Hexahedral Mesh Generation, Optimization, and Visualization

Hexahedral (or hex-) meshes are preferred in many medical, engineering, and scientific simulation applications due to their desired numerical properties (e.g., natural tensor embedding, large tolerance for anisotropy deformation, and less numerical stiffness), which usually lead to faster and more accurate simulations for those applications. In the general hex-meshing pipeline (i.e. hex-mesh generation and post-processing), my dissertation addresses the following problems. First, most existing techniques aim at generating a hex-mesh with a uniform element size, which may not be suitable for simulations that require higher accuracy in certain regions of interest. To address this issue, a new framework is proposed to generate a hex-mesh with varying element sizes according to the input surface feature, while still having a simple structure. Second, a new framework to untangle hex-meshes with inverted elements effectively is introduced. The proposed method first untangles the hex-mesh and improves its quality via optimizing each edge-angle to its ideal degree. The same framework can be used to further improve the hex-mesh element quality after untangling. Third, an effective framework to calculate and visualize the complexity of a hex-mesh structure is introduced. This framework enables us to decode the configurations in hex-meshes structures in a multi-level fashion. It also introduces a first comprehensive complexity metric for the measurement of the quality of hex-mesh structure, which will benefit the future optimization and manipulation of hex-mesh structure to achieve a desired configuration. Fourth, based on the knowledge obtained from the above structure analysis, a new structure simplification method is proposed, which cancels groups of singularities in a semi-global fashion. This strategy has been implemented for quad-mesh simplification and has a high potential to be extended to simplifying hex-meshes. In summary, this dissertation contributes to all steps of hex-meshing pipeline, including its generation, optimization, and visualization. Based on this dissertation, a robust and automatic framework for the generation of high-quality meshes with structures that adapt to the needs of applications can be developed.