# zbMATH — the first resource for mathematics

An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces. (English) Zbl 1047.76558
Summary: A technique is presented for the numerical simulation of high-speed multimaterial impact. Of particular interest is the interaction of solid impactors with targets. The computations are performed on a fixed Cartesian mesh by casting the equations governing material deformation in Eulerian conservation law form. The advantage of the Eulerian setting is the disconnection of the mesh from the boundary deformation allowing for large distortions of the interfaces. Eigenvalue analysis reveals that the system of equations is hyperbolic for the range of materials and impact velocities of interest. High-order accurate ENO shock-capturing schemes are used along with interface tracking techniques to evolve sharp immersed boundaries. The numerical technique is designed to tackle the following physical phenomena encountered during impact: (1) high velocities of impact leading to large deformations of the impactor as well as targets; (2) nonlinear wave-propagation and the development of shocks in the materials; (3) modeling of the constitutive properties of materials under intense impact conditions and accurate numerical calculation of the elasto-plastic behavior described by the models; (4) phenomena at multiple interfaces (such as impactor-target, target-ambient and impactor-ambient), i.e. both free surface and surface-surface dynamics. Comparison with Lagrangian calculations is made for the elasto-plastic deformation of solid material. The accuracy of convex ENO scheme for shock capturing, with the Mie-Gruneisen equation of state for pressure, is closely examined. Good agreement of the present finite difference fixed grid results is obtained with exact solutions in 1D and benchmarked moving finite element solutions for axisymmetric Taylor impact.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 76N15 Gas dynamics (general theory) 76L05 Shock waves and blast waves in fluid mechanics
##### Keywords:
Moving boundaries; Impact; Multimaterial flows; High-strain rate
Full Text:
##### References:
 [1] Zukas, J.A.; Nicholas, T.; Swift, H.F.; Gresczuk, L.B.; Curran, D.R., Impact dynamics, (1982), Wiley New York [2] Meyers, M.A., Dynamic behavior of materials, (1994), Wiley New York · Zbl 0893.73002 [3] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high-order accurate essentially non-oscillatory schemes, III, J. comp. phys., 131, 3-47, (1997) · Zbl 0866.65058 [4] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comp. phys., 77, 439-471, (1988) · Zbl 0653.65072 [5] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. comp. phys., 83, 32-78, (1989) · Zbl 0674.65061 [6] Liu, X.-D.; Osher, S., Convex ENO high order schemes without field-by-field decomposition or staggered grids, J. comp. phys., 142, 304-330, (1998) · Zbl 0941.65082 [7] Ye, T.; Mittal, R.; Udaykumar, H.S.; Shyy, W., An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. comp. phys., 156, 209-240, (1999) · Zbl 0957.76043 [8] Udaykumar, H.S.; Mittal, R.; Rampunggoon, P.; Khanna, A., An eulerian – lagrangian Cartesian grid method for simulating flows with complex moving boundaries, J. comp. phys., 174, 1-36, (2001) · Zbl 1106.76428 [9] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. meth. appl. mech. eng., 99, 235-395, (1992) · Zbl 0763.73052 [10] Liu, W.-K.; Belytschko, T.; Chang, H., An arbitrary lagrangian – eulerian finite element method for path-dependent materials, Comput. meth. appl. mech. eng., 58, 227-245, (1986) · Zbl 0585.73117 [11] Camacho, G.T.; Ortiz, M., Computational modeling of impact damage in brittle materials, Int. J. solids struct., 33, 2899-2938, (1996) · Zbl 0929.74101 [12] Camacho, G.T.; Ortiz, M., Adaptive Lagrangian modeling of ballistic penetration of metallic targets, Comput. meth. appl. mech. eng., 142, 269-301, (1997) · Zbl 0892.73056 [13] Duarte, A.; Oden, J.T., An h-p adaptive method using clouds, Comput. meth. appl. mech. eng., 139, 237-262, (1996) · Zbl 0918.73328 [14] Johnson, G.R.; Stryk, R.A.; Beissel, S.R., SPH for high velocity impact computations, Comput. meth. appl. mech. eng., 139, 347-373, (1996) · Zbl 0895.76069 [15] Liu, W.-K.; Hao, S.; Belytschko, T.; Li, S.; Chang, C.T., Multi-scale methods, Int. J. numer. meth. eng., 47, 7, (2000) [16] Belytschko, T.; Guo, Y.; Liu, W.-K.; Xiao, S.P., Unified stability analysis of meshless particle methods, Int. J. numer. meth. eng., 48, 9, (2000) [17] Dolbow, J.; John; Moes, N.; Belytschko, T., Discontinuous enrichment in finite elements with a partition of unity method, Finite elements anal. des., 36, 3, (2000) · Zbl 0981.74057 [18] Moes, N.; Dolbow, J.; Belytschko, T., Finite element method for crack growth without remeshing, Int. J. numer. meth. eng., 46, 1, 131-150, (1999) · Zbl 0955.74066 [19] Sukumar, N.; Moes, N.; Moran, B.; Belytschko, T., Extended finite element method for three-dimensional crack modelling, Int. J. numer. meth. eng., 48, 1549-1570, (2000) · Zbl 0963.74067 [20] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comp. phys., 39, 201, (1981) · Zbl 0462.76020 [21] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comp. phys., 25, 220-252, (1977) · Zbl 0403.76100 [22] Glimm, J.; Grove, J.; Lindquist, B.; McBryan, O.A.; Tryggvason, G., The bifurcation of tracked scalar waves, SIAM J. sci. stat. comput., 1, 61-79, (1988) · Zbl 0636.65132 [23] Braes, H.; Wriggers, P., Arbitrary Lagrangian Eulerian finite element analysis of free surface flow, Comput. meth. appl. mech. eng., 95-109, (2000) · Zbl 0967.76053 [24] Lucy, L.B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 12, 1013-1024, (1977) [25] Monaghan, J.J., An introduction to SPH, Comput. phys. commun., 48, 89-96, (1988) · Zbl 0673.76089 [26] Dilts, G.A., Moving-least-squares-particle hydrodynamics I. consistency and stability, Int. J. numer. meth. eng., 44, 1115-1155, (1999) · Zbl 0951.76074 [27] Randles, P.W.; Libersky, L.D., Normalized SPH with stress points, Int. J. numer. meth. eng., 48, 1445-1462, (2000) · Zbl 0963.74079 [28] Vignjevic, R.; Campbell, J.; Libersky, L., A treatment of zero-energy modes in the smoothed particle hydrodynamics method, Comput. meth. appl. mech. eng., 184, 67-85, (2000) · Zbl 0989.74079 [29] Chen, J.K.; Beraum, J.E.; Jih, C.J., An improvement for tensile instability in smoothed particle hydrodnamics, Comput. mech., 23, 279-287, (1999) · Zbl 0949.74078 [30] Aluru, N.R., A reproducing kernel particle method for meshless analysis of microelectromechanical systems, Comput. mech., 23, 324-338, (1999) · Zbl 0949.74077 [31] Chen, J.K.; Beraun, J.E.; Carney, T.C., A corrective smoothed particle method for boundary value problems in heat conduction, Int. J. numer. meth. eng., 46, 231-252, (1999) · Zbl 0941.65104 [32] Dilts, G.A., Moving least-squares particle hydrodynamics II: conservation and boundaries, Int. J. numer. meth. eng., 48, 1503-1524, (2000) · Zbl 0960.76068 [33] Morris, J.P.; Fox, P.J.; Zhu, Y., Modeling low Reynolds number incompressible flows using SPH, J. comp. phys., 136, 214-226, (1997) · Zbl 0889.76066 [34] Scardovelli, R.; Zaleski, S., Direct numerical simulation of free surface and interfacial flow, Ann. rev. fluid mech., 31, 567, (1999) [35] Beckermann, C.; Diepers, H.J.; Steinbach, I.; Karma, A.; Tong, X., Modeling melt convection in phase-field simulations of solidification, J. comp. phys., 154, 468-496, (1999) · Zbl 0960.82015 [36] Trangenstein, J.A., A second-order algorithm for the dynamic response of soils, Impact computing sci. eng., 2, 1-39, (1990) [37] Trangenstein, J.A., A second-order algorithm for two-dimensional solid mechanics, Comput. mech., 13, 343-359, (1994) · Zbl 0793.73102 [38] Trangenstein, J.A., Adaptive mesh refinement for wave propagation in nonlinear solids, SIAM J. sci. comput., 16, 819-939, (1995) · Zbl 0831.35104 [39] Trangenstein, J.A.; Pember, R.B., The Riemann problem for longitudinal motion in an elastic – plastic bar, SIAM J. sci. stat. comput., 12, 180-207, (1991) · Zbl 0714.73028 [40] Miller, G.H.; Colella, P., A high-order Eulerian Godunov method for elastic – plastic flow in solids, J. comp. phys., 167, 1, 131-176, (2001) · Zbl 0997.74078 [41] Benson, D.J., A multi-material Eulerian formulation for the efficient solution of impact and penetration problems, Comput. mech., 15, 558-571, (1995) · Zbl 0831.73011 [42] Cooper, S.R.; Benson, D.J.; Nesterenko, V.F., A numerical exploration of the role of void geometry on void collapse and hot spot formation in ductile materials, Int. J. plasticity, 16, 525-540, (2000) · Zbl 1043.74519 [43] Anderson, D.M.; McFadden, G.B.; Wheeler, A.A., Diffuse interface methods in fluid mechanics, Ann. rev. fluid mech., 30, 139-165, (1998) · Zbl 1398.76051 [44] Benson, D.J., A mixture theory for contact in multi-material Eulerian formulations, Comput. meth. appl. mech. eng., 140, 59-86, (1997) · Zbl 0892.73048 [45] R. Menikoff, E. Kober, Compaction waves in granular HMX, Los Alamos National Lab Report, LA-13456-MS, 1999 [46] Menikoff, R., Errors when shock waves interact due to numerical shock width, SIAM J. sci. stat. comput., 15, 5, 1227-1242, (1994) · Zbl 0826.35069 [47] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. comput. phys., 135, 1, 8-29, (1997) · Zbl 0889.65133 [48] Hou, T.Y.; Li, Z.; Osher, S.; Zhao, H., A hybrid method for moving interface problems with application to the hele – shaw flow, J. comp. phys., 134, 2, 236-247, (1997) · Zbl 0888.76067 [49] Liu, X.-D.; Fedkiw, R.P.; Kang, M., A boundary condition capturing method for poisson’s equation on irregular domains, J. comput. phys., 160, 1, 151-178, (2000) · Zbl 0958.65105 [50] Udaykumar, H.S.; Shyy, W.; Rao, M.M., Elafint: a mixed eulerian – lagrangian method for fluid flows with complex and moving boundaries, Int. J. numer. meth. fluids, 22, 691, (1996) · Zbl 0887.76059 [51] Udaykumar, H.S.; Mittal, R.; Shyy, W., Solid – liquid phase front computations in the sharp interface limit on fixed grids, J. comput. phys., 153, 535-574, (1999) · Zbl 0953.76071 [52] Fedkiw, R.P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457-492, (1999) · Zbl 0957.76052 [53] Fedkiw, R.P., Coupling an eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method, J. comput. phys., 175, 200-224, (2002) · Zbl 1039.76050 [54] Kang, M.; Fedkiw, R.; Liu, X.-D., A boundary condition capturing method for multiphase incompressible flow, J. sci. comput., 15, 323-360, (2000) · Zbl 1049.76046 [55] Leveque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 4, 1019-1044, (1994) · Zbl 0811.65083 [56] Osher, S.; Sethian, J.A., Fronts propagating with curvature dependent speed: algorithms based in hamilton – jacobi formulations, J. comp. phys., 79, 12-49, (1988) · Zbl 0659.65132 [57] M. Arienti, E. Morano, J. Shepherd, Nonreactive Euler flows with Mie-Gruneisen equation of state for high explosives, 1999. Available from www.caltech.edu/ eric/Papers/FM99-8.pdf [58] Fedkiw, R.P.; Marquina, A.; Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. comp. phys., 148, 545-578, (1999) · Zbl 0933.76075 [59] Glaister, P., An approximate Riemann solver for the Euler equations for real gases, J. comp. phys., 74, 382-408, (1988) · Zbl 0632.76079 [60] Dukowicz, J.K., A general, non-iterative Riemann solver for godunov’s method, J. comp. phys., 61, 119-137, (1985) · Zbl 0629.76074 [61] Miller, G.H.; Puckett, E.G., A high-order Godunov method for multiple condensed phases, J. comp. phys., 128, 134-164, (1996) · Zbl 0861.65117 [62] K.J. Vanden, Characteristic analysis of the uniaxial stress and strain governing equations with thermal-elastic and Mie-Gruneison equation of state, Technical Memorandum, AFRL, Eglin AFB, Eglin, FL, 1998 [63] Trangenstein, J.A.; Collela, P., A high-order Godunov method for modeling finite deformation in elastic – plastic solids, Comm. pure appl. math., 44, 41-100, (1991) · Zbl 0714.73027 [64] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comp. phys., 126, 202-228, (1996) · Zbl 0877.65065 [65] Donat, R.; Marquina, A., Capturing shock reflections: an improved flux formula, J. comp. phys., 125, 42-58, (1996) · Zbl 0847.76049
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.