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A Second-Order Cell-Centered Lagrangian Method for Two-Dimensional Elastic-Plastic Flows

Part of: Hyperbolic equations and systems Coupling of solid mechanics with other effects Special kinds of problems Equations of mathematical physics and other areas of application Plastic materials, materials of stress-rate and internal-variable type

Published online by Cambridge University Press:  31 October 2017

Jun-Bo Cheng ,
Yueling Jia ,
Song Jiang ,
Eleuterio F. Toro  and
Ming Yu
Jun-Bo Cheng*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China
Yueling Jia*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China
Song Jiang*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China
Eleuterio F. Toro*
Affiliation:
Laboratory of Applied Mathematics, University of Trento, Trento, Italy
Ming Yu*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China
*
*Corresponding author. Email addresses:jia_yueling@iapcm.ac.cn(Y. L. Jia), cheng_junbo@iapcm.ac.cn(J.-B. Cheng), jiang@iapcm.ac.cn(S. Jiang), eleuterio.toro@unitn.it(E. F. Toro), Yu_ming@iapcm.ac.cn(M. Yu)
*Corresponding author. Email addresses:jia_yueling@iapcm.ac.cn(Y. L. Jia), cheng_junbo@iapcm.ac.cn(J.-B. Cheng), jiang@iapcm.ac.cn(S. Jiang), eleuterio.toro@unitn.it(E. F. Toro), Yu_ming@iapcm.ac.cn(M. Yu)
*Corresponding author. Email addresses:jia_yueling@iapcm.ac.cn(Y. L. Jia), cheng_junbo@iapcm.ac.cn(J.-B. Cheng), jiang@iapcm.ac.cn(S. Jiang), eleuterio.toro@unitn.it(E. F. Toro), Yu_ming@iapcm.ac.cn(M. Yu)
*Corresponding author. Email addresses:jia_yueling@iapcm.ac.cn(Y. L. Jia), cheng_junbo@iapcm.ac.cn(J.-B. Cheng), jiang@iapcm.ac.cn(S. Jiang), eleuterio.toro@unitn.it(E. F. Toro), Yu_ming@iapcm.ac.cn(M. Yu)
*Corresponding author. Email addresses:jia_yueling@iapcm.ac.cn(Y. L. Jia), cheng_junbo@iapcm.ac.cn(J.-B. Cheng), jiang@iapcm.ac.cn(S. Jiang), eleuterio.toro@unitn.it(E. F. Toro), Yu_ming@iapcm.ac.cn(M. Yu)

Abstract

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For 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises’ yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von Mises’ yielding condition make the construction of the solution to the Riemann problem a challenging task. In this paper, we first analyze the wave structure of the Riemann problem and develop accordingly a Four-Rarefaction wave approximate Riemann Solver with Elastic waves (FRRSE). In the construction of FRRSE one needs to use an iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implementation and efficient. Based on FRRSE as a building block, we propose a 2nd-order cell-centered Lagrangian numerical scheme. Numerical results with smooth solutions show that the scheme is of second-order accuracy. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. For shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al., while it is more accurate in resolving rarefaction waves.

Keywords

Cell-centered Lagrangian schemehigh-order schemehypo-elastic constitutive modelfour-rarefaction Riemann solver with elastic waves

MSC classification

Secondary: 35L50: Initial-boundary value problems for first-order hyperbolic systems 35Q74: PDEs in connection with mechanics of deformable solids 74C05: Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials) 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74M20: Impact

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1224 - 1257
Copyright
Copyright © Global-Science Press 2017 

References

[1] Wilkins, M.L., Methods in computational physics, Volume 3, Chapter Calculation of elasticplastic flow, Academic press, 1964, pp.211–263.Google Scholar
[2] Burton, D.E., Carney, T.C., Morgan, N.R., Sambasivan, S.K., M.J., and Shashkov, , A cell-centered Lagrangian Godunov-like method for solid dynamics, Comput. Fluids, 83(2013), 33–47; http://dx.doi.org/10.1016/j.compfluid. 2012.09.08.Google Scholar
[3] Kluth, G., Després, B., Discretization of hyperelasticity on unstructured mesh with a cellcentered Lagrangian scheme, J. Comput. Physics., 229(2010), 9092–9118.Google Scholar
[4] Maire, P.-H., Abgrall, R., Breil, J., Loubère, R., Rebourcet, B., A nominally second-order cellcentered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids, J. Comput. Physics., 235(2013), 626–665.CrossRefGoogle Scholar
[5] Sambasivan, S. K., Loubère, R. and Shashkov, M.J., a finite volume lagrangian cell-centered mimetic approach for computing elasto-plastic deformation of solids in general unstructed grids, 6th European Congress on Computational Methods in Applied Sciences and Engineering 2012 (ECCOMAS 2012), Vienna, Austria, 10 September 2012 through 14 September 201.Google Scholar
[6] Després, B.. A geometrical approach to non-conservative shocks and elastoplastic shocks. Arch. Rational Mech. Anal., 186(2)(2007), 175–308.Google Scholar
[7] Gavrilyuk, S.L., Favrie, N. and Saurel, R.. Modeling wave dynamics of compressible elastic materials. J. Comput. Physics., 227(2008), 2941–2969.Google Scholar
[8] Hill, D.J., Pullin, D., Ortiz, M. and Meiron, D.. An Eulerian hybrid WENO centered-difference solver for elastic-plastic solids. J. Comput. Physics., 229(2010), 9503-9072.Google Scholar
[9] Miller, G.H. and Collela, P.. A high-order Eulerian Godunov method for elastic-plastic flow in solids, J. Comput. Physics., 167(2001), 131–176.CrossRefGoogle Scholar
[10] Trangenstein, J.A. and Collela, P.. A high-order Godunov method for modeling finite deformation in elastic-plastic solids, Communications on Pure and Applied mathematics, XLIV, 1991, 41–100.CrossRefGoogle Scholar
[11] Barton, P.T., Drikakis, D., Romenski, E. and Titarev, V.A.. Exact and approximate solutions of Riemann problems in non-linear elasticity, J. Comput. Physics., 228(2009), 7046–7068.CrossRefGoogle Scholar
[12] Titarev, V.A., Romenski, E. and Toro, E. F.. MUSTA-type upwind fluxes for non-linear elasticity, International Journal for Numerical Methods in Engineering., 73(2008), 897–926.CrossRefGoogle Scholar
[13] Kahaner, David, Moler, Cleve, Nash, Stephen. Numerical methods and software. Prentice-Hall, ISBN 978-0-13-627258-8, 1989.Google Scholar
[14] Cheng, Jun-bo, Toro, Eleuterio F., Jiang, Song, Yu, Ming, and Tang, Weijun. A high-order cellcentered Lagrangian scheme for one-dimensional elastic-plastic problems. Computer and Fluids, 122(2015), 136–152, Doi: http://10.1016/j.compfluid.2015.08.029, 2015.Google Scholar
[15] Roache, P. J.. Code verification by the method of manufactured solutions. Trans. ASME, J. Fluids Engineering, 124(2002), 4–10.Google Scholar
[16] Salari, K. and Knupp, P.. Code verification by the method of manufactured solutions. Sandia Report, Sandia National Laboratories, SAND2000-1444, 2000.Google Scholar
[17] Howell, B.P. and Ball, G.J.. A free-Lagrange augmented Godunov method for the simulation of elastic-plastic solids. J. Comput. Phys., 175(2002), 128–167.Google Scholar
[18] Toro, E F. Riemann solvers and numerical methods for fluid dynamics, Springer, 3rd edition, 2009.Google Scholar