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Hexahedral Mesh Generation from Volumetric Data by Dual Interval Volume

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Hexahedral Mesh Generation from Volumetric Data by Dual Interval Volume Hexahedral Mesh Generation from Volumetric Data by Dual Interval Volume Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Fei Xiao Graduate Program in Computer Science and Engineering The Ohio State University 2018 Thesis Committee Dr. Rephael Wenger, Advisor Dr. Tamal Dey Copyrighted by Fei Xiao 2018 Abstract Finite element methods play an important role in the field of scientific research and engineering applications. An important requirement of numerical methods is to discretize the model into a mesh composed of simple elements. In three-dimensional numerical analysis, tetrahedral and hexahedral elements are usually used. Tetrahedral meshes have the advantage of high efficiency, easy implementation, flexibility for adaptive mesh generation and (relatively) easy mesh regeneration. However, hexahedral meshes have an advantage over tetrahedron element meshes regarding the analysis accuracy and total number of elements. This makes hexahedral meshes an attractive choice for numerical analysis. In medical and industrial applications, X-ray computed tomography (CT) or magnetic resonance imaging (MRI) scanners are widely used in medical and industrial diagnostics. The CT and MRI produce a regular grid with scalar values at each grid vertex. This regular grid is called volumetric data. Hexahedral mesh generation for volumetric data provides an opportunity to exploit the structure provided in the volumetric data. One way to generate hexahedral meshes from the volumetric data is to extract the surface geometry first and then mesh the object. However, this two-step approach takes extra time and effort for mesh generation. Also, this approach does not take the advantage of the regular grid from which the surface is generated. i The advantage of generating hexahedral meshes directly from a regular grid is that the underlying grid can guide hexahedral mesh generation. There are usually one or two hexahedral vertices placed in each grid cube. This property keeps the vertices well separated. Previous work constructing hexahedral meshes directly from volumetric data placed a single mesh vertex in each grid cube. That algorithm required adaptive refinement to avoid non-manifold conditions. This paper proposes a new algorithm, called Dual Interval Volume, to generate an all hexahedral mesh directly from volumetric data. Our algorithm allows two or more hexahedral mesh vertices to be placed in the same grid cube such that some of the non- manifold issues can be avoided. Special rules are used to guarantee that the output meshes are manifolds. Smoothing methods are implemented to improve the mesh quality. The algorithm is tested with real volumetric data and the results are compared with commercial software. ii Acknowledgments First and foremost, I would like to express my deep gratitude to my advisor, Dr. Rephael Wenger, for his support and guidance throughout last two years. By working with him, I have learned to solve problems in computer science. He is a rigorous researcher. I feel very lucky that learning with Dr. Wenger is such a pleasant journey. I am grateful to Dr. Tamal Dey for serving on the thesis committee, as well as for his Geometric Modeling class. Among all the classes I have ever taken, his class is the most important one to this thesis. Dr. Dey is good at encouraging students asking questions, and then answering the questions themselves. This process helped me understand the geometric concepts better and longer. The best thing for a student like me is learning from such a remarkable teacher. At last, I give my appreciation to my wife and my parents. Without their support and encouragement, I cannot finish this thesis. It is power of love keeps me progressing. iii Vita Jul 2013 . .B. Sc. Naval Architecture and Ocean Engineering Shanghai Jiao Tong University. Shanghai, China May 2015 . M.Sc. Ocean Engineering Texas A&M University College Station, Texas, USA Aug 2015 – Present . ..M.Sc Student, Computer Science and Engineering The Ohio State University. Columbus, OH, USA Publications Soghrati, S., Xiao, F., & Nagarajan, A. (2017). A conforming to interface structured adaptive mesh refinement technique for modeling fracture problems. Computational Mechanics, 59(4), 667-684. Fields of Study Major Field: Computer Science and Engineering iv Table of Contents Abstract ................................................................................................................................ i Acknowledgments.............................................................................................................. iii Vita ..................................................................................................................................... iv List of Tables .................................................................................................................... vii List of Figures .................................................................................................................. viii Chapter 1. Introduction ....................................................................................................... 1 Chapter 2. Background ...................................................................................................... 4 2.1 Isosurface extraction ............................................................................................. 5 2.2 Mesh Generation ................................................................................................... 9 2.2.1 Definition of Mesh ..................................................................................... 9 2.2.2 Triangle/Tetrahedral Mesh Generation .................................................... 11 2.2.3 Hexahedral Mesh ............................................................................................. 12 2.3 Mesh Quality Improvement ................................................................................ 14 Chapter 3. Dual Interval Volume ..................................................................................... 16 3.1 Definitions and Notations ................................................................................... 16 3.2 Dual Marching Cubes ......................................................................................... 17 3.3 DIVol .................................................................................................................. 22 3.3.1 Background .............................................................................................. 22 3.3.2 Construction of the Interval Volume Lookup Table ................................ 23 3.4 Manifold .............................................................................................................. 29 Chapter 4 Hexahedral Mesh Quality ................................................................................. 45 v 4.1 Hexahedral Mesh Quality Metrics ...................................................................... 45 4.2 Mesh Quality Improvement ............................................................................. 48 Chapter 5 Experimental Results........................................................................................ 51 5.1 Manifold Check of Random Data ....................................................................... 51 5.2 Manifold Check on Real Data ............................................................................. 52 5.3 Mesh Quality Check on Real Data for Interval Volume [σ0, +∞] .................... 53 5.3.1 Mesh Quality at Internal Vertices ............................................................ 53 5.3.2 Mesh Quality at All Vertices ................................................................... 58 5.4 Mesh Quality Check on Real Data for Interval Volume [σ0, σ1] ...................... 59 5.4.1 Mesh Quality at Internal Vertices ............................................................ 59 5.4.2 Mesh Quality at All Vertices ................................................................... 60 5.5 Mesh Quality Improvement ................................................................................ 61 5.5.1 Short Edge Length Improvement ............................................................. 61 5.5.2 Scaled Jacobian Improvement for All Vertices ....................................... 64 5.5.3 Scaled Jacobian Improvement for Range [σ0, σ1] .................................. 66 5.6 Mesh Quality Comparison with Bolt 2.0 ............................................................ 69 Chapter 6 Conclusions ...................................................................................................... 77 Bibliography ..................................................................................................................... 79 Appendix A. ...................................................................................................................... 86 vi List of Tables Table 1 Mesh Quality Tests of Neghip at Internal Vertices ............................................. 54 Table 2 Comparison of CPU Time between DIVol and Bolt 2.0 ..................................... 70 Table 3 Comparison of Normalized Scaled Jacobian of DIVol and Bolt 2.0 at Internal Vertices ............................................................................................................................. 75 Table 4 Comparison of Normalized Scaled Jacobian of DIVol and Bolt 2.0 at All Vertices ............................................................................................................................
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