Tetrahedral Meshes in Biomedical Applications: Generation, Boundary Recovery and Quality Enhancements

Mesh generation is a fundamental precursor to finite element implementations for solution of partial differential equations in engineering and science. This dissertation advances the field in three distinct but coupled areas. A robust and fast three dimensional mesh generator for arbitrarily shaped geometries was developed. It deploys nodes throughout the domain based upon user-specified mesh density requirements. The system is integer and pixel based which eliminates round off errors, substantial memory requirements and cpu intensive calculations. Linked, but fully detachable, to the mesh generation system is a physical boundary recovery routine. Frequently, the original boundary topology is required for specific boundary condition applications or multiple material constraints. Historically, this boundary preservation was not available. An algorithm was developed, refined and optimized that recovers the original boundaries, internal and external, with fidelity. Finally, a node repositioning algorithm was developed that maximizes the minimum solid angle of tetrahedral meshes. The highly coveted 2D Delaunay property that maximizes the minimum interior angle of a triangle mesh does not extend to its 3D counterpart, to maximize the minimum solid angle of a tetrahedron mesh. As a consequence, 3D Delaunay created meshes have unacceptable sliver tetrahedral elements albeit composed of 4 high quality triangle sides. These compromised elements are virtually unavoidable and can foil an otherwise intact mesh. The numerical optimization routine developed takes any preexisting tetrahedral mesh and repositions the nodes without changing the mesh topology so that the minimum solid angle of the tetrahedrons is maximized. The overall quality enhancement of the volume mesh might be small, depending upon the initial mesh. However, highly distorted elements that create ill-conditioned global matrices and foil a finite element solver are enhanced significantly

Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

Granite: A scientific database model and implementation

The principal goal of this research was to develop a formal comprehensive model for representing highly complex scientific data. An effective model should provide a conceptually uniform way to represent data and it should serve as a framework for the implementation of an efficient and easy-to-use software environment that implements the model. The dissertation work presented here describes such a model and its contributions to the field of scientific databases. In particular, the Granite model encompasses a wide variety of datatypes used across many disciplines of science and engineering today. It is unique in that it defines dataset geometry and topology as separate conceptual components of a scientific dataset. We provide a novel classification of geometries and topologies that has important practical implications for a scientific database implementation. The Granite model also offers integrated support for multiresolution and adaptive resolution data. Many of these ideas have been addressed by others, but no one has tried to bring them all together in a single comprehensive model. The datasource portion of the Granite model offers several further contributions. In addition to providing a convenient conceptual view of rectilinear data, it also supports multisource data. Data can be taken from various sources and combined into a unified view. The rod storage model is an abstraction for file storage that has proven an effective platform upon which to develop efficient access to storage. Our spatial prefetching technique is built upon the rod storage model, and demonstrates very significant improvement in access to scientific datasets, and also allows machines to access data that is far too large to fit in main memory. These improvements bring the extremely large datasets now being generated in many scientific fields into the realm of tractability for the ordinary researcher. We validated the feasibility and viability of the model by implementing a significant portion of it in the Granite system. Extensive performance evaluations of the implementation indicate that the features of the model can be provided in a user-friendly manner with an efficiency that is competitive with more ad hoc systems and more specialized application specific solutions

7th International Meshing Roundtable '98

The goal of the 7th International Meshing Roundtable is to bring together researchers and developers from industry, academia, and government labs in a stimulating, open environment for the exchange of technical information related to the meshing process. In the past, the Roundtable has enjoyed significant participation from each of these groups from a wide variety of countries

Generalized swap operation for tetrahedrizations

Abstract. Mesh optimization of 2D and 3D triangulations is used in multiple applications extensively. For example, mesh optimization is crucial in the context of adaptively discretizing geometry, typically representing the geometrical boundary conditions of a numerical simulation, or adaptively discretizing the entire space over which various dependent variables of a numerical simulation must be approximated. Together with operations applied to the vertices the so-called edge or face swap operations are the building block of all optimization approaches. To speed up the optimization or to avoid local minima of the function measuring overall mesh quality these swaps are combined to generalized swap operations with a less local impact on the triangulation. Despite the fact that these swap operations change only the connectivity of a triangulation, it depends on the geometry of the triangulation whether the generalized swap will generate inconsistently oriented or degenerate simplices. Because these are undesirable for numerical reasons, this paper is concerned with geometric criteria that guarantee the generalized swaps for a 3D triangulation to yield only valid, non-degenerate triangulations. Key words: 3D triangulation; geometric conditions; swap operations.