Adaptation de maillages non structurés pour des problèmes instationnaires

This Note deals with the adaptation of unstructured meshes for transient CFD problems. The proposed approach is based on a new mesh adaptation algorithm and a metric intersection in time procedure suitable to capture such phenomena. More precisely, a new specific loop is inserted in the main adaptation loop to solve a transient fixed point problem. The mesh adaptation stage consists in optimizing the current mesh so as to obtain a unit mesh with respect to this metric. A 2D example is provided to emphasize the efficiency of the proposed method. To cite this article: F. Alauzet et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 773–778.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Abridged English version Unstructured mesh adaptation has proved very efficient for steady computations in CFD, as it allows one to capture the behavior of the physical phenomena while getting the desired accuracy for the numerical solution. Moreover, this method allows one to substantially reduce the computational cost of the numerical simulation by reducing the mesh size [3]. However, when dealing with unsteady configurations, it is of the utmost importance to follow the evolution of the physical phenomena (for instance a moving shock Adresses e-mail : [email protected] (F. Alauzet); [email protected] (B. Mohammadi).  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés S1631-073X(02)02553-0/FLA 773 F. Alauzet et al. / C. R. Acad. Sci. Paris, Ser. I 335 (2002) 773–778 in the computational domain). It is thus necessary to mesh, in an adequate manner, all the regions where phenomena evolve. However, attention must be paid so as to avoid remeshing a part too large of the domain with a dense mesh or remeshing too often thus impacting the overall computational cost. Currently, only a few papers have specifically addressed this type of problems [4,6,7]. In this context, we propose a new mesh adaptation algorithm, based on the resolution of a transient fixed point problem for the couple mesh-solution and likely suitable to predict the evolution of the physical phenomena. This algorithm is composed of two embedded loops. At each iteration of the main adaptation loop, a time period [t, t + t] is considered where the solution evolves. In the inner loop, the transient fixed point problem is solved. At each internal iteration step (from t to t + t), a new adapted mesh is created based on the metric associated with the solution at t + t and the computation is re-started with the same initial solution at t . The inner process is repeated until the convergence (i.e., the desired accuracy of the solution) is achieved at t + t . Then, we resume the outer adaptation loop at t + t and the whole process is iterated [1]. Unstructured mesh adaptation is based on the computation of the edge lengths with respect to a discrete metric [2]. Specific algorithms have been designed so as to reduce the impact of mesh modifications at each mesh adaptation step (most of the current mesh entities are preserved in the areas where the solution is not changing). The anisotropic metric is defined using an a posteriori error estimate based on a discrete approximation of the Hessian of the solution. The aim is to equi-distribute the interpolation error along the mesh edges. In order to properly mesh all regions where the solution evolves, a metric intersection in time is introduced in the metric definition. In three dimensions specifically, this such-defined computational metric is intersected with the discrete geometric metric (based on the intrinsic properties of the surface) so as to preserve the domain geometry. In this Note, we will mainly focuss on the metric construction and the adaptation scheme for unsteady problems. A 2D numerical example is provided so as to emphasize the efficiency of the proposed approach.