One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh

There are 174 Subdivisions of the Hexahedron into Tetrahedra

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian

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

Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.Comment: Preprint submitted to CAD (26th IMR special issue

Testing Scenarios on Geological Models: Local Interface Insertion in a 2D Mesh and its Impact on Seismic Wave Simulation

In this work, we consider a relatively simple case of fluid monitoring in a subsurface gas reservoir. Seismic wave velocities for porous rocks vary depending on fluid saturation, and our objective is to quantitatively evaluate the impact of the water/gas contact depth on elastic wave propagation. To efficiently test different contact depth scenarios and assess their impact on wave propagation, we propose to locally modify a 2D geological model and run time-dependent elastic simulations. The input model is a triangulated surface conforming to geological structures and representing physical properties. The 2D meshed model is locally updated, meaning that only a given region is modified and that the other parts of the mesh remain identical. To create several models by modifying only the reservoir layer, we insert a water/gas contact defined by a level-set at several depths with MMG. During the insertion, specific care is taken to maintain the conformity of the output mesh. As compared to global remeshing, the local modification reduces the cost of recomputing physical properties over the updated mesh. We run the numerical simulations by using Hou10ni2D code, which is based on a Discontinuous Galerkin method. Our results on a gas reservoir show a consistent behavior: we observe a correlation between the depth difference and L2-norm, the larger the distance from the reference depth contact, the higher the L2-norm. This approach could therefore be integrated into an inversion loop to determine the position of the fluid contact and reduce uncertainties in the reservoir model

3D geomodelling combining implicit surfaces and Voronoi-based remeshing: A case study in the Lorraine Coal Basin (France)

International audienceIn this paper we demonstrate how recent geomodelling techniques can be combined and used to build a 3D geological model on a real case study: the former coal mine of Merlebach (France), that is targeted to be exploited for low-temperature geothermal energy production. From geological maps, cross-sections, borehole and mine exploitation data, we build a 3D model in which are identified the rocks and infrastructures having significantly different permeabilities. First, a structural model of the main geological interfaces in our area of interest (2 horizons and 13 faults) is built with classical geomodelling techniques. Then, we propose to model by surfaces the 71 irregularly stacked, very close and very thin, subvertical coal beds. To ease their construction, we use an implicit method which represents 3D surfaces as isovalues of a scalar field defined in a 3D tetrahedral grid of the area. The corresponding triangulated surfaces are remeshed with a recently proposed method based on Voronoi diagrams so that the exploited parts of the coal beds, now filled by sand, can be computed. The 3D surface-based geological model, in which infrastructures can be inserted as piecewise lines, can be volumetrically meshed. It is available for download as supplemental material, as well as a volumetric grid

Short-term telomere dynamics is associated with glucocorticoid levels in wild populations of roe deer

International audienceWhile evidence that telomere length is associated with health and mortality in humans and birds is accumulating, a large body of research is currently seeking to identify factors that modulate telomere dynamics. We tested the hypothesis that high levels of glucocorticoids in individuals under environmental stress should accelerate telomere shortening in two wild populations of roe deer (Capreolus capreolus) living in different ecological contexts. From two consecutive annual sampling sessions, we found that individuals with faster rates of telomere shortening had higher concentrations of fecal glucocorticoid metabolites, suggesting a functional link between glucocorticoid levels and telomere attrition rate. This relationship was consistent for both sexes and populations. This finding paves the way for further studies of the fitness consequences of exposure to environmental stressors in wild vertebrates

Prise en compte de la complexité géométrique des modèles structuraux dans des méthodes de maillage fondées sur le diagramme de Voronoï

Depending on the specific method used to build a 3D structural model, and on the exact purpose of this model, its mesh must be adapted so that it enforces criteria on element types, maximum number of elements, and mesh quality. Meshing methods developed for applications others than geomodeling forbid any modification of the input model, that may be desirable in geomodeling to better control the number of elements in the final mesh and their quality. The objective of this thesis is to develop meshing methods that fulfill this requirement to better manage the geometrical complexity of B-Rep geological structural models. An analysis of the sources of geometrical complexity in those models is first proposed. The introduced measures are a first step toward the definition of tools allowing objective comparisons of structural models and permit to characterize the model zones that are more complicated to mesh. We then introduce two original meshing methods based on Voronoi diagrams: the first for surface remeshing, the second for hybrid gridding. The key ideas of these methods are identical: (1) the use of a centroidal Voronoi optimization to have a globally controlled number of elements of good quality, and (2) combinatorial considerations to locally build the final mesh while sometimes modifying the initial model. The surface remeshing method is automatic and permits to simplify a model at a given resolution. The gridding method generates a hybrid volumetric mesh. Prisms and pyramids fill the very thin layers of the model while the remaining regions are filled with tetrahedra.Selon la méthode utilisée pour construire un modèle structural en trois dimensions et selon l'application à laquelle il est destiné, son maillage, en d'autres termes sa représentation informatique, doit être adapté afin de respecter des critères de type, de nombre et de qualité de ses éléments. Les méthodes de maillage développées dans d'autres domaines que la géomodélisation ne permettent pas de modifier le modèle d'entrée. Ceci est souhaitable en géomodélisation afin de mieux contrôler le nombre d'éléments du maillage et leur qualité. L'objectif de cette thèse est de développer des méthodes de maillage permettant de remplir ces objectifs afin de gérer la complexité géométrique des modèles structuraux définis par frontières. Premièrement, une analyse des sources de complexité géométrique dans ces modèles est proposée. Les mesures développées constituent une première étape dans la définition d'outils permettant la comparaison objective de différents modèles et aident à caractériser précisément les zones plus compliquées à mailler dans un modèle. Ensuite, des méthodes originales de remaillage surfacique et de maillage volumique fondées sur l'utilisation des diagrammes de Voronoï sont proposées. Les fondements de ces deux méthodes sont identiques : (1) une optimisation de type Voronoï barycentrique est utilisée pour globalement obtenir un nombre contrôlé d'éléments de bonne qualité et (2) des considérations combinatoires permettant de construire localement le maillage final, éventuellement en modifiant le modèle initial. La méthode de remaillage surfacique est automatique et permet de simplifier un modèle à une résolution donnée. L'originalité de la méthode de maillage volumique est que les éléments générés sont de types différents. Des prismes et pyramides sont utilisés pour remplir les zones très fines du modèle, tandis que le reste du modèle est rempli avec des tétraèdres

Prise en compte de la complexité géométrique des modèles structuraux dans des méthodes de maillage fondées sur le diagramme de Voronoï

Depending on the specific method used to build a 3D structural model, and on the exact purpose of this model, its mesh must be adapted so that it enforces criteria on element types, maximum number of elements, and mesh quality. Meshing methods developed for applications others than geomodeling forbid any modification of the input model, that may be desirable in geomodeling to better control the number of elements in the final mesh and their quality. The objective of this thesis is to develop meshing methods that fulfill this requirement to better manage the geometrical complexity of B-Rep geological structural models. An analysis of the sources of geometrical complexity in those models is first proposed. The introduced measures are a first step toward the definition of tools allowing objective comparisons of structural models and permit to characterize the model zones that are more complicated to mesh. We then introduce two original meshing methods based on Voronoi diagrams: the first for surface remeshing, the second for hybrid gridding. The key ideas of these methods are identical: (1) the use of a centroidal Voronoi optimization to have a globally controlled number of elements of good quality, and (2) combinatorial considerations to locally build the final mesh while sometimes modifying the initial model. The surface remeshing method is automatic and permits to simplify a model at a given resolution. The gridding method generates a hybrid volumetric mesh. Prisms and pyramids fill the very thin layers of the model while the remaining regions are filled with tetrahedra.Selon la méthode utilisée pour construire un modèle structural en trois dimensions et selon l'application à laquelle il est destiné, son maillage, en d'autres termes sa représentation informatique, doit être adapté afin de respecter des critères de type, de nombre et de qualité de ses éléments. Les méthodes de maillage développées dans d'autres domaines que la géomodélisation ne permettent pas de modifier le modèle d'entrée. Ceci est souhaitable en géomodélisation afin de mieux contrôler le nombre d'éléments du maillage et leur qualité. L'objectif de cette thèse est de développer des méthodes de maillage permettant de remplir ces objectifs afin de gérer la complexité géométrique des modèles structuraux définis par frontières. Premièrement, une analyse des sources de complexité géométrique dans ces modèles est proposée. Les mesures développées constituent une première étape dans la définition d'outils permettant la comparaison objective de différents modèles et aident à caractériser précisément les zones plus compliquées à mailler dans un modèle. Ensuite, des méthodes originales de remaillage surfacique et de maillage volumique fondées sur l'utilisation des diagrammes de Voronoï sont proposées. Les fondements de ces deux méthodes sont identiques : (1) une optimisation de type Voronoï barycentrique est utilisée pour globalement obtenir un nombre contrôlé d'éléments de bonne qualité et (2) des considérations combinatoires permettant de construire localement le maillage final, éventuellement en modifiant le modèle initial. La méthode de remaillage surfacique est automatique et permet de simplifier un modèle à une résolution donnée. L'originalité de la méthode de maillage volumique est que les éléments générés sont de types différents. Des prismes et pyramides sont utilisés pour remplir les zones très fines du modèle, tandis que le reste du modèle est rempli avec des tétraèdres