zbMATH — the first resource for mathematics

An interface-capturing Godunov method for the simulation of compressible solid-fluid problems. (English) Zbl 1452.74033
Summary: In this paper a new three-dimensional Eulerian interface-capturing model is proposed for the simulation of compressible solid-fluid problems. The model is a development of a well-established five equation multi-fluid model to include material strength, by incorporating a new kinematic evolution equation for the elastic stretch tensor, and augmenting the Mie-Grüneisen equation-of-state to include a contribution from elastic strain. Principal to the development of the model is the fact that a great number of solid materials can be described by the particular general equation-of-state framework, which reduces to the Mie-Grüneisen equation-of-state in the limit of zero strength and is thus equally applicable to a wide class of fluids. The constitutive models are founded on hyperelastic theory and are therefore thermodynamically compatible. Only one kinematic equation is required for arbitrary numbers of components by assuming that mixtures are described by a common deviatoric strain tensor. The system of evolution equations can be written in conservation law form, and are therefore suitable for application of Godunov’s method; specifically an HLLD Riemann solver is formulated for the calculation of numerical fluxes. Interfaces are sharpened by using the THINC method in conjunction with MUSCL reconstruction. A time operator splitting strategy is used to divide the time integration into an explicit elastic update, which uses the third order TVD Runge-Kutta method, followed by an implicit plastic update. The method is verified using a number of challenging solid-fluid problems, including a high-velocity impact of a viscoplastic strain-hardening cylinder in air. The potential of the model for practical problems is demonstrated through three-dimensional simulation of an explosive buried in solid ground material.
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S10 Finite volume methods applied to problems in solid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics, general
Full Text: DOI
[1] Allaire, G.; Clerc, S.; Kokh, S., A five equation model for the simulation of interfaces between compressible fluids, J. Comput. Fluids, 181, 577-616 (2002) · Zbl 1169.76407
[2] Banerjee, B., An Evaluation of Plastic Flow Stress Models for the Simulation of High Temperature and High Strain Rate Deformation of Metals, 1-43 (2005), University of Utah
[3] 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, 7046 (2009) · Zbl 1172.74032
[4] Barton, P. T.; Drikakis, D.; Romenski, E., An Eulerian finite-volume scheme for large elastoplastic deformations in solids, Int. J. Numer. Methods Eng., 81, 453 (2010) · Zbl 1183.74331
[5] Barton, P. T.; Drikakis, D., An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces, J. Comput. Phys., 229, 5518 (2010) · Zbl 1346.74179
[6] Barton, P. T.; Obadia, B.; Drikakis, D., A conservative level-set based method for compressible solid/fluid problems on fixed grids, J. Comput. Phys., 230, 7867 (2011) · Zbl 1432.74060
[7] Barton, P. T.; Romenski, E., On computational modelling of strain-hardening material dynamics, Commun. Comput. Phys., 11, 1525-1546 (2012)
[8] Barton, P. T.; 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)
[9] Barton, P. T., An Eulerian method for finite deformation anisotropic damage with application to high strain-rate problems, Int. J. Plast., 83, 225-251 (2016)
[10] Barton, P. T., A level-set based Eulerian method for simulating problems involving high strain-rate fracture and fragmentation, Int. J. Impact Eng., 117, 75-84 (2018)
[11] Batten, P.; Clarke, N.; Lambert, C.; Causen, D. M., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput., 18, 1553-1570 (1997) · Zbl 0992.65088
[12] Bažant, Z. P., Easy-to-compute tensors with symmetric inverse approximating Hencky finite strain and its rate, J. Eng. Mater. Technol., 120, 131-136 (1998)
[13] Bruhns, O. T.; Xiao, H.; Meyers, A., Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky’s logarithmic strain tensor, Proc. R. Soc. Lond. Ser. A, 457, 2207 (2001) · Zbl 1048.74505
[14] Cook, A. W.; Ghaisas, N. S.; Subramaniam, A.; Lele, S. K., Evaluation of an Eulerian Multi-Material Mixture Formulation Based on a Single Inverse Deformation Gradient Tensor Field (2017), Lawrence Livermore National Laboratory, Tech. Rep. LLNL-JRNL-741479
[15] Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics (2016), Springer · Zbl 1364.35003
[16] Deng, X.; Inaba, S.; Xie, B.; Shyue, K.-M.; Xiao, F., High fidelity discontinuity-resolving reconstruction for compressible multiphase flows with moving interfaces, J. Comput. Phys., 371, 945-966 (2018)
[17] Dorovskii, V. N.; Iskol’dskii, A. M.; Romenskii, E. I., Dynamics of impulsive metal heating by a current and electrical explosion of conductors, J. Appl. Mech. Tech. Phys., 24, 454 (1984)
[18] Ehrgott, J. Q.; Akers, S. A.; Windham, J. E.; Rickman, D. D.; Danielson, K. T., The influence of soil parameters on the impulse and airblast overpressure loading above surface-laid and shallow-buried explosives, Shock Vib., 18, 857-874 (2011)
[19] Favrie, N.; Gavrilyuk, S. L.; Saurel, R., Solid – fluid diffuse interface model in cases of extreme deformations, J. Comput. Phys., 228, 6037 (2009) · Zbl 1280.74013
[20] Favrie, N.; Gavrilyuk, S. L., Diffuse interface model for compressible fluid - compressible elastic-plastic solid interaction, J. Comput. Phys., 231, 2695-2723 (2012) · Zbl 1430.74036
[21] Garrick, D. P.; Hagan, W. A.; Regele, J. D., An interface capturing scheme for modeling atomization in compressible flows, J. Comput. Phys., 344, 260-280 (2017)
[22] Godunov, S. K.; Romenskii, E. I., Elements of Continuum Mechanics and Conservation Laws (2003), Kluwer Academic/Plenum Publishers · Zbl 1031.74004
[23] Grove, J.; Menikoff, R., The anomalous reflection of a shock wave at a material interface, J. Fluid Mech., 219, 313-336 (1990)
[24] Hank, S.; Gavrilyuk, S.; Favrie, N.; Massoni, Jacques, Impact simulation by an Eulerian model for interaction of multiple elastic-plastic solids and fluids, Int. J. Impact Eng., 109, 104-111 (2017)
[25] Hank, S.; Favrie, N.; Massoni, Jacques, Modeling hyperelasticity in non-equilibrium multiphase flows, J. Comput. Phys., 330, 65-91 (2017) · Zbl 1378.74022
[26] Hill, D. J.; Pullin, D.; Ortiz, M.; Meiron, D., An Eulerian hybrid WENO centered-difference solver for elastic – plastic solids, J. Comput. Phys., 229, 9053 (2010) · Zbl 1427.74028
[27] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202 (1996) · Zbl 0877.65065
[28] Johnsen, E.; Colonius, T., Implementation of WENO schemes in compressible multicomponent flow problems, J. Comput. Phys., 219, 715-732 (2006) · Zbl 1189.76351
[29] Johnson, G. R.; Cook, W. H., Fracture characteristics of three metals subjected to various strains, strain-rates, temperatures and pressures, Eng. Fract. Mech., 21, 31-48 (1985)
[30] Kokh, S.; Lagoutiere, F., An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five equation model, J. Comput. Phys., 229, 2773-2809 (2010) · Zbl 1302.76129
[31] Lee, E. H., Elastic-plastic deformation at finite strains, J. Appl. Mech., 36, 1-6 (1969) · Zbl 0179.55603
[32] Lesuer, D. R.; Kay, G. J.; LeBlanc, M. M., Modeling Large-Strain, High-Rate Deformation in Metals (2001), Lawrence Livermore National Laboratory Report UCRL-JC-134118
[33] López Ortega, A.; Lombardini, M.; Pullin, D. I.; Meiron, D. I., Numerical simulation of elastic-plastic solid mechanics using an Eulerian stretch tensor approach and HLLD Riemann solver, J. Comput. Phys., 257, 414-441 (2014) · Zbl 1349.65463
[34] Michael, L.; Nikiforakis, N., A multi-physics methodology for the simulation of reactive flow and elastoplastic structural response, J. Comput. Phys., 367, 1-27 (2018) · Zbl 1415.76473
[35] Miller, G. H.; Colella, P., A high-order Eulerian Godunov method for elastic-plastic flow in solids, J. Comput. Phys., 167, 131 (2001) · Zbl 0997.74078
[36] Miller, G. H.; Colella, P., A conservative three-dimensional Eulerian method for coupled solid-fluid shock capturing, J. Comput. Phys., 183, 26 (2002) · Zbl 1057.76558
[37] Miyoshi, T.; Kusano, K., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 315-344 (2005) · Zbl 1114.76378
[38] Ndanou, S.; Favrie, N.; Gavrilyuk, S., Multi-solid and multi-fluid diffuse interface model: applications to dynamic fracture and fragmentation, J. Comput. Phys., 295, 523-555 (2015) · Zbl 1349.74377
[39] Ottosen, N. S.; Ristinmaa, M., The Mechanics of Constitutive Modeling (2005), Elsevier
[40] Youngs, D. L., An Interface Tracking Method for a 3d Eulerian Hydrodynamics Code (1984), Technical Report 44/92/35, AWRE
[41] Plohr, B. J.; Sharp, D. H., A conservative formulation for plasticity, Adv. Appl. Math., 13, 462 (1992) · Zbl 0771.73022
[42] Quirk, J. J.; Karni, S., On the dynamics of a shock – bubble interaction, J. Fluid Mech., 318, 129-163 (1996) · Zbl 0877.76046
[43] Radhakrishnan, K.; Hindmarsh, A., Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations (1993), Lawrence Livermore National Laboratory, Tech. Rep. UCRL-ID-113855, NASA Reference Publication 1327
[44] Romenskii, E. I., Hyperbolic equations of Maxwell’s nonlinear model of elastoplastic heat-conducting media, Sib. Math. J., 30, 606 (1989) · Zbl 0741.73022
[45] Saurel, R.; Pantano, C., Diffuse-interface capturing methods for compressible two-phase flows, Annu. Rev. Fluid Mech., 50, 105-130 (2018) · Zbl 1384.76054
[46] Shukla, R. K.; Pantano, C.; Freund, J. B., An interface capturing method for the simulation of multi-phase compressible flows, J. Comput. Phys., 229, 7411-7439 (2010) · Zbl 1425.76289
[47] Shukla, R. K., Nonlinear preconditioning for efficient and accurate interface capturing in simulation of multicomponent compressible flows, J. Comput. Phys., 276, 508-540 (2014) · Zbl 1349.65110
[48] Shyue, K.-M., A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions, J. Comput. Phys., 215, 219-244 (2006) · Zbl 1140.76401
[49] Shyue, K-M.; Xiao, F., An Eulerian interface sharpening algorithm for compressible two-phase flow: the algebraic THINC approach, J. Comput. Phys., 268, 326-354 (2014) · Zbl 1349.76388
[50] So, K. K.; Hu, X. Y.; Adams, N. A., Anti-diffusion interface sharpening technique for two-phase compressible flow simulations, J. Comput. Phys., 231, 4304-4323 (2012) · Zbl 1426.76428
[51] Schoch, S.; Nordin-Bates, K.; Nikiforakis, N., An Eulerian algorithm for coupled simulations of elastoplastic-solids and condensed-phase explosives, J. Comput. Phys., 252, 163-194 (2013) · Zbl 1349.74346
[52] Titarev, V. A.; Romenski, E.; Toro, E. F., MUSTA-type upwind fluxes for non-linear elasticity, Int. J. Numer. Methods Eng., 73, 897 (2008) · Zbl 1159.74046
[53] Tran, L. B.; Udaykumar, H. S., A particle-level set-based sharp interface cartesian grid method for impact, penetration, and void collapse, J. Comput. Phys., 193, 469-510 (2004) · Zbl 1109.74376
[54] Van Leer, B., Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23, 263-275 (1997) · Zbl 0339.76039
[55] Wilkins, M. L., Calculation of Elastic-Plastic Flow (1963), UCRL Technical Report UCRL-7322
[56] Wilkins, M. L.; Guinan, M. W., Impact of cylinders on a rigid boundary, J. Appl. Phys., 44, 1200-1206 (1973)
[57] Wilkins, M. L., Computer Simulation of Dynamic Phenomena (1999), Springer · Zbl 0926.76001
[58] Xiao, H.; Bruhns, T.; Meyers, A., Logarithmic strain, logarithmic spin and logarithmic rate, Acta Mech., 124, 89-105 (1997) · Zbl 0909.73006
[59] Xiao, F.; Honma, Y.; Kono, T., A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Methods Fluids, 48, 1023-1040 (2005) · Zbl 1072.76046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.