Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms

Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the W1p norm and we consider Lagrange finite elements of arbitrary polynomial order m-1. We establish sharp asymptotic error estimates as N tends to infinity when the optimal anisotropic triangulation is used. A similar problem has been studied earlier, but with the error measured in the Lp norm. The extension of this analysis to the W1p norm is crucial in order to match more closely the needs of numerical PDE analysis, and it is not straightforward. In particular, the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle, a second feature that does not appear for the Lp error norm. Our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We discuss the extension of our results to finite elements on simplicial partitions of a domain of arbitrary dimension, and we provide with some numerical illustration in two dimensions.Comment: 37 pages, 6 figure

Galerkin projection of discrete fields via supermesh construction

Interpolation of discrete FIelds arises frequently in computational physics. This thesis focuses on the novel implementation and analysis of Galerkin projection, an interpolation technique with three principal advantages over its competitors: it is optimally accurate in the L2 norm, it is conservative, and it is well-defined in the case of spaces of discontinuous functions. While these desirable properties have been known for some time, the implementation of Galerkin projection is challenging; this thesis reports the first successful general implementation. A thorough review of the history, development and current frontiers of adaptive remeshing is given. Adaptive remeshing is the primary motivation for the development of Galerkin projection, as its use necessitates the interpolation of discrete fields. The Galerkin projection is discussed and the geometric concept necessary for its implementation, the supermesh, is introduced. The efficient local construction of the supermesh of two meshes by the intersection of the elements of the input meshes is then described. Next, the element-element association problem of identifying which elements from the input meshes intersect is analysed. With efficient algorithms for its construction in hand, applications of supermeshing other than Galerkin projections are discussed, focusing on the computation of diagnostics of simulations which employ adaptive remeshing. Examples demonstrating the effectiveness and efficiency of the presented algorithms are given throughout. The thesis closes with some conclusions and possibilities for future work

Adaptation de maillages anisotropes

RÉSUMÉ En simulation numérique, l'adaptation de maillages se révèle être un outil essentiel à l'obtention de résultats crédibles. Un maillage mal adapté à une solution entraîne généralement une mauvaise définition de cette dernière et donc une perte de précision. Généralement, plus le maillage est fin, meilleure sera la précision de la solution. Toutefois, cette finesse a un prix et les maillages résultants peuvent devenir très gros en matière de nombre de noeuds. Plus le nombre de noeuds est important, plus le temps de calcul de la solution est long. Les maillages anisotropes sont formés d'éléments étirés et orientés de manière à minimiser l'erreur d'interpolation sur le domaine tout en respectant un certain nombre de noeuds. Ces maillages particuliers permettent d'obtenir des solutions de précision équivalentes à celles obtenues sur des maillages classiques isotropes, mais comptent souvent beaucoup moins de noeuds. Ces économies en termes de noeuds et d'éléments ne sont pas négligeables en ce qui a trait aux temps de calcul de la solution et à l'espace mémoire requis. La méthode sur laquelle est basée l'adaptation de maillages anisotropes présentée dans ce travail fait appel aux métriques optimales multi-échelle afin de minimiser l'erreur d'interpolation en norme Lp pour un nombre donné de noeuds N. La métrique est un tenseur d'ordre deux qui définit une transformation affine de l'espace physique vers un espace virtuel. Un élément anisotrope, qui est étiré dans l'espace physique, devient isotrope une fois transformé dans l'espace virtuel. L'étirement et l'orientation de l'élément dans l'espace physique assurent l'équirépartition de l'erreur d'interpolation dans toutes les directions, et minimisent cette dernière pour un élément triangulaire linéaire ayant une certaine aire. La taille, l'étirement et l'orientation des éléments anisotropes sont déterminés en chaque noeud du maillage par la métrique. Celle-ci est calculée à partir de la matrice hessienne, qui est la matrice des dérivées secondes de la solution. Le ratio des racines carrées des valeurs propres de cette matrice définit l'étirement de l'élément. Les tailles respectives aux deux directions principales sont définies par l'inverse des racines carrées de ces valeurs propres.----------Mesh adaptation is essential in numerical simulation to obtain reliable results. A mesh poorly adapted to a solution will generate a wrong definition of the solution, and consequently it will lack precision. Generally, when the mesh is finer, the precision of the solution will be better. However, this increase of resolution has its price and the resulting meshes could become very large in terms of number of nodes. The computation time necessary for a solution will increase as the number of nodes on the mesh. Anisotropic meshes are formed of stretched elements oriented such as to minimize the interpolation error on the domain for a fixed number of nodes. These specials meshes can produce solutions of equivalent precision as the ones obtained from isotropic classic meshes, but with a lot less nodes. This reduction of nodes and elements is not negligible when considering solution processing time and necessary memory space. The method on which is based the following anisotropic mesh adaptation technique refers to a multi-scale optimal metric minimizing the Lp norm of the interpolation error for a fixed number of nodes N. The metric is an order 2 tensor defining an affine transformation from a physical space to a virtual space. An anisotropic element, stretched in the physical space, becomes isotropic when transformed in the virtual space. The element stretching and orientation in the physical space guarantee equidistribution of the interpolation error in every direction, and minimize it for a triangular element of fixed area. The size, the tretching and the orientation of anisotropic elements are determined at every node of the mesh by the metric. This metric is calculated from the hessian matrix, which is the second derivatives matrix of the solution. The ratio of eigenvalue square root of this matrix defines the element stretching. The sizes along the eigenvectors directions are defined by the inverse of the corresponding eigenvalue square root. The eigenvector associated to the eigenvalue smallest absolute value defines the stretching principal direction. The metric is then represented by the hessian matrix modified to be symmetrical positive defined such as a metric tensor. The optimal metric is finally obtained by averaging the nodal metrics over the domain. The Lp norm used to calculate the metric controls the mesh nodes concentration. As the norm is lower, the mesh generated offers a better definition for low amplitude structures of the solution. Therefore the nodes are then more evenly distributed on the domain. When higher norms are used, the nodes are concentrated on anisotropic structures, such as shock waves or boundary layers