Archive of Applied Mechanics

, Volume 80, Issue 12, pp 1429–1447 | Cite as

Velocity-based reciprocal theorems in elastodynamics and BIEM implementation issues

  • Christos G. Panagiotopoulos
  • G. D. ManolisEmail author


Reciprocal theorems in elastodynamics are introduced as extensions of respective theorems from elastostatics. Inasmuch as the latter is a subset of the former, the aim here is to present an elastodynamic reciprocal theorem that also includes elastostatics as a special case when the time variable becomes irrelevant. This is accomplished by introducing a velocity-based reciprocal theorem, whose basic properties are subsequently explored. The next step is to use this theorem and formulate a numerical approach based on boundary integral equation statements and compare them with existing formulations based on conventional reciprocity relations. The applications presented here involve the standard mechanical oscillator and the unidimensional axial element as two simple, yet important problems of structural dynamics. Along with the numerical results, a thorough stability analysis of the corresponding time-stepping algorithms is formulated. In both cases, the superior performance of the methodologies built on velocity-based reciprocal theorems is clearly demonstrated.


Boundary elements Time domain formulations Stability analysis Reciprocal theorems Velocity-based integral formulations Wave propagation Dynamic response 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Love A.E.H.: Mathematical Theory of Elasticity. Cambridge University Press, New York (1927)zbMATHGoogle Scholar
  2. 2.
    Paraskevopoulos, E.A., Panagiotopoulos, C.G., Manolis, G.D.: Imposition of time-dependent boundary conditions in FEM formulations for elastodynamics: critical assessment of penalty-type methods. Comput. Mech. (2009, accepted)Google Scholar
  3. 3.
    Manolis G.D., Beskos D.E.: Boundary Element Methods in Elastodynamics. Computational Mechanics Publications/Unwin Hyman, Ltd, London (1988)Google Scholar
  4. 4.
    Achenbach J.D.: Wave Propagation in Elastic Solids. Elsevier, Amsterdam (1999)Google Scholar
  5. 5.
    Achenbach J.D.: Reciprocity in Elastodynamics. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  6. 6.
    Dominguez J.: Boundary Elements in Dynamics. Computational Mechanics Publications/Elsevier Applied Science, UK (1993)zbMATHGoogle Scholar
  7. 7.
    Mansur, W.J.: A time-stepping technique to solve wave propagation problems using the boundary element method. Ph.D. thesis, University of Southampton (1983)Google Scholar
  8. 8.
    Rihter C.: A Green’s Function Time Domain BEM of Elastodynamics. Computational Mechanics Publications, Southampton (1997)Google Scholar
  9. 9.
    Mansur W.J., Carrer J.A.M., Siqueira E.F.N.: Time discontinuous linear traction approximation in time-domain bem scalar wave propagation analysis. Int. J. Numer. Methods Eng. 42, 667–683 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Carrer J.A.M., Mansur W.J.: Time discontinuous linear traction approximation in time-domain bem: 2-d elastodynamics. Int. J. Numer. Methods Eng. 49, 833–848 (2000)zbMATHCrossRefGoogle Scholar
  11. 11.
    Frangi A.: “Causal” shape functions in the time domain boundary element method. Comput. Mech. 25, 533–541 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Araujo F.C., Mansur W.J., Nishikava L.K.: A linear time-marching algorithm in 3D BEM formulation for elastodynamics. Eng. Anal. Bound. Elem. 23, 825–833 (1999)zbMATHCrossRefGoogle Scholar
  13. 13.
    Costabel M.: Time-dependent problems with the boundary integral equation method. In: Stein, E., Borst, R., Hughes, T.J.R. (eds) Encyclopedia of Computational Mechanics, Wiley, Chichester (2004)Google Scholar
  14. 14.
    Graffi D.: Sul teorema di reciprocita nella dinamica dei corpi elastici. Memoric della Reale Accademia delle Scinze dell’Istituto di Bologna 10, 103–111 (1946)MathSciNetGoogle Scholar
  15. 15.
    Wheeler L.T., Stenberg E.: Some theorems in classical elastodynamics. Arch. Ration. Mech. Anal. 31, 51–90 (1968)Google Scholar
  16. 16.
    Carbonaro B., Russo R.: On Graffi’s reciprocal theorem in unbounded domains. J. Elast. 15, 35–42 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Betti E.: Teori della elasticita. Il Nuove Ciemento Series 4, 7–10 (1872)Google Scholar
  18. 18.
    Achenbach J.D.: Reciprocity and Related Topics in Elastodynamics. Appl. Mech. Rev. 59, 13–32 (2006)CrossRefGoogle Scholar
  19. 19.
    de Hoop A.T.: Handbook of Radiation and Scattering of Waves. Academic Press, London (1995)Google Scholar
  20. 20.
    Eringen A.C., Suhubi E.S.: Elastodynamics, Volume II: Linear Theory. Academic Press, New York (1974)Google Scholar
  21. 21.
    Rayleigh L.: Some general theorems relating to vibrations. Proc. Lond. Math. Soc. 4, 357–368 (1873)Google Scholar
  22. 22.
    Lamb H.: On reciprocal theorems in dynamics. Proc. Lond. Math. Soc. 19, 144–151 (1888)CrossRefGoogle Scholar
  23. 23.
    Kellog O.D.: Foundations of Potential Theory. Dover, New York (1929)Google Scholar
  24. 24.
    Ahmad A., Banerjee P.K.: Multi-domain bem for two dimensional problems of elastodynamics. Int. J. Numer. Methods Eng. 26, 891–911 (1988)zbMATHCrossRefGoogle Scholar
  25. 25.
    Birgison B., Crouch S.L.: Elastodynamic boundary element method for piecewise homogeneous media. Int. J. Numer. Methods Eng. 42, 1045–1069 (1998)CrossRefGoogle Scholar
  26. 26.
    Holl H.J., Belyaev A.K., Irschik H.: A numerical algorithm for nonlinear dynamic problems based on BEM. Eng. Anal. Bound. Elem. 23, 503–513 (1996)CrossRefGoogle Scholar
  27. 27.
    Panagiotopoulos C.G.: Reciprocal theorems in structural dynamics including initial conditions. Earthq. Eng. Struct. Dyn. 35, 653–656 (2006)CrossRefGoogle Scholar
  28. 28.
    Clough R.W., Penzien J.: Dynamics of Structures, 3rd edn. McGraw-Hill, New York (1993)Google Scholar
  29. 29.
    Cole D.M., Kosloff D.D., Minster J.B.: A numerical boundary integral equation method for elastodynamics. I. Bull. Seismol. Soc. Am. 68, 1331–1357 (1978)Google Scholar
  30. 30.
    Pierce A., Siebrits E.: Stability Analysis of Model Problems for Elastodynamic Boundary Element Discretizations. Numer. Methods Partial Differ. Equ. 12, 585–613 (1996)CrossRefGoogle Scholar
  31. 31.
    Mitchell A.R., Griffiths D.F.: The Finite Difference Method in Partial Differential Equations. Wiley, New York (1988)Google Scholar
  32. 32.
    Belytschko T., Hughes T.J.R.: Computational Methods for Transient Analysis. North-Holland, Amsterdam (1983)zbMATHGoogle Scholar
  33. 33.
    Maxima, a Computer Algebra System,
  34. 34.
    Greenberg M.D.: Application of Green’s Functions in Science and Engineering. Prentice-Hall, Inc., Englewood Cliffs (1971)Google Scholar
  35. 35.
    Moser W., Antes H., Beer G.: A Duhamel integral based approach to one-dimensional wave propagation analysis in layered media. Comput. Mech. 35, 115–126 (2005)zbMATHCrossRefGoogle Scholar
  36. 36.
    de Langre, E., Axisa, F., Guilbaund, D.: Forced flexural vibrations of beams using a time-stepping boundary element method. In: Brebbia, C.A., Chaudouet-Miranda, A. (eds.) Boundary Elements in Mechanical and Electrical Engineering. Computational Mechanics Publications, West Germany (1990)Google Scholar
  37. 37.
    Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. Mathematische Annalen 100, 32–74 (1928, in German). Republished in English translation in IBM journal, pp. 215–238 (1967)Google Scholar
  38. 38.
    Keller H.B.: Propagation of stress discontinuities in inhomogeneous elastic media. SIAM Rev. 6, 356–382 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Bathe K.J.: Finite Element Procedures. Prentice Hall, Upper Saddle River (1996)Google Scholar
  40. 40.
    Nemesis, an experimental finite element code,

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Civil EngineeringAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations