Singularity classification as a design tool for multiblock grids

A major stumbling block in interactive design of 3-D multiblock grids is the difficulty of visualizing the design as a whole. One way to make this visualization task easier is to focus, at least in early design stages, on an aspect of the grid which is inherently easy to present graphically, and to conceptualize mentally, namely the nature and location of singularities in the grid. The topological behavior of a multiblock grid design is determined by what happens at its edges and vertices. Only a few of these are in any way exceptional. The exceptional behaviors lie along a singularity graph, which is a 1-D construct embedded in 3-D space. The varieties of singular behavior are limited enough to make useful symbology on a graphics device possible. Furthermore, some forms of block design manipulation that appear appropriate to the early conceptual-modeling phase can be accomplished on this level of abstraction. An overview of a proposed singularity classification scheme and selected examples of corresponding manipulation techniques is presented

Design of meso-scale cellular structure for rapid manufacturing

Customized cellular material is a relatively new area made possible by advancements in rapid manufacturing technologies. Rapid manufacturing is ideal for the production of customized cellular structure, especially on the meso scale, due to the size and complexity of the design. The means to produce this type of structure now exist, but the processes to design the structure are not well developed. The manual design of customized cellular material is not realistic due to the large number of features. Currently there are few tools available that aid in the design of this type of material. In this thesis, an automated tool to design customized cellular structure is presented.M.S.Committee Chair: Rosen, David; Committee Member: Choi, Seung-Kyum; Committee Member: Sitaraman, Sures

Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips

We present transfer matrices for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and M\"obius strips of the square, triangular, and honeycomb lattices of width $L_y$ and arbitrarily great length $L_x$. We relate these results to our earlier exact solutions for square-lattice strips with $L_y=3,4,5$, triangular-lattice strips with $L_y=2,3,4$, and honeycomb-lattice strips with $L_y=2,3$ and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a M\"obius strip of a lattice $\Lambda$ and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width $L_y$. New results are presented for the $L_y=5$ strip of the triangular lattice and the $L_y=4$ and $L_y=5$ strips of the honeycomb lattice. Using these results and taking the infinite-length limit $L_x \to \infty$, we determine the continuous accumulation locus of the zeros of the above partition function in the complex $q$ plane, including the maximal real point of nonanalyticity of the degeneracy per site, $W$ as a function of $q$.Comment: 62 pages, latex, 6 eps figures, includes additional results, e.g., loci ${\cal B}$, requested by refere

Towards recovery of complex shapes in meshes using digital images for reverse engineering applications

When an object owns complex shapes, or when its outer surfaces are simply inaccessible, some of its parts may not be captured during its reverse engineering. These deficiencies in the point cloud result in a set of holes in the reconstructed mesh. This paper deals with the use of information extracted from digital images to recover missing areas of a physical object. The proposed algorithm fills in these holes by solving an optimization problem that combines two kinds of information: (1) the geometric information available on the surrounding of the holes, (2) the information contained in an image of the real object. The constraints come from the image irradiance equation, a first-order non-linear partial differential equation that links the position of the mesh vertices to the light intensity of the image pixels. The blending conditions are satisfied by using an objective function based on a mechanical model of bar network that simulates the curvature evolution over the mesh. The inherent shortcomings both to the current holefilling algorithms and the resolution of the image irradiance equations are overcom

Loopy Cuts: Surface-Field Aware Block Decomposition for Hex-Meshing.

We present a new fully automatic block-decomposition hexahedral meshing algorithm capable of producing high quality meshes that strictly preserve feature curve networks on the input surface and align with an input surface cross-field. We produce all-hex meshes on the vast majority of inputs, and introduce localized non-hex elements only when the surface feature network necessitates those. The input to our framework is a closed surface with a collection of geometric or user-demarcated feature curves and a feature-aligned surface cross-field. Its output is a compact set of blocks whose edges interpolate these features and are loosely aligned with this cross-field. We obtain this block decomposition by cutting the input model using a collection of simple cutting surfaces bounded by closed surface loops. The set of cutting loops spans the input feature curves, ensuring feature preservation, and is obtained using a field-space sampling process. The computed loops are uniformly distributed across the surface, cross orthogonally, and are loosely aligned with the cross-field directions, inducing the desired block decomposition. We validate our method by applying it to a large range of complex inputs and comparing our results to those produced by state-of-the-art alternatives. Contrary to prior approaches, our framework consistently produces high-quality field aligned meshes while strictly preserving geometric or user-specified surface features

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences