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

Efficient Computation of the Extrema of Algebraic Quality Measures for Curvilinear Finite Elements

The development of high-order computational methods for solving partial differential equations on unstructured grids has been underway for many years. Such methods critically depend on the availability of high-quality curvilinear meshes, as one bad element can degrade the solution in the whole domain. The usual way of generating curved meshes is to first generate a (high-quality) straight-sided mesh. Then, mesh entities that are classified on the boundaries of the domain are curved. This operation introduces a "shape-distortion" that should be controlled. Quality measures allow to quantify to which point an element is well-shaped. They also provide tools to improve the quality of meshes through optimization. In this work we propose an efficient method to compute several quality measures for curved elements, based on the Jacobian of the mapping between the straight-sided elements and the curved ones. Contrary to the approach presented in "A. Gargallo-Peiró, X. Roca, J. Peraire, and J. Sarrate. Distortion and quality measures for validating and generating high-order tetrahedral meshes. Engineering with Computers, pages 1–15, 2014.", which relies on an L2-norm over the elements, we compute the actual minimum and maximum of the local quality measure for each element. The method is an extension of previous works on the validity of those elements (A. Johnen et al., 2013). The key feature is that we can adaptively expand functions based on the Jacobian matrix and its determinant in terms of Bézier functions. Bézier functions have both properties of boundedness and positivity, which allow sharp computation of minimum or maximum of the interpolated functions

Indirect quadrangular mesh generation and validation of curved finite elements

Among the different types of 3D finite element meshes, hexahedral meshes present properties that can be highly desirable, such as alignment with physical features or a lower computational cost. For this reason and despite the maturity of the tetrahedral mesh generators, hexahedral mesh generation has always been a prolific research domain. Yet, there exists currently no robust algorithm capable of generating conformal all-hexahedral meshes with prescribed input size field on any arbitrary geometry. One difficulty that remains is that there exists no method to robustly assert that a hexahedron is valid. Indeed, linear hexahedra can be folded (tangled) in the same way than curvilinear tetrahedra. This thesis addresses two subjects. First, two original quadrangular mesh generation techniques are investigated, with the aim to generalize them to 3D. Both are indirect methods and thus consider the problem of combining pairs of triangles of an initial input triangular mesh. The first technique, called Blossom-Quad, computes the optimal solution of this problem with respect to a given quality criterion. As for any indirect method, the quality of the solution strongly depends on the location of the nodes in the initial triangular mesh. The generalization to 3D is however unclear and a second technique is investigated. This one aims at computing a near-optimal solution by using a look-ahead tree technique. The corresponding algorithm allows tuning the quality of the final mesh by choosing the depth of the tree as a parameter. This technique gives a promising way forward, especially as it is directly applicable in 3D. The second subject concerns the development of a method that permits to compute, with respect to any prescribed tolerance, the extrema of Jacobian-based quantities defined on finite elements of any order and type. Applied to the Jacobian determinant, this method allows to assert the validity of any (curvi-)linear finite element. This method is also applied to a quality measure that quantifies the pointwise anisotropy of the elements. Besides being very attractive for hexahedral mesh generation, this method is especially useful for the analysis of curvilinear finite element meshes. It can moreover be an important component of optimization techniques for achieving robustness.Parmi les différents types de maillages d’éléments finis 3D, les maillages hexaédriques présentent des propriétés qui peuvent être extrêmement attrayantes telles que l'alignement avec les caractéristiques physiques ou un coût de calcul plus faible. Pour cette raison, et malgré la maturité des algorithmes de génération de maillages tétraédriques, la génération de maillage hexaédrique a toujours été un domaine de recherche prolifique. Pourtant, il n'existe pas encore d'algorithme robuste capable de générer des maillages totalement hexaédriques, conformes, sur des géométries arbitraires, tout en respectant un champ de taille imposée. Une difficulté qui subsiste est qu'il n'existe pas de méthode pour affirmer de manière robuste qu'un hexaèdre est valide. En effet, les hexaèdres linéaires peuvent être repliés sur eux-mêmes au même titre que peuvent l'être les tétraèdres courbes. Cette thèse traite de deux sujets. Premièrement, deux techniques originales de génération de maillages quadrangulaires sont examinées avec le but de pouvoir les généraliser au problème 3D. Ces deux techniques prennent l'approche indirecte qui consiste à combiner des paires de triangles dans un maillage triangulaire initial d'entrée. La première technique, appelée Blossom-Quad, calcule la solution optimale de ce problème par rapport à un critère de qualité donné. Comme pour n'importe quelle autre méthode indirecte, la qualité de la solution dépend fortement de la localisation des nœuds dans le maillage triangulaire initial. La généralisation au problème 3D est toutefois incertaine et une seconde technique est examinée. Celle-ci vise à calculer une solution presque optimale en utilisant une technique d'arbre de décision. L'algorithme correspondant permet de commander la qualité du maillage final en choisissant la profondeur de l'arbre comme paramètre. Cette technique donne une voie à suivre prometteuse, particulièrement de par le fait qu'elle soit directement applicable au problème 3D. Le second sujet traite du développement d'une méthode qui permet de calculer avec la précision voulue les valeurs extrêmes de quantités basées sur le jacobien qui sont par essence valables pour les éléments finis de n'importe quel type et n'importe quel ordre. Appliquée au déterminant jacobien, cette méthode, permet d'affirmer la validité des éléments finis linéaires ou courbes. Cette méthode est aussi appliquée à une mesure de qualité qui quantifie l'anisotropie ponctuelle des éléments. En plus d'être très intéressante pour la génération de maillage hexaédrique, cette méthode est particulièrement utile pour l'analyse de maillages d'éléments finis courbes. Elle peut également être un composant important des techniques d'optimisation afin d'accroître leur robustesse

Sequential decision-making approach for quadrangular mesh generation

A new indirect quadrangular mesh generation algorithm which relies on sequential decision-making techniques to search for optimal triangle recombinations is presented. In contrast to the state-of-art Blossom-quad algorithm, this new algorithm is a good candidate for addressing the 3D problem of recombining tetrahedra into hexahedra.DOMHE

Efficient evaluation of the geometrical validity of curvilinear finite elements

The development of high-order numerical techniques on unstructured grids has been underway for many years. The accuracy of these methods strongly depends of the accuracy of the geometrical discretization, and thus depends on the availability of quality curvilinear meshes. The usual way of building such curvilinear meshes is to first generate a straight sided mesh. Then, mesh entities that are classified on the curved boundaries of the domain are curved accordingly. Some internal mesh entities may be curved as well. If we assume that the straight sided mesh is composed of well shaped elements, curving elements introduces a kind of "shape distortion" that should be controlled so that the final curvilinear mesh is also composed of well shaped elements. In this work we propose a method to analyze curvilinear meshes in terms of their elementary jacobians. The method does not deal with the actual generation of the high order mesh. Instead, it provides an efficient way to guarantee that a curvilinear element is geometrically valid, i.e., that its jacobian is strictly positive in all its reference domain. It also provides a way to measure the distortion of the curvilinear element. The key feature of the method is to adaptively expand the elementary jacobians in a polynomial basis, built using Bézier functions, that has both properties of boundedness and positivity. The algorithm has been implemented in the open-source mesh generator Gmsh, and allows to control the geometrical validity of curvilinear meshes made of triangles, quadrangles, tetrahedra, hexahedra and prisms of any order

Efficient Computation of the Minimum of Shape Quality Measures on Curvilinear Finite Elements

We present a method for computing robust shape quality measures defined for any order of finite elements. All type of elements are considered, including pyramids. The measures are defined as the minimum of the pointwise quality of curved elements. The computation of the minimum, based on previous work presented by Johnen et al. (2013) [1] and [2], is very efficient. The key feature is to expand polynomial quantities into Bézier bases which allows to compute sharp bounds on the minimum of the pointwise quality measures