Space-time finite element methods for second-order hyperbolic equations☆

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Abstract

Space-time finite element methods are presented to accurately solve elastodynamics problems that include sharp gradients due to propagating waves. The new methodology involves finite element discretization of the time domain as well as the usual finite element discretization of the spatial domain. Linear stabilizing mechanisms are included which do not degrade the accuracy of the space-time finite element formulation. Nonlinear discontinuity-capturing operators are used which result in more accurate capturing of steep fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. The space-time finite element method possesses a firm mathematical foundation in that stability and convergence of the method have been proved. In addition, the formulation has been extended to structural dynamics problems and can be extended to higher-order hyperbolic systems.

References (92)

  • R. Bonnerot et al.

    A third order accurate discontinuous finite element method for the one-dimensional Stefan problem

    J. Comput. Phys.

    (1979)
  • T.J.R. Hughes et al.

    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.

    (1987)
  • C. Johnson et al.

    Finite element methods for linear hyperbolic problems

    Comput. Methods Appl. Mech. Engrg.

    (1984)
  • T.J.R. Hughes et al.

    Space-time finite element methods for elastodynamics: Formulations and error estimates

    Comput. Methods Appl. Mech. Engrg.

    (1988)
  • T.J.R. Hughes et al.

    A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advection-diffusion systems

    Comput. Methods Appl. Mech. Engrg.

    (1986)
  • T.J.R. Hughes et al.

    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.

    (1987)
  • T.J.R. Hughes et al.

    A mixed finite element formulation for Reissner-Mindlin plate theory: Uniform convergence of all high-order spaces

    Comput. Methods Appl. Mech. Engrg.

    (1988)
  • L.P. Franca et al.

    Two classes of mixed finite element methods

    Comput. Methods Appl. Mech. Engrg.

    (1988)
  • A.F.D. Loula et al.

    Stability, convergence and accuracy of a new finite element method for the circular arch problem

    Comput. Methods Appl. Mech. Engrg.

    (1987)
  • A.F.D. Loula et al.

    Mixed Petrov-Galerkin methods for the Timoshenko beam problem

    Comput. Methods Appl. Mech. Engrg.

    (1987)
  • T.J.R. Hughes et al.

    A new finite element formulation for computational fluid dynamics: II. Beyond SUPG

    Comput. Methods Appl. Mech. Eng.

    (1986)
  • T.J.R. Hughes et al.

    A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems

    Comput. Methods Appl. Mech. Engrg.

    (1986)
  • A.C. Galeão et al.

    A consistent approximate upwind Petrov-Galerkin methods for convection-dominated problems

    Comput. Methods Appl. Mech. Engrg.

    (1988)
  • H.M. Hilber

    Analysis and design of numerical integration methods in structural dynamics

  • H.M. Hilber et al.

    Improved numerical dissipation for time integration algorithms in structural dynamics

    Earthquake Engrg. Struct. Dyn.

    (1977)
  • H.M. Hilber et al.

    Collocation, dissipation and ‘overshoot’ for time integration schemes in structural dynamics

    Earthquake Engrg. Struct. Dyn.

    (1978)
  • T.J.R. Hughes

    Analysis of transient algorithms with particular reference to stability behavior

  • T.J.R. Hughes

    The Finite Element Method: Linear Static and Dynamic Finite Element Analysis

    (1987)
  • P. Hansbo

    Adaptivity and streamline diffusion procedures in the finite element method

  • I. Fried

    Finite element analysis of time-dependent phenomena

    AIAA J.

    (1969)
  • J.T. Oden

    A general theory of finite elements II. Applications

    Internat. J. Numer. Methods in Engrg.

    (1969)
  • M.E. Gurtin

    Variational principles for linear initial value problems

    Quart. Appl. Math.

    (1964)
  • C.D. Bailey

    Application of Hamilton's law of varying action

    AIAA J.

    (1975)
  • M. Baruch et al.

    Hamilton's principle, Hamilton's law—6n correct formulations

    AIAA J.

    (1982)
  • R.E. Nickell et al.

    Approximate solutions in linear, coupled thermoelasticity

    J. Appl. Mech.

    (1968)
  • R. Riff et al.

    Stability of time finite elements

    AIAA J.

    (1984)
  • R. Riff et al.

    Time finite element discretization of Hamilton's law of varying action

    AIAA J.

    (1984)
  • T.E. Simkins

    Unconstrained variational statements for initial and boundary-value problems

    AIAA J.

    (1978)
  • T.E. Simkins

    Finite elements for initial value problems in dynamics

    AIAA J.

    (1981)
  • D.R. Smith et al.

    When is Hamilton's principle an extremum principle?

    AIAA J.

    (1974)
  • C.V. Smith

    Discussion on ‘Hamilton, Ritz and plastodynamics’

    J. Appl. Mech.

    (1977)
  • J.R. Yu et al.

    Analysis of heat conduction in solids by space-time finite element method

    Internat. J. Numer. Methods Engrg.

    (1985)
  • D.A. Peters et al.

    hp-version finite elements for the space-time domain

    Comput. Mech.

    (1988)
  • C. Bajer

    Triangular and tetrahedral space-time finite elements in vibrational analysis

    Internat. J. Numer. Methods Engrg.

    (1986)
  • C. Bajer

    Notes on the stability of non-rectangular space-time finite elements

    Internat. J. Numer. Methods Engrg.

    (1987)
  • R. Bonnerot et al.

    A second order finite element method for the one-dimensional Stefan problem

    Internat. J. Numer. Methods Engrg.

    (1974)

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    ☆

    This research was sponsored by the U.S. Office of Naval Research under Contract Number N00014-84-K-0600.

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    Copyright © 1990 Published by Elsevier B.V.