Non conformal adaptation and mesh smoothing for compressible Lagrangian fluid dynamics

ESAIM: Proc., 2008, Vol. 24, pp. 111-129
DOI: 10.1051/proc:2008033

Non conformal adaptation and mesh smoothing for compressible Lagrangian fluid dynamics

Hoch P.¹, Marchal S.², Vasilenko Y.³ and Feiz A. A.⁴

¹ CEA-DIF, Bruyères-Le-Châtel 91297, Arpajon Cedex, France,
² Université Henri Poincaré Nancy 1., Institut Elie Cartan Nancy (Mathématiques), B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France,
³ University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics, Univerzitni 22, Plzen 306 14, Czech Republic,
⁴ Université Paris-Est Marne-la-Vallée, LETEM, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France,

philippe.hoch@cea.fr
Sandrine.Marchal@iecn.u-nancy.fr
yana@kma.zcu.cz
feiz@univ-mlv.fr

Published online: 10 October 2008

Abstract
The context of the present work is the numerical approximation of two-dimensional compressible Lagrangian fluid dynamics. The equation on $\rho$ , U and E respectively the density, the velocity and the total energy describes the conservation of mass, momentum, and total energy of the fluid. For a numerical approximation, the starting point is a first order non direct ALE (Arbitrary Lagrangian Eulerian method) that consists of two stages. The first step is a pure Lagrangian step with a centered scheme [1] or [19] (for staggered schemes see [15], [7], [4], but we did not use them in this work). The second step is the union of a rezoning step (mesh is smoothed to cure pathological cells and/or adapted to concentrate nodes near singularities, but the connectivity is fixed here) [18], [2], [10], [16], and a remapping step, defining the old values on this new mesh in a conservative way (see [12], [21], [16]). In some cases, we need to refine/derefine locally the moving mesh, so that this adaptation step takes place between existing Lagrangian/ALE formulations, making the entire process a three steps scheme. We propose an adaptation strategy based on a variant of local non conformal cells. More precisely, as we will see the notion of “non conformal” is only partial, meaning that the new nodes created by adaptation can be a degree of freedom exactly as the master nodes, we call them semi-slaves. With a different approach for local non conformal quadrangular based refinement (see [8]), in our case unlike their Lagrangian step, we consider all nodes as degree of freedom meaning that we do not consider hanging nodes. The refinement step is generic with respect to the polygonal mesh, the number of nodes in a cell is arbitrary and these are not restricted to be a constant (only triangles and/or quadrilaterals), the conservation is straightforward. The derefinement step is done by local non conformity and the conservation can be obtain either by self-intersection volumes fluxing (line clipping) or by polygonal clipping (more precise than coarse volume averaging). Our approach naturally leads to define this notion of non conformity as a special object inside the description of a generic unstructured polygonal mesh. The development and the simulations have been done with the GO++ package. keywords: Polygonal Unstructured ALE, Hybrid Mesh Representation, Local non Conforming Cell, Dynamic Adaptation, Centered Lagrangian Hydrodynamic, C++.