Structural Engineering and Mechanics

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Solution of the two-dimensional scalar wave equation by the time-domain boundary element method: Lagrange truncation strategy in time integration

  • Carrer, J.A.M. (Programa de Pos-Graduacao em Metodos Numericos em Engenharia, Universidade Federal do Parana) ;
  • Mansur, W.J. (Programa de Engenharia Civil, COPPE/UFRJ, Universidade Federal do Rio de Janeiro)
  • Received : 2005.08.24
  • Accepted : 2006.02.21
  • Published : 2006.06.20
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Abstract

This work presents a time-truncation scheme, based on the Lagrange interpolation polynomial, for the solution of the two-dimensional scalar wave problem by the time-domain boundary element method. The aim is to reduce the number of stored matrices, due to the convolution integral performed from the initial time to the current time, and to keep a compromise between computational economy and efficiency and the numerical accuracy. In order to verify the accuracy of the proposed formulation, three examples are presented and discussed at the end of the article.

Keywords

References

  1. Carrer, J.A.M. and Mansur, W.J. (1996), ' Time-domain BEM analysis for the 2D scalar wave equation: Initial conditions contributions to space and time derivatives', Int. J. Numer. Meth. Eng., 39, 2167-2188
  2. Carrer, J.A.M. and Mansur, W.J. (2002), ' Time-dependent fundamental solution generated by a not impulsive source in the boundary element method analysis of the 2D scalar wave equation ', Commun. Numer. Meth.Eng., 18, 277-285 https://doi.org/10.1002/cnm.487
  3. 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
  4. Demirel, V. and Wang, S. (1987),' Efficient boundary element method for two-dimensional transient wave propagation problems ', Appl. Mathematical Modelling, 11, 411-416 https://doi.org/10.1016/0307-904X(87)90165-X
  5. Dominguez, J. (1993), Boundary Elements in Dynamics, Computational Mechanics Publications, Southampton, Boston
  6. Gaul, L. and Schanz, M. (1999), ' A comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains ', Comput. Method Appl. Mech. Eng., 179,111-123 https://doi.org/10.1016/S0045-7825(99)00032-8
  7. Hadamard, J. (1952), Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York
  8. Hatzigeorgiou, G.D. and Beskos, D.E. (2001), ' Transient dynamic response of 3-D elastoplastic structures by the DIBEM ', Proc. of the XXIII International Conf.on the Boundary Element Method, (eds. D.E. Beskos, C.A Brebbia, J.T. Katsikadelis, G.D. Manolis), Lemnos, Greece
  9. 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
  10. de Lacerda, L.A, Wrobel, L.C. and Mansur, W.J. (1996), ' A boundary integral formulation for two-dimensional acoustic radiation in a subsonic uniform flow ', J. of the Acoustical Society of America, 100, 98-107 https://doi.org/10.1121/1.415871
  11. Manolis, G.D. (1983), ' A comparative study on three boundary element method approaches to problems in elastodynamics ', Int. J. Numer. Meth. Eng., 19, 73-91 https://doi.org/10.1002/nme.1620190109
  12. 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
  13. Mansur, J.W. and Carrer, J.A.M. (1993),' Two-dimensional transient BEM analysis for the scalar wave equation: Kernels ', Engineering Analysis with Boundary Elements, 12, 283-288 https://doi.org/10.1016/0955-7997(93)90055-P
  14. Mansur, W.J., Carrer, J.A.M. and Siqueira, E.F.N. (1998), ' Time discontinuous linear traction approximation in time domain BEM scalar wave propagation analysis ', Int. J. Numer. Meth. Eng., 42, 667-683 https://doi.org/10.1002/(SICI)1097-0207(19980630)42:4<667::AID-NME380>3.0.CO;2-8
  15. Mansur, W.J. and de Lima-Silva, W. (1992), ' Efficient time truncation in two-dimensional BEM analysis of transient wave propagation problems ', Earthq. Eng. Struct. Dyn., 21, 51-63 https://doi.org/10.1002/eqe.4290210104
  16. Morse, P.M. and Ingard, K.V. (1968), Theoretical Acoustics, McGraw-Hill, London
  17. Partridge, P.W., Brebbia, C.A. and Wrobel, L.C. (1992), The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, Boston
  18. Schanz, M. (2001), Wave Propagation in Viscoelastic and Poroelastic Continua, Lectures Notes in Applied Mechanics, Vol. 2, Springer-Verlag, Berlin, Heidelberg
  19. Soares Jr., D. and Mansur, W.J. (2004),' Compression of time generated matrices in two-dimensional timedomain elastodynamic BEM analysis ', Int. J. Numer. Meth. Eng., 61, 1209-1218 https://doi.org/10.1002/nme.1111
  20. Yu, G, Mansur,W.J., Carrer, J.A.M. and Gong, L. (1998),' A linear e method applied to 2D time-domain BEM ', Commun. Numer. Meth. Eng., 14, 1171-1179 https://doi.org/10.1002/(SICI)1099-0887(199812)14:12<1171::AID-CNM217>3.0.CO;2-G

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