Space-time finite element methods for elastodynamics: Formulations and error estimates. (English) Zbl 0616.73063

Space-time finite element methods are developed for classical elastodynamics. The approach employs the discontinuous Galerkin method in time and incorporates stabilizing terms of least-squares type. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Optimal error estimates are predicted, and confirmed numerically, for arbitrary combinations of displacement and velocity interpolations. The procedures developed are easily generalized to structural dynamics and a wide class of second-order hyperbolic problems.


74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
Full Text: DOI


[1] Argyris, J.H.; Scharpf, D.W., Finite elements in time and space, Nucl. engrg. design, 10, 456-464, (1969)
[2] Dahlquist, G., A special stability problem for linear multistep methods, Bit, 3, 27-43, (1963) · Zbl 0123.11703
[3] Delfour, M.; Hager, W.; Trochu, F., Discontinuous Galerkin methods for ordinary differential equations, Math. comp., 36, 455-473, (1981) · Zbl 0469.65053
[4] 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
[5] Eriksson, K.; Johnson, C.; Lennblad, J., Optimal error estimates and adaptive time and space step control for linear parabolic problems, () · Zbl 0626.65101
[6] L.P. Franca, T.J.R. Hughes, A.F.D. Loula and I. Miranda, A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation, Numer. Math. (to appear). · Zbl 0656.73036
[7] Fried, I., Finite element analysis of time-dependent phenomena, Aiaa j., 7, 1170-1173, (1969) · Zbl 0179.55001
[8] T.J.R. Hughes, Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Internat. J. Numer. Meths. Fluids (to appear. · Zbl 0638.76080
[9] Hughes, T.J.R.; Belytschko, T.; Liu, W.K., Convergence of an element-partitioned subcycling algorithm for the semidiscrete heat equation, Numer. meths. partial differential equations, 3, 131-137, (1987) · Zbl 0653.65054
[10] 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 multi-dimensional advective-diffusive systems, Comput. meths. appl. mech. engrg., 63, 97-112, (1987) · Zbl 0635.76066
[11] Hughes, T.J.R.; Hilber, H.M.; Taylor, R.L., A reduction scheme for problems of structural dynamics, Internat. J. solids and structures, 12, 749-767, (1976) · Zbl 0348.73029
[12] Hughes, T.J.R.; Marsden, J.E., Classical elastodynamics as a linear symmetric hyperbolic system, J. elasticity, 8, 97-110, (1978) · Zbl 0373.73015
[13] 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
[14] John, F., Finite amplitude waves in a homogeneous isotropic elastic solid, Comm. pure appl. math., 30, 421-446, (1977) · Zbl 0404.73023
[15] Johnson, C., Error estimates and automatic time step control for numerical methods for stiff ordinary differential equations, ()
[16] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. meths. appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[17] 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, ()
[18] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, () · Zbl 0618.65105
[19] Johnson, C.; Szepessy, A., On the convergence of streamline diffusion finite element methods for hyperbolic conservation laws, (), 75-91
[20] Lesaint, P.; Raviart, P.-A., On a finite element method for solving the neutron transport equation, (), 89-123
[21] 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. meths. appl. mech. engrg., 63, 281-303, (1987) · Zbl 0607.73077
[22] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin methods for the Timoshenko beam problem, Comput. meths. appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076
[23] 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, ()
[24] Mizukami, A., Variable explicit finite element methods for unsteady heat conduction equations, Comput. meths. appl. mech. engrg., 59, 101-109, (1986) · Zbl 0595.73130
[25] Nävert, U., A finite element method for convection-diffusion problems, ()
[26] Oden, J.T., A general theory of finite elements II. applications, Internat. J. numer. meths. engrg., 1, 247-259, (1969) · Zbl 0263.73048
[27] Stakgold, I., Green’s functions and boundary value problems, (1979), Wiley New York · Zbl 0421.34027
[28] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall, Englewood Cliffs, NJ · Zbl 0278.65116
[29] Wilson, E.L.; Nickell, R.E., Application of finite element method to heat conduction analysis, Nucl. engrg. design, 4, 1-11, (1966)
[30] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill London · Zbl 0435.73072
[31] Zienkiewicz, O.C., A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach, Earthquake engrg. structural dynamics, 5, 413-418, (1977)
[32] Zienkiewicz, O.C.; Parekh, C.J., Transient field problems—two and three dimensional analysis by isoparametric finite elements, Internat. J. numer. meths. engrg., 2, 61-71, (1970) · Zbl 0262.73072
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.