- La métrique d'un élément est une métrique dans laquelle l'élément est régulier, avec des arêtes de longueur unité (on trouve une formulation similaire dans ). Cette notion est liée en particulier aux problèmes de génération de maillages unité ([3,6,11], etc.) et d'erreur d'interpolation d'un champ donné ([1,2,10] , etc.). Dans cette note, on va développer le formalisme sous-jacent dans le cadre d'un élément géométrique simplicial (triangle, tétraèdre, simplexe de dimension quelconque) et décrire quelques propriétés liées.
Métrique et qualité d'un simplexe
[Show abstract] [Hide abstract] ABSTRACT: Ce papier introduit la notion de métrique liée à un élément géométrique simplicial (un simplexe). Cette métrique est utilisée dans la résolution par la méthode des éléments finis d'équations aux dérivées partielles, à la fois pour la génération de maillages et les estimateurs d'erreur basés sur l'erreur d'interpolation. En outre, cette métrique renseigne sur la géométrie de l'élément et, en particulier, sur sa qualité en forme dans un espace euclidien quelconque.
- After solving analytically this optimization problem (see e.g. [34, 22, 14, 23, 5], we get the unique optimal (M L p (x)) x∈Ω as:
Anisotropic norm-oriented mesh adaptation for a Poisson problem
[Show abstract] [Hide abstract] ABSTRACT: We present a novel formulation for the mesh adaptation of the approximation of a Partial Differential Equation (PDE). The discussion is restricted to a Poisson problem. The proposed norm-oriented formulation extends the goal-oriented formulation since it is equation-based and uses an adjoint. At the same time, the norm-oriented formulation somewhat supersedes the goal-oriented one since it is basically a solution-convergent method. Indeed, goal-oriented methods rely on the reduction of the error in evaluating a chosen scalar output with the consequence that, as mesh size is increased (more degrees of freedom), only this output is proven to tend to its continuous analog while the solution field itself may not converge. A remarkable quality of goal-oriented metric-based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization, in the well-identified set of metrics, of a well-defined functional. In the new proposed formulation, we amplify this advantage. We search, in the same well-identified set of metrics, the minimum of a norm of the approximation error. The norm is prescribed by the user and the method allows addressing the case of multi-objective adaptation like, for example in aerodynamics, adaptating the mesh for drag, lift and moment in one shot. In this work, we consider the basic linear finite-element approximation and restrict our study to L 2 norm in order to enjoy second-order convergence. Numerical examples for the Poisson problem are computed.
- First, not only many computations are performed in much better conditions than with traditional methods but, also, they allow computations which were simply not feasible without anisotropic adaptation, like the propagation of a sonic boom from aircraft to ground . Second, anisotropic mesh adaptors provide mesh convergence at high-order for singular problems . For non-singular problems but rather heterogeneous problems, non-adaptative methods will produce higher order convergence only with very fine meshes.
A mesh-adaptative metric-based full multigrid for the Poisson problem: A mesh-adaptative metric-based full multigrid for the Poisson problem
[Show abstract] [Hide abstract] ABSTRACT: This paper studies the combination of the Full-Multi-Grid (FMG) algorithm with an anisotropic metric-based mesh adaptation algorithm. For the sake of simplicity, the case of an elliptic two-dimentional Partial Differential Equation (PDE) is studied. Meshes are unstructured and non-embedded, defined through the metric-based parametrisation. A rather classical MG preconditionner is applied, in combination with a quasi-Newton fixed point. An anisotropic metric-based mesh adaptation loop is introduced inside the FMG algorithm. FMG convergence stopping test is re-visited. Applications to a few 2D continuous and discontinuous-coefficient elliptic model problems show the efficiency of this combination. This article is protected by copyright. All rights reserved.
- A general analysis of this favorable behavior is not yet available for the multi-dimensional case. Although not mathematically demonstrated, the property of early capturing and early asymptotic convergence has been observed in all numerical experiments, see .
Adaptive metric-based multigrid for a Poisson problem with discontinuous coefficients
[Show abstract] [Hide abstract] ABSTRACT: In order to solve the linear partial differential equation Au = f, we
combine two methods: Full-Multigrid method and Hessian-based mesh adaptation. First, we
define independently an Hessian-based mesh adaptation loop and a FMG algorithm where, at
each phase, the equation is solved by a preconditioned GMRES with multigrid as
preconditioner. Then we insert the adaptive loop between the FMG phases. We use this new
algorithm and we compare its results with those obtained with non-adaptive FMG.
- The parameter of choice is then the mesh complexity N, and the metric (Eq. (11)) is used to build a sequence of meshes with the same orientation and anisotropic coefficients that satisfy the following error estimate :
Anisotropic mesh adaptation on Lagrangian Coherent Structures
[Show abstract] [Hide abstract] ABSTRACT: The finite-time Lyapunov exponent (FTLE) is extensively used as a criterion to reveal fluid flow structures, including unsteady separation/attachment surfaces and vortices, in laminar and turbulent flows. However, for large and complex problems, flow structure identification demands computational methodologies that are more accurate and effective. With this objective in mind, we propose a new set of ordinary differential equations to compute the flow map, along with its first (gradient) and second order (Hessian) spatial derivatives. We show empirically that the gradient of the flow map computed in this way improves the pointwise accuracy of the FTLE field. Furthermore, the Hessian allows for simple interpolation error estimation of the flow map, and the construction of a continuous optimal and multiscale LpLp metric. The Lagrangian particles, or nodes, are then iteratively adapted on the flow structures revealed by this metric. Typically, the L1L1 norm provides meshes best suited to capturing small scale structures, while the L∞L∞ norm provides meshes optimized to capture large scale structures. This means that the mesh density near large scale structures will be greater with the L∞L∞ norm than with the L1L1 norm for the same mesh complexity, which is why we chose this technique for this paper. We use it to optimize the mesh in the vicinity of LCS. It is found that Lagrangian Coherent Structures are best revealed with the minimum number of vertices with the L∞L∞ metric.
- In fact, during the remeshing operations, the elements are much more volatile than the mesh nodes and therefore defining fields on a continuous basis ease their reconstruction, interpolation or extrapolation . From a theoretical point of view, a continuous metric field could be a direct way to represent the underlying Riemannian space, in which the measurement of length varies at each point and direction [15,1,5]. However, a flat element P 1 can be regarded as a representation of the metric space of the underlying tangent space.
Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing
[Show abstract] [Hide abstract] ABSTRACT: Metric tensors play a key role to control the generation of unstructured anisotropic meshes. In practice, the most well established error analysis enables to calculate a metric tensor on an element basis. In this paper, we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools. First, the unit mesh metric is defined and well justified on a node basis, by using the statistical concept of length distribution tensors. Then, the interpolation error analysis is performed on the projected approximate scalar field along the edges. The error estimate is established on each edge whatever the dimension is. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. Several examples of interpolation error are proposed as well as preliminary results of anisotropic adaptation for interface and free surface problem using a level set method.