Archive of Applied Mechanics

, Volume 80, Issue 12, pp 1429–1447

Velocity-based reciprocal theorems in elastodynamics and BIEM implementation issues

Original

Abstract

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.

Keywords

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

References

1. 1.
Love A.E.H.: Mathematical Theory of Elasticity. Cambridge University Press, New York (1927)
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)
6. 6.
Dominguez J.: Boundary Elements in Dynamics. Computational Mechanics Publications/Elsevier Applied Science, UK (1993)
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)
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)
11. 11.
Frangi A.: “Causal” shape functions in the time domain boundary element method. Comput. Mech. 25, 533–541 (2000)
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)
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)
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)
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)
19. 19.
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)
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)
25. 25.
Birgison B., Crouch S.L.: Elastodynamic boundary element method for piecewise homogeneous media. Int. J. Numer. Methods Eng. 42, 1045–1069 (1998)
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)
27. 27.
Panagiotopoulos C.G.: Reciprocal theorems in structural dynamics including initial conditions. Earthq. Eng. Struct. Dyn. 35, 653–656 (2006)
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)
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)
33. 33.
Maxima, a Computer Algebra System, http://maxima.sourceforge.net/
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)
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)
39. 39.
Bathe K.J.: Finite Element Procedures. Prentice Hall, Upper Saddle River (1996)Google Scholar
40. 40.
Nemesis, an experimental finite element code, http://www.nemesis-project.org/