Three-dimensional transient elastodynamic analysis by a space and time-discontinuous Galerkin finite element method

This study presents a space and time-discontinous Galerkin (TDG) finite element method for analyzing three-dimensional linear transient elastodynamic problems. This novel method uses both the displacements and velocities as basic unknowns and approximates them as piecewise linear functions which are continuous in space and discontinuous in time. The improved algorithm employs the Gauss-Seidel method to calculate iteratively the solutions that exist in the numerical implementation. Stability analyses of TDG method reveal that such a method retains the unconditionally stable behavior with greater efficiency than other direct time integration algorithms. In addition, numerical examples are presented, demonstrating that the proposed method is more accurate than several commonly used algorithms in structural dynamic applications.

[1]  John C. Houbolt,et al.  A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft , 1950 .

[2]  F. B. Ellerby,et al.  Numerical solutions of partial differential equations by the finite element method , by C. Johnson. Pp 278. £40 (hardback), £15 (paperback). 1988. ISBN 0-521-34514-6, 34758-0 (Cambridge University Press) , 1989, The Mathematical Gazette.

[3]  G. Hulbert.  Time finite element methods for structural dynamics , 1992 .

[4]  Isaac Fried,et al.  Finite-element analysis of time-dependent phenomena. , 1969 .

[5]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[6]  G. Hulbert.  A unified set of single-step asymptotic annihilation algorithms for structural dynamics , 1994 .

[7]  Thomas J. R. Hughes,et al.  Space-time finite element methods for second-order hyperbolic equations , 1990 .

[8]  T. Hughes,et al.  Space-time finite element methods for elastodynamics: formulations and error estimates , 1988 .

[9]  C. Chien,et al.  A particular integral BEM/time-discontinuous FEM methodology for solving 2-D elastodynamic problems , 2001 .

[10]  H. Saunders.  Book Reviews : NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS K.-J. Bathe and E.L. Wilson Prentice-Hall, Inc, Englewood Cliffs, NJ , 1978 .

[11]  J. Tinsley Oden,et al.  A GENERAL THEORY OF FINITE ELEMENTS II. APPLICATIONS , 1969 .

[12]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[13]  K. Bathe.  Finite Element Procedures , 1995 .

[14]  Edward L. Wilson,et al.  Nonlinear dynamic analysis of complex structures , 1972 .

[15]  T.-Y. Wu,et al.  An improved predictor/multi-corrector algorithm for a time-discontinuous Galerkin finite element method in structural dynamics , 2000 .

[16]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[17]  Nils-Erik Wiberg,et al.  STRUCTURAL DYNAMIC ANALYSIS BY A TIME‐DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD , 1996 .

[18]  R. B. Deal,et al.  Modern mathematics for the engineer , 1958 .

[19]  O. C. Zienkiewicz,et al.  The finite element method, fourth edition; volume 2: solid and fluid mechanics, dynamics and non-linearity , 1991 .

[20]  John Argyris,et al.  Finite Elements in Time and Space , 1969, The Aeronautical Journal (1968).