Finite Element Analysis of the Effect of Acoustic Wavelength to Hierarchical Side Length and Facet Area for Elastic Scattering from Polygonal Rings and Geodesic Spheres

In this work, the frequency response from two-dimensional polygon and three-dimensional geodesic spheres is numerically simulated using a coupled structural acoustic finite element model. The model is composed of a submerged thin-walled elastic shell structure surrounded by an infinite acoustic air domain. Infinite elements are used to simulate the far-field acoustic radiation condition. Results for the faceted polygon and geodesic sphere are compared to their canonical counterpart viz. a circle/ring and a spherical shell. A unique feature of this study is to compare results as the number of facets in the polygon or geodesic is increased, such that the surface area converges in the limit of a large number of facet sides approaching the geometry of a circle or sphere. In this work the ratio of acoustic wavelength to the local geometric parameter of edge length in 2-D, and facet area in 3-D is proposed and varied to quantify the comparison between the faceted shapes with that of the corresponding reference circle or sphere. A threshold ratio is proposed, up to which scattering response of a polygonal/geodesic spherical scatterer matches the scattering response of a circle/sphere which has the same diameter as the circumscribing circle/sphere of the polygon/geodesic sphere. This ratio is an approximation and can be considered as a guide rule for design. Conversely, this ratio can be used for the inverse scattering problem, where from a known scattering response, the faceted geometry can be predicted without prior knowledge. The geodesic sphere was invented by Buckminster Fuller in the early 1950’s, has been of interest in architecture due to the larger open interior spaces which can be constructed. Of particular interest in this work is the hierarchical geometric structure of the geodesic sphere which increasingly approximates a spherical surface as the hierarchy (degree) increases. The geodesic sphere has been modelled by taking an icosahedron and projecting the triangular faces onto a surface of the sphere using vector geometry. The scattering response of elastic structures in the mid-frequency resonance band depends strongly on the total mass. For comparisons, the natural frequency of the hierarchical geometries generated in 2-D and 3-D, are designed to have the same total mass. Using this approach, differences in natural frequency and scattering response are driven primarily by changes in overall stiffness and stiffness distribution, and to a lesser degree, by changes in mass distribution. To give a wide range of frequency response, natural vibration frequencies for the different elastic shells have been extracted up to 3000 Hz corresponding to the nondimensional frequency ka = 55, where k is the wavenumber defined by the circular frequency over the acoustic wave speed (speed of sound in air), and a is the diameter of the circle/shell which circumscribes the scatterer. Convergence with the natural frequencies of ring/sphere is observed as the hierarchy in polygons (number of sides) and the geodesic sphere (degree) increases. The target strength is calculated at the important front and back locations on the surface of the elastic scatterer subject to an incoming plane acoustic wave along the major axis aligned with the geometry. More frequency data points near the natural frequencies are used to provide increased resolution needed to capture the peak amplitudes in the response at resonance. Target strength at the same location, calculated for the circle/spherical scatterer is compared and quantified by the ratio of wavelength to facet dimension. Scattering from rigid bodies has been studied to validate the elastic scattering response in air

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

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

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

Field D* pathfinding in weighted simplicial complexes

Includes abstract.Includes bibliographical references.The development of algorithms to efficiently determine an optimal path through a complex environment is a continuing area of research within Computer Science. When such environments can be represented as a graph, established graph search algorithms, such as Dijkstra’s shortest path and A*, can be used. However, many environments are constructed from a set of regions that do not conform to a discrete graph. The Weighted Region Problem was proposed to address the problem of finding the shortest path through a set of such regions, weighted with values representing the cost of traversing the region. Robust solutions to this problem are computationally expensive since finding shortest paths across a region requires expensive minimisation. Sampling approaches construct graphs by introducing extra points on region edges and connecting them with edges criss-crossing the region. Dijkstra or A* are then applied to compute shortest paths. The connectivity of these graphs is high and such techniques are thus not particularly well suited to environments where the weights and representation frequently change. The Field D* algorithm, by contrast, computes the shortest path across a grid of weighted square cells and has replanning capabilites that cater for environmental changes. However, representing an environment as a weighted grid (an image) is not space-efficient since high resolution is required to produce accurate paths through areas containing features sensitive to noise. In this work, we extend Field D* to weighted simplicial complexes – specifically – triangulations in 2D and tetrahedral meshes in 3D