A conservative slide line method for cell-centered semi-Lagrangian and ALE schemes in 2D
Bertoluzza, Silvia1 ; Del Pino, Stéphane2 ; Labourasse, Emmanuel3
1 Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Pavia, Italy.
2 CEA, DAM, DIF 91297, Arpajon France.
3 CEA, DAM, DIF 91297, Arpajon France.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 187-214.

In this paper, we propose a new cell-center method to treat sliding of compressible fluid domains. We describe at first the theoretical framework based on [S. Del Pino, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1027–1032]. We introduce the notion of slide lines thanks to a mortar-like approach. We propose and analyze a P 1 -P 0 discretization of the theoritical method. We also describe a simple ALE procedure that preserves the slide line Lagrangian so that no mixed-cells model is necessary. Finally we present a set of representative numerical tests.

DOI : 10.1051/m2an/2015037
Classification : 65Z05, 35L65, 65N08, 65N30, 76N15
Mots-clés : Compressible gas, Lagrange, ALE, slide lines, finite-volumes, finite-elements, mortar
Bertoluzza, Silvia 1 ; Del Pino, Stéphane 2 ; Labourasse, Emmanuel 3

1 Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Pavia, Italy.
2 CEA, DAM, DIF 91297, Arpajon France.
3 CEA, DAM, DIF 91297, Arpajon France. 
@article{M2AN_2016__50_1_187_0,
     author = {Bertoluzza, Silvia and Del Pino, St\'ephane and Labourasse, Emmanuel},
     title = {A conservative slide line method for cell-centered {semi-Lagrangian} and {ALE} schemes in {2D}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {187--214},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {1},
     year = {2016},
     doi = {10.1051/m2an/2015037},
     zbl = {1382.76181},
     mrnumber = {3460106},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015037/}
}
TY  - JOUR
AU  - Bertoluzza, Silvia
AU  - Del Pino, Stéphane
AU  - Labourasse, Emmanuel
TI  - A conservative slide line method for cell-centered semi-Lagrangian and ALE schemes in 2D
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 187
EP  - 214
VL  - 50
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015037/
DO  - 10.1051/m2an/2015037
LA  - en
ID  - M2AN_2016__50_1_187_0
ER  - 
%0 Journal Article
%A Bertoluzza, Silvia
%A Del Pino, Stéphane
%A Labourasse, Emmanuel
%T A conservative slide line method for cell-centered semi-Lagrangian and ALE schemes in 2D
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 187-214
%V 50
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015037/
%R 10.1051/m2an/2015037
%G en
%F M2AN_2016__50_1_187_0
Bertoluzza, Silvia; Del Pino, Stéphane; Labourasse, Emmanuel. A conservative slide line method for cell-centered semi-Lagrangian and ALE schemes in 2D. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 187-214. doi : 10.1051/m2an/2015037. http://www.numdam.org/articles/10.1051/m2an/2015037/

I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179–192. | DOI | MR | Zbl

A.L. Bauer, D.E. Burton, E.J. Caramana, R. Loubère, M.J. Shashkov and P.P. Whalen, The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics. J. Comput. Phys. 218 (2006) 572–593. | DOI | MR | Zbl

F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173–199. | DOI | MR | Zbl

D.J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Meth. Appl. Mech. Engrg. 99 (1992) 235–394. | DOI | MR | Zbl

C. Bernardi, Y. Maday and A.T. Patera, A New Nonconforming Approach to Domain Decomposition: The Mortar Element Method. Nonlin. Partial Differ. Equ. Appl. Edited by H. Brezis and J. L. Lions. Pitman, New York (1994) 13–51. | MR | Zbl

N.G. Bourago and V.N. Kukudzhanov, A Review of Contact Algorithms. The Institute for Problems in Mechanics of RAS. Izv. RAN, MTT Translation into english (2005) 45–87.

J.P. Braeunig, B. Desjardin and J.M. Ghidaglia, A totally Eulerian finite volume solver for multi-material fluid flows. Eur. J. Mech. B/Fluids 28 (2009) 475–485. | DOI | MR | Zbl

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer−Verlag, New York (1991). | MR | Zbl

F. Brezzi and L.D. Marini, Macro Hybrid Elements and Domain Decomposition Methods. In Vol. 89 of Optimisation et Contrôle, Meeting in honour of J. Céa, edited by J.D. et al. CÉPADUÈS-Edition, Toulouse (1993) (1992). | MR | Zbl

E.J. Caramana, The implementation of slide lines as a combined force and velocity boundary condition. J. Comput. Phys. 228 (2009) 3911–3916. | DOI | Zbl

E.J. Caramana, D.E. Burton, M.J. Shashkov and P.P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. J. Comput. Phys. 146 (1998) 227–262. | DOI | MR | Zbl

G. Carré, S. Del Pino, B. Després and E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension. J. Comput. Phys. 228 (2009) 5160–5183. | DOI | MR | Zbl

G. Clair, B. Després and E. Labourasse, A one-mesh method for the cell-centered discretization of sliding. Comput. Meth. Appl. Mech. Engrg. 269 (2014) 315–333. | DOI | MR | Zbl

A. Claisse, P. Rouzier and J.M. Ghidaglia, A 2D Sliding Algorithm for Eulerian Multimaterial Simulations. In ECCOMAS 6th European Congress on Computational Methods in Applied Sciences and Engineering (2012).

S. Del Pino, A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1027–1032. | DOI | MR | Zbl

B. Després and E. Labourasse, Stabilization of cell-centered compressible Lagrangian methods using subzonal entropy. J. Comput. Phys. 231 (2012) 6559–6595. | DOI | MR | Zbl

B. Després and C. Mazeran, Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Rational Mech. Anal. 178 (2005) 327–372. | DOI | MR | Zbl

V. Dyadeshko and M. Shashkov, Reconstruction of multi-material interfaces from moment data. J. Comput. Phys. 227 5361–5384 (2008) | DOI | MR | Zbl

J.M. Escobar, E. Rodríguez, R. Motenegro and G.M. Montero, G.Y.J., Simultaneous untangling and smoothing of tetrahedral meshes. Comput. Meth. Appl. Mech. Engrg. 192 (2003) 2775–2787. | DOI | Zbl

G. Folzan, Modélisation multi-matériaux multi-vitesse en dynamique rapide. Under the direction of P. Le Tallec and J.-P. Perlat (in french). Ph.D. thesis, École Poytechnique (2013).

G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd edition. John Hopkins University Press (1996). | MR | Zbl

C.W. Hirt and B.D. Nichols, Volume Of Fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1981) 201–225. | DOI | Zbl

B.I. Jun, A modified equipotential method for grid relaxation. Tech. Rep. UCRL-JC-138277. Lawrence Livermore National Laboratory (2000)

M. Kucharik, R. Loubère, L. Bednárik and R. Liska, Enhancement of Lagrangian Slide Lines as a Combined for and Velocity Boundary Condition. Comput. Fluids (2012). | MR | Zbl

X.S. Li, An Overview of SuperLU: Algorithms, Implementation and User Interface. In Vol. 31 (2005) 302–325. | MR | Zbl

P.H. Maire, R. Abgrall, J. Breil and J. Ovadia, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems. SIAM J. Sci. Comput. 29 (2007) 1781–1824. | DOI | MR | Zbl

C. Mazeran, Sur la structure mathématique et l’approximation numérique de l’hydrodynamique Lagrangienne bidimensionelle. Under the direction of B. Després (in french). Ph.D. thesis, Université Bordeaux I (2007).

N.R. Morgan, M.A. Kenamond, D.E. Burton, T.C. Carney and D.J. Ingraham, An approach for treating contact surfaces in Lagrangian cell-centered hydrodynamics. J. Comput. Phys. 250 (2013) 527–554. http://www.sciencedirect.com/science/article/pii/S002199911300346X | DOI | MR

J. Von Neumann and R.D. Richtmyer, A method for the calculation of hydrodynamics shocks. J. Appl. Phys. 21 (1950) 232–237. | DOI | MR | Zbl

O. Steinbach, On a generalized L 2 projection and some related stability estimates in Sobolev spaces. Numer. Math. 90 (2002) 775–786. | DOI | MR | Zbl

R. Tipton, Grid optimization by equipotential relaxation. Unpublished manuscript (1990).

M.L. Wilkins, Calculation of Elastic-Plastic Flow. In Vol. 3 of Meth. Comput. Phys. Academic Press (1964) 211–263.

D.L. Youngs, Time dependent Multi-Material Flow with Large Fluid Distortion. In Numer. Methods Fluid Dyn. Edited by K.W. Morton, M.J. Baines (1982) 273–285 | Zbl

Y.B. Zel’dovich and Y.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Vol. 1. Academic Press, New York and London (1966).

Cité par Sources :