Generation and Optimization of Quadrilateral and Hexahedral Meshes

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Structured quadrilateral (quad) meshes are preferred in many engineering and medical applications due to their desired numerical properties. Despite the extensive research, automatic generation of quad- and hex- meshes with optimal structure remains a challenge. In-lieu-of generating a quad-mesh directly from an arbitrary input, structure simplification can be performed on quad-meshes obtained from triangular meshes. Several techniques utilizing local and global operations for structure simplification have been proposed in this regard. Despite promising results, robustness, scalability, feature preservation and limitations of both local and global simplifications call for a novel and comprehensive framework that addresses the aforementioned limiting factors. In this work, we introduce a few semi-global simplification operations and devise a new framework that utilizes separatrix based operations along with local and global operations for structure simplification of planar and surface quadrilateral meshes. We also employ an automated singularity movement technique to further simplify the mesh structure. In addition, we propose an angle-based mesh optimization algorithm to enhance the quality of simplified meshes. We provide a comprehensive comparison of the results obtained through our simplification framework with existing techniques on a variety of models and demonstrate that our framework is robust and can effectively reduce the mesh singularities to a large extent while preserving the features of quad meshes with complex structures. We also attempt to extend the framework to address the structure simplification of unstructured hexahedral (hex-) meshes that contain more complex scenarios due to the additional dimension. Even though the initial simplification results on some simple hex-meshes are promising, more works need to be done to extend the operations used in quad-meshes simplification for hex-meshes.

Geometry Processing, Computational Geometry, Mesh Optimization, Mesh Generation
Portions of this document appear in: Xu, K., Akram, M. N., & Chen, G. (2020). Semi-global Quad Mesh Structure Simplification via Separatrix Operations. SA'20: SIGGRAPH Asia 2020 Technical Communications; and in: Akram, M. N., Si, L., & Chen, G. (2021). An Embedded Polygon Strategy for Quality Improvement of 2D Quadrilateral Meshes with Boundaries. In VISIGRAPP (1: GRAPP) (pp. 177-184); and in: Akram, M. N., Xu, K., & Chen, G. (2022). Structure simplification of planar quadrilateral meshes. Computers & Graphics, 109, 1-14.