MAMERN'09: 3rd International Conference on Approximation Methods and numerical Modeling in Environment and Natural Resources June 8-11, 2009 Pau - France MAMERN'09 http://lma.univ-pau.fr/meet/mamern09/ contact:
[email protected] Minisymposium Title: Adaptive Anisotropic Mesh Generation: Theory and Practical Aspects Organizers : A. Agouzal & N. Debit, University of Lyon 1, France The adaptive mesh methods significantly improve accuracy of simulations and allow to solve large problems appearing in engineering applications. The majority of these methods use meshes with regular shaped elements. However, it was shown that anisotropic simplexes may significantly improve accuracy of simulations, once you get a robust method for adaptive generation of unstructured anisotropic meshes. The goal of this minisymposium is to discuss modern trends in adaptive anisotropic mesh generation. The contributed talks will cover both the implementation issues and theoretical aspects of adaptivity. Edge-based a Posteriori Error Estimators for Generation of d-Dimensional Quasi-Optimal Meshes Yuri Vassilevski, Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow Abdellatif Agouzal, University of Lyon 1, France Konstantin Lipnikov, Los Alamos National Laboratory, USA We consider optimal meshes that minimize the error of the piecewise linear interpolation over all simplicial meshes with a fixed number of cells. Theoretical results on asymptotic dependences of L_p-norms of the errors on the number of mesh cells are presented. Both the interpolation error and the gradient of interpolation error are analysed. In practice, the conventional adaptive procedures produce meshes close to optimal. Such meshes are called quasi-optimal. They give a slightly higher errors but the same asymptotic rate of error reduction. Quasi-optimal meshes are uniform or quasiuniform in an appropriate continuous tensor metric [1,2]. Metric recovery is the cornerstone of the mesh adaptation. It is usually based on either a discrete Hessian recovery or a posteriori error estimators. It is known that the accuracy of the Hessian recovery is very low although the method exhibits surprisingly good behavior in practice. A posteriori error estimators may provide a reliable alternative for metric recovery [3]. We present a new method of metric recovery targeted to the minimization of L_p-norms of the interpolation error or the gradient error of interpolation. The core of the method is robust and reliable edge-based a posteriori error estimator. The method is analysed for conformal simplicial meshes in spaces of arbitrary dimension d. Numerical results on the adapted meshes confirm the predicted asymptotic rates of both the interpolation error and the gradient error of interpolation. References 1. Y.Vassilevski, A.Agouzal, An unified asymptotical analysis of interpolation errors for optimal meshes. Doklady Mathematics, 72, 879-882, 2005. 2. W.Huang, Metric tensors for anisotropic mesh generation. J. Comp.Phys., 204, 633-665, 2005. 3. A.Agouzal, K.Lipnikov, Y.Vassilevski, Generation of quasi-optimal meshes based on a posteriori error estimates. Proceedings of 16th International Meshing Roundtable, 2007.
MAMERN'09: 3rd International Conference on Approximation Methods and numerical Modeling in Environment and Natural Resources June 8-11, 2009 Pau - France MAMERN'09 http://lma.univ-pau.fr/meet/mamern09/ contact:
[email protected]
Anisotropic Goal-Oriented Mesh Optimisation Adrien Loseille(*), Frédéric Alauzet(*), Alain Dervieux(**) (*) INRIA, Domaine de Voluceau, Le Chesnay, 78153, France (**) INRIA, 2004 Route des Lucioles, 06902 Sophia-Antipolis, France Supersonic transportation is among the ones who has provoked the earliest and strongest reactions against environmental impact. Mastering the sonic boom impact remains a challenge, and this needs studies and accurate simulations. The complex wave which produces the ground sonic boom is made of strong discontinuities, (shocks and contact discontinuities) combined with smoother transitions. Paradoxally, this discontinuous example was a good experimental test for continuous Pl finite-element-type approximations, as far as they are combined with mesh adaptation. ln our mesh adaptation study, we look for continuous models of the mesh under the form of a metric field described by a symmetric nxn positive matrix defined in the whole computational domain. With this model, we can analyse the convergence order of the mesh adaptive method for singular solutions. Metrics can be obtained as well from Hessian of a field as well as from a goal oriented analysis using an adjoint system. Application to Sonic boom prediction computations will illustrate the accuracy of these methods. References A. Loseille, A. Dervieux, P.J. Frey and F. Alauzet, "Achievement of global second-order mesh convergence for discontinuous flows with adapted unstructured meshes", AIAA paper 2007-4186, Miami, FL, USA, June 2007 F. Alauzet, "High-Order method and mesh adaptation for Euler equations", Int. J. Numer. Meth. Fluids, Vol. 56, pp. 1069-1076, 2008. Error Indicator and Mesh Adaption in FreeFem ++ Frédéric Hecht and R. Kuate Laboratoire J.L. Lions - UPMC/Paris 6, INRIA Projet Gamma, France The main goal of this lecture is to present new and old techniques of mesh adaption under classical assumption : The mesh generation needs a metric to define the "mesh size" constraint [3]. The Hessian error indicator gives naturally the metric, but the construction is not obvious for P1 finite element and Lp norm. We will show how to build anisotropic error indicator for high order finite elements [6]. The other error indicators (residual or hierarchical [5]) just give a level of error. So how to build a metric from this local error indicator ? by using a priori error estimates, the new mesh size is almost given by the following formula: h(n+1)(x)= hn(x)/fn(En(x)) where En(x) is the level of error at point x given by the local error indicator, hn is the previous “mesh size” field, and fn a user-defined function, which is just linear in general. We will give numerical results to illustrate the implementation of these techniques using FreeFem ++ software, see [1, 2].
MAMERN'09: 3rd International Conference on Approximation Methods and numerical Modeling in Environment and Natural Resources June 8-11, 2009 Pau - France MAMERN'09 http://lma.univ-pau.fr/meet/mamern09/ contact:
[email protected]
References [1] F.Hecht : freefem++ documentation, on the web at http://www . freefem. org/ff++. [2] I. Danaila, F. Hecht, O. Pironneau: Simulation numérique en C++, Dunod, 2003. [3] P.L. George: Automatic triangulation, Wiley 1996. [4] F. Hecht: The mesh adapting software: bamg. INRIA report 1998. [5] R. Verfürth - A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley & Teubner (1996). [6] F. Hecht, R. Kuate - Error indicator and mesh adaption in FreeFem++ , 17 th Mesh Round Table. Pittsburgh, Pennsylvania, http://www . imr. sandia. Gov , 2008. Adaptive Anisotropic Parallel Mesh Adaptation with Applications to Interface Capturing Problem T. Coupez, H. Digonnet, Y. Mesri Cemef, Centre de Mise en Forme des Matériaux, Sophia-Antipolis, France We present a dynamic parallel mesh adaptation procedure. It is based on the definition of an anisotropic a posteriori error estimator, the search of the optimal mesh (metric) that minimizes the error estimator under the constraint of a given number of nodes and the use of the serial mesh generator (MTC) in a parallel context. The parallelization strategy consists in balancing dynamically the work flow by repartitioning the mesh after each re-meshing stage. Numerical 2D and 3D applications are presented here to show that the proposed anisotropic error estimator gives an accurate representation of the exact error. We will show also, that the optimal adaptive mesh procedure provides a mesh refinement and element stretching which appropriately captures interfaces for practical application problems. Finally, the anisotropic adapted meshes provide highly accurate solutions that are shown to be unreachable with a globally-refined meshes. A Posteriori Error Analysis for an Anisotropic Elliptic Problem using the Crouzeix-Raviart Element Boujemâa Achchab, LM2CE, FSJES, Université Hassan1, Settat, Maroc In this work, we derive a posteriori error estimates for the nonconforming finite element approximations of an elliptic problem, with anisotropic diffusion. The estimator yields an global upper and lower local bounds on the error. The constants of equivalence between the error and the estimator are independents of the anisotropy of the problem. We adopt the error in constitutive law approach.
