Skip to main content
SpringerLink
Log in

A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A new fast multipole formulation for the hypersingular BIE (HBIE) for 2D elasticity is presented in this paper based on a complex-variable representation of the kernels, similar to the formulation developed earlier for the conventional BIE (CBIE). A dual BIE formulation using a linear combination of the developed CBIE and HBIE is applied to analyze multi-domain problems with thin inclusions or open cracks. Two pre-conditioners for the fast multipole boundary element method (BEM) are devised and their effectiveness and efficiencies in solving large-scale problems are discussed. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM using the dual BIE formulation. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale 2D multi-domain elasticity problems. The method can be applied to study composite materials, functionally-graded materials, and micro-electro-mechanical-systems with coupled fields, all of which often involve thin shapes or thin inclusions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Rokhlin V (1985). Rapid solution of integral equations of classical potential theory. J Comp Phys 60: 187–207

    Article  MATH  MathSciNet  Google Scholar 

  2. Greengard LF and Rokhlin V (1987). A fast algorithm for particle simulations. J Comput Phys 73(2): 325–348

    Article  MATH  MathSciNet  Google Scholar 

  3. Greengard LF (1988). The rapid evaluation of potential fields in particle systems. MIT Press, Cambridge

    MATH  Google Scholar 

  4. Peirce AP and Napier JAL (1995). A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics. Int J Numer Meth Eng 38: 4009–4034

    Article  MATH  MathSciNet  Google Scholar 

  5. Gomez JE and Power H (1997). A multipole direct and indirect BEM for 2D cavity flow at low Reynolds number. Eng Anal Bound Elem 19: 17–31

    Article  Google Scholar 

  6. Fu Y, Klimkowski KJ, Rodin GJ, Berger E, Browne JC, Singer JK, Geijn RAVD and Vemaganti KS (1998). A fast solution method for three-dimensional many-particle problems of linear elasticity. Int J Numer Meth Eng 42: 1215–1229

    Article  MATH  Google Scholar 

  7. Nishimura N, Yoshida K and Kobayashi S (1999). A fast multipole boundary integral equation method for crack problems in 3D. Eng Anal Bound Elem 23: 97–105

    Article  MATH  Google Scholar 

  8. Mammoli AA and Ingber MS (1999). Stokes flow around cylinders in a bounded two-dimensional domain using multipole-accelerated boundary element methods. Int J Numer Meth Eng 44: 897–917

    Article  MATH  MathSciNet  Google Scholar 

  9. Nishimura N (2002). Fast multipole accelerated boundary integral equation methods. Appl Mech Rev 55(4): 299–324

    Article  Google Scholar 

  10. Greengard LF, Kropinski MC and Mayo A (1996). Integral equation methods for Stokes flow and isotropic elasticity in the plane. J Comput Phys 125: 403–414

    Article  MATH  MathSciNet  Google Scholar 

  11. Greengard LF and Helsing J (1998). On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites. J Mech Phys Solids 46(8): 1441–1462

    Article  MATH  MathSciNet  Google Scholar 

  12. Richardson JD, Gray LJ, Kaplan T and Napier JA (2001). Regularized spectral multipole BEM for plane elasticity. Eng Anal Bound Elem 25: 297–311

    Article  MATH  Google Scholar 

  13. Fukui T (1998) Research on the boundary element method—development and applications of fast and accurate computations. Ph.D. dissertation (in Japanese), Department of Global Environment Engineering, Kyoto University

  14. Fukui T, Mochida T, Method K (1997) Crack extension analysis in system of growing cracks by fast multipole boundary element method (in Japanese), Seventh BEM technology conference (JASCOME, Tokyo, 1997) pp 25–30

  15. Liu YJ and Nishimura N (2006). The fast multipole boundary element method for potential problems: a tutorial. Eng Anal Bound Elem 30(5): 371–381

    Article  Google Scholar 

  16. Liu YJ (2005). A new fast multipole boundary element method for solving large-scale two-dimensional elastostatic problems. Int J Numer Meth Eng 65(6): 863–881

    Article  Google Scholar 

  17. Wang P and Yao Z (2006). Fast multipole DBEM analysis of fatigue crack growth. Computat Mech 38: 223–233

    Article  MathSciNet  Google Scholar 

  18. Yamada Y, Hayami K (1995) A multipole boundary element method for two dimensional elastostatics. Report METR 95–07, Department of Mathematical Engineering and Information Physics, University of Tokyo

  19. Yao Z, Kong F, Wang H and Wang P (2004). 2D simulation of composite materials using BEM. Eng Anal Bound Elem 28(8): 927–935

    Article  MATH  Google Scholar 

  20. Wang J, Crouch SL and Mogilevskaya SG (2005). A fast and accurate algorithm for a Galerkin boundary integral method. Comput Mech 37(1): 96

    Article  MATH  MathSciNet  Google Scholar 

  21. Rizzo FJ (1967). An integral equation approach to boundary value problems of classical elastostatics. Q Appl Math 25: 83–95

    MATH  Google Scholar 

  22. Mukherjeec S (1982). Boundary element methods in creep and fracture. Applied Science Publishers, New York

    Google Scholar 

  23. Cruse TA (1988). Boundary element analysis in computational fracture mechanics. Kluwer, Dordrecht

    MATH  Google Scholar 

  24. Brebbia CA and Dominguez J (1989). Boundary elements—an introductory course. McGraw-Hill, New York

    MATH  Google Scholar 

  25. Banerjee PK (1994). The boundary element methods in engineering, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  26. Krishnasamy G, Rizzo FJ, Rudolphi TJ (1991) Hypersingular boundary integral equations: their occurrence, interpretation, regularization and computation. In: Banerjee PK et al (eds) Developments in boundary element methods, Chap 7. Elsevier, London

    Google Scholar 

  27. Sladek V, Sladek J (eds) (1998) Singular integrals in boundary element methods. In: Brebbia CA, Aliabadi MH (eds) Advances in boundary element series, Computational Mechanics Publications, Boston

  28. Mukherjee S (2000). Finite parts of singular and hypersingular integrals with irregular boundary source points. Eng Anal Bound Elem 24: 767–776

    Article  MATH  Google Scholar 

  29. Liu YJ and Rizzo FJ (1992). A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems. Comp Meth Appl Mech Eng 96: 271–287

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu YJ and Rizzo FJ (1993). Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions. Comp Meth Appl Mech Eng 107: 131–144

    Article  MATH  MathSciNet  Google Scholar 

  31. Liu YJ, Rizzo FJ (1997) Scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation. J Acoust Soc Am 102(2), No. Pt.1, August, 926–932

    Google Scholar 

  32. Shen L and Liu YJ (2007). An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation. Comput Mech 40(3): 461–472

    Article  Google Scholar 

  33. Liu YJ (2006). Dual BIE approaches for modeling electrostatic MEMS problems with thin beams and accelerated by the fast multipole method. Eng Anal Bound Elem 30(11): 940–948

    Article  Google Scholar 

  34. Liu YJ and Shen L (2007). A dual BIE approach for large-scale modeling of 3-D electrostatic problems with the fast multipole boundary element method. Int J Numer Meth Eng 71(7): 837–855

    Article  Google Scholar 

  35. Liu YJ (2008). A new fast multipole boundary element method for solving 2D Stokes flow problems based on a dual BIE formulation. Eng Anal Bound Elem 32(2): 139–151

    Article  Google Scholar 

  36. Muskhelishvili NI (1958). Some basic problems of mathematical theory of elasticity. Noordhoff, Groningen

    Google Scholar 

  37. Sokolnikoff IS (1956). Mathematical theory of elasticity. McGraw Hill, New York

    MATH  Google Scholar 

  38. Mogilevskaya SG and Linkov AM (1998). Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 22(1): 88–92

    Article  MATH  MathSciNet  Google Scholar 

  39. Linkov AM and Mogilevskaya SG (1998). Complex hypersingular BEM in plane elasticity problems. In: Sladek, V and Sladek, J (eds) Singular integrals in boundary element method, pp 299–364. Computational Mechanics Publications, Southampton

    Google Scholar 

  40. Liu YJ, Nishimura N, Otani Y, Takahashi T, Chen XL and Munakata H (2005). A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model. J Appl Mech 72(1): 115–128

    Article  MATH  Google Scholar 

  41. Liu YJ, Nishimura N and Otani Y (2005). Large-scale modeling of carbon-nanotube composites by the boundary element method based on a rigid-inclusion model. Comput Mater Sci 34(2): 173– 187

    Article  Google Scholar 

  42. Liu YJ, Xu N and Luo JF (2000). Modeling of interphases in fiber-reinforced composites under transverse loading using the boundary element method. J Appl Mech 67(1): 41–49

    Article  MATH  Google Scholar 

  43. Krishnasamy G, Rizzo FJ and Liu YJ (1994). Boundary integral equations for thin bodies. Int J Numer Meth Eng 37: 107–121

    Article  MATH  MathSciNet  Google Scholar 

  44. Wang J, Mogilevskaya SG and Crouch SL (2003). Benchmarks results for the problem of interaction between a crack and a circular inclusion. J Appl Mech 70: 619–621

    Article  MATH  Google Scholar 

  45. Mogilevskaya SG and Crouch SL (2004). A Galerkin boundary integral method for multiple circular elastic inclusions with uniform interphase layers. Int J Solids Struct 41: 1285–1311

    Article  MATH  Google Scholar 

  46. Wang J, Mogilevskaya SG and Crouch SL (2005). An embedding method for modeling micromechanical behavior and macroscopic properties of composite materials. Int J Solids Struct 42: 4588–4612

    Article  MATH  Google Scholar 

  47. Gross D and Seelig T (2006). Fracture mechanics with an introduction to micromechanics. Springer, The Netherlands

    MATH  Google Scholar 

  48. Gomez JE and Power H (2000). A parallel multipolar indirect boundary element method for the Neumann interior Stokes flow problem. Int J Numer Meth Eng 48(4): 523–543

    Article  MATH  Google Scholar 

  49. Yoshida K, Nishimura N and Kobayashi S (2001). Application of fast multipole Galerkin boundary integral equation method to crack problems in 3D. Int J Numer Meth Eng 50: 525–547

    Article  MATH  Google Scholar 

  50. Shen L and Liu YJ (2007). An adaptive fast multipole boundary element method for three-dimensional potential problems. Comput Mech 39(6): 681–691

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Cincinnati, P.O. Box 210072, Cincinnati, OH, 45221-0072, USA

    Y. J. Liu

Authors
  1. Y. J. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. J. Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y.J. A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation. Comput Mech 42, 761–773 (2008). https://doi.org/10.1007/s00466-008-0274-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-008-0274-2

Keywords

Search

Navigation