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

Frame Fields for Hexahedral Mesh Generation

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

Workshop on the Integration of Finite Element Modeling with Geometric Modeling

The workshop on the Integration of Finite Element Modeling with Geometric Modeling was held on 12 May 1987. It was held to discuss the geometric modeling requirements of the finite element modeling process and to better understand the technical aspects of the integration of these two areas. The 11 papers are presented except for one for which only the abstract is given

The receding front method applied to hexahedral mesh generation of exterior domains

Two of the most successful methods to generate unstructured hexahedral meshes are the grid-based methods and the advancing front methods. On the one hand, the grid-based methods generate high-quality hexahedra in the inner part of the domain using an inside–outside approach. On the other hand, advancing front methods generate high-quality hexahedra near the boundary using an outside–inside approach. To combine the advantages of both methodologies, we extend the receding front method: an inside–outside mesh generation approach by means of a reversed advancing front. We apply this approach to generate unstructured hexahedral meshes of exterior domains. To reproduce the shape of the boundaries, we first pre-compute the mesh fronts by combining two solutions of the Eikonal equation on a tetrahedral reference mesh. Then, to generate high-quality elements, we expand the quadrilateral surface mesh of the inner body towards the unmeshed external boundary using the pre-computed fronts as a guide.Two of the most successful methods to generate unstructured hexahedral meshes are the grid-based methods and the advancing front methods. On the one hand, the grid-based methods generate high-quality hexahedra in the inner part of the domain using an inside–outside approach. On the other hand, advancing front methods generate high-quality hexahedra near the boundary using an outside–inside approach. To combine the advantages of both methodologies, we extend the receding front method: an inside–outside mesh generation approach by means of a reversed advancing front. We apply this approach to generate unstructured hexahedral meshes of exterior domains. To reproduce the shape of the boundaries, we first pre-compute the mesh fronts by combining two solutions of the Eikonal equation on a tetrahedral reference mesh. Then, to generate high-quality elements, we expand the quadrilateral surface mesh of the inner body towards the unmeshed external boundary using the pre-computed fronts as a guide.Peer ReviewedPostprint (author's final draft

Paving the path towards automatic hexahedral mesh generation

Esta tesis versa sobre el desarrollo de las tecnologías para la generación de mallas de hexaedros. El proceso de generar una malla de hexaedros no es automático y su generación requiere varias horas te trabajo de un ingeniero especializado. Por lo tanto, es importante desarrollar herramientas que faciliten dicho proceso de generación. Con este fin, se presenta y desarrolla un método de proyección de mallas, una técnica de sweeping o barrido, un algoritmo para la obtención de mallas por bloques, y un entorno de generación de mallas. Las implementaciones más competitivas del método de sweeping utilizan técnicas de proyección de mallas basadas en métodos afines. Los métodos afines más habituales presentan varios problemas relacionados con la obtención de sistemas de ecuaciones normales de rango deficiente. Para solucionar dichos problemas se presenta y analiza un nuevo método afín que depende de dos parámetros vectoriales. Además, se detalla un procedimiento automático para la selección de dichos vectores. El método de proyección resultante preserva la forma de las mallas proyectadas. Esta proyección es incorporada también en una nueva herramienta de sweeping. Dicha herramienta genera capas de nodos internos que respetan la curvatura de las superficies inicial y final. La herramienta de sweeping es capaz de mallar geometrías de extrusión definidas por trayectorias curvas, secciones no constantes a lo largo del eje de sweeping, y superficies inicial y final con diferente forma y curvatura.En las últimas décadas se han propuesto varios ataques para la generación automática de mallas de hexahedros. Sin embargo, todavía no existe un algoritmo rápido y robusto que genere automáticamente mallas de hexaedros de alta calidad. Se propone un nuevo ataque para la generación de mallas por bloques mediante la representación de la geometría y la topología del dual de una malla de hexaedros. En dicho ataque, primero se genera una malla grosera de tetraedros. Después, varió polígonos planos se añaden al interior de los elementos de la malla grosera inicial. Dichos polígonos se denotan como contribuciones duales locales y representan una versión discreta del dual de una malla de hexaedros. En el último paso, la malla por bloques se obtiene como el dual de la representación del dual generada. El algoritmo de generación de mallas por bloques es aplicado a geometrías que presentan diferentes características geométricas como son superficies planas, superficies curvas, configuraciones delgadas, agujeros, y vértices con valencia mayor que tres.Las mallas se generan habitualmente con la ayuda de entornos interactivos que integran una interfaz CAD y varios algoritmos de generación de mallas. Se presenta un nuevo entorno de generación de mallas especializado en la generación de cuadriláteros y hexaedros. Este entorno proporciona la tecnología necesaria para implementar les técnicas de generación de mallas de hexaedros presentadas en esta tesis.This thesis deals with the development of hexahedral mesh generation technology. The process of generating hexahedral meshes is not fully automatic and it is a time consuming task. Therefore, it is important to develop tools that facilitate the generation of hexahedral meshes. To this end, a mesh projection method, a sweeping technique, a block-meshing algorithm, and an interactive mesh generation environment are presented and developed. Competitive implementations of the sweeping method use mesh projection techniques based on affine methods. Standard affine methods have several drawbacks related to the statement of rank deficient sets of normal equations. To overcome these drawbacks a new affine method that depends on two vector parameters is presented and analyzed. Moreover, an automatic procedure that selects these two vector parameters is detailed. The resulting projection procedure preserves the shape of projected meshes. Then, this procedure is incorporated in a new sweeping tool. This tool generates inner layers of nodes that preserve the curvature of the cap surfaces. The sweeping tool is able to mesh extrusion geometries defined by non-linear sweeping trajectories, non-constant cross sections along the sweep axis, non-parallel cap surfaces, and cap surfaces with different shape and curvature. In the last decades, several general-purpose approaches to generate automatically hexahedral meshes have been proposed. However, a fast and robust algorithm that automatically generates high-quality hexahedral meshes is not available. A novel approach for block meshing by representing the geometry and the topology of a hexahedral mesh is presented. The block-meshing algorithm first generates an initial coarse mesh of tetrahedral elements. Second, several planar polygons are added inside the elements of the initial coarse mesh. These polygons are referred as local dual contributions and represent a discrete version of the dual of a hexahedral mesh. Finally, the dual representation is dualized to obtain the final block mesh. The block-meshing algorithm is applied to mesh geometries that present different geometrical characteristics such as planar surfaces, curved surfaces, thin configurations, holes, and vertices with valence greater than three.Meshes are usually generated with the help of interactive environments that integrate a CAD interface and several meshing algorithms. An overview of a new mesh generation environment focused in quadrilateral and hexahedral mesh generation is presented. This environment provides the technology required to implement the hexahedral meshing techniques presented in this thesis.Postprint (published version

Three-dimensional modeling of natural heterogeneous objects

En la medicina y otros campos relacionados cuando se va a estudiar un objeto natural, se toman imágenes de tomografía computarizada a través de varios cortes paralelos. Estos cortes se apilan en datos de volumen y se reconstruyen en modelos computacionales con el fin de estudiar la estructura de dicho objeto. Para construir con éxito modelos tridimensionales es importante la identificación y extracción precisa de todas las regiones que comprenden el objeto heterogéneo natural. Sin embargo, la construcción de modelos tridimensionales por medio del computador a partir de imágenes médicas sigue siendo un problema difícil y plantea dos problemas relacionados con las inexactitudes que surgen de, y son inherentes al proceso de adquisición de datos. El primer problema es la aparición de artefactos que distorsionan el límite entre las regiones. Este es un problema común en la generación de mallas a partir de imágenes médicas, también conocido como efecto de escalón. El segundo problema es la extracción de mallas suaves 3D que se ajustan a los límites de las región que conforman los objetos heterogéneos naturales descritos en las imágenes médicas. Para resolver estos problemas, se propone el método CAREM y el método RAM. El énfasis de esta investigación está puesto en la exactitud y fidelidad a la forma de las regiones necesaria en las aplicaciones biomédicas. Todas las regiones representadas de forma implícita que componen el objeto heterogéneo natural se utilizan para generar mallas adaptadas a los requisitos de los métodos de elementos finitos a través de un enfoque de modelado de ingeniería reversa, por lo tanto, estas regiones se consideran como un todo en lugar de piezas aisladas ensambladas.In medicine and other related fields when a natural object is going to be studied, computed tomography images are taken through several parallel slices. These slices are then stacked in volume data and reconstructed into 3D computer models. In order to successfully build 3D computer models of natural heterogeneous objects, accurate identification and extraction of all regions comprising the natural heterogeneous object is important. However, building 3D computer models of natural heterogeneous objects from medical images is still a challenging problem, and poses two issues related to the inaccuracies which arise from and are inherent to the data acquisition process. The first issue is the appearance of aliasing artifacts in the boundary between regions, a common issue in mesh generation from medical images, also known as stair-stepped artifacts. The second issue is the extraction of smooth 3D multi-region meshes that conform to the region boundaries of natural heterogeneous objects described in the medical images. To solve these issues, the CAREM method and the RAM method are proposed. The emphasis of this research is placed on accuracy and shape fidelity needed for biomedical applications. All implicitly represented regions composing the natural heterogeneous object are used to generate meshes adapted to the requirements of finite element methods through a reverse engineering modeling approach, thus these regions are considered as whole rather than loosely assembled parts.Doctor en IngenieríaDoctorad

Mesh adaptation for high-order flow simulations

Mesh adaptation has only been considered for high-order flow simulations in recent years and many techniques are still to be made more robust and efficient with curvilinear meshes required by these high-order methods. This thesis covers the developments made to improve the mesh generation and adaptation capabilities of the open-source spectral/hp element framework Nektar++ and its dedicated mesh utility NekMesh. This thesis first covers the generation of quality initial meshes typically required before an iterative adaptation procedure can be used. For optimal performance of the spectral/hp element method, quadrilateral and hexahedral meshes are preferred and two methods are presented to achieve this, either entirely or partially. The first method, inspired from cross field methods, solves a Laplace problem to obtain a guiding field from which a valid two-dimensional quadrilateral block decomposition can be automatically obtained. In turn, naturally curved meshes are generated. The second method takes advantage of the medial axis to generate structured partitions in the boundary layer region of three-dimensional domains. The method proves to be robust in generating hybrid high-order meshes with boundary layer aligned prismatic elements near boundaries and tetrahedral elements elsewhere. The thesis goes on to explore the adaptation of high-order meshes for the simulation of flows using a spectral/hp element formulation. First a new approach to moving meshes, referred to here as r-adaptation, based on a variational framework, is described. This new r-adaptation module is then enhanced by p-adaptation for the simulation of compressible inviscid flows with shocks. Where the flow is smooth, p-adaptation is used to benefit from the spectral convergence of the spectral/hp element methods. Where the flow is discontinuous, e.g. at shock waves, r-adaptation clusters nodes together to better capture these field discontinuities. The benefits of this dual, rp-adaptation approach are demonstrated through two-dimensional benchmark test cases.Open Acces