Homeomorphic Tetrahedralization of Multi-material Images with Quality and Fidelity Guarantees

We present a novel algorithm for generating three-dimensional unstructured tetrahedral meshes of multi-material images. The algorithm produces meshes with high quality since it provides a guaranteed dihedral angle bound of up to 19.47° for the output tetrahedra. In addition, it allows for user-specified guaranteed bounds on the two-sided Hausdorff distance between the boundaries of the mesh and the boundaries of the materials. Moreover, the mesh boundary is proved to be homeomorphic to the object surface. The algorithm is fast and robust, it produces a sufficiently small number of mesh elements that comply with these guarantees, as compared to other software. The theory and effectiveness of our method are illustrated with the experimental evaluation on synthetic and real medical data

Scalable Parallel Delaunay Image-to-Mesh Conversion for Shared and Distributed Memory Architectures

Mesh generation is an essential component for many engineering applications. The ability to generate meshes in parallel is critical for the scalability of the entire Finite Element Method (FEM) pipeline. However, parallel mesh generation applications belong to the broader class of adaptive and irregular problems, and are among the most complex, challenging, and labor intensive to develop and maintain. In this thesis, we summarize several years of the progress that we made in a novel framework for highly scalable and guaranteed quality mesh generation for finite element analysis in three dimensions. We studied and developed parallel mesh generation algorithms on both shared and distributed memory architectures. In this thesis we present a novel two-level parallel tetrahedral mesh generation framework capable of delivering and sustaining close to 6000 of concurrent work units (cores). We achieve this by leveraging concurrency at two different granularity levels by using a hybrid message passing and multi-threaded execution model which is suitable to the hierarchy of the hardware architecture of the distributed memory clusters. An end-user productivity and scalability study was performed on up to 6000 cores, and indicated very good end-user productivity with about 300 million tets per second and about 3600 weak scaling speedup. Both of these results suggest that: compared to the best previous algorithm, we have seen an improvement of more than 7000 times in performance, measured in terms of speed (elements per second) by using about 180 times more CPUs, for geometries that are by many orders of magnitude more complex

Automatic Linear and Curvilinear Mesh Generation Driven by Validity Fidelity and Topological Guarantees

Image-based geometric modeling and mesh generation play a critical role in computational biology and medicine. In this dissertation, a comprehensive computational framework for both guaranteed quality linear and high-order automatic mesh generation is presented. Starting from segmented images, a quality 2D/3D linear mesh is constructed. The boundary of the constructed mesh is proved to be homeomorphic to the object surface. In addition, a guaranteed dihedral angle bound of up to 19:47o for the output tetrahedra is provided. Moreover, user-specified guaranteed bounds on the distance between the boundaries of the mesh and the boundaries of the materials are allowed. The mesh contains a small number of mesh elements that comply with these guarantees, and the runtime is compatible in performance with other software. Then the curvilinear mesh generator allows for a transformation of straight-sided meshes to curvilinear meshes with C1 or C2 smooth boundaries while keeping all elements valid and with good quality as measured by their Jacobians. The mathematical proof shows that the meshes generated by our algorithm are guaranteed to be homeomorphic to the input images, and all the elements inside the meshes are guaranteed to be with good quality. Experimental results show that the mesh boundaries represent the objects\u27 shapes faithfully, and the accuracy of the representation is improved compared to the corresponding linear mesh

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains