Extension of ALE methodology to unstructured conical meshes

Abstract

Abstract. We propose a bi-dimensional nite volume extension of a continuous ALE method on unstructured cells whose edges are parameterized by rational quadratic Bezier curves. For each edge, the control point possess a weight that permits to represent any conic (see for example [LIGACH]) and thanks to [WAGUSEDE, WAGU], we are able to compute the exact area of our cells. We then give an extension of scheme for remapping step based on volume uxing [MARSHA] and selfintersection ux [ALE2DHAL]. For the rezoning phase, we propose a three step process based on moving nodes, followed by control point and weight re-adjustment. Finally, for the hydrodynamic step, we present the GLACE scheme [GLACE] extension (at rst-order) on conic cell using the same formalism. We only propose some preliminary rst-order simulations for each steps: Remap, Pure Lagrangian and nally ALE (rezoning and remapping).Résumé. Nous proposons une extension volumes nis bi-dimensionnelle d'une méthode ALE continue sur des cellules non structurées dont les bords sont paramétrés par des courbes de Bézier quadratiques rationnelles. Pour chaque arête, le point de contrôle possède un poids qui permet de représenter n'importe quelle conique [LIGACH] et grâce à [WAGUSEDE, WAGU], nous pouvons calculer l'aire exacte de nos cellules. Pour la phase de remapping, on donne l'extension de deux schéma, l'un basé sur le calcul de ux de volumes [MARSHA] et l'autre par ux avec autointersection [ALE2DHAL]. Pour la phase de lissage de maillage, nous proposons un processus en trois étapes basées sur le déplacement des noeuds, suivi de celui des points de contrôle puis nalement du réajustement du poids. Enn, pour la phase hydrodynamique, on présente l'extension du schéma GLACE [GLACE] (à l'ordre un) sur les cellules coniques en utilisant le même formalisme. Nous montrons seulement des simulations préliminaires à l'ordre 1 sur chaque étape : Remap, Lagrange pur et ALE (rezoning et remapping).

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part II: The two-dimensional case

“…where J reads as the cofactor matrix of tensor J. By the use of Nanson formula (6) along with the mass conservation ρ |J| = ρ 0 , it is then obvious to see the perfect equivalence between the updated and total Lagrangian formulations, systems (3) and (5).…”

“…The differences will then arise from the definition of the numerical flux F and the treatment of the boundary integral ∂ωc . Consequently, to avoid the positivity-preserving study to be too scheme specific, we present and make use here of a general first-order finite volume formulation that will prove to fit different existing cell-centered Lagrangian schemes, such as those presented in [6,32,5,41,43]. The general scheme relies on general polygonal cells defined either by straight line edges, of quadrature rules or some approximations, the cell boundary integral present in equation (12) can be expressed as a combination of control point contributions.…”

“…In the end, satisfying such relation will allow us to define explicitly the velocity for grid control points. Finally, the differences in the first-order schemes presented for instance in [6,32,5,43] will arise from the type of cells considered, the definition of the control point set Q c and the way condition (18) is ensured.…”

“…In this subsection, a general first-order scheme has been introduced. Particular choices in the type of cells considered, Figure 1, in the definition of the control point set Q c , or in the way the conservation condition (18) is ensured will enable us to recover the schemes presented in [6,32,5,43]. To emphasize that, brief specifications of the schemes will now be given.…”

“…GLACE extension on conic cells. In [5], Boutin et al introduced an extension of the GLACE scheme on conical mesh, see Figure 1(b). In this configuration, a point x located on the conical face f pp + is defined through…”