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An Eulerian hybrid WENO centered-difference solver for elastic-plastic solids. (English) Zbl 1427.74028
Summary: We present a finite-difference based solver for hyper-elastic and viscoplastic systems using a hybrid of the weighted essentially non-oscillatory (WENO) schemes combined with explicit centered difference to solve the equations of motion expressed in an Eulerian formulation. By construction our approach minimizes both numerical dissipation errors and the creation of curl-constraint violating errors away from discontinuities while avoiding the calculation of hyperbolic characteristics often needed in general finite-volume schemes. As a result of the latter feature, the formulation allows for a wide range of constitutive relations and only an upper-bound on the speed of sound at each time is required to ensure a stable timestep is chosen. Several one- and two-dimensional examples are presented using a range of constitutive laws with and without additional plastic modeling. In addition we extend the reflection technique combined with ghost-cells to enforce fixed boundaries with a zero tangential stress condition (i.e. free-slip).

74B20 Nonlinear elasticity
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74S20 Finite difference methods applied to problems in solid mechanics
Full Text: DOI
[1] Marsden, J.; Hughes, T., Mathematical foundations of elasticity, (1994), Dover Publications
[2] Trangenstein, J.; Colella, P., A higher-order Godunov method for modeling finite deformation in elastic – plastic solids, Commun. pure appl. math., 44, 1, 41-100, (1991) · Zbl 0714.73027
[3] Trangenstein, J.; Pember, R., Numerical algorithms for strong discontinuities in elastic – plastic solids, J. comput. phys., 103, 1, 63-89, (1992) · Zbl 0779.73080
[4] Trangenstein, J., A 2nd-order Godunov algorithm for 2-dimensional solid mechanics, Comput. mech., 13, 5, 343-359, (1994) · Zbl 0793.73102
[5] Weiner, J., Statistical mechanics of elasticity, (2002), Dover Publications · Zbl 1032.82001
[6] Zel’dovich, Y.B.; Raizer, Y.P., Physics of shock waves and high-temperature hydrodynamics phenomena, (2002), Dover Publications
[7] Miller, G.; Colella, P., A high-order Eulerian Godunov method for elastic – plastic flow in solids, J. comput. phys., 167, 1, 131-176, (2001) · Zbl 0997.74078
[8] Miller, G.; Colella, P., A conservative three-dimensional Eulerian method for coupled solid – fluid shock capturing, J. comput. phys., 183, 1, 26-82, (2002) · Zbl 1057.76558
[9] Barton, P.T.; Drikakis, D., An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces, J. comput. phys., 229, 15, 5518-5540, (2010) · Zbl 1346.74179
[10] Miller, G., An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics, J. comput. phys., 193, 1, 198-225, (2004) · Zbl 1047.65069
[11] Titarev, V.A.; Romenski, E.; Toro, E.F., Musta-type upwind fluxes for non-linear elasticity, Int. J. numer. methods eng., 73, 7, 897-926, (2008) · Zbl 1159.74046
[12] Barton, P.T.; Drikakis, D.; Romenski, E.; Titarev, V.A., Exact and approximate solutions of Riemann problems in non-linear elasticity, J. comput. phys., 228, 18, 7046-7068, (2009) · Zbl 1172.74032
[13] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065
[14] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111
[15] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43, 1, 89-112, (2001) · Zbl 0967.65098
[16] Hindmarsh, A., ODEPACK, A systematized collection of ODE solvers, Scientific computing, (1983), North-Holland, pp. 55-64
[17] K. Radhakrishnan, A. Hindmarsh, Description and use of lsode, the livermore solver for ordinary differential equations, Tech. Rep. UCRL-ID-113855 NASA/Reference Publication 1327, Lawrence Livermore National Laboratory, 1993.
[18] Hill, D.J.; Pullin, D.I., Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks, J. comput. phys., 194, 2, 435-450, (2004) · Zbl 1100.76030
[19] Pantano, C.; Deiterding, R.; Hill, D.; Pullin, D., A low numerical dissipation patch-based adaptive mesh refinement method for large-eddy simulation of compressible flows, J. comput. phys., 221, 1, 63-87, (2007) · Zbl 1125.76034
[20] Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM rev., 51, 82-126, (2009) · Zbl 1160.65330
[21] V. Weirs, G. Candler, Optimization of weighted ENO schemes for DNS of compressible turbulence, AIAA, No. 97-1940, 1997, pp. 1-11.
[22] Deiterding, R., A virtual test facility for the efficient simulation of solid material response under strong and detonation wave loading, Eng. comput., 22, 325-347, (2006)
[23] Landau, L.; Liftshitz, E., Fluid mechanics, (1987), Pergamon Press, pp. 378-382 (section 101)
[24] Blatz, P.; Ko, W., Application of finite elasticity to the deformation of rubbery materials, Trans. soc. rheol., 6, 223-251, (1962)
[25] Plohr, J.; Plohr, B., Linearized analysis of richtmyer – meshkov flow for elastic materials, J. fluid mech., 537, 55-89, (2005) · Zbl 1074.74037
[26] López Ortega, A.; Hill, D.J.; Pullin, D.I.; Meiron, D.I., Linearized richtmyer – meshkov flow analysis for impulsively accelerated incompressible solids, Phys. rev. E, 81, 6, 066305, (2010)
[27] Deiterding, R., Construction and application of an AMR algorithm for distributed memory computers, (), 361-372 · Zbl 1065.65114
[28] Deiterding, R.; Radovitzky, R.; Mauch, S.P.; Noels, L.; Cummings, J.C.; Meiron, D.I., A virtual test facility for the efficient simulation of solid material response under strong shock and detonation wave loading, Eng. comput., 22, 3-4, 325-347, (2007)
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