160 research outputs found

    Finding Hexahedrizations for Small Quadrangulations of the Sphere

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    This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201

    HexBox: Interactive Box Modeling of Hexahedral Meshes

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    We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid- based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques

    Integration of geometric modeling and advanced finite element preprocessing

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    The structure to a geometry based finite element preprocessing system is presented. The key features of the system are the use of geometric operators to support all geometric calculations required for analysis model generation, and the use of a hierarchic boundary based data structure for the major data sets within the system. The approach presented can support the finite element modeling procedures used today as well as the fully automated procedures under development

    Frame Fields for Hexahedral Mesh Generation

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    As a discretized representation of the volumetric domain, hexahedral meshes have been a popular choice in computational engineering science and serve as one of the main mesh types in leading industrial software of relevance. The generation of high quality hexahedral meshes is extremely challenging because it is essentially an optimization problem involving multiple (conflicting) objectives, such as fidelity, element quality, and structural regularity. Various hexahedral meshing methods have been proposed in past decades, attempting to solve the problem from different perspectives. Unfortunately, algorithmic hexahedral meshing with guarantees of robustness and quality remains unsolved. The frame field based hexahedral meshing method is the most promising approach that is capable of automatically generating hexahedral meshes of high quality, but unfortunately, it suffers from several robustness issues. Field based hexahedral meshing follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a feature-aligned frame field and (2) generation of an integer-grid map that best aligns with the frame field. The main robustness issue stems from the fact that smooth frame fields frequently exhibit singularity graphs that are inappropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The thesis aims at analyzing the gap between the topologies of frame fields and hexahedral meshes and developing algorithms to realize a more robust field based hexahedral mesh generation. The first contribution of this work is an enumeration of all local configurations that exist in hexahedral meshes with bounded edge valence and a generalization of the Hopf-Poincaré formula to octahedral (orthonormal frame) fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large non-linear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field. In the collaboration work with colleagues [BRK+22], the dataset HexMe consisting of practically relevant models with feature tags is set up, allowing a fair evaluation for practical hexahedral mesh generation algorithms. The extendable and mutable dataset remains valuable as hexahedral meshing algorithms develop. The results of the standard field based hexahedral meshing algorithms on the HexMesh dataset expose the fragility of the automatic pipeline. The major contribution of this thesis improves the robustness of the automatic field based hexahedral meshing by guaranteeing local meshability of general feature aligned smooth frame fields. We derive conditions on the meshability of frame fields when feature constraints are considered, and describe an algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite the fact that local meshability is only a necessary but not sufficient condition for the stronger requirement of meshability, our algorithm increases the 2% success rate of generating valid integer-grid maps with state-of-the-art methods to 57%, when compared on the challenging HexMe dataset

    Combinatorial meshing for mechanical FEM

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    Diese Dissertation führt die Forschung zur Erzeugung von FEM Netzen für mechanische Simulationen fort. Zur zielgerichteten Steuerung der weiteren Forschung in diesem Feld wurde eine Umfrage zur Identifikation der Kerninteressen der Anwender durchgef¨uhrt. Das vorgestellte Verfahren des Combinatorial Meshing ist ein neuartiges Konzept im Bereich Grid Based Meshing. Im Gegensatz zu den kartesischen Gittern, die im Grid Based Meshing Anwendung finden wird ein an das Problem angepasstes Gitter genutzt. Dieses Precursor Mesh wird durch Analyse des CAD Strukturbaums der Geometrie gewählt. Die Zellen des Precursor Mesh werden mit vorberechneten Netzsegmenten – sogenannten Superelementen gefüllt. Die Wahl passender Superelemente wird als combinatorisches Optimierungsproblem modelliert. Dieses wird mit Hilfe von Answer Set Programming (ASP) und einem alternativen heuristischen Ansatz gelöst. Beide Verfahren werden in Hinblick auf Zeitkomplexität und Ergebnisqualität verglichen. Das resultierende Netz ist eine Grobe Näherung der Zielgeometrie, die an geometrische Elemente angebunden werden muss. Für diesen Prozess wird ein neuer Algorithmus vorgestellt, der automatisch identifizieren kann, an welche Geometrieelemente Oberflächenknoten des Netzes gebunden werden müssen um die Zielgeometrie möglichst exakt abzubilden. Für die Erzeugung der Superelemente wird ein neues Verfahren auf Basis von ASP entwickelt. Um die Generierung von FEM Netzen mit ASP zu ermöglichen, wird das Problem der Netzgenerierung als graphentheoretisches Problem modelliert. Dieses ist die Wahl eines optimalen Subgraphen aus einem Primärgraph. Dieses Problem wird mit einem ASP Solver für verschiedene Optimierungsziele gelöst. Die Graphenformulierung ist zudem ein Fortschritt im theoretischen Verständnis der Komplexität der Netzgenerierung.his dissertation advances the research of mesh generation for Finite Element Method simulation for mechanical applications. In order to target further research at user needs, a survey is conducted to identify the most pressing issues in FEM software. The concept of Combinatorial Meshing is proposed as a novel approach to grid based meshing. While conventional grid based meshing works on trivial Cartesian grids, the use of a Precursor Mesh instead of a grid is proposed. Appropriate Precursor Meshes are selected by analyzing the internal feature structure of the provided CAD data. The cells of this Precursor Mesh are then filled with precomputed mesh templates – called Super Elements. The selection of appropriate Super Elements is modeled as a combinatorial optimization problem. To solve this problem, Answer Set programming (ASP) and a heuristic approach are compared with respect to their time complexity and result quality. The resulting mesh is a rough approximation of the target geometry which then has to be fitted to the geometric entities. For this process a novel algorithm is presented which is able to automatically identify the geometric entities on which the surface nodes of the mesh have to be drawn in order to generate high quality meshes and correctly approximate the desired geometry. For the generation of Super Element Meshes, a novel approach based on ASP is developed. In order to enable meshing with ASP, a graph representation of a mesh is developed and the meshing process is formulated as a graph selection problem. It is then solved with an ASP solver for multiple optimization goals. The graph formulation will also aid the theoretical understanding of meshing complexity

    Design and Optimization of a 3-D Plasmonic Huygens Metasurface for Highly-Efficient Flat Optics

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    For miniaturization of future USAF unmanned aerial and space systems to become feasible, accompanying sensor components of these systems must also be reduced in size, weight and power (SWaP). Metasurfaces can act as planar equivalents to bulk optics, and thus possess a high potential to meet these low-SWaP requirements. However, functional efficiencies of plasmonic metasurface architectures have been too low for practical application in the infrared (IR) regime. Huygens-like forward-scattering inclusions may provide a solution to this deficiency, but there is no academic consensus on an optimal plasmonic architecture for obtaining efficient phase control at high frequencies. This dissertation asks the question: what are the ideal topologies for generating Huygens-like metasurface building blocks across a full 2π phase space? Instead of employing any a priori assumption of fundamental scattering topologies, a genetic algorithm (GA) routine was developed to optimize a “blank slate” grid of binary voxels inside a 3D cavity, evolving the voxel bits until a near-globally optimal transmittance (T) was attained at a targeted phase. All resulting designs produced a normalized T≥80 across the entire 2π range, which is the highest metasurface efficiency reported to-date for a plasmonic solution in the IR regime
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