Greedy bisection generates optimally adapted triangulations

We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f. The refinement procedure picks the triangle which maximizes the local Lp approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the Lp norm when the algorithm is applied to C2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations.Comment: 24 page

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

Continuous Mesh Model and Well-Posed Continuous Interpolation Error Estimation

Rapport de recherche INRIAIn the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density and stretching of anisotropic meshes. Such structures are used to compute lengths in adaptive mesh generators. In this report, a Riemannian metric space is shown to be more than a way to compute a distance. It is proven to be a reliable continuous mesh model. In particular, we demonstrate that the linear interpolation error can be derived continuously for a continuous mesh. In its tangent space, a Riemannian metric space reduces to a constant metric tensor so that it simply spans a metric space. Metric tensors are then used to continuously model discrete elements. On this basis, geometric invariants have been extracted. They connect a metric tensor to the set of all the discrete elements which can be represented by this metric. As the behavior of a Riemannian metric space is obtained by patching together the behavior of each of its tangent spaces, the global mesh model arises from gathering together continuous element models. We complete the continuous-discrete analogy by providing a continuous interpolation error estimate and a well-posed definition of the continuous linear interpolate. The later is based on an exact relation connecting the discrete error to the continuous one. From one hand, this new continuous framework freed the analysis of the topological mesh constraints. On the other hand, powerful mathematical tools are available and well defined on the space of continuous meshes: calculus of variations, differentiation, optimization, ..., whereas these tools are not defined on the space of discrete meshes

Impact of triangle shapes using high-order discretizations and direct mesh adaptation for output error

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 95-101).The impact of triangle shapes, including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for high-order discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and high-order discretizations. In addition, large angles are not problematic for the conditioning of the linear systems arising from discretization. They can be overcome through small increases in preconditioning costs. A direct adaptation scheme that controls the output error via mesh operations and mesh smoothing is also developed. The decision of mesh operations is solely based on output error distribution without any a priori assumption on error convergence rate. Anisotropy is introduced by evaluating the error changes due to potential edge split, and thus the anisotropies of both primal and dual solutions are taken into account. This scheme is demonstrated to produce grids with fewer degrees of freedom for a specified error level than the existing metric-based approach.by Huafei Sun.S.M

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 par un estimateur d'erreur hiérarchique

Dans cette thèse, nous présentons un nouvel estimateur d’erreur de type hiérarchique utilisable dans un algorithme d’adaptation de maillages afin d’obtenir une approximation plus précise d’une équation aux dérivées partielles. Nous décrivons les avantages que possèdent ce nouvel estimateur d’erreur versus ceux qui existent déjà dans la littérature et nous justifions sa construction. Plusieurs résultats numériques seront présentés dans les cas uni, bi et tridimensionnels. Nous montrons des exemples académiques (où la solution analytique est connue) pour mesurer l’efficacité et la précision du nouvel estimateur d’erreur. Nous montrons également des exemples d’adaptation de maillages pour des équations modélisant des phénomènes physiques comme l’écoulement d’un fluide autour d’un cylindre, la diffusion instationnaire et le contact entre des corps élastiques déformables. Ces exemples montrent que le nouvel estimateur d’erreur est utilisable pour une très grande classe de problèmes.In this thesis, we present a new hierarchical error estimator that can be used in a mesh adaptation algorithm to obtain a more accurate approximation to the solution of a partial differential equation. This error estimator has many advantages that other existing error estimators do not have or lack of. For instance, it is, by construction, independant of the differential operator used to model a certain physical phenomena. It is also naturally generalisable to the case of approximations of arbitrary order, and this, without any specific treatment to the underlying theory. Finally, it is efficient, optimal in a sense that will be defined and permits the elements to stretch in a priviledged direction (anisotropy) in order to obtain high accuracy against regularly refined meshes. Many examples are given in the one, two and three dimensional cases. Analytical examples (the solution is known) is given to measure the effiency and precision of the new error estimator. Other examples of mesh adaptation for equations modeling different physical phenomena like the flow of a fluid around a cylinder, unsteady diffusion and contact between deformable elastic bodies are presented. These examples show that the new error estimator can be used for a wide variety of problems

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

An optimization framework for adaptive higher-order discretizations of partial differential equations on anisotropic simplex meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 271-281).Improving the autonomy, efficiency, and reliability of partial differential equation (PDE) solvers has become increasingly important as powerful computers enable engineers to address modern computational challenges that require rapid characterization of the input-output relationship of complex PDE governed processes. This thesis presents work toward development of a versatile PDE solver that accurately predicts engineering quantities of interest to user-prescribed accuracy in a fully automated manner. We develop an anisotropic adaptation framework that works with any localizable error estimate, handles any discretization order, permits arbitrarily oriented anisotropic elements, robustly treats irregular features, and inherits the versatility of the underlying discretization and error estimate. Given a discretization and any localizable error estimate, the framework iterates toward a mesh that minimizes the error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using element-wise local solves; synthesis of the local errors to construct a surrogate error model based on an affine-invariant metric interpolation framework; and optimization of the surrogate model to drive the mesh toward optimality. The combination of the framework with a discontinuous Galerkin discretization and an a posteriori output error estimate results in a versatile PDE solver for reliable output prediction. The versatility and effectiveness of the adaptive framework are demonstrated in a number of applications. First, the optimality of the method is verified against anisotropic polynomial approximation theory in the context of L2 projection. Second, the behavior of the method is studied in the context of output-based adaptation using advection-diffusion problems with manufactured primal and dual solutions. Third, the framework is applied to the steady-state Euler and Reynolds-averaged Navier-Stokes equations. The results highlight the importance of adaptation for high-order discretizations and demonstrate the robustness and effectiveness of the proposed method in solving complex aerodynamic flows exhibiting a wide range of scales. Fourth, fully-unstructured space-time adaptivity is realized, and its competitiveness is assessed for wave propagation problems. Finally, the framework is applied to enable spatial error control of parametrized PDEs, producing universal optimal meshes applicable for a wide range of parameters.by Masayuki Yano.Ph.D