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Space-time finite element methods for second-order hyperbolic equations. (English) Zbl 0754.73085
A modification of finite element method to solve elastodynamics problems is presented. The authors use finite elments to discretize the temporal domain as well as the spatial domain. Linear stabilizing mechanisms are included and nonlinear discontinuity-capturing operators are used to improve the performance of the algorithm in regions where the solution exhibits large gradients. Stability and convergence of the method are proved.
The method is illustrated by solving two test problems: the first is the problem of the impact against the rigid wall of a one-dimensional homogeneous elastic bar, the another is the wave propagation problem for a bar consisting of two different materials.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Hilber, H.M., Analysis and design of numerical integration methods in structural dynamics, ()
[2] Hilber, H.M.; Hughes, T.J.R.; Taylor, R.L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake engrg. struct. dyn., 5, 283-292, (1977)
[3] Hilber, H.M.; Hughes, T.J.R., Collocation, dissipation and ‘overshoot’ for time integration schemes in structural dynamics, Earthquake engrg. struct. dyn., 6, 99-118, (1978)
[4] Hughes, T.J.R., Analysis of transient algorithms with particular reference to stability behavior, (), 67-155
[5] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[6] Hansbo, P., Adaptivity and streamline diffusion procedures in the finite element method, () · Zbl 0874.76036
[7] Argyris, J.H.; Scharpf, D.W., Finite elements in time and space, Nucl. engrg. des., 10, 456-464, (1969)
[8] Fried, I., Finite element analysis of time-dependent phenomena, Aiaa j., 7, 1170-1173, (1969) · Zbl 0179.55001
[9] Oden, J.T., A general theory of finite elements II. applications, Internat. J. numer. methods in engrg., 1, 247-259, (1969) · Zbl 0263.73048
[10] Gurtin, M.E., Variational principles for linear initial value problems, Quart. appl. math., 22, 252-256, (1964) · Zbl 0173.37602
[11] Bailey, C.D., Application of Hamilton’s law of varying action, Aiaa j., 13, 1154-1157, (1975) · Zbl 0323.70020
[12] Bailey, C.D., The method of Ritz applied to the equations of Hamilton, Comput. methods appl. mech. engrg., 7, 235-247, (1976) · Zbl 0322.70015
[13] Baruch, M.; Riff, R., Hamilton’s principle, Hamilton’s law—6^n correct formulations, Aiaa j., 20, 687-692, (1982) · Zbl 0484.70016
[14] Borri, M.; Ghiringhelli, G.L.; Lanz, M.; Mantegazza, P.; Merlini, T., Dynamic response of mechanical systems by a weak Hamiltonian formulation, Comput. & structures, 20, 495-508, (1985) · Zbl 0574.73091
[15] Howard, G.F.; Penny, J.E.T., The accuracy and stability of time domain finite element solutions, J. sound vibration, 61, 585-595, (1978) · Zbl 0407.73061
[16] Nickell, R.E.; Sackman, J.L., Approximate solutions in linear, coupled thermoelasticity, J. appl. mech., 35, 255-266, (1968) · Zbl 0159.55605
[17] Riff, R.; Baruch, M., Stability of time finite elements, Aiaa j., 22, 1171-1173, (1984) · Zbl 0568.73081
[18] Riff, R.; Baruch, M., Time finite element discretization of Hamilton’s law of varying action, Aiaa j., 22, 1310-1318, (1984) · Zbl 0562.73060
[19] Simkins, T.E., Unconstrained variational statements for initial and boundary-value problems, Aiaa j., 16, 559-563, (1978) · Zbl 0377.73006
[20] Simkins, T.E., Finite elements for initial value problems in dynamics, Aiaa j., 19, 1357-1362, (1981) · Zbl 0471.70008
[21] Smith, D.R.; Smith, C.V., When is Hamilton’s principle an extremum principle?, Aiaa j., 12, 1573-1576, (1974) · Zbl 0296.70010
[22] Smith, C.V., Discussion on ‘hamilton, Ritz and plastodynamics’, J. appl. mech., 44, 796-797, (1977)
[23] Wilson, E.L.; Nickell, R.E., Application of finite element method to heat conduction analysis, Nucl. engrg. des., 4, 276-286, (1966)
[24] Yu, J.R.; Hsu, T.R., Analysis of heat conduction in solids by space-time finite element method, Internat. J. numer. methods engrg., 21, 2001-2012, (1985) · Zbl 0571.73119
[25] Peters, D.A.; Izadpanah, A.P., hp-version finite elements for the space-time domain, Comput. mech., 3, 73-88, (1988) · Zbl 0627.73081
[26] Bajer, C., Triangular and tetrahedral space-time finite elements in vibrational analysis, Internat. J. numer. methods engrg., 23, 2031-2048, (1986) · Zbl 0597.73080
[27] Bajer, C., Notes on the stability of non-rectangular space-time finite elements, Internat. J. numer. methods engrg., 24, 1721-1739, (1987) · Zbl 0625.73084
[28] Bonnerot, R.; Jamet, P., A second order finite element method for the one-dimensional Stefan problem, Internat. J. numer. methods engrg., 8, 811-820, (1974) · Zbl 0285.65071
[29] Bonnerot, R.; Jamet, P., Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements, J. comput. phys., 25, 163-181, (1977) · Zbl 0364.65091
[30] Bruch, J.C.; Zyvoloski, G., A finite element weighted residual solution to one-dimensional field problems, Internat. J. numer. methods engrg., 6, 577-585, (1973) · Zbl 0258.76064
[31] Bruch, J.C.; Zyvoloski, G., Transient two-dimensional heat conduction problems solved by the finite element method, Internat. J. numer. methods engrg., 8, 481-494, (1974) · Zbl 0281.65060
[32] Morandi Cecchi, M.; Cella, A., A Ritz-Galerkin approach to heat conduction: method and results, (), 767-768
[33] Cella, A.; Lucchesi, M., Space-time finite elements for the wave propagation problem, Meccanica, 10, 168-170, (1975) · Zbl 0365.73018
[34] Cella, A.; Lucchesi, M.; Pasquinelli, G., Space-time elements for the shock wave propagation problem, Internat. J. numer. methods engrg., 15, 1475-1488, (1980) · Zbl 0466.76062
[35] Cheung, Y.K.; Tham, L.G., Time-space finite elements for unsaturated flow through porous media, (), 251-256
[36] Chung, K.S., The fourth-dimension concept in the finite element analysis of transient heat transfer problems, Internat. J. numer. methods engrg., 17, 315-325, (1981) · Zbl 0445.65112
[37] Jamet, P.; Bonnerot, R., Numerical solution of the Eulerian equations of compressible flow by a finite element method which follows the free boundary and the interfaces, J. comput. phys., 18, 21-45, (1975) · Zbl 0303.76030
[38] Kacprzyk, Z.; Lewiński, T., Comparison of some numerical integration methods for the equations of motion of systems with a finite number of degrees of freedom, Engrg. transactions, 31, 213-240, (1983) · Zbl 0542.73105
[39] Kok, A.W.M., Pulses in finite elements, (), 286-301
[40] Lewis, D.L.; Lund, J.; Bowers, K.L., The space-time sinc-Galerkin method for parabolic problems, Internat. J. numer. methods engrg., 24, 1629-1644, (1987) · Zbl 0643.65068
[41] Nguyen, H.; Reynen, J., A space-time least-square finite element scheme for advection-diffusion equations, Comput. methods appl. mech. engrg., 42, 331-342, (1984) · Zbl 0517.76089
[42] Papanastasiou, A.C.; Scriven, L.E.; Macosko, C.W., Bubble growth and collapse in viscoelastic liquids analyzed, J. non-Newtonian fluid mech., 16, 53-75, (1984) · Zbl 0559.76012
[43] Varoḡlu, E.; Finn, W.D.L., Space-time finite elements incorporating characteristics for the Burgers’ equation, Internat. J. numer. methods engrg., 16, 171-184, (1980) · Zbl 0449.76076
[44] Argyris, J.H.; Vaz, L.E.; Willam, K.J., Higher order methods for transient diffusion analysis, Comput. methods appl. mech. engrg., 12, 243-278, (1977) · Zbl 0365.65061
[45] Kawahara, M.; Hasegawa, K., Periodic Galerkin finite element method of tidal flow, Internat. J. numer. methods engrg., 12, 115-127, (1978) · Zbl 0368.76049
[46] Kujawski, J.; Desai, C.S., Generalized time finite element algorithm for non-linear dynamic problems, Engrg. computations, 1, 247-251, (1984)
[47] Zienkiewicz, O.C.; Parekh, C.J., Transient field problems—two and three dimensional analysis by isoparametric finite elements, Internat. J. numer. methods engrg., 2, 61-71, (1970) · Zbl 0262.73072
[48] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill London · Zbl 0435.73072
[49] Zienkiewicz, O.C., A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach, Earthquake engrg. struct. dyn., 5, 413-418, (1977)
[50] Reed, W.H.; Hill, T.R., Triangular mesh methods for the neutron transport equation, ()
[51] Lesaint, P.; Raviart, P.-A., On a finite element method for solving the neutron transport equation, (), 89-123
[52] Bonnerot, R.; Jamet, P., A third order accurate discontinuous finite element method for the one-dimensional Stefan problem, J. comput. phys., 32, 145-167, (1979) · Zbl 0415.65058
[53] Delfour, M.; Hager, W.; Trochu, F., Discontinuous Galerkin methods for ordinary differential equations, Math. comp., 36, 455-473, (1981) · Zbl 0469.65053
[54] Hughes, T.J.R.; Franca, L.P.; Mallet, M., A new finite element formulation for computational fluid dynamics: VI. convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. methods appl. mech. engrg., 63, 97-112, (1987) · Zbl 0635.76066
[55] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M.; Johan, Z.; Shakib, F., The Galerkin/least-squares method for advective-diffusive equations, (), 75-99
[56] Jamet, P., Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. numer. anal., 15, 912-928, (1978) · Zbl 0434.65091
[57] 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
[58] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, () · Zbl 0618.65105
[59] Johnson, C., Streamline diffusion methods for problems in fluid mechanics, (), 251-261
[60] Johnson, C.; Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. comp., 47, 1-18, (1986) · Zbl 0609.76020
[61] Johnson, C., Numerical solutions of partial differential equations by the finite element method, (1987), Cambridge Univ. Press Cambridge
[62] Nävert, U., A finite element method for convection-diffusion problems, ()
[63] Shakib, F., Finite element analysis of the compressible Euler and Navier-Stokes equations, ()
[64] Thomée, V., Galerkin finite element methods for parabolic problems, (1984), Springer New York · Zbl 0528.65052
[65] Johnson, C., Error estimates and automatic time step control for numerical methods for stiff ordinary differential equations, ()
[66] Eriksson, K.; Johnson, C.; Lennblad, J., Optimal error estimates and adaptive time and space step control for linear parabolic problems, () · Zbl 0626.65101
[67] Eriksson, K.; Johnson, C., Error estimates and automatic time step control for nonlinear parabolic problems, I, SIAM J. numer. anal., 24, 12-23, (1987) · Zbl 0618.65104
[68] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems: I. A linear model problem, () · Zbl 0732.65093
[69] Johnson, C.; Nie, Y.-Y.; Thomée, V., An a posteriori error estimate and automatic time step control for a backward Euler discretization of a parabolic problem, ()
[70] Johnson, C.; Szepessy, A., On the convergence of streamline diffusion finite element methods for hyperbolic conservation laws, (), 75-91
[71] Johnson, C.; Szepessy, A., On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. comp., 49, 427-444, (1987) · Zbl 0634.65075
[72] Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, () · Zbl 0685.65086
[73] Szepessy, A., Convergence of the streamline diffusion finite element method for conservation laws, () · Zbl 0751.65061
[74] Hughes, T.J.R.; Marsden, J.E., Classical elastodynamics as a linear symmetric hyperbolic system, J. elasticity, 8, 97-110, (1978) · Zbl 0373.73015
[75] John, F., Finite amplitude waves in a homogeneous isotropic elastic solid, Commun. pure appl. math., 30, 421-446, (1977) · Zbl 0404.73023
[76] Hughes, T.J.R.; Hulbert, G.M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comput. methods appl. mech. engrg., 66, 339-363, (1988) · Zbl 0616.73063
[77] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimensional advection-diffusion systems, Comput. methods appl. mech. engrg., 58, 305-328, (1986) · Zbl 0622.76075
[78] Hughes, T.J.R.; Franca, L.P.; Harari, I.; Mallet, M.; Shakib, F.; Spelce, T.E., Finite element method for high-speed flows: consistent calculation of boundary flux, ()
[79] Hughes, T.J.R., Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Internat. J. numer. methods fluids, 7, 1261-1275, (1987) · Zbl 0638.76080
[80] Hughes, T.J.R.; Franca, L.P., A new finite element method for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. methods appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067
[81] Hughes, T.J.R.; Franca, L.P., A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all high-order spaces, Comput. methods appl. mech. engrg., 67, 223-240, (1988) · Zbl 0611.73077
[82] Franca, L.P.; Hughes, T.J.R., Two classes of mixed finite element methods, Comput. methods appl. mech. engrg., 69, 89-129, (1988) · Zbl 0651.65078
[83] Loula, A.F.D.; Franca, L.P.; Hughes, T.J.R.; Miranda, I., Stability, convergence and accuracy of a new finite element method for the circular arch problem, Comput. methods appl. mech. engrg., 63, 281-303, (1987) · Zbl 0607.73077
[84] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin methods for the Timoshenko beam problem, Comput. methods appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076
[85] Loula, A.F.D.; Miranda, I.; Hughes, T.J.R.; Franca, L.P., A successful mixed formulation for axisymmetric shell analysis employing discontinuous stress fields of the same order as the displacement field, (), 581-599
[86] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. methods appl. mech. eng., 54, 341-355, (1986) · Zbl 0622.76074
[87] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. methods appl. mech. engrg., 58, 329-336, (1986) · Zbl 0587.76120
[88] Johnson, C.; Szepessy, A., Shock-capturing streamline diffusion finite element methods for nonlinear conservation laws, (), 101-108
[89] do Carmo, E.G.Dutra; Galeão, A.C., A consistent formulation of the finite element method to solve convective-diffusive transport problems, Rev. brasileira ciênc. mec., 4, 309-340, (1986), (in Portuguese).
[90] Galeão, A.C.; do Carmo, E.G.Dutra, A consistent approximate upwind Petrov-Galerkin methods for convection-dominated problems, Comput. methods appl. mech. engrg., 68, 83-95, (1988) · Zbl 0626.76091
[91] Hulbert, G.M., Space-time finite element methods for second-order hyperbolic equations, () · Zbl 0769.70002
[92] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
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