Structural Engineering and Mechanics

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A fourth order finite difference method applied to elastodynamics: Finite element and boundary element formulations

  • Souza, L.A. (CT/UEL - Universidade Estadual de Londrina) ;
  • Carrer, J.A.M. (COPPE/UFRJ - Universidade Federal do Rio de Janeiro) ;
  • Martins, C.J. (COPPE/UFRJ - Universidade Federal do Rio de Janeiro)
  • Received : 2003.04.02
  • Accepted : 2003.11.25
  • Published : 2004.06.25
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Abstract

This work presents a direct integration scheme, based on a fourth order finite difference approach, for elastodynamics. The proposed scheme was chosen as an alternative for attenuating the errors due to the use of the central difference method, mainly when the time-step length approaches the critical time-step. In addition to eliminating the spurious numerical oscillations, the fourth order finite difference scheme keeps the advantages of the central difference method: reduced computer storage and no requirement of factorisation of the effective stiffness matrix in the step-by-step solution. A study concerning the stability of the fourth order finite difference scheme is presented. The Finite Element Method and the Boundary Element Method are employed to solve elastodynamic problems. In order to verify the accuracy of the proposed scheme, two examples are presented and discussed at the end of this work.

Keywords

References

  1. Abramowitz, M. and Stegun, I.A. (1984), Handbooks of Mathematical Functions, Dover Publications, Inc, New York.
  2. Bathe, K.J. (1996), Finite Element Procedures, New Jersey, Prentice Hall Inc.
  3. Beskos, D.E. (1977), "Boundary element methods in dynamic analysis: Part II 1986-1996", Applied Mechanics Reviews, 50, 149-197. https://doi.org/10.1115/1.3101695
  4. Carrer, J.A.M. and Telles, J.C.F. (1992), "A boundary element formulation to solve transient dynamic elastoplastic problems", Comput. Struct., 45, 707-713. https://doi.org/10.1016/0045-7949(92)90489-M
  5. Cohen, G. and Joly, P. (1990), "Fourth order schemes for the heterogeneous acoustics equation", Comput. Meth. Eng., 80, 397-407. https://doi.org/10.1016/0045-7825(90)90044-M
  6. Cook, R.D., Malkus, D.S. and Plesha, M.E. (1989), Concepts and Applications of Finite Element Analysis, New York, John Wiley and Sons.
  7. Hartmann, F. (1980), "Computing C-matrix in non-smooth boundary points", in C.A. Brebbia (ed.), New Developments in Boundary Element Methods, 367-379, CML Publications Limited, Southampton.
  8. Hatzigeorgiou, G.D. and Beskos, D.E. (2001), "Transient dynamic response of 3-D elastoplastic structures by the D/BEM", Proc. XXIII Int. Conf. on the Boundary Element Method, (eds. D.E. Beskos, C.A. Brebbia, J.T. Katsikadelis, G.D. Manolis), Lemnos, Greece.
  9. Hilber, H.M., Hughes, T.J.R. and Taylor, R.L. (1977), "Improved numerical dissipation for time integration algorithms in structural dynamics", Int. J. Earthq. Eng. Struct. Dyn., 5, 283-292. https://doi.org/10.1002/eqe.4290050306
  10. Houbolt, J.C. (1974), "A recurrence matrix solution for the dynamic response of elastic aircraft", Journal of the Aeronautical Sciences, 17, 540-550.
  11. Kontoni, D.P.N. and Beskos, D.E. (1993), "Transient dynamic elastoplastic analysis by the dual reciprocity BEM", Engineering Analysis with Boundary Elements, 12, 1-16. https://doi.org/10.1016/0955-7997(93)90063-Q
  12. Kreyszig, E. (1999), Advanced Engineering Mathematics, John Wiley & Sons, Inc., 8th edition.
  13. Mansur, W.J. (1983), "A time-stepping technique to solve wave propagation problems using the boundary element method", Ph.D. Thesis, University of Southampton, England.
  14. Newmark, N.M. (1959), "A method of computation for structural dynamics", J. Eng. Mech. Div., ASCE, 85, 67- 94.
  15. Partridge, P.W., Brebbia, C.A. and Wrobel, L.C. (1992), The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, Boston.
  16. Souza, L.A. and Moura, C.A. (1997), "Fourth order finite difference for explicit integration in the time-domain of elastodynamic problems (in portuguese)", XVIII CILAMCE, Brasília, 1, 263-272.
  17. Telles, J.C.F. (1983), "On the application of the boundary element method to inelastic problems", Ph.D. Thesis, University of Southampton, England.
  18. Weaver, W. Jr. and Johnston, P.R. (1987), Structural Dynamics by Finite Elements, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
  19. Wilson, E.L., Farhoomand, I. and Bathe, K.J. (1973), "Nonlinear dynamic analysis of complex structures", Int. J. Earthq. Eng. Struct. Dyn., 1, 241-252.

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