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A Time-Domain Finite Element Formulation for Transient Dynamic Linear Elasticity

과도 선형 동탄성 문제의 시간영역 유한요소해석

  • Published : 2001.04.01

Abstract

Transient linear elastodynamic problems are numerically analyzed in a time-domain by the Finite Element Method, for which the variational formulation based upon the equations of motion in convolution integral is newly derived. This formulation is implicit and does not include the time derivative terms so that the computation procedure is simple and less assumptions are required comparing to the conventional time-domain dynamic numerical algorithms, being able to get the improved numerical accuracy and stability. That formulation is expanded using the semi-discrete approximation to obtain the finite element equations. In the temporal approximation, the time axis is divided equally and constant and linear time variations are assumed in those intervals. It is found that unconditionally stable numerical results are obtained in case of the constant time variation. Some numerical examples are given to show the versatility of the presented formulation.

Keywords

Transient;Dynamic Elasticity;Finite Element Method;Elastic Wave

References

  1. Timoshenko, S.P. and Goodier, J.N., 1970, Theory of Elasticity, 3rd Edn., McGraw-Hill, New York
  2. Mansur, W.J., 1983, A Time-stepping Technique to Solve Wave Propagation Problems Using the Boundary Element Method, Ph.D. Dissertation, University of Southhampton, U.K.
  3. Israil, A.S.M. and Banerjee, P.K., 1990, 'Advanced Development of Time-Domain BEM for Two-Dimensional Scalar Wave Propagation,' Int. J. Num. Eng., Vol. 29, pp. 1003-1020 https://doi.org/10.1002/nme.1620290507
  4. Chou, P.E. and Koenig, H.A., 1966, 'A Unified Approach to Cylindrical and Spherical Elastic Waves by Method of Characteristics,' Trans. ASME, J. Appl. Mech., pp. 159-167
  5. 김진석, 1998, '라플라스 변환을 이용한 단순한 지지보의 동적 변형률 해석,' 대한기계학회논문집, 제22권, 제10호, pp. 1858-1865
  6. Bedford, A. and Drumheller, D. S., 1994, Introduction to Elastic Wave Propagation, John Wiley, Chichester, Chap. 4
  7. Gurtin, M. E., 1964, 'Variational Principles for Linear Elastodynamics,' Archive for Rational Mechanics and Analysis, Vol. 16, pp. 234-250 https://doi.org/10.1007/BF00248489
  8. Achenbach, J. D., 1975, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, Chap. 2
  9. Lee, T.W. and Sim, W.J., 1992, 'Efficient Time-Domain Finite Element Analysis for Dynamic Coupled Thermoelasticity,' Computers and Structures, Vol. 45, No. 4, pp. 785-793 https://doi.org/10.1016/0045-7949(92)90496-M
  10. Ahmad, S., 1986, Linear and Nonlinear Dynamic Analysis by Boundary Element Method, Ph.D. Thesis, SUNY at Buffalo, Chap. VIII
  11. Reddy, J.N., 1993, An Introduction to the Finite Element Method, McGraw-Hill, New York, Chap. 10
  12. Zienkiewicz, O.C. and Taylor, R.L., 1991, The Finite Element Method(4th edn), Vol. 2, Dynamics and Nonlinearity, McGraw-Hill, London, Chap. 9-10
  13. Fu, C.C., 1970, 'A Method for the Numerical Integration of the Equations of Motion Arising From a Finite-Element Analysis,' Trans. ASME, J. Appl. Mech. pp. 599-605