Adaptive all-hexahedral meshing algorithms have many desirable features. These algorithms provide a mesh that is efficient for analysis by providing a high element density in specific locations, such as areas of high stress gradient or high curvature and reduced mesh density in other areas of less importance. In addition, inside-out hexahedral grid based schemes, using Cartesian structured grids for the base mesh, have shown great promise in accommodating automatic all-hexahedral algorithms. In these algorithms mesh refinement is generally used to capture geometric features. Unfortunately, most adaptive mesh generation algorithms employ a 3-refinement method. This method, although easy to employ, provides a mesh that is coarse in most areas and highly refined in other areas. Because a single refined hex is subdivided into 27 new hexes, regardless of the desired refinement, there is little control on mesh density. This paper will present an adaptive all-hexahedral grid-based meshing algorithm that employs a 2-refinement insertion method. 2-refinement is based on dividing a hex to be refined into eight new hexes. This allows greater control on mesh density which in turn increases computational efficiency.

In this work, we propose using edge valence as a quality predictor when used as a driver for adapting all hexahedral meshes. Edge valence, for hexahedra, is defined as the number of faces attached to an edge. It has shown to be a more reliable quality predictor than node valence for hexahedral meshes. An edge valence of 3, 4, or 5 within the volume of a hexahedral mesh has provided at least a positive scaled Jacobian for all observed meshes, without the presence of over constraining geometry. It is often desirable to adapt an existing mesh through sheet operations such as column collapse, sheet insertion, or sheet extraction. Examples of hexahedral mesh adaptation include refining and coarsening. This work presents a general algorithm for a priori prediction of edge valence when used with column collapse and sheet extraction operations. Using the predicted edge valence we present a method for guiding the mesh adaptation procedure which will result in an overall higher quality mesh than when driven by mesh quality alone. Other quality metrics such as the Jacobian are unfit for predictive algorithms because of their heavy dependence on node positioning instead of hex topology. Results have been derived from application of the algorithm towards the localized coarsening process.

The development of algorithms for the automatic creation of finite element meshes composed entirely of hexahedra (all-hex) is an active area of research. All-hex meshes are desirable for their characteristic of high accuracy with a low node count. Sweeping is one of the most widely used algorithms for generating all-hex meshes. A limitation of sweeping, however, is that it can currently be applied only to prismatic or extruded geometry types. This thesis develops a method to combine sweeping with another algorithm known as "Grafting". Grafting adjusts the mesh on one volume to conform to a second volume. In this manner it is useful for meshing multi-axis geometry in that a single axis can be meshed with sweeping and then secondary axes can be grafted on. By creating an algorithm for automatically performing these processes, the base set of geometry that can be automatically meshed with these methods is greatly increased. This new algorithm is called Graft-Sweeping. With the combination of sweeping and Grafting, geometry that contains multiple source surfaces, multiple target surfaces, and multiple sweep axes can be meshed. The results of this algorithm on several example geometries are given showing the strengths and weaknesses of this technique. From the results it can be seen that the Graft-Sweep algorithm can produce a finite element mesh in about half the time of manual Grafting and sweeping operations with similar mesh quality. When compared to sweeping alone, Graft-Sweeping is several times faster but the quality is usually reduced. A second area of research for this thesis is to determine when Grafting can be used to enhance the meshing process. It is shown that the best results are obtained when Grafting is used on structured meshes and the mesh size is considerably smaller than the size of the feature that is being grafted.

Hexahedral refinement increases the density of an all-hexahedral mesh in a specified region, improving numerical accuracy. Previous research using solely sheet refinement theory made the implementation computationally expensive and unable to effectively handle multiply-connected transition elements and self-intersecting hexahedral sheets. The Selective Approach method is a new procedure that combines two diverse methodologies to create an efficient and robust algorithm able to handle the above stated problems. These two refinement methods are: 1) element by element refinement and 2) directional refinement. In element by element refinement, the three inherent directions of a hexahedron are refined in one step using one of seven templates. Because of its computational superiority over directional refinement, but its inability to handle multiply-connected transition elements, element by element refinement is used in all areas of the specified region except regions local to multiply-connected transition elements. The directional refinement scheme refines the three inherent directions of a hexahedron separately on a hexahedron by hexahedron basis. This differs from sheet refinement which refines hexahedra using hexahedral sheets. Directional refinement is able to correctly handle multiply-connected transition elements. A ranking system and propagation scheme allow directional refinement to work within the confines of the Selective Approach Algorithm.

Finite element mesh adaptation methods can be used to improve the efficiency and accuracy of solutions to computational modeling problems. For many finite element modeling applications, a conforming all-hexahedral mesh is preferred. When adapting a hexahedral mesh, localized modifications that preserve topologic conformity are often desired. Effective hexahedral refinement methods that satisfy these criteria have recently become available. However, due to hexahedral mesh topology constraints, little progress has been made in the area of hexahedral coarsening. This thesis presents a new method to locally coarsen conforming all-hexahedral meshes. The method works on both structured and unstructured meshes and is not based on undoing previous refinement. Building upon recent developments in quadrilateral coarsening, the method utilizes hexahedral sheet and column operations, including pillowing, column collapsing, and sheet extraction. A general algorithm for automated coarsening is presented and examples of models that have been coarsened are shown. While results are promising, further work is needed to improve the automated process.

The development of algorithms that effectively modify all-hexahedral finite element mesh is currently an active area of research. Mesh modification can be used to improve mesh quality reduce the time required to mesh a model, and improve the finite element analysis results. However, general modification of all-hexahedral meshes has proven difficult because of the global effects of local modifications. This thesis explains the global constraints on modifying all-hexahedral meshes and then presents three mesh modification techniques that make it possible to do local modifications while accounting for the global effects. These techniques are sheet insertion, sheet extraction, and mesh cutting. Sheet insertion is used to refine a mesh by inserting sheets of hexahedral elements into existing meshes. Sheet extraction coarsens existing meshes by deleting sheets of elements from the mesh. Mesh cutting is used to modify a simple mesh to fit it to complex geometric feature. The mesh modification techniques are covered in detail with representative graphics. Examples are given that demonstrate the application of each technique to the mesh generation process.

This thesis presents the development and evaluation of the knife finite element which is a degenerate case of a hexahedral element. The knife connectivity is an artifact of automatic all-hexahedral mesh generators. Currently, knives are propagated to the surface of the mesh for removal. However since this disturbs the surface mesh, other alternatives are needed. This thesis investigates the option of leaving the knife connectivity in the mesh and treating it as a valid finite element. The shape functions and stiffness matrix for the knife element are derived and evaluated using theoretical and practical evaluations. It is concluded that the knife finite element is a viable element and should be used in finite element analysis when the knife connectivity occurs. Using the knife element reduces the work involved with fixing the knife connectivity by propagation or other means and will produce acceptable results in most cases.

In this thesis, quadratic hexahedral edge elements have been applied to the three dimensional for open region electromagnetic scattering problems. For this purpose, a semi-automatic all-hexahedral mesh generation algorithm is developed and implemented. Material properties inside the elements and along the edges are also determined and prescribed during the mesh generation phase in order to be used in the solution phase. Based on the condition number quality metric, the generated mesh is optimized by means of the Particle Swarm Optimization (PSO) technique. A framework implementing hierarchical hexahedral edge elements is implemented to investigate the performance of linear and quadratic hexahedral edge elements. Perfectly Matched Layers (PMLs), which are implemented by using a complex coordinate transformation, have been used for mesh truncation in the software. Sparse storage and relevant efficient matrix ordering are used for the representation of the system of equations. Both direct and indirect sparse matrix solution methods are implemented and used. Performance of quadratic hexahedral edge elements is deeply investigated over the radar cross-sections of several curved or flat objects with or without patches. Instead of the de-facto standard of 0.1 wavelength linear element size, 0.3-0.4 wavelength quadratic element size was observed to be a new potential criterion for electromagnetic scattering and radiation problems.

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In this thesis, three mesh generation software packages have been developed and implemented. The first two were based on structured mesh generation algorithms and used to solve structured surface and volume mesh generation problems of three-dimensional domains. Structured mesh generation algorithms were based on the concept of isoparametric coordinates. In structured surface mesh generation software, quadrilateral mesh elements were generated for complex three-dimensional surfaces and these elements were then triangulated in order to obtain high-quality triangular mesh elements. Structured volume mesh generation software was used to generate hexahedral mesh elements for volumes. Tetrahedral mesh elements were constructed from hexahedral elements using hexahedral node insertion method. The results, which were produced by the mesh generation algorithms, were converted to a required format in order to be saved in output files. The third software package is an unstructured quality tetrahedral mesh generator and was used to generate exact Delaunay tetrahedralizations, constrained (conforming) Delaunay tetrahedralizations and quality conforming Delaunay tetrahedralizations. Apart from the mesh generation algorithms used and implemented in this thesis, unstructured mesh generation techniques that can be used to generate quadrilateral, triangular, hexahedral and tetrahedral mesh elements were also discussed.