A Proposal for a Dissertation on Tetrahedral/Hexahedral Finite Element Meshing from Volumetric Imaging Data

The proposed research for the Computational Engineering and Sciences (CES) option of the Computational and Applied Mathematics (CAM) Ph.D. program is in the area of tetrahedral/hexahedral finite element meshing from volumetric imaging data with defined boundaries. The proposed research work is to extract adaptive tetrahedral and hexahedral finite element meshes with guaranteed quality directly from volumetric imaging data, such as Computed Tomography (CT), Magnetic Resonance Imaging (MRI) and Signed Distance Function (SDF) data. The extracted meshes should conform to a level set boundary defined as {(x,y,z)| f (x,y,z) = 0 or f ′(x,y,z) = 0}, where f ′ is a modified function of f by segmentation techniques. Both the interior and exterior volume of the boundary needs to be tetrahedralized or hexahedralized, and hanging nodes are prohibited. A 3D mesh generation software called LBIE-Mesh (Level set Boundary and Interior-Exterior Mesher) will be developed. The proposed approach to generate the required finite element meshes is as follows: (i) Anisotropic diffusion with bilaterail prefiltering and the contour spectrum based interval volume selection are set as a preprocessing. The preprocessing step removes noise in the scanned imaging data, and helps to choose boundary isosurfaces if necessary. Segmentation techniques can detect level set boundaries, so they are proposed to be a part of preprocessing especially for scanned soft tissue structures. (ii) A top-down octree subdivision coupled with the dual contouring method is used to rapidly extract adaptive 3D finite element meshes with correct topology from volumetric imaging data. The dual contouring method is extended to crack-free 3D meshing with feature sensitive adaptation. (iii) The proposed final step is to improve the quality of the extracted tetrahedral and hexahedral meshes. For tetrahedral meshes, the edge-ratio and JoeLiu parameter are chosen to measure the quality. The coarsening and refinement method removes tetrahedra with bad edge-ratios, face/edge swapping and the smoothing method improves the mesh quality measured by the Joe-Liu parameters. The smoothing method can also be used to improve the quality of hexahedral meshes measured by the Jacobian determinants. The discretization of the Laplacian operator and physicallybased quality improvement will be studied. Anisotropic meshes can be obtained by the necessary refinement along the required direction and the extended Laplacian smoothing techniques. Preliminary work includes the development of adaptive tetrahedral mesh extraction and uniform hexahedral mesh construction. The edge contraction method and a simple Laplacian smoothing method are implemented to improve the quality of the extracted tetrahedral meshes. The preliminary version of LBIEMesh has been used to construct 3D meshes for some objects. For example, adaptive and quality tetrahedral meshes of the molecule mouse acetylcholinesterase (mAChE) are generated and have been successfully used in solving the diffusion problem using the finite element method (FEM). We also use the software to construct tetrahedral meshes of the human heart, which can be used for studying cardiac fluid dynamics. Sometimes people would like to choose hexahedral meshes instead of tetrahedra and guaranteed-quality tetra/hexa meshes are required in finite element calculations, therefore the proposed future work includes adaptive hexahedral mesh generation, and the algorithm development in generating guaranteed-quality meshes. The current version of LBIE-Mesh supports interactive operations, so it is memory consuming. People may need finite element meshes from a very large volumetric data. For example, the molecule microtubule has more than one million particles. In the future work, the data structure of LBIE-Mesh needs to be improved for arbitrary large datasets. Parallelization is also necessary in the performance improvement. The finite element meshing of the patient-specific human heart is very important for predictive medicine such as the heart valve replacement and cardiovascular disease. Therefore the proposed future work also includes editing a known heart model according to the segmentation results of scanned data to construct