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Cell centered direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity. (English) Zbl 1390.76399
Summary: This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by I. Peshkov and E. Romenski [Contin. Mech. Thermodyn. 28, No. 1–2, 85–104 (2016; Zbl 1348.76046)], which is based on the theory of nonlinear hyperelasticity of S. K. Godunov and E. I. Romenskii [“Nonstationary equations of nonlinear elasticity theory in eulerian coordinates”, J. Appl. Mech. Tech. Phys. 13, No. 6, 868–884 (1972; doi:10.1007/bf01200547); Elements of continuum mechanics and conservation laws. Translation from the 1998 Russian original. New York, NY: Kluwer Academic/Plenum Publishers (2003; Zbl 1031.74004)], further denoted by GPR model. Notably, the governing PDE system is symmetric hyperbolic and fully consistent with the first and the second principle of thermodynamics. The nonlinear system of governing equations of the GPR model is overdetermined, large and includes stiff source terms as well as non-conservative products. In this paper, we solve this model for the first time on moving unstructured meshes in multiple space dimensions by employing high order accurate one-step ADER-WENO finite volume schemes in the context of cell-centered direct arbitrary-Lagrangian-Eulerian (ALE) algorithms. The numerical method is based on a WENO polynomial reconstruction operator on moving unstructured meshes, a fully-discrete one-step ADER scheme that is able to deal with stiff sources [the second author et al., J. Comput. Phys. 227, No. 8, 3971–4001 (2008; Zbl 1142.65070)], a nodal solver with relaxation to determine the mesh motion, and a path-conservative technique of Castro & Parés for the treatment of non-conservative products [C. Parés, SIAM J. Numer. Anal. 44, No. 1, 300–321 (2006; Zbl 1130.65089); M. Castro et al., Math. Comput. 75, No. 255, 1103–1134 (2006; Zbl 1096.65082)]. We present numerical results obtained by solving the GPR model with ADER-WENO-ALE schemes in the stiff relaxation limit, showing that fluids (Euler or Navier-Stokes limit), as well as purely elastic or elasto-plastic solids can be simulated in the framework of nonlinear hyperelasticity with the same system of governing PDE. The obtained results are in good agreement when compared to exact or numerical reference solutions available in the literature.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 74B20 Nonlinear elasticity
HE-E1GODF
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##### References:
 [1] Aboiyar, T.; Georgoulis, E.; Iske, A., Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction, SIAM J Sci Comput, 32, 3251-3277, (2010) · Zbl 1221.65236 [2] Balsara, D.; Shu, C., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J Comput Phys, 160, 405-452, (2000) · Zbl 0961.65078 [3] Barth, T.; Frederickson, P., Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction., 28th aerospace sciences meeting, (1990) [4] Barton, P.; Romenski, E., On computational modelling of strain-hardening material dynamics, Commun Comput Phys, 11(5), 1525-1546, (2012) [5] Barton, P.; Deiterding, R.; Meiron, D.; Pullin, D., Eulerian adaptive finite-difference method for high-velocity impact and penetration problems, J Comput Phys, 240, 76-99, (2013) [6] Barton, P.; Drikakis, D.; Romenski, E.; Titarev, V., Exact and approximate solutions of Riemann problems in non-linear elasticity, J Comput Phys, 228, 18, 7046-7068, (2009) · Zbl 1172.74032 [7] Barton, P.; Drikakis, D.; Romenski, E., An Eulerian finite-volume scheme for large elastoplastic deformations in solids, Int J Numer Methods Eng, 81, 453-484, (2010) · Zbl 1183.74331 [8] Barton, P.; Obadia, B.; Drikakis, D., A conservative level-set based method for compressible solid/fluid problems on fixed grids, J Comput Phys, 230, 7867-7890, (2011) · Zbl 1432.74060 [9] Becker, R., Stosswelle und detonation, Physik, 8, 321, (1923) [10] Boscheri, W.; Balsara, D.; Dumbser, M., Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers, J. Comput. Phys., 267, 112-138, (2014) · Zbl 1349.76309 [11] Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun Comput Phys, 14, 1174-1206, (2013) · Zbl 1388.65075 [12] Boscheri, W.; Dumbser, M., A direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and nonconservative hyperbolic systems in 3D, J Comput Phys, 275, 484-523, (2014) · Zbl 1349.76310 [13] Boscheri, W.; Dumbser, M., An efficient quadrature-free formulation for high order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes, J Sci Comput, 66, 240-274, (2016) · Zbl 1338.65219 [14] Boscheri, W.; Dumbser, M.; Balsara, D., High order Lagrangian ADER-WENO schemes on unstructured meshes - application of several node solvers to hydrodynamics and magnetohydrodynamics, Int J Numer Methods Fluids, 76, 737-778, (2014) [15] Boscheri W., Loubère R., M.Dumbser. Multi-dimensional direct arbitrary-lagrangian-eulerian ader-mood high order finite volume schemes for non-conservative hyperbolic systems with stiff source terms. submitted 2016. · Zbl 1349.76310 [16] Burton, D.; Carney, T.; Morgan, N.; Sambasivan, S.; Shashkov, M., A cell-centered Lagrangian Godunov-like method for solid dynamics, Comput Fluids, 83, 33-47, (2013) · Zbl 1290.76095 [17] Burton, D.; Morgan, N.; Carney, T.; Kenamond, M., Reduction of dissipation in Lagrange cell-centered hydrodynamics (cch) through corner gradient reconstruction(cgr), J Comput Phys, 299, 229-280, (2015) · Zbl 1351.76099 [18] Castro, M.; Gallardo, J.; López, J.; Parés, C., Well-balanced high order extensions of godunov’s method for semilinear balance laws, SIAM J Numer Anal, 46, 1012-1039, (2008) · Zbl 1159.74045 [19] Castro, M.; Gallardo, J.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math Comput, 75, 1103-1134, (2006) · Zbl 1096.65082 [20] Cheng, J.; Shu, C., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J Comput Phys, 227, 1567-1596, (2007) · Zbl 1126.76035 [21] Cheng, J.; Toro, E. F.; Jiang, S.; Tang, W., A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme, J Comput Phys, 251, 53-80, (2013) · Zbl 1349.65279 [22] Cockburn, B.; Karniadakis, G. E.; Shu, C., Discontinuous Galerkin methods, Lecture Notes in Computational Science and Engineering;, (2000), Springer [23] Cravero, I.; Semplice, M., On the accuracy of weno and cweno reconstructions of third order on nonuniform meshes, J. Sci. Comput., (2015) · Zbl 1343.65116 [24] Després, B.; Mazeran, C., Lagrangian gas dynamics in two-dimensions and Lagrangian systems, Arch Rational Mech Anal, 178, 327-372, (2005) · Zbl 1096.76046 [25] Diot, S.; Loubère, R.; Clain, S., The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems, Int J Numer Methods Fluids, 73, 362-392, (2013) [26] Dobrev, V. A.; Kolev, T. V.; Rieben, R. N., High order curvilinear finite elements for elastic plastic Lagrangian dynamics, J Comput Phys, 257, Part B, 1062-1080, (2014) · Zbl 1351.76057 [27] Dubiner, M., Spectral methods on triangles and other domains, J Sci Comput, 6, 345-390, (1991) · Zbl 0742.76059 [28] Dumbser, M.; Balsara, D.; Toro, E.; Munz, C., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J Comput Phys, 227, 8209-8253, (2008) · Zbl 1147.65075 [29] Dumbser, M.; Boscheri, W., High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows, Computers and Fluids, 86, 405-432, (2013) · Zbl 1290.76081 [30] Dumbser, M.; Castro, M.; Parés, C.; Toro, E., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Comput Fluids, 38, 1731-1748, (2009) · Zbl 1177.76222 [31] Dumbser, M.; Enaux, C.; Toro, E., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J Comput Phys, 227, 3971-4001, (2008) · Zbl 1142.65070 [32] Dumbser, M.; Hidalgo, A.; Castro, M.; Parés, C.; Toro, E., FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput Methods Appl Mech Eng, 199, 625-647, (2010) · Zbl 1227.76043 [33] Dumbser, M.; Kaeser, M.; Titarev, V.; Toro, E., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J Comput Phys, 226, 204-243, (2007) · Zbl 1124.65074 [34] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J Comput Phys, 221, 693-723, (2007) · Zbl 1110.65077 [35] Dumbser, M.; Peshkov, I.; Romenski, E.; Zanotti, O., High order ader schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids, J Comput Phys, 314, 824-862, (2016) · Zbl 1349.76324 [36] Dumbser, M.; Toro, E. F., A simple extension of the osher Riemann solver to non-conservative hyperbolic systems, J Sci Comput, 48, 70-88, (2011) · Zbl 1220.65110 [37] Dumbser, M.; Uuriintsetseg, A.; Zanotti, O., On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws, Commun Comput Phys, 14, 301-327, (2013) · Zbl 1373.76126 [38] Dumbser, M.; Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, J Comput Phys, 228, 6991-7006, (2009) · Zbl 1261.76028 [39] Dumbser, M.; Zanotti, O.; Loubère, R.; Diot, S., A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J Comput Phys, 278, 47-75, (2014) · Zbl 1349.65448 [40] Favrie, N.; Gavrilyuk, S. L.; Saurel, R., Solid-fluid diffuse interface model in cases of extreme deformations, J Comput Phys, 228, 16, 6037-6077, (2009) · Zbl 1280.74013 [41] Favrie, N.; Gavrilyuk, S., Diffuse interface model for compressible fluid-compressible elastic-plastic solid interaction, J Comput Phys, 231, 2695-2723, (2012) · Zbl 1430.74036 [42] Frenkel, J., Kinetic theory of liquids, (1955), Dover · Zbl 0063.01447 [43] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J Comput Phys, 144, 194-212, (1998) · Zbl 1392.76048 [44] Galera, S.; Maire, P.; Breil, J., A two-dimensional unstructured cell-centered multi-material ale scheme using vof interface reconstruction, J Comput Phys, 229, 5755-5787, (2010) · Zbl 1346.76105 [45] Gavrilyuk, S.; Favrie, N.; Saurel, R., Modelling wave dynamics of compressible elastic materials, J Comput Phys, 227, 2941-2969, (2008) · Zbl 1155.74020 [46] Godunov, S., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math USSR - Sbornik, 47, 271-306, (1959) · Zbl 0171.46204 [47] Godunov, S.; Peshkov, I., Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium, Comput Math Math Phys, 50(8), 1409-1426, (2010) · Zbl 1224.74017 [48] Godunov, S.; Romenski, E., Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates, J Appl Mech Tech Phys, 13, 868-885, (1972) [49] Godunov, S.; Romenski, E., Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media, Comput Fluid Dyn Rev, 95, 19-31, (1995) · Zbl 0875.73025 [50] Godunov, S.; Romenski, E., Thermodynamics, conservation laws, and symmetric forms of differential equations in mechanics of continuous media, Computational fluid dynamics review 95, 19-31, (1995), John Wiley, NY · Zbl 0875.73025 [51] Godunov, S.; Romenski, E., Elements of continuum mechanics and conservation laws, (2003), Kluwer Academic/ Plenum Publishers New York [52] Hidalgo, A.; Dumbser, M., ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J Sci Comput, 48, 173-189, (2011) · Zbl 1221.65231 [53] Hu, C.; Shu, C., Weighted essentially non-oscillatory schemes on triangular meshes, J Comput Phys, 150, 97-127, (1999) · Zbl 0926.65090 [54] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp element methods in CFD, (1999), Oxford University Press Oxford · Zbl 0954.76001 [55] Käser, M.; Iske, A., ADER schemes on adaptive triangular meshes for scalar conservation laws, J Comput Phys, 205, 486-508, (2005) · Zbl 1072.65116 [56] Kluth, G.; Desprs, B., Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme, J Comput Phys, 229, 24, 9092-9118, (2010) · Zbl 1427.74029 [57] Liu, W.; Cheng, J.; Shu, C., High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations, J Comput Phys, 228, 8872-8891, (2009) · Zbl 1287.76181 [58] Loubère, R.; Dumbser, M.; Diot, S., A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun Comput Phys, 16, 718-763, (2014) · Zbl 1373.76137 [59] Maire, P., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes., J Comput Phys, 228, 2391-2425, (2009) · Zbl 1156.76434 [60] Maire, P.-H.; Abgrall, R.; Breil, J.; LoubèRe, R.; Rebourcet, B., A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids, J Comput Phys, 235, 626-665, (2013) · Zbl 1291.74186 [61] Ndanou, S.; Favrie, N.; Gavrilyuk, S., Criterion of hyperbolicity in hyperelasticity in the case of the stored energy in separable form, J Elasticity, 115(1), 1-25, (2014) · Zbl 1302.35254 [62] Olliver-Gooch, C.; Altena, M. V., A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, J Comput Phys, 181, 729-752, (2002) · Zbl 1178.76251 [63] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J Numer Anal, 44, 300-321, (2006) · Zbl 1130.65089 [64] Peshkov, I.; Grmela, M.; Romenski, E., Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid-fluid transitions, Continuum Mech Thermodyn, 27, 6, 905-940, (2015) · Zbl 1341.80023 [65] Peshkov, I.; Romenski, E., A hyperbolic model for viscous Newtonian flows, Continuum Mech Thermodyn, 28, 85-104, (2016) · Zbl 1348.76046 [66] Rhebergen, S.; Bokhove, O.; van der Vegt, J., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J Comput Phys, 227, 1887-1922, (2008) · Zbl 1153.65097 [67] Romenski, E., Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Math Comput Model, 28(10), 115-130, (1998) · Zbl 1076.74501 [68] Semplice, M.; Coco, A.; Russo, G., Adaptive mesh refinement for hyperbolic systems based on third-order compact weno reconstruction, J Sci Comput, 66, 2, 692-724, (2016) · Zbl 1335.65077 [69] Shi, J.; Hu, C.; Shu, C., A technique of treating negative weights in WENO schemes, J Comput Phys, 175, 108-127, (2002) · Zbl 0992.65094 [70] S. Sambasivan, A Lagrangian cell centered mimetic formulation for computing elsto-plastic deformation of solids, MULTIMAT conference, Arcachon, France, (2011) [71] S. Sambasivan; M. Shashkov; D. E.Burton, A finite volume cell-centered Lagrangian hydrodynamics approach for solids in general unstructured grids, Int J Numer Methods Fluids, 72, 7, (2013) [72] Stroud, A., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013 [73] Titarev, V.; Romenski, E.; Toro, E., MUSTA-type upwind fluxes for non-linear elasticity, Int J Numer Methods Eng, 73, 897-926, (2008) · Zbl 1159.74046 [74] Titarev, V.; Toro, E., ADER schemes for three-dimensional nonlinear hyperbolic systems, J Comput Phys, 204, 715-736, (2005) · Zbl 1060.65641 [75] Titarev, V.; Tsoutsanis, P.; Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun Comput Phys, 8, 585-609, (2010) · Zbl 1364.76121 [76] Toro, E. F.; Titarev, V. A., Derivative Riemann solvers for systems of conservation laws and ADER methods, J Comput Phys, 212, 1, 150-165, (2006) · Zbl 1087.65590 [77] Toro, E., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, (2009), Springer Heidelberg · Zbl 1227.76006 [78] Toro, E.; Titarev, V., ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions, J Comput Phys, 202, 196-215, (2005) · Zbl 1061.65103 [79] Toumi, I., A weak formulation of roe’s approximate Riemann solver, J Comput Phys, 102, 360-373, (1992) · Zbl 0783.65068 [80] Tsoutsanis, P.; Titarev, V.; Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J Comput Phys, 230, 1585-1601, (2011) · Zbl 1210.65160 [81] Udaykumar, H.; Tran, L.; Belk, D.; Vanden, K., An Eulerian method for computation of multimaterial impact with {ENO} shock-capturing and sharp interfaces, J Comput Phys, 186, 1, 136-177, (2003) · Zbl 1047.76558 [82] Wilkins, M. L., Calculation of elastic-plastic flow, Methods Comput Phys, 3, 211-263, (1964) [83] Zanotti, O.; Fambri, F.; Dumbser, M.; Hidalgo, A., Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiting, Comput Fluids, 118, 204-224, (2015) · Zbl 1390.76381 [84] Zel’Dovich, Y. B.; Raizer, Y. P.; Probstein, R. F.; Hayes, W. D., Physics of shock waves and high-temperature hydrodynamic phenomena, 2, (1967), Academic press London, New York, Sydney [85] Zhang, Y.; Shu, C., Third order WENO scheme on three dimensional tetrahedral meshes, Commun Comput Phys, 5, 836-848, (2009) · Zbl 1364.65177
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