Champs de repères pour les modèles CAO

National audienceMaillage quadrangulaire de modèles CAO obtenus avec notre champ de repères. Le champ de repères le plus lisse, obtenu par une méthode usuelle, aurait une topologie incompatible avec un maillage quadrangulaire dans les zones montrées par les flèches en orange. L'impact de ces incohérences n'est pas local : cela produirait des quadrilatères dégénérés sur l'ensemble des zones surlignées en orange

Frame Fields for CAD models

International audienceGiven a triangulated surface, a unit length tangent vector field can be used to orient entities located on the surface, such as glyphs or strokes. When these entities are invariant under a π/2 rotation (squares, or curvature hatching), the orientation can be represented by a frame field i.e. four orthogonal tangent unit vectors at each point of the surface. The generation of such fields is a key component of recent quad meshing algorithms based on global parameterization, as it defines the orientation of the final facets. State-of-the-art methods are able to generate smooth frame fields subject to some hard constraints (direction and topology) or smooth constraints (matching the curvature direction). When we have a surface triangular mesh, and a vector defined on each facet, we can't directly know if all the vectors are colinear. We first have to define the (so called) parallel transport of every edge to compare the vectors on a common plan. When dealing with CAD models, the field must be aligned with feature edges. A problem occurs when there is a low angle corner formed by two colliding feature edges. Our solution not only defines the parallel transport to obtain smoothed frame fields on a surface triangular mesh, it also redefines the parallel transport wherever there is a low angle corner, to smooth a frame field as if these corners' angles were π/2

Maillage à dominante Polycube

This thesis studies polycube methods for the generation of hexahedral meshes. These meshes, and more generally block structured meshes, are very much in demand for numerical simulations of physical phenomena (nuclear fission, flow, aerospace...). However, there is no industrially viable method to generate them. We are studying a family of methods that show great promise in filling this gap: global parametrization methods. To date, there are still many robustness problems, which we try to correct. For this purpose, we focus on the subfamily of polycube methods.To obtain a hexahedral mesh of a domain with polycubes - clusters of unit cubes -, the procedure is the following: the boundary of the domain is colored, then deformed according to these colors to have its boundary aligned with the axes, which makes it a polycuboid. This polycuboid is then intersected with a grid to give a polycube. Applying the inverse deformation on this polycube gives us a hexahedral mesh.We make two major contributions to this procedure: first, a method that allows to compute the deformation efficiently and with guarantees that it will be of good quality. Second, we introduce a new robust method to perform the grid intersection and deformation inversion steps. Finally, we present avenues of study for the difficult step of coloring, the last brick necessary to have robust methods for generating polycube-based hexahedral meshes.Cette thèse étudie les méthodes de polycubes pour la génération de maillages hexaédriques. Ces maillages, et plus généralement les maillages structurés par blocs, sont très recherchés pour effectuer des simulations numériques de phénomènes physiques (fission nucléaire, écoulement, aérospatiale…). Or, il n’existe pas de méthode industriellement viable pour les générer. Nous étudions une famille de méthodes très prometteuses pour combler ce vide : les méthodes de paramétrisation globales. À ce jour, il subsiste de nombreux problèmes de robustesse, que nous essayons de corriger. Pour cela, nous nous focalisons sur la sous-famille des méthodes de polycubes. Pour obtenir un maillage hexaédrique d’un domaine avec des polycubes - amas de cubes unités -, la procédure est la suivante : le bord du domaine est colorié, puis déformé en fonction de ces couleurs pour avoir son bord aligné avec les axes, ce qui en fait un polycuboid. Ce polycuboid est ensuite intersecté avec une grille pour donner un polycube. Appliquer la déformation inverse sur ce polycube nous donne un maillage hexaédrique. Nous apportons deux contributions majeures à cette procédure : d’abord une méthode qui permet de calculer la déformation efficacement et avec des garanties que celle-ci sera de bonne qualité. Ensuite, nous introduisons une nouvelle méthode robuste pour effectuer les étapes d’intersection avec la grille et d’inversion de la déformation. Nous présentons enfin des pistes d’études pour l’étape difficile de la coloration, dernière brique nécessaire pour avoir des méthodes robustes de génération de maillages hexaédriques à base de polycubes

Maillage à dominante Polycube

This thesis studies polycube methods for the generation of hexahedral meshes. These meshes, and more generally block structured meshes, are very much in demand for numerical simulations of physical phenomena (nuclear fission, flow, aerospace...). However, there is no industrially viable method to generate them. We are studying a family of methods that show great promise in filling this gap: global parametrization methods. To date, there are still many robustness problems, which we try to correct. For this purpose, we focus on the subfamily of polycube methods.To obtain a hexahedral mesh of a domain with polycubes - clusters of unit cubes -, the procedure is the following: the boundary of the domain is colored, then deformed according to these colors to have its boundary aligned with the axes, which makes it a polycuboid. This polycuboid is then intersected with a grid to give a polycube. Applying the inverse deformation on this polycube gives us a hexahedral mesh.We make two major contributions to this procedure: first, a method that allows to compute the deformation efficiently and with guarantees that it will be of good quality. Second, we introduce a new robust method to perform the grid intersection and deformation inversion steps. Finally, we present avenues of study for the difficult step of coloring, the last brick necessary to have robust methods for generating polycube-based hexahedral meshes.Cette thèse étudie les méthodes de polycubes pour la génération de maillages hexaédriques. Ces maillages, et plus généralement les maillages structurés par blocs, sont très recherchés pour effectuer des simulations numériques de phénomènes physiques (fission nucléaire, écoulement, aérospatiale…). Or, il n’existe pas de méthode industriellement viable pour les générer. Nous étudions une famille de méthodes très prometteuses pour combler ce vide : les méthodes de paramétrisation globales. À ce jour, il subsiste de nombreux problèmes de robustesse, que nous essayons de corriger. Pour cela, nous nous focalisons sur la sous-famille des méthodes de polycubes. Pour obtenir un maillage hexaédrique d’un domaine avec des polycubes - amas de cubes unités -, la procédure est la suivante : le bord du domaine est colorié, puis déformé en fonction de ces couleurs pour avoir son bord aligné avec les axes, ce qui en fait un polycuboid. Ce polycuboid est ensuite intersecté avec une grille pour donner un polycube. Appliquer la déformation inverse sur ce polycube nous donne un maillage hexaédrique. Nous apportons deux contributions majeures à cette procédure : d’abord une méthode qui permet de calculer la déformation efficacement et avec des garanties que celle-ci sera de bonne qualité. Ensuite, nous introduisons une nouvelle méthode robuste pour effectuer les étapes d’intersection avec la grille et d’inversion de la déformation. Nous présentons enfin des pistes d’études pour l’étape difficile de la coloration, dernière brique nécessaire pour avoir des méthodes robustes de génération de maillages hexaédriques à base de polycubes

Limits and prospects of polycube labelings

International audiencePolycubes have been a fruitful approach for all-hexahedral mesh generation, thanks to an attractive robustness/quality trade-off. Starting from a tetrahedral mesh of the input shape, a polycube can be easily represented by associating one of the six signed orthogonal direction ±{X, Y, Z} to boundary triangles, called labeling. Not all labelings induce polycubes, therefore validity criteria have been proposed. Despite satisfactory in most cases, they are neither necessary nor sufficient. By presenting failure cases, we open the discussion towards new approaches to discriminate between valid and invalid polycube labelings

Limits and prospects of polycube labelings

International audiencePolycubes have been a fruitful approach for all-hexahedral mesh generation, thanks to an attractive robustness/quality trade-off. Starting from a tetrahedral mesh of the input shape, a polycube can be easily represented by associating one of the six signed orthogonal direction ±{X, Y, Z} to boundary triangles, called labeling. Not all labelings induce polycubes, therefore validity criteria have been proposed. Despite satisfactory in most cases, they are neither necessary nor sufficient. By presenting failure cases, we open the discussion towards new approaches to discriminate between valid and invalid polycube labelings

Evocube: a Genetic Labeling Framework for Polycube-Maps

International audiencePolycube-maps are used as base-complexes in various fields of computational geometry, including the generation of regular all-hexahedral meshes free of internal singularities. However, the strict alignment constraints behind polycube-based methods make their computation challenging for CAD models used in numerical simulation via Finite Element Method (FEM). We propose a novel approach based on an evolutionary algorithm to robustly compute polycube-maps in this context.We address the labeling problem, which aims to precompute polycube alignment by assigning one of the base axes to each boundary face on the input. Previous research has described ways to initialize and improve a labeling via greedy local fixes. However, such algorithms lack robustness and often converge to inaccurate solutions for complex geometries. Our proposed framework alleviates this issue by embedding labeling operations in an evolutionary heuristic, defining fitness, crossover, and mutations in the context of labeling optimization. We evaluate our method on a thousand smooth and CAD meshes, showing Evocube converges to valid labelings on a wide range of shapes. The limitations of our method are also discussed thoroughly

Evocube: a Genetic Labeling Framework for Polycube-Maps

Polycube-maps are used as base-complexes in various fields of computational geometry, including the generation of regular all-hexahedral meshes free of internal singularities. However, the strict alignment constraints behind polycube-based methods make their computation challenging for CAD models used in numerical simulation via Finite Element Method (FEM). We propose a novel approach based on an evolutionary algorithm to robustly compute polycube-maps in this context.We address the labeling problem, which aims to precompute polycube alignment by assigning one of the base axes to each boundary face on the input. Previous research has described ways to initialize and improve a labeling via greedy local fixes. However, such algorithms lack robustness and often converge to inaccurate solutions for complex geometries. Our proposed framework alleviates this issue by embedding labeling operations in an evolutionary heuristic, defining fitness, crossover, and mutations in the context of labeling optimization. We evaluate our method on a thousand smooth and CAD meshes, showing Evocube converges to valid labelings on a wide range of shapes. The limitations of our method are also discussed thoroughly

Evocube: a Genetic Labeling Framework for Polycube-Maps

International audiencePolycube-maps are used as base-complexes in various fields of computational geometry, including the generation of regular all-hexahedral meshes free of internal singularities. However, the strict alignment constraints behind polycube-based methods make their computation challenging for CAD models used in numerical simulation via Finite Element Method (FEM). We propose a novel approach based on an evolutionary algorithm to robustly compute polycube-maps in this context.We address the labeling problem, which aims to precompute polycube alignment by assigning one of the base axes to each boundary face on the input. Previous research has described ways to initialize and improve a labeling via greedy local fixes. However, such algorithms lack robustness and often converge to inaccurate solutions for complex geometries. Our proposed framework alleviates this issue by embedding labeling operations in an evolutionary heuristic, defining fitness, crossover, and mutations in the context of labeling optimization. We evaluate our method on a thousand smooth and CAD meshes, showing Evocube converges to valid labelings on a wide range of shapes. The limitations of our method are also discussed thoroughly

On Local Invertibility and Quality of Free-boundary Deformations

International audienceMesh untangling is still a hot topic in applied mathematics. Tangled or folded meshes appear in many applications involving mappings or deformations. Despite the fact that a large number of mesh untangling strategies was proposed during the last decades, this problem still persists. Recently we have proposed a numerical optimization scheme [1] that provably untangles 2d and 3d meshes with inverted elements by partially solving a finite number of unconditional minimization problems. The method is robust for fixed boundary mesh untangling problems, and it can be applied to some extent to free boundary untangling. The problem, however, is that the absence of inverted elements does not guarantee invertibility of the deformation (map). The invertibility is lost if the mesh gets caught in a k-covering trap, i.e. in a local minimum of the deformation energy where all mesh elements are not inverted but total angle around certain vertex is above 2π for 2D and above 4π for 3D. This problem is particularly vexing when partially constrained mesh deformation problems are considered. In this paper we show how to improve the method suggested in [1]. Namely, we show the way to guarantee absence of k-covering folds, and so, the local invertibility is assured. We demonstrate enhanced stability of suggested untangling technique which has a potential to make untangling a routine operation over meshes