic analysis of mechanisms, bio-mechanical engineering, computational fluid dynamics, and physically-based simulations. Recently, the generation of a high quality all-hex mesh of a given volume has gained much attention, where a hex-mesh should have high surface conformity, regular element shapes, and simple global structure. This dissertation investigates the problem of obtaining a high quality hex-mesh with respect to the above quality requirements and makes the following contributions:

Firstly, I introduce a volumetric partitioning strategy based on a generalized sweeping framework to seamlessly partition the volume enclosed by an input triangle mesh into a small number of deformed cube-like components. This is achieved by a user-designed volumetric harmonic function that guides the decomposition of the input volume into a skeletal structure aligning with features of the input object. This pipeline has been applied to a variety of 3D objects to demonstrate its utility.

Secondly, I present a first and complete pipeline to reduce the complexity of the global structure of an input hex-mesh by aligning mis-matched singularities. Specifically, I first remove redundant cube-like components to reduce the complexity of the structure while maintaining singularities unchanged, and then perform a structure-aware optimization to improve the geometric fidelity of the resulting hex-mesh.

Thirdly, I propose the first practical framework to simplify the global structure of any valid all-hex meshes. My simplification was achieved by procedurally removing base complex sheets and base complex chords that constitute the base complex of a hex-mesh. To maintain the surface geometric features, I introduced a parameterization based collapsing strategy for the removal operations. Given a user-specified level of simplicity, I identified the inversion-free hex-mesh with the optimal simplified structure using a binary search strategy from the obtained all-hex structure hierarchy.

Finally, given that there currently does not exist a widely accepted guideline for the selection of proper element quality metrics for hex-meshes, I performed the first comprehensive study on the correlation among available quality metrics for hex-meshes. My analysis first computed the linear correlation coefficients between pairs of metrics. Then, the most relevant metrics were identified for three selected applications -- the linear elasticity, Poisson, and Stoke problems, respectively. To address the need of a large set of sampled meshes well-distributed in the metric space, I proposed a two-level noise insertion strategy. Results of this work can be used as preliminary yet practical guidelines for the development of effective hex-mesh generation and optimization techniques.