Finite Element Modeling Driven by Health Care and Aerospace Applications

This thesis concerns the development, analysis, and computer implementation of mesh generation algorithms encountered in finite element modeling in health care and aerospace. The finite element method can reduce a continuous system to a discrete idealization that can be solved in the same manner as a discrete system, provided the continuum is discretized into a finite number of simple geometric shapes (e.g., triangles in two dimensions or tetrahedrons in three dimensions). In health care, namely anatomic modeling, a discretization of the biological object is essential to compute tissue deformation for physics-based simulations. This thesis proposes an efficient procedure to convert 3-dimensional imaging data into adaptive lattice-based discretizations of well-shaped tetrahedra or mixed elements (i.e., tetrahedra, pentahedra and hexahedra). This method operates directly on segmented images, thus skipping a surface reconstruction that is required by traditional Computer-Aided Design (CAD)-based meshing techniques and is convoluted, especially in complex anatomic geometries. Our approach utilizes proper mesh gradation and tissue-specific multi-resolution, without sacrificing the fidelity and while maintaining a smooth surface to reflect a certain degree of visual reality. Image-to-mesh conversion can facilitate accurate computational modeling for biomechanical registration of Magnetic Resonance Imaging (MRI) in image-guided neurosurgery. Neuronavigation with deformable registration of preoperative MRI to intraoperative MRI allows the surgeon to view the location of surgical tools relative to the preoperative anatomical (MRI) or functional data (DT-MRI, fMRI), thereby avoiding damage to eloquent areas during tumor resection. This thesis presents a deformable registration framework that utilizes multi-tissue mesh adaptation to map preoperative MRI to intraoperative MRI of patients who have undergone a brain tumor resection. Our enhancements with mesh adaptation improve the accuracy of the registration by more than 5 times compared to rigid and traditional physics-based non-rigid registration, and by more than 4 times compared to publicly available B-Spline interpolation methods. The adaptive framework is parallelized for shared memory multiprocessor architectures. Performance analysis shows that this method could be applied, on average, in less than two minutes, achieving desirable speed for use in a clinical setting. The last part of this thesis focuses on finite element modeling of CAD data. This is an integral part of the design and optimization of components and assemblies in industry. We propose a new parallel mesh generator for efficient tetrahedralization of piecewise linear complex domains in aerospace. CAD-based meshing algorithms typically improve the shape of the elements in a post-processing step due to high complexity and cost of the operations involved. On the contrary, our method optimizes the shape of the elements throughout the generation process to obtain a maximum quality and utilizes high performance computing to reduce the overheads and improve end-user productivity. The proposed mesh generation technique is a combination of Advancing Front type point placement, direct point insertion, and parallel multi-threaded connectivity optimization schemes. The mesh optimization is based on a speculative (optimistic) approach that has been proven to perform well on hardware-shared memory. The experimental evaluation indicates that the high quality and performance attributes of this method see substantial improvement over existing state-of-the-art unstructured grid technology currently incorporated in several commercial systems. The proposed mesh generator will be part of an Extreme-Scale Anisotropic Mesh Generation Environment to meet industries expectations and NASA\u27s CFD visio

Adaptive Mesh Refinement for Electromagnetic Simulation

We consider problems related to initial meshing and adaptive mesh refinement for the electromagnetic simulation of various structures. The quality of the initial mesh and the performance of the adaptive refinement are of great importance for the finite element solution of the Maxwell equations, since they directly affect the accuracy and the computational time. In this paper, we describe the complete meshing workflow, which allows the simulation of arbitrary structures. Test simulations confirm that the presented approach allows to reach the quality of the industrial simulation software

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

Locally optimal Delaunay-refinement and optimisation-based mesh generation

The field of mesh generation concerns the development of efficient algorithmic techniques to construct high-quality tessellations of complex geometrical objects. In this thesis, I investigate the problem of unstructured simplicial mesh generation for problems in two- and three-dimensional spaces, in which meshes consist of collections of triangular and tetrahedral elements. I focus on the development of efficient algorithms and computer programs to produce high-quality meshes for planar, surface and volumetric objects of arbitrary complexity. I develop and implement a number of new algorithms for mesh construction based on the Frontal-Delaunay paradigm - a hybridisation of conventional Delaunay-refinement and advancing-front techniques. I show that the proposed algorithms are a significant improvement on existing approaches, typically outperforming the Delaunay-refinement technique in terms of both element shape- and size-quality, while offering significantly improved theoretical robustness compared to advancing-front techniques. I verify experimentally that the proposed methods achieve the same element shape- and size-guarantees that are typically associated with conventional Delaunay-refinement techniques. In addition to mesh construction, methods for mesh improvement are also investigated. I develop and implement a family of techniques designed to improve the element shape quality of existing simplicial meshes, using a combination of optimisation-based vertex smoothing, local topological transformation and vertex insertion techniques. These operations are interleaved according to a new priority-based schedule, and I show that the resulting algorithms are competitive with existing state-of-the-art approaches in terms of mesh quality, while offering significant improvements in computational efficiency. Optimised C++ implementations for the proposed mesh generation and mesh optimisation algorithms are provided in the JIGSAW and JITTERBUG software libraries

Real-Time High-Quality Image to Mesh Conversion for Finite Element Simulations

Technological Advances in Medical Imaging have enabled the acquisition of images accurately describing biological tissues. Finite Element (FE) methods on these images provide the means to simulate biological phenomena such as brain shift registration, respiratory organ motion, blood flow pressure in vessels, etc. FE methods require the domain of tissues be discretized by simpler geometric elements, such as triangles in two dimensions, tetrahedra in three, and pentatopes in four. This exact discretization is called a mesh . The accuracy and speed of FE methods depend on the quality and fidelity of the mesh used to describe the biological object. Elements with bad quality introduce numerical errors and slower solver convergence. Also, analysis based on poor fidelity meshes do not yield accurate results specially near the surface. In this dissertation, we present the theory and the implementation of both a sequential and a parallel Delaunay meshing technique for 3D and ---for the first time--- 4D space-time domains. Our method provably guarantees that the mesh is a faithful representation of the multi-tissue domain in topological and geometric sense. Moreover, we show that our method generates graded elements of bounded radius-edge and aspect ratio, which renders our technique suitable for Finite Element analysis. A notable feature of our implementation is speed and scalability. The single-threaded performance of our 3D code is faster than the state of the art open source meshing tools. Experimental evaluation shows a more than 82% weak scaling efficiency for up to 144 cores, reaching a rate of more than 14.3 million elements per second. This is the first 3D parallel Delaunay refinement method to achieve such a performance, on either distributed or shared-memory architectures. Lastly, this dissertation is the first to develop and examine the sequential and parallel high-quality and fidelity meshing of general space-time 4D multi-tissue domains

Parallel generalized Delaunay mesh refinement

The modeling of physical phenomena in computational fracture mechanics, computational fluid dynamics and other fields is based on solving systems of partial differential equations (PDEs). When PDEs are defined over geometrically complex domains, they often do not admit closed form solutions. In such cases, they are solved approximately using discretizations of domains into simple elements like triangles and quadrilaterals in two dimensions (2D), and tetrahedra and hexahedra in three dimensions (3D). These discretizations are called finite element meshes. Many applications, for example, real-time computer assisted surgery, or crack propagation from fracture mechanics, impose time and/or mesh size constraints that cannot be met on a single sequential machine. as a result, the development of parallel mesh generation algorithms is required.;In this dissertation, we describe a complete solution for both sequential and parallel construction of guaranteed quality Delaunay meshes for 2D and 3D geometries. First, we generalize the existing 2D and 3D Delaunay refinement algorithms along with theoretical proofs of mesh quality in terms of element shape and mesh gradation. Existing algorithms are constrained by just one or two specific positions for the insertion of a Steiner point inside a circumscribed disk of a poorly shaped element. We derive an entire 2D or 3D region for the selection of a Steiner point (i.e., infinitely many choices) inside the circumscribed disk. Second, we develop a novel theory which extends both the 2D and the 3D Generalized Delaunay Refinement methods for the concurrent and mathematically guaranteed independent insertion of Steiner points. Previous parallel algorithms are either reactive relying on implementation heuristics to resolve dependencies in parallel mesh generation computations or require the solution of a very difficult geometric optimization problem (the domain decomposition problem) which is still open for general 3D geometries. Our theory solves both of these drawbacks. Third, using our generalization of both the sequential and the parallel algorithms we implemented prototypes of practical and efficient parallel generalized guaranteed quality Delaunay refinement codes for both 2D and 3D geometries using existing state-of-the-art sequential codes for traditional Delaunay refinement methods. On a heterogeneous cluster of more than 100 processors our implementation can generate a uniform mesh with about a billion elements in less than 5 minutes. Even on a workstation with a few cores, we achieve a significant performance improvement over the corresponding state-of-the-art sequential 3D code, for graded meshes

Mesh generation for voxel -based objects

A new physically-based approach to unstructured mesh generation via Monte-Carlo simulation is proposed. Geometrical objects to be meshed are represented by systems of interacting particles with a given interaction potential. A new way of distributing nodes in complex domains is proposed based on a concept of dynamic equilibrium ensemble, which represents a liquid state of matter. The algorithm is simple, numerically stable and produces uniform node distributions in domains of complex geometries and different dimensions. Well-shaped triangles or tetrahedra can be created by connecting a set of uniformly-spaced nodes. The proposed method has many advantages and potential applications.;The new method is applied to the problem of meshing of voxel-based objects. By customizing system potential energy function to reflect surface features, particles can be distributed into desired locations, such as sharp corners and edges. Feature-preserved surface mesh can then be constructed by connecting the node set.;A heuristic algorithm using an advancing front approach is proposed to generate triangulated surface meshes on voxel-based objects. The resultant surface meshes do not inherit the anisotropy of the underlying hexagonal grid. However, the important surface features, such as edges and corners may not be preserved in the mesh.;To overcome this problem, surface features such as edges, corners need to be detected. A new approach of edge capturing is proposed and demonstrated. The approach is based on a Laplace solver with incomplete Jacobi iterations, and as such is very simple and efficient. This edge capturing approach combined with the mesh generation methods above forms a simple and robust technique of unstructured mesh generation on voxel-based objects.;A graphical user interface (GUI) capable of complex geometric design and remote simulation control was implemented. The GUI was used in simulations of large fuel-cell stacks. It enables one to setup, run and monitor simulations remotely through secure shell (SSH2) connections. A voxel-based 3D geometrical modeling module is built along with the GUI. The flexibility of voxel-based geometry representation enables one to use this technique for both geometric design and visualization of volume data

Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees

pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach

Boundary recovery for 3D Delaunay triangulation

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