Acta Mechanica

, Volume 229, Issue 4, pp 1893–1910 | Cite as

Transient dynamic analysis of generally anisotropic materials using the boundary element method

  • R. Q. Rodríguez
  • A. F. Galvis
  • P. Sollero
  • C. L. Tan
  • E. L. Albuquerque
Original Paper


This work presents the use of an explicit-form Green’s function for 3D general anisotropy in conjunction with the dual reciprocity boundary element method and the radial integration method to analyse elastodynamic problems using BEM. The latter two schemes are to treat the inertial loads in the time-domain formulation of the dynamic problem. The direct analysis with the Houbolt’s algorithm is used to perform the transient analysis. The efficient and accurate computation of the fundamental solutions has been a subject of great interest for 3D general anisotropic materials due to their mathematical complexity. A recently proposed scheme based on double Fourier series representation of these solutions is used to this end. Numerical examples are presented to demonstrate the feasibility and successful implementation of applying these schemes in synthesis to treat 3D elastodynamic problems of anisotropic materials.


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The authors would like to thank the Center for Computational Engineering and Sciences (CCES-CEPID/UNICAMP) and the National Center for High Performance Computing in São Paulo (CENAPAD-SP) for the support of computational facilities. Also to the São Paulo Research Foundation (FAPESP) Grant No. 2015/22199-9 and the National Council for the Scientific and Technological Development (CNPq) Grant No. 54283/2014-2 for the financial support of this research.


  1. 1.
    Lifshitz, I.M., Rozenzweig, L.N.: Construction of the green tensor fot the fundamental equation of elasticity theory in the case of unbounded elastic anisotropic medium. Z. Éksp. Teor. Fiz. 17, 783–791 (1947)Google Scholar
  2. 2.
    Wilson, R., Cruse, T.: Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis. Int. J. Numer. Methods Eng. 12, 1383–1397 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Sales, M.A., Gray, L.J.: Evaluation of the anisotropic Green’s function and its derivatives. Comput. Struct. 69, 247–254 (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Tonon, F., Pan, E., Amadei, B.: Green’s functions and boundary element method formulation for 3D anisotropic media. Comput. Struct. 79, 469–482 (2001)CrossRefGoogle Scholar
  5. 5.
    Phan, P.V., Gray, L.J., Kaplan, T.: On the residue calculus evaluation of the 3-D anisotropic elastic green’s function. Commun. Numer. Methods Eng. 20, 335–341 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Wang, C.Y., Denda, M.: 3D BEM for general anisotropic elasticity. Int. J. Solids Struct. 44, 7073–7091 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ting, T.C.T., Lee, V.G.: The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. Q. J. Mech. Appl. Math. 50, 407–426 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lee, V.: Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials. Mech. Res. Commun. 30, 241–249 (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Tavara, L., Ortiz, J.E., Mantic, V., Paris, R.: Unique real-variable expression of displacement and traction fundamental solutions covering all transversely isotropic materials for 3D BEM. Int. J. Numer. Methods Eng. 74, 776–798 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shiah, Y.C., Tan, C.L., Lee, V.G.: Evaluation of explicit-form fundamental solutions for displacements and stresses in 3D anisotropic elastic solids. CMES Comput. Model. Eng. Sci. 34, 205–226 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Tan, C.L., Shiah, Y.C., Lin, C.W.: Stress analysis of 3D generally anisotropic elastic solids using the boundary element method. CMES Comput. Model. Eng. Sci. 41, 195–214 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Tan, C.L., Shiah, Y.C., Wang, C.Y.: Boundary element elastic stress analysis of 3D generally anisotropic solids using fundamental solutions based on fourier series. Int. J. Solids Struct. 50, 2701–2711 (2013)CrossRefGoogle Scholar
  13. 13.
    Shiah, Y.C., Tan, C.L.: The boundary integral equation for 3D general anisotropic thermoelasticity. Comput. Model. Eng. Sci. 102(6), 425–447 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Saez, A., Dominguez, J.: BEM analysis of wave scattering in transversely isotropic solids. Int. J. Numer. Methods Eng. 44, 1283–1300 (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Dominguez, J.: Boundary Elements in Dynamics. Computational Mechanics Publications, Southampton (1993)zbMATHGoogle Scholar
  16. 16.
    Venturini, W.: A study of boundary element method and its application on engineering problems. Professorial Thesis, Sao Carlos, University of Sao Paulo (1988)Google Scholar
  17. 17.
    Gao, X.W.: The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng. Anal. Bound. Elem. 26, 905–916 (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Albuquerque, E., Sollero, P., Venturini, W., Aliabadi, M.: Boundary element method analysis of anisotropic Kirchhoff plates. Int. J. Solids Struct. 43, 4029–4046 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Albuquerque, E.L., Sollero, P., de Paiva, W.P.: The BEM and the RIM in the dynamic analysis of symmetric laminate composite plates. In: Alves, M., da Costa Mattos, H.S. (eds.) Mechanics of Solids in Brazil 2007. Brazilian Society of Mechanical Sciences and Engineering, Sao Paulo (2007)Google Scholar
  20. 20.
    Albuquerque, E., Sollero, P., Aliabadi, M.: The boundary element method applied to time dependent problems in anisotropic materials. Int. J. Solids Struct. 39(5), 1405–1422 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Albuquerque, E., Sollero, P., Fedelinski, P.: Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems. Comput. Struct. 81, 1703–1713 (2003)CrossRefGoogle Scholar
  22. 22.
    Albuquerque, E., Sollero, P., Fedelinski, P.: Free vibration analysis of anisotropic material structures using the boundary element method. Eng. Anal. Bound. Elem. 27, 977–985 (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Albuquerque, E., Sollero, P., Aliabadi, M.: Dual boundary element method for anisotropic dynamic fracture mechanics. Int. J. Numer. Methods Eng. 59, 1187–1205 (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Galvis, A., Sollero, P.: 2D analysis of intergranular dynamic crack propagation in polycrystalline materials a multiscale cohesive zone model and dual reciprocity boundary elements. Comput. Struct. 164, 1–14 (2016)CrossRefGoogle Scholar
  25. 25.
    Kögl, M., Gaul, L.: A 3D boundary element method for dynamic analysis of anisotropic elastic solids. Comput. Model. Eng. Sci. 1(4), 27–43 (2000)zbMATHGoogle Scholar
  26. 26.
    Kögl, M., Gaul, L.: A boundary element method for transient piezoelectric analysis. Eng. Anal. Bound. Elem. 24, 591–598 (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Kögl, M., Gaul, L.: Free vibration analysis of anisotropic solids with the boundary element method. Eng. Anal. Bound. Elem. 27, 107–114 (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Partridge, P.W., Brebbia, C.A., Wrobel, L.C.: The Dual Reciprocity Boundary Element Method. Elsevier, Amsterdam (1992)zbMATHGoogle Scholar
  29. 29.
    Gaul, L., Kögl, M., Wagner, M.: Boundary Element Methods for Engineers and Scientists. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  30. 30.
    Carrer, J., Fleischfresser, S., Garcia, L., Mansur, W.: Dynamic analysis of Timoshenko beams by the boundary element method. Eng. Anal. Bound. Elem. 37, 1602–1616 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Useche, J., Harnish, C.: A boundary element method formulation for modal analysis of doubly curved thick shallows shells. Appl. Math. Model. 40, 3591–3600 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shiah, Y.C., Tan, C.L., Wang, C.Y.: An efficient numerical scheme for the evaluation of the fundamental solution and its derivatives in 3D generally anisotropic elasticity. In: Advances in Boundary Element and Meshless Techniques XIII, pp. 190–199 (2012)Google Scholar
  33. 33.
    Lee, V.G.: Derivatives of the three-dimensional Green’s function for anisotropic materials. Int. J. Solids Struct. 46, 3471–3479 (2009)CrossRefzbMATHGoogle Scholar
  34. 34.
    Shiah, Y.C., Tan, C.L., Lee, R.F.: Internal point solutions for displacements and stresses in 3D anisotropic elastic solids using the boundary element method. CMES Comput. Model. Eng. Sci. 69, 167–197 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Nardini, D., Brebbia, C.A.: A New Approach to Free Vibration Analysis Using Boundary Elements. Springer, Berlin (1982)CrossRefzbMATHGoogle Scholar
  36. 36.
    Wrobel, L.C., Brebbia, C.A.: The dual reciprocity boundary element formulation for non-linear diffusion problems. Comput. Methods Appl. Mech. Eng. 65(2), 147–164 (1987)CrossRefzbMATHGoogle Scholar
  37. 37.
    Grundemann, H.: A general procedure transferring domain integrals onto boundary integrals in BEM. Eng. Anal. Bound. Elem. 6(4), 214–222 (1989)CrossRefGoogle Scholar
  38. 38.
    Atkinson, K.E.: The numerical evaluation of particular solutions for Poisson’s equation. IMA J. Numer. Anal. 5, 319–338 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Golberg, M.A.: The numerical evaluation of particular solutions in the BEM—a review. Bound. Elem. Commun. 6, 99–106 (1995)Google Scholar
  40. 40.
    Schclar, N.A.: Anisotropic Analysis Using Boundary Elements. Computational Mechanics Publications, Southampton (1994)zbMATHGoogle Scholar
  41. 41.
    Gao, X.W.: Boundary only integral equations in boundary element analysis. In: Proceedings of the International Conference on Boundary Element Techniques (2001)Google Scholar
  42. 42.
    Wood, W.L.: Practical Time Stepping Schemes. Clarendon Press, Oxford (1990)zbMATHGoogle Scholar
  43. 43.
    Houbolt, J.C.: A recurrence matrix solution for the dynamic response of elastic aircraft. J. Aeronaut. Sci. 17, 540–550 (1950)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85, 67–94 (1959)Google Scholar
  45. 45.
    Park, K.C.: An improved stiffly stable method for direct integration of nonlinear structural dynamic equations. ASME J. Appl. Mech. 42, 464–470 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Loeffler, C.F., Mansur, W.J.: Analysis of time integration schemes for boundary element applications to transient wave propagation problems. In: Boundary Element Techniques: Applications in Stress Analysis and Heat transfer. Computational Mechanics Publications, Southampton (1987)Google Scholar
  47. 47.
    Agnantiaris, J.P., Polyzos, D., Beskos, D.E.: Some studies on dual reciprocity BEM for elastodynamic analysis. Comput. Mech. 17, 270–277 (1996)CrossRefzbMATHGoogle Scholar
  48. 48.
    Timoshenko, S., Goodier, J.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
  49. 49.
    Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963)zbMATHGoogle Scholar
  50. 50.
    Clough, R.W., Penzien, J.: Dynamics of Structures, 3rd edn. Computer & Structures, Inc., Berkeley (2003)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • R. Q. Rodríguez
    • 1
    • 4
  • A. F. Galvis
    • 1
  • P. Sollero
    • 1
  • C. L. Tan
    • 2
  • E. L. Albuquerque
    • 3
  1. 1.Department of Computational Mechanics, School of Mechanical EngineeringUniversity of CampinasCampinasBrazil
  2. 2.Department of Mechanical and Aerospace EngineeringCarleton UniversityOttawaCanada
  3. 3.Department of Mechanical EngineeringFaculty of Technology, University of BrasiliaBrasíliaBrazil
  4. 4.School of Civil EngineeringState University of Mato Grosso Campus Tangará da SerraTangará da SerraBrazil

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