In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Xifeng Gao

will defend his dissertation

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

Abstract

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, generating a high quality all-hex mesh of a given volume has gained much attention, where a hex-mesh should have high geometric fidelity, regular element shapes, and simple global structure. This dissertation tries to tackle the problem of obtaining a high quality hex-mesh with respect to the quality requirements through making the following contributions:

Firstly, we introduce a volumetric partitioning strategy based on a generalized sweeping framework to seamlessly partition the volume of an input triangle mesh into an as few as possible collection of deformed 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, we present a first and complete pipeline to reduce the complexity of the global structure of an input hex-mesh by correcting mis-aligned singularities. Specifically, we first remove redundant components to reduce the complexity of the structure while maintaining singularities unchanged, and then propose a structure-aware optimization strategy to improve the geometric fidelity of the resulting hex-mesh.

Thirdly, we propose the first simplification framework to simplify the global structure of any valid all-hex meshes. Our simplification is achieved by procedurally removing base complex sheets and base complex chords that composing the base-complex of a hex-mesh. To maintain the surface geometric feature, we introduce a parameterization based collapsing strategy for the removal operations. Given a user-specified level of simplicity, we choose 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 shape quality metrics for hex-meshes, we perform a first comprehensive study on the correlation among available quality metrics for hex-meshes. Our analysis first computes the linear correlation coefficients between pairs of metrics. Then, identifies the most relevant metrics 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, we propose 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.

Date: Monday, April 11, 2016
Time: 4:00 PM
Place: PGH 550
Advisor: Prof. Guoning Chen and Zhigang Deng

Faculty, students, and the general public are invited.