Optimal Dual Schemes for Adaptive Grid Based Hexmeshing

Hexahedral meshes are an ubiquitous domain for the numerical resolution of partial differential equations. Computing a pure hexahedral mesh from an adaptively refined grid is a prominent approach to automatic hexmeshing, and requires the ability to restore the all hex property around the hanging nodes that arise at the interface between cells having different size. The most advanced tools to accomplish this task are based on mesh dualization. These approaches use topological schemes to regularize the valence of inner vertices and edges, such that dualizing the grid yields a pure hexahedral mesh. In this paper we study in detail the dual approach, and propose four main contributions to it: (i) we enumerate all the possible transitions that dual methods must be able to handle, showing that prior schemes do not natively cover all of them; (ii) we show that schemes are internally asymmetric, therefore not only their implementation is ambiguous, but different implementation choices lead to hexahedral meshes with different singular structure; (iii) we explore the combinatorial space of dual schemes, selecting the minimum set that covers all the possible configurations and also yields the simplest singular structure in the output hexmesh; (iv) we enlarge the class of adaptive grids that can be transformed into pure hexahedral meshes, relaxing one of the tight requirements imposed by previous approaches, and ultimately permitting to obtain much coarser meshes for same geometric accuracy. Last but not least, for the first time we make grid-based hexmeshing truly reproducible, releasing our code and also revealing a conspicuous amount of technical details that were always overlooked in previous literature, creating an entry barrier that was hard to overcome for practitioners in the field

Finding Hexahedrizations for Small Quadrangulations of the Sphere

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

Adaptive mesh refinement techniques for high-order finite-volume WENO schemes

This paper demonstrates the capabilities of Adaptive Mesh Refinement Techniques (AMR) on 2D hybrid unstructured meshes, for high order finite volume WENO methods. The AMR technique developed is a conformal adapting unstructured hybrid quadrilaterals and triangles (quads & tris) technique for resolving sharp flow features in accurate manner for steady-state and time dependent flow problems. In this method, the mesh can be refined or coarsened which depends on an error estimator, making decision at the parent level whilst maintaining a conformal mesh, the unstructured hybrid mesh refinement is done hierarchically.When a numerical method can work on a fixed conformal mesh this can be applied to do dynamic mesh adaptation. Two Refinement strategies have been devised both following a H-P refinement technique, which can be applied for providing better resolution to strong gradient dominated problems. The AMR algorithm has been tested on cylindrical explosion test and forward facing step problems

Large-scale Geometric Data Decomposition, Processing and Structured Mesh Generation

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations and simulations on 2D/3D regions or on curved geometric objects, discretizing or approximating the geometric data using a polygonal or polyhedral meshes is always the first step of the procedure. The quality of this tessellation often dictates the subsequent computation accuracy, efficiency, and numerical stability. When compared with unstructured meshes, the structured meshes are favored in many scientific/engineering tasks due to their good properties. However, generating high-quality structured mesh remains challenging, especially for complex or large-scale geometric data. In industrial Computer-aided Design/Engineering (CAD/CAE) pipelines, the geometry processing to create a desirable structural mesh of the complex model is the most costly step. This step is semi-manual, and often takes up to several weeks to finish. Several technical challenges remains unsolved in existing structured mesh generation techniques. This dissertation studies the effective generation of structural mesh on large and complex geometric data. We study a general geometric computation paradigm to solve this problem via model partitioning and divide-and-conquer. To apply effective divide-and-conquer, we study two key technical components: the shape decomposition in the divide stage, and the structured meshing in the conquer stage. We test our algorithm on vairous data set, the results demonstrate the efficiency and effectiveness of our framework. The comparisons also show our algorithm outperforms existing partitioning methods in final meshing quality. We also show our pipeline scales up efficiently on HPC environment

Combinatorial meshing for mechanical FEM

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