×

Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua. (English) Zbl 0626.73076

The fundamental arbitrary Lagrangian-Eulerian (ALE) mechanics and its finite element formulation are given. The tangential stiffness matrix, which is shown to be composed of the linearized material response matrix, the geometrical stiffness matrix, and the ALE transport matrix are derived from a consistent linearization procedure. Various numerical methods for the ALE finite element equations are then presented, and several examples are analyzed to examine some features of the method.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
76M99 Basic methods in fluid mechanics
74B20 Nonlinear elasticity
74-04 Software, source code, etc. for problems pertaining to mechanics of deformable solids

Software:

DYNA3D; LS-DYNA
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Belytschko, T.; Stolarski, H.; Liu, W.K.; Carpenter, N.; Ong, S.-J., Stress projection for membrane and shear locking in shell finite elements, Comput. meths. appl. engrg., 51, 221-258, (1985) · Zbl 0581.73091
[2] Belytschko, T.; Ong, S.-J.; Liu, W.K.; Kennedy, J.M., Hourglass control in linear and nonlinear problems, Comput. meths. appl. mech. engrg., 43, 251-276, (1984) · Zbl 0522.73063
[3] Belytschko, T., An overview of semidiscretization and time integration procedures, (), 1-63
[4] Belytschko, T.; Flanagan, D.P.; Kennedy, J.M., Finite element methods with user-controlled meshes for fluid-structure interaction, Comput. meths. appl. mech. engrg., 33, 669-688, (1982) · Zbl 0492.73089
[5] Belytschko, T.; Kennedy, J.M.; Schoeberle, D.F., Quasi-Eulerian finite element formulation for fluidstructure interaction, ASME J. pressure vessel technol., 102, 62-69, (1980)
[6] Belytschko, T.; Kennedy, J.M., Computer models for subassembly simulation, Nuclear engrg. design, 49, 17-38, (1978)
[7] 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. meths. appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[8] Christie, I.; Griffiths, D.F.; Mitchell, F.R.; Zienkiewicz, O.C., Finite element methods for second order differential equations with significant first derivatives, Internat. J. numer. meths. engrg., 10, 1389-1396, (1976) · Zbl 0342.65065
[9] Dendy, J.E., Two methods of Galerkin type achieving optimum L2 rates of convergence for first order hyperbolics, Internat. J. numer. meths. engrg., 11, 637-653, (1974) · Zbl 0293.65077
[10] Derbalian, K.A.; Lee, E.H.; Mallet, R.L.; McMeeking, R.M., Finite element metal forming analysis with spacially fixed mesh, (), 39-47
[11] Donea, J., Arbitrary Lagrangian-Eulerian finite element methods, (), 473-516
[12] Haber, R.B.; Koh, H.M., Explicit expressions for energy release rates using virtual crack extensions, Internat. J. numer. meths. engrg., 21, 301-315, (1985) · Zbl 0551.73091
[13] Haber, R.B.; Abel, J.F., Contact-slip analysis using mixed displacements, ASCE J. engrg. mech., 109, 411-429, (1983)
[14] Hallquist, J.O.; Benson, D.J., ()
[15] Heinrich, J.C.; Huyakorn, P.S.; Zienkiewicz, O.C.; Mitchell, A.R., An upwind finite element scheme for two-dimensional convective transport, Internat. J. numer. meths. engrg., 11, 131-145, (1977) · Zbl 0353.65065
[16] Hirt, C.W.; Amsden, A.A.; Cook, J.L., An arbitrary Lagrangian Eulerian computing method for all flow speeds, J. comput. phys., 14, 227-253, (1974) · Zbl 0292.76018
[17] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. meths. appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093
[18] Hughes, T.J.R.; Liu, W.K.; Zimmerman, T.K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. meths. appl. mech. engrg., 29, 329-349, (1981) · Zbl 0482.76039
[19] Hughes, T.J.R.; Winget, J.M., Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis, Internat. J. numer. meths. engrg., 16, 1862-1867, (1980) · Zbl 0463.73081
[20] Hughes, T.J.R.; Liu, W.K., Implicit-explicit finite elements in transient analysis, J. appl. mech., 45, 371-378, (1978) · Zbl 0392.73076
[21] Hughes, T.J.R., A simple scheme for developing upwind finite elements, Internat. J. numer. meths. engrg., 12, 1359-1365, (1978) · Zbl 0393.65044
[22] Krieg, R.D.; Key, S.W., Implementation of a time independent plasticity theory into structural computer programs, (), 125-137
[23] Liu, W.K.; Ong, J.S.; Uras, R.A., Finite element stabilization matrices—a unification approach, Comput. meths. appl. mech. engrg., 53, 13-46, (1985) · Zbl 0553.73065
[24] Liu, W.K.; Belytschko, T.; Ong, J.S.; Law, E.S., The use of stabilization matrices in nonlinear analysis, (), Engrg. comput., 2, 47-55, (1985)
[25] Liu, W.K., Notes on advanced finite element methods, (1984), Department of Mechanical and Nuclear Engineering, Northwestern University Evanston, IL
[26] Liu, W.K.; Ma, D., Computer implementation aspects for fluid structure interaction problems, Comput. meths. appl. mech. engrg., 31, 129-148, (1982) · Zbl 0478.73061
[27] Liu, W.K., Finite element procedures for fluid-structure interactions and applications to liquid storage tanks, Nuclear engrg. design, 65, 221-238, (1981)
[28] Liu, W.K.; Gvildys, J., Fluid-structure interaction of tanks with an eccentric core barrel, Comput. meths. appl. mech. engrg., 58, 51-57, (1986) · Zbl 0595.73045
[29] Liu, W.K.; Law, E.S.; Lam, D.; Belytschko, T., Resultant-stress degenerated-shell element, Comput. meths. appl. mech. engrg., 55, 259-300, (1986) · Zbl 0587.73113
[30] Liu, W.K.; Belytschko, T.; Chang, H., An arbitrary Lagrangian-Eulerian finite element method for path-dependent materials, Comput. meths. appl. mech. engrg., 58, 227-246, (1986) · Zbl 0585.73117
[31] Lohner, R.; Morgan, K.; Zienkiewicz, O.C., The solution of nonlinear hyperbolic equation systems by the finite element method, Internat. J. numer. meths. fluids, 4, 1043-1063, (1984) · Zbl 0551.76002
[32] Malvern, L.E., Introduction to the mechanics of a continuous medium, (1965), Prentice-Hall Englewood Cliffs, NJ · Zbl 0044.40001
[33] Morton, K.W.; Parrott, A.K., Generalized Galerkin methods for first-order hyperbolic equations, J. comput. phys., 36, 249-270, (1980) · Zbl 0458.65098
[34] Noh, W.F., CEL: A time-dependent two-space-dimensional coupled eulerian-Lagrangian code, ()
[35] Pracht, W.E., Calculating three-dimensional fluid flows at all flow speeds with an eulerian-Lagrangian computing mesh, J. comput. phys., 17, 132-159, (1975) · Zbl 0294.76016
[36] Graff, K.F., Wave motion in elastic solids, (1975), Ohio State University Press Columbus, OH · Zbl 0314.73022
[37] Richtmyer, R.D.; Morton, K.W., Difference methods for initial-value problems, (1967), Interscience New York · Zbl 0155.47502
[38] Roache, P.J., Computational fluid dynamics, (1972), Hermosa Publishers Albuquerque, NM · Zbl 0251.76002
[39] Spalding, D.B., A novel finite difference formulation for differential expressions involving both first and second derivatives, Internat. J. numer. meths. engrg., 4, 551-559, (1972)
[40] Stein, L.R.; Gentry, R.A.; Hirt, C.W., Computational simulation of transient blast loading on three-dimensional structures, Comput. meths. appl. mech. engrg., 11, 57-74, (1977) · Zbl 0366.76027
[41] Trulio, J.G., ()
[42] Ziekiewicz, O.C., The finite element method, (1977), McGraw-Hill New York
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.