Computational methods in Lagrangian and Eulerian hydrocodes. (English) Zbl 0763.73052

Summary: Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.


74S05 Finite element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
Full Text: DOI


[1] Johnson, W.E.; Anderson, C.E., History and application of hydrocodes in hypervelocity impact, Internat. J. impact engrg., 5, 423-439, (1987)
[2] Anderson, C.E., An overview of the theory of hydrocodes, Internat. J. impact engrg., 5, 33-59, (1987)
[3] Zukas, J.A.; Nicholas, T.; Swift, H.F.; Greszczuk, L.B.; Curran, D.R., Impact dynamics, (1982), Wiley New York
[4] Malvern, L.E., Introduction to the mechanics of a continuous medium, (1969), Prentice Hall Englewood Cliffs, NJ · Zbl 0181.53303
[5] Hancock, S., ()
[6] Addessio, F.L.; Carroll, D.E.; Dukowicz, J.K.; Harlow, F.H.; Johnson, J.N.; Kaskiwa, B.A.; Maltrud, M.E.; Ruppel, H.M., CAVEAT: A computer code for fluid dynamics problems with large distortion and internal slip, Los alamos national laboratory, UC-32, (1988)
[7] Noh, W.F., CEL: A time-dependent, two-space-dimensional, coupled Eulerian-Lagrange code, (), 117-179
[8] Hughes, T.J.R.; Liu, W.K.; Zimmerman, T.K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. methods appl. mech. engrg., 29, 329-349, (1981) · Zbl 0482.76039
[9] Donea, J., Arbitrary Lagrangian-Eulerian finite element methods, (), 473-516
[10] Noh, W.F., Numerical methods in hydrodynamic calculations, Lawrence livermore national laboratory, UCRL-52112, (1976)
[11] Courant, R.; Friedrichs, K.O., Supersonic flow and shock waves, (1976), Springer New York · Zbl 0365.76001
[12] Sutcliffe, W.G., BBC hydrodynamics, Lawrence livermore national laboratory, UCIR-716, (1973)
[13] DeBar, R.B., Fundamentals of the KRAKEN code, Lawrence livermore laboratory, UCIR-760, (1974)
[14] Holian, K.S.; Mandell, D.A.; Adams, T.F.; Addessio, F.L.; Baumgardner, J.R.; Mosso, S.J., MESA: A 3-D computer code for armor/anti-armor applications, ()
[15] Trulio, J.G.; Trigger, K.R., Numerical solution of the one-dimensional Lagrangian hydrodynamic equations, Lawrence radiation laboratory, UCRL-6267, (1961)
[16] Maenchen, G.; Sack, S., The TENSOR code, ()
[17] Wilkins, M., Calculation of elastic-plastic flow, (), 211-263
[18] Krieg, R.D.; Key, S.W., Implementation of a time dependent plasticity theory into structural programs, (), 125-137
[19] Godunov, S.K., Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 271-306, (1959) · Zbl 0171.46204
[20] Hughes, T.J.R., The finite element method, linear, static and dynamic finite element analysis, (1987), Prentice Hall Englewood Cliffs, NJ
[21] Zienkiewicz, O.C., The finite element method, (1987), McGraw-Hill New York · Zbl 0623.65120
[22] Bathe, K.J., Finite element procedures in engineering analysis, (1982), Prentice Hall Englewood Cliffs, NJ · Zbl 0528.65053
[23] Cook, R.D.; Malkus, D.S.; Plesha, M.E., Concepts and applications of finite element analysis, (1989), Wiley New York · Zbl 0696.73039
[24] Johnson, C., Numerical solutions of partial differential equations by the finite element method, (1987), Cambridge Univ. Press Cambridge
[25] Rank, E.; Katz, C.; Werner, H., On the importance of the discrete maximum principle in transient analysis using finite element methods, Internat. J. numer. methods engrg., 19, 1771-1782, (1983) · Zbl 0526.65080
[26] Hallquist, J.O., User’s manual for DYNA2D—an explicit two-dimensional hydrodynamic finite element code with interactive rezoning, Lawrence livermore national laboratory, UCID-18756, (1984), Rev. 2
[27] Hallquist, J.O., DYNA3D course notes, Lawrence livermore national laboratory, UCID-19899, (1987), Rev. 2
[28] Benson, D.J., Vectorizing the right-hand side assembly in an explicit finite element program, Comput. methods appl. mech. engrg., 73, 147-152, (1989) · Zbl 0689.68023
[29] Johnson, G.R.; Stryk, R.A., User instructions for the EPIC-2 code, Air force armament laboratory, AFATL-TR-86-51, (1986)
[30] Nagtegaal, J.C.; Parks, D.M.; Rice, J.R., On numerically accurate finite element solutions in the fully plastic range, Comput. methods appl. mech. engrg., 4, 153-177, (1974) · Zbl 0284.73048
[31] Johnson, G.R.; Stryk, R.A., Eroding interface and improved tetrahedral element algorithms for high-velocity impact computations in three dimensions, Internat. J. impact engrg., 5, 411-421, (1987)
[32] Demuth, R.B.; Margolin, L.G.; Nichols, B.D.; Adams, T.F.; Smith, B.W., SHALE: A computer program for solid dynamics, Los alamos national laboratory, LA-10236, (1985)
[33] Amdsden, A.; Ruppel, H.M.; Hirt, C.W., SALE: A simplified ALE computer program for fluid flow at all speeds, (1980), Los Alamos Scientific Laboratory
[34] Amsden, A.A.; Ruppel, H.M., SALE-3D: A simplified ALE computer program for calculating three-dimensional fluid flow, (1981), Los Alamos Scientific Laboratory
[35] Amsden, A.A.; Hirt, C.W., YAQUI: an arbitrary Lagrangian-Eulerian computer program for fluid flow at all speeds, Los alamos scientific laboratory, LA-5100, (1973) · Zbl 0255.76045
[36] Belytschko, T.B.; Kennedy, J.M.; Schoeberle, D.F., On finite element and difference formulations of transient fluid-structure problems, (), IV39-IV54
[37] Hallquist, J.O., NIKE2D—A vectorized implicit, finite deformation finite element code for analyzing the static and dynamic response of 2-D solids with interactive rezoning and graphics, Lawrence livermore national laboratory, UCID-19677, (1986), Rev. 1
[38] Truesdell, C.; Noll, W., The non-linear field theories of mechanics, () · Zbl 0779.73004
[39] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice Hall Englewood Cliffs, NJ · Zbl 0545.73031
[40] Billington, E.W.; Tate, A., The physics of deformation and flow, (1981), McGraw-Hill New York · Zbl 0499.73001
[41] Gurtin, M.E., An introduction to continuum mechanics, () · Zbl 0203.26802
[42] Johnson, G.C.; Bammann, D.J., A discussion of stress rates in finite deformation problems, Internat. J. solids and structures, 20, 725-737, (1984) · Zbl 0546.73031
[43] Polukhin, P.; Gorelik, S.; Vorontsov, V., Physical principles of plastic deformation, (1983), Mir Moscow
[44] Taylor, L.M.; Flanagan, D.P., PRONTO 2D: A two-dimensional transient solid dynamics program, Sandia national laboratories, SAND86-0594, (1987)
[45] S. Hancock, Personal communication, 1991.
[46] D. Bammann, Personal communication, Sandia National Laboratories, 1983.
[47] Hoger, A.; Carlson, D.E., Determination of the stretch and rotation in the polar decomposition of the deformation gradient, Quart. appl. math., 42, 113-117, (1984) · Zbl 0551.73004
[48] Dienes, J.K., On the analysis of rotation and stress rate in deforming bodies, Acta mech., 32, 217-232, (1979) · Zbl 0414.73005
[49] Hughes, T.J.R.; Winget, J., Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, Internat. J. numer. methods engrg., 15, 1862-1867, (1980) · Zbl 0463.73081
[50] Margolin, L.G., A centered artificial viscosity for cells with large aspect ratio, Lawrence livermore national laboratory, UCRL-53882, (1988)
[51] Benson, D.J., A new two-dimensional flux-limited shock viscosity, Comput. methods appl. mech. engrg., 93, 39-95, (1991) · Zbl 0850.73050
[52] D.E. Burton, Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity, Lawrence Livermore National Laboratory, UCRL-JC-104258 preprint.
[53] Von Neumann, J.; Richtmyer, R.D., A method for the numerical calculation of hydrodynamic shocks, J. appl. phys., 21, (1950) · Zbl 0037.12002
[54] Crowley, W.P., Numerical methods in fluid dynamics, Lawrence livermore national laboratory, UCRL-51824, (1975) · Zbl 0581.76075
[55] Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. pure appl. math., 7, 159-193, (1954) · Zbl 0055.19404
[56] Harten, A.; Hyman, J.M.; Lax, P.D., On finite-difference approximations and entropy conditions for shocks, Comm. pure appl. math., 29, 297-322, (1976) · Zbl 0351.76070
[57] Noh, W.F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, Lawrence livermore national laboratory, UCRL-53669, (1985) · Zbl 0619.76091
[58] Wilkins, M.L., Use of artificial viscosity in multidimensional fluid dynamic calculations, J. comput. phys., 36, 381-403, (1980)
[59] Landshoff, R., A numerical method for treating fluid flow in the presence of shocks, Los alamos scientific laboratory, LA-1930, (1955)
[60] Richards, G.T., Derivation of a generalized von Neumann pseudo-viscosity with directional properties, Lawrence radiation laboratory, UCRL-14244, (1965)
[61] Kuropatenko, V.F., ()
[62] R. Christensen, Personal communication, Lawrence Livermore National Laboratory, 1989.
[63] R. Christensen, Personal communication, Lawrence Livermore National Laboratory, 1990.
[64] Van Leer, B., Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection, J. comput. phys., 23, 276-299, (1977) · Zbl 0339.76056
[65] Cherry, J.T.; Sack, S.; Maenchen, G.; Kransky, V., Two-dimensional stress-induced adiabatic flow, Lawrence livermore national laboratory, UCRL-50987, (1970)
[66] Burton, D.E.; Lettis, L.A.; Bryan, J.B.; Frary, N.R., Physics and numerics of the TENSOR code [incomplete preliminary documentation], Lawrence livermore national laboratory, UCID-19428, (1982)
[67] Schulz, W.D., Tensor artificial viscosity for numerical hydrodynamics, J. math. phys., 5, 1, (1964) · Zbl 0208.53703
[68] Washizu, K., Variational methods in elasticity and plasticity, (1982), Pergamon Oxford, UK · Zbl 0164.26001
[69] Flanagan, D.P.; Belytschko, T., A uniform strain heaxhedron and quadrilateral and orthogonal hourglass control, Internat. J. numer. methods engrg., 17, 679-706, (1981) · Zbl 0478.73049
[70] Belytschko, T.; Bachrach, W.E., Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Comput. methods appl. mech. engrg., 54, 279-301, (1986) · Zbl 0579.73075
[71] Margolin, L.G.; Pyun, J.J., A method for treating hourglass patterns, ()
[72] Kosloff, D.; Frazier, G., Treatment of hourglass patterns in low order finite element codes, Numer. anal. methods geomech., 2, 57-72, (1978)
[73] Belytschko, T.; Bindeman, L.P., Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems, Comput. methods appl. mech. engrg., 88, 311-340, (1991) · Zbl 0742.73019
[74] Simo, J.C.; Hughes, T.J.R., On the variational foundations of assumed strain methods, J. appl. mech., 53, 51-54, (1986) · Zbl 0592.73019
[75] Thompson, S.L., CSQ—A two dimensional hydrodynamic program with energy flow and material strength, Sandia laboratories, SAND74-0122, (1975)
[76] McGlaun, J.M.; Thompson, S.L.; Elrick, M.G., CTH: A three-dimensional shock wave physics code, ()
[77] Kikuchi, N.; Oden, J.T., Contact problems in elasticity: A study of variational inequalities and finite element methods for a class of contact problems in elasticity, SIAM stud., 8, (1986) · Zbl 0685.73002
[78] Benson, D.J.; Hallquist, J.O., A single surface contact algorithm for the post-buckling analysis of shell structures, Comput. methods appl. mech. engrg., 78, 141-163, (1990) · Zbl 0708.73079
[79] Chaudhary, A.B.; Bathe, K.J., A Lagrange multiplier segment procedure for solution of three dimensional contact problems, U.S. army ballistic research laboratory, BRL-CR-544, (1985)
[80] Zhong, Z.H., On contact-impact problems, ()
[81] Belytschko, T.; Neal, M.O., Contact-impact by the pinball algorithm with penalty, projection, and Lagrangian methods, () · Zbl 0825.73984
[82] Simo, J.C.; Wriggers, P.; Taylor, R.L., A perturbed Lagrangian formulation for the finite element solution of contact problems, Comput. methods appl. mech. engrg., 50, 163-180, (1985) · Zbl 0552.73097
[83] R. Christensen, Personal communication, Lawrence Livermore National Laboratory, 1990.
[84] Belytschko, T.; Lin, J.I., A new interaction algorithm with erosion for EPIC-3, U.S. army ballistic research laboratory, BRL-CR-540, (1985)
[85] Carpenter, N.J.; Taylor, R.L.; Katona, M.G., Lagrange constraints for transient surface contact, Internat. J. numer. methods engrg., 32, 103-128, (1991) · Zbl 0763.73053
[86] T. Belytschko, Personal communication, Northwestern University, 1990.
[87] Nilsson, L.; Zhong, Z.H.; Oldeburg, M., Analysis of shell structures subjected to contacts-impacts, ()
[88] Hallquist, J.O.; Goudreau, G.L.; Benson, D.J., Sliding interfaces with contact-impact in large-scale Lagrangian computations, Comput. methods appl. mech. engrg., 51, 107-137, (1985) · Zbl 0567.73120
[89] Bertholf, L.D.; Benzley, S.E., TOODY II, A computer program for two-dimensional wave propagation, ()
[90] Johnson, N.J.; Addessio, F.L.; Baumgardner, J.R.; Kashiwa, B.A.; Rauenzahn, R.M., CAVEAT; current extensions and future capabilities, (1990), Los Alamos National Laboratory
[91] A finite element method for a class of contact-impact problems, Comput. methods appl. mech. engrg., 8, 249-276, (1976) · Zbl 0367.73075
[92] Key, S.W., HONDO—A finite element computer program for the large deformation response of axisymmetric solids, Sandia national laboratory, report 74-0039, (1974)
[93] Stecher, F.P.; Johnson, G.C., Lagrangian computations for projectile penetration into thick plates, (), G00240.
[94] Hicks, D.L., Stability analysis of WONDY for a special case of Maxwell’s law, Mech. comp., 32, 1123, (1978) · Zbl 0388.76006
[95] Hallquist, J.O.; Benson, D.J., DYNA3D User’s manual (nonlinear dynamic analysis of structures in three dimensions), Lawrence livermore national laboratory, UCID-19592, (1987), Rev. 3
[96] J.A. Trangenstein and P. Colella, A higher-order Godunov method for modeling finite deformation in elastic-plastic solids, Comm. Pure Appl. Math., submitted. · Zbl 0714.73027
[97] Colella, P.; Woodward, P., The piecewise parabolic method (PPM) for gas dynamical simulations, J. comput. phys., 54, 174-201, (1984) · Zbl 0531.76082
[98] Dukowicz, J.K., A general, non-iterative Riemann solver for Godunov’s method, J. comput. phys., 61, 119-137, (1985) · Zbl 0629.76074
[99] Harten, A., ENO schemes with subcell resolution, J. comput. phys., 83, 148-184, (1989) · Zbl 0696.65078
[100] Addessio, F.L.; Johnson, J.N., CAVEAT material model, Los alamos national laboratory memorandum, (7 August 1985)
[101] Trangestein, J.A., A second-order algorithm for the dynamic response of soils, ()
[102] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 171-209, (1990) · Zbl 0694.65041
[103] Dukowicz, J.K.; Cline, M.C.; Addessio, F.L., A general topology Godunov method, J. comput. phys., 82, 29-63, (1989) · Zbl 0665.76032
[104] Dukowicz, J.K.; Meltz, B.J.A., Vorticity error in multidimensional Lagrangian codes, Los alamos national laboratory, LA-UR-90-697, (1990), (revised)
[105] Roe, P.L., Characteristic-based schemes for the Euler equations, Annu. rev. fluid mech., 18, 337-365, (1986) · Zbl 0624.76093
[106] Hancock, S., Numerical studies of a one-dimensional Lagrangian Godunov scheme, ()
[107] Kashiwa, B.; Lee, W.H., Comparisons between the cell-centered and staggered mesh Lagrangian hydrodynamics, (), submitted.
[108] Addessio, F.L.; Baumgardner, J.R.; Dukowicz, J.K.; Johnson, N.L.; Kashiwa, B.A.; Rauenzahn, R.M.; Zemach, C., CAVEAT: A computer code for fluid dynamics problems with large distortion and internal slip, (1990), Los Alamos National Laboratory
[109] Lax, P.; Wendroff, B., Systems of conservation laws, Comm. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802
[110] Davis, S.F., SIAM J. sci. statist. comput., 8, 1-18, (1987)
[111] Particle methods in fluid dynamics and plasma physics, (), (1)
[112] Harlow, F.H., PIC and its progeny, Comput. phys. comm., 48, 1-10, (1988)
[113] Brackbill, J.U.; Kothe, D.B.; Ruppel, H.M., FLIP: A low-dissipation, particle-in-cell method for fluid flow, Comput. phys. comm., 48, 25-38, (1988)
[114] Benz, W., Smooth particle hydrodynamics: A review, Harvard-Smithsonian center for astrophysics preprint series no. 2884, (1989)
[115] Fritts, M.J.; Crowley, W.P.; Trease, H., The free-Lagrange method, () · Zbl 0573.00014
[116] Hageman, L.J.; Wilkins, D.E.; Sedgwick, R.T.; Waddell, J.L., HELP, A multimaterial Eulerian program for compressible fluid and elastic-plastic flows in two space dimensions and time, System, science and software, SSS-R-75-2654, (1975)
[117] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. methods appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093
[118] Sod, G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 27, 1-31, (1978) · Zbl 0387.76063
[119] Roache, P.J., Computational fluid dynamics, (1976), Hermosa Publishers
[120] Chorin, A.J., Flame advection and propagation algorithms, J. comput. phys., 35, 1-11, (1980) · Zbl 0425.76086
[121] Couch, R.; Albright, E.; Alexander, N., The JOY computer code, Lawrence livermore national laboratory, UCID-19688, (1983)
[122] Benson, D.J., An efficient, accurate, simple ALE method for nonlinear finite element programs, Comput. methods appl. mech. engrg., 72, 305-350, (1989) · Zbl 0675.73037
[123] Sharp, R.W.; Barton, R.T., HEMP advection model, Lawrence livermore national laboratory, UCID-17809, (1981), Rev. 1
[124] Liu, W.K.; Chang, H.; Chen, J.-S.; Belytschko, T., Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua, Comput. methods appl. mech. engrg., 68, 259-310, (1988) · Zbl 0626.73076
[125] Woodward, P.R.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, Lawrence livermore national laboratory, UCRL-86952, (1982)
[126] Chorin, A.; Hughes, T.J.R.; McCracken, M.F.; Marsden, J.E., Product formulas and numerical algorithms, Comm. pure appl. math., 31, 205-256, (1978) · Zbl 0358.65082
[127] Simo, J.C.; Ortiz, M., A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comput. methods appl. mech. engrg., 49, 221-245, (1985) · Zbl 0566.73035
[128] Winslow, A.M., Equipotential zoning of two-dimensional meshes, Lawrence radiation laboratory, UCRL-7312, (1963)
[129] Predebon, W.W.; Anderson, C.E.; Walker, J.D., Inclusion of evolutionary damage measures in Eulerian wavecodes, (1990), preprint · Zbl 0735.73087
[130] Ghosh, S.; Kikuchi, N., An arbitrary Lagrangian-Eulerian finite element method for large deformation analysis of elastic-viscoplastic solids, Comput. methods appl. mech. engrg., 86, 127-188, (1991) · Zbl 0825.73687
[131] Liu, W.K.; Belytschko, T.; Chang, H., An arbitrary Lagrangian-Eulerian finite element method for path-dependent materials, Comput. methods appl. mech. engrg., 58, 227-245, (1986) · Zbl 0585.73117
[132] Lasaint, P.; Raviart, P.A., On a finite element method for solving the neutron transport equation, (), 89-123 · Zbl 0341.65076
[133] Chavent, G.G.; Cockburn, B.; Cohen, G.; Jaffre, J., A discontinuous finite element method for nonlinear hyperbolic equations, (), 337-342 · Zbl 0605.65055
[134] Caussignac, P.; Touzani, R., Solution of three-dimensional boundary layer equations by a discontinuous finite element method, part I: numerical analysis of a linear model problem, Comput. methods appl. mech. engrg., 78, 249-271, (1990) · Zbl 0706.76037
[135] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[136] Hughes, T.J.R.; Mallet, M.; Mizukami, M., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. methods appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074
[137] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[138] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[139] Mizukami, A.; Hughes, T.J.R., A Petrov-Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle, Comput. methods appl. mech. engrg., 50, 181-193, (1985) · Zbl 0553.76075
[140] D.J. Benson, Momentum advection on a staggered mesh, J. Comput. Phys., in press. · Zbl 0758.76038
[141] McGlaun, Personal communication, Sandia National Laboratories, 1990.
[142] Book, D.L.; Boris, J.P., Flux-corrected transport I. SHASTA, A transport algorithm that works, J. comput. phys., 11, 38-69, (1973) · Zbl 0251.76004
[143] Van Leer, B., Toward the ultimate conservative difference scheme. II. monotonicity and conservation combined in a second-order scheme, J. comput. phys., 14, 361-370, (1974) · Zbl 0276.65055
[144] Roe, P.L., Some contributions to the modeling of discontinuous flows, (), 354-359
[145] Chäkravarthy, S.R.; Osher, S., Computing with high resolution upwind schemes for hyperbolic equations, large-scale computations in fluid mechanics, Lectures in appl. math., 22, 57-86, (1985)
[146] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357-393, (1983) · Zbl 0565.65050
[147] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. numer. anal., 21, 995-1011, (1984) · Zbl 0565.65048
[148] Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. numer. anal., 21, 217-235, (1984) · Zbl 0592.65069
[149] Dahlquist, G.; Bjorck, A., Numerical methods, (1974), Prentice Hall Englewood Cliffs, NJ
[150] Shu, C.W., Numerical experiments on the accuracy of ENO and modified ENO schemes, J. sci. comput., 5, 127-149, (1990) · Zbl 0732.65085
[151] Rogerson, A.M.; Meiburg, E., A numerical study of the convergence properties of ENO schemes, J. sci. comput., 5, 151-167, (1990) · Zbl 0732.65086
[152] Harten, A.; Osher, S., Uniformly high-order accurate nonoscillatory schemes, SIAM J. numer. anal., 24, 279-309, (1987) · Zbl 0627.65102
[153] Huynh, H.T., Second-order accurate nonoscillatory schemes for scalar conservation laws, NASA technical memorandum 102010, (1989) · Zbl 0728.76074
[154] Huynh, H.T., Accurate monotone cubic interpolation, NASA technical memorandum 103789, (1991) · Zbl 0772.65004
[155] Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, J. comput. phys., 31, 335-362, (1979) · Zbl 0416.76002
[156] Löhner, R., An adaptive finite element scheme for transient problems in CFD, Comput. methods appl. mech. engrg., 61, 323-338, (1987) · Zbl 0611.73079
[157] Oden, J.T.; Strouboulis, T.; Devloo, P., Adaptive finite element methods for the analysis of inviscid compressible flow: part I. fast refinement/unrefinement and moving mesh methods for unstructured meshes, Comput. methods appl. mech. engrg., 59, 327-362, (1986) · Zbl 0593.76080
[158] Smolarkiewicz, P.K., A fully multidimensional positive definite advection transport algorithm with small implicit diffusion, J. comput. phys., 54, 325-362, (1984)
[159] Margolin, L.G.; Smolarkiewicz, P.K., Antidiffusive velocities for multipass donor cell advection, Lawrence livermore national laboratory, UCID-21866, (1989) · Zbl 0922.76255
[160] Bell, J.B.; Dawson, C.N.; Shibin, G.R., An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions, J. comput. phys., 74, 1-24, (1988) · Zbl 0684.65088
[161] Eidelman, S.; Colella, P.; Shreeve, R.P.S., Application of the Godunov method and higher order extensions of the Godunov method to cascade flow modeling, () · Zbl 0555.76016
[162] Van Leer, B., Multidimensional explicit difference schemes for hyperbolic conservation laws, (), 493
[163] Benson, D.J., Vectorization techniques for explicit arbitrary Lagrangian-Eulerian calculations, Comput. methods appl. mech. engrg., 96, 303-328, (1992)
[164] L.G. Margolin, Personal communication, 1989.
[165] Margolin, L.G.; Beason, C.W., Remapping on the staggered mesh, ()
[166] Trefethen, L.N., Group velocity in finite difference schemes, SIAM rev., 24, 2, (1982) · Zbl 0487.65055
[167] Sandier, I.S.; Rubin, D., An algorithm and a modular subroutine for the cap model, Internat. J. numer. anal. methods geomech., 3, 173-186, (1979) · Zbl 0393.73002
[168] Hyman, J.M., Numerical methods for tracking interfaces, Physics D, 12, 396-407, (1984) · Zbl 0604.65092
[169] Oran, E.S.; Boris, J.P., Numerical simulation of reactive flow, (1987), Elsevier New York · Zbl 0762.76098
[170] Johnson, W.E., TOIL (A two-material version of the OIL code), General atomic report GAMD-8073, (1967)
[171] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. phys., 39, 201-225, (1981) · Zbl 0462.76020
[172] Noh, W.F.; Woodward, P., SLIC (simple line interface calculation), () · Zbl 0382.76084
[173] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[174] McMaster, W.H., Computer codes for fluid-structure interactions, ()
[175] Youngs, D.L., Time dependent multi-material flow with large fluid distortion, (), 273-285 · Zbl 0537.76071
[176] Youngs, D.L., An interface tracking method for a 3D Eulerian hydrodynamics code, AWRE design mathematics division, AWRE/44/92/35, (1987)
[177] Zienkiewicz, O.C.; Xi-Kui, L.; Nakazawa, S., Iterative solution of mixed problems and stress recovery procedures, Comput. appl. numer. methods, 1, 3-9, (1985) · Zbl 0586.73127
[178] Milgram, M.S., Does a point Lie inside a polygon, J. comput. phys., 84, 134-144, (1989) · Zbl 0682.68054
[179] Dukowicz, J.K., Conservative rezoning (remapping) for general quadrilateral meshes, J. comput. phys., 54, 411-424, (1984) · Zbl 0534.76008
[180] Dukowicz, J.K.; Kodis, J.W., Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations, SIAM J. sci. statist. comput., 5, 305-321, (1987) · Zbl 0644.76085
[181] Thorn, B.J.; Holdridge, D.B., The TOOREZ Lagrangian rezoning code, Sandia national laboratories, SLA-73-1057, (1973)
[182] Lee, W.H.; Kwak, D., PIC method for a two-dimensional elastic-plastic-hydro code, Comput. phys. comm., 48, 11-16, (1986)
[183] Lee, W.H.; Painter, J.W., Material void opening computation using particle method, ()
[184] Belytschko, T.; Fish, J.; Engelmann, B.E., A finite element with embedded localization zones, Comput. methods appl. mech. engrg., 70, 59-89, (1988) · Zbl 0653.73032
[185] Thompson, J.F.; Warsi, Z.U.A.; Mastin, C.W., Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review, J. comput. phys., 47, 1-108, (1982) · Zbl 0492.65011
[186] Thompson, J.F.; Warsi, Z.U.A.; Mastin, C.W., Numerical grid generation foundations and applications, (1985), North-Holland New York · Zbl 0598.65086
[187] Bell, J.B.; Colella, P.; Trangenstein, J.A.; Welcome, M., Adaptive mesh refinement on moving quadrilateral grids, ()
[188] ()
[189] Winslow, A.M.; Barton, R.T., Rescaling of equipotential smoothing, Lawrence livermore national laboratory, UCID-19486, (1982)
[190] Winslow, A.M., Adaptive mesh zoning by the equipotential method, Lawrence livermore national laboratory, UCID-19062, (1981)
[191] Brackbill, J.U.; Saltzman, J.S., Adaptive zoning for singular problems in two dimensions, J. comput. phys., 46, 3, (1982) · Zbl 0489.76007
[192] Giannakopoulos, A.E.; Engel, A.J., Directional control in grid generation, J. comput. phys., 74, 422-439, (1988) · Zbl 0637.65120
[193] Fish, J.; Belytschko, T., Elements with embedded localization zones for large deformation problems, Comput. & structures, 30, 247-256, (1988) · Zbl 0667.73033
[194] C.E. Anderson, P.E. O’Donoghue and D. Skerhut, A mixture theory approach for the shock response of composite materials, J. Composite Mater., in press.
[195] N. Johnson, Personal communication, Los Alamos National Laboratory, 1990.
[196] Harlow, F.H.; Amsden, A.A., Flow of interpenetrating material phases, J. comput. phys., 18, 440-464, (1975) · Zbl 0322.76041
[197] Drumheller, D.S., Hypervelocity impact of mixtures, (), (1-4)
[198] Johnson, G.R.; Stryk, R.A.; Dodd, J.G., Dynamic Lagrangian computations for solids, with variable nodal connectivity for severe distortions, Internat. J. numer. methods engrg., 23, 509-522, (1986) · Zbl 0585.73118
[199] Evans, M.E.; Harlow, F.H., The particle-in-cell method for hydrodynamic calculations, Los alamos national laboratory, LA-2139, (1957)
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.