Skip to main content
SpringerLink
Log in

An alternative collocation boundary element method for static and dynamic problems

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

Abstract

A collocation boundary element formulation is presented which is based on a mixed approximation formulation similar to the Galerkin boundary element method presented by Steinbach (SIAM J Numer Anal 38:401–413, 2000) for the solution of Laplace’s equation. The method is also applicable to vector problems such as elasticity. Moreover, dynamic problems of acoustics and elastodynamics are included. The resulting system matrices have an ordered structure and small condition numbers in comparison to the standard collocation approach. Moreover, the employment of Robin boundary conditions is easily included in this formulation. Details on the numerical integration of the occurring regular and singular integrals and on the solution of the arising systems of equations are given. Numerical experiments have been carried out for different reference problems. In these experiments, the presented approach is compared to the common nodal collocation method with respect to accuracy, condition numbers, and stability in the dynamic case.

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. Achenbach JD (2005) Wave propagation in elastic solids. Amsterdam, North-Holland

    Google Scholar 

  2. Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, 3rd edn

  3. Bebendorf M, Rjasanow S (2003) Adaptive low-rank approximation of collocation matrices. Computing 70: 1–24

    Article  MATH  MathSciNet  Google Scholar 

  4. Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14: 1–137

    Article  MATH  MathSciNet  Google Scholar 

  5. Beskos DE (1987) Boundary element methods in dynamics analysis. Appl Mech Rev 40: 1–23

    Article  Google Scholar 

  6. Beskos DE (1997) Boundary element methods in dynamic analysis: part II (1986–1996). Appl Mech Rev 50(3): 149–197

    Article  Google Scholar 

  7. Courant R, Friedrichs K, Lewy H (1928) Über die partiellen Differenzengleichungen der mathematischen Physik. Math Ann 100: 32–74

    Article  MATH  MathSciNet  Google Scholar 

  8. Duffy MG (1982) Quadrature over a pyramid or cube of inte- grands with a singularity at a vertex. SIAM J Numer Anal 19: 1260–1262

    Article  MATH  MathSciNet  Google Scholar 

  9. Dunavant DA (1985) High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int J Numer Methods Eng 21: 1129–1148

    Article  MATH  MathSciNet  Google Scholar 

  10. Gaul L, Kögl M, Wagner M (2003) Boundary element methods for engineers and scientists. Springer, Heidelberg

    MATH  Google Scholar 

  11. Golub GH, van Loan CF (1996) Matrix computations. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  12. Graff KF (1991) Wave motions in elastic solids. Dover, New York

    Google Scholar 

  13. Hartmann F (1989) Introduction to boundary elements. Theory and applications. Springer, Heidelberg

    MATH  Google Scholar 

  14. Higham NJ (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation. ACM Trans Math Softw 14: 381–396

    Article  MATH  MathSciNet  Google Scholar 

  15. Johnston PR (2000) Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals. Int J Numer Methods Eng 47: 1709–1730

    Article  MATH  MathSciNet  Google Scholar 

  16. Kielhorn L, Schanz M (2007) CQM based symmetric Galerkin BEM: regularization of strong and hypersingular kernels in 3-d elastodynamics. Int J Numer Methods Eng. doi:10.1002/nme.2381

  17. Krommer AR, Ueberhuber CW (1998) Computational integration. SIAM

  18. Kupradze VD, Gegelia TG, Basheleishvili MO, Burchuladze TV (1979) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. Amsterdam, North-Holland

    MATH  Google Scholar 

  19. Lachat JC, Watson JO (1976) Effective numerical treatment of boundary integral equations: a formulation for three-dimensional elastostatics. Int J Numer Methods Eng 10: 991–1005

    Article  MATH  Google Scholar 

  20. Lambert JD (1990) Numerical methods for ordinary differential systems. Wiley, New York

    Google Scholar 

  21. Love AEH (1944) Treatise on the mathematical theory of elasticity. Dover, New York

    MATH  Google Scholar 

  22. Lubich C (1988) Convolution quadrature and discretized operational calculus I & II. Numer Math 52:129–145; 413–425

    Google Scholar 

  23. Mansur WJ (1983) A time-stepping technique to solve wave propagation problems using the boundary element method. PhD thesis, University of Southampton

  24. Mantič V (1993) A new formula for the c-matrix in the Somigliana identity. J Elast 33: 193–201

    Google Scholar 

  25. Of G (2006) BETI-Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen. PhD thesis, University of Stuttgart

  26. París F, Cañas J (1997) Boundary element method. Oxford University Press, Oxford

    MATH  Google Scholar 

  27. Patterson C, Elsebai NAS (1982) A regular boundary method using non-conforming elements for potentials in three dimensions. In: Brebbia CA (ed) Boundary element methods in engineering, pp 112–126

  28. Patterson C, Sheikh MA (1981) Non-conforming boundary elements for stress analysis. In: Brebbia CA (ed) Boundary element methods, pp 137–152

  29. Patterson C, Sheikh MA (1984) Interelement continuity in the boundary element method. In: Brebbia CA (eds) Topics in boundary element research. Springer, Heidelberg, pp 121–141

    Google Scholar 

  30. Pekeris CL (1955) The seismic surface pulse. Proc Natl Am Soc 41: 469–480

    Article  MATH  MathSciNet  Google Scholar 

  31. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2002) Numerical Recipes in C++. Cambridge University Press, Cambridge

    Google Scholar 

  32. Rjasanow S, Steinbach O (2007) The fast solution of boundary integral equations. Springer, Heidelberg

    MATH  Google Scholar 

  33. Rüberg T (2008) Non-conforming FEM/BEM coupling in time domain, volume 3 of computation in engineering and science. Verlag der Technischen Universität Graz

  34. Schanz M (2001) Wave propagation in viscoelastic and poroelastic continua—A boundary element approach. Springer, Heidelberg

    MATH  Google Scholar 

  35. Schanz M, Antes H (1997) A new visco- and elastodynamic time domain boundary element formulation. Comput Mech 20: 452–459

    Article  MATH  MathSciNet  Google Scholar 

  36. Schanz M, Kielhorn L (2005) Dimensionless variables in a Poroe lastodynamic time domain bounday element formulation. Build Res J 53(2–3): 175–189

    Google Scholar 

  37. Schwab C, Wendland WL (1992) On numerical cubature of singular surface integrals in boundary element methods. Numer Math 62: 343–369

    Article  MATH  MathSciNet  Google Scholar 

  38. Steinbach O (2003) Stability estimates for hybrid domain decomposition methods. Springer, Heidelberg

    MATH  Google Scholar 

  39. Steinbach O (2000) Mixed approximations for boundary elements. SIAM J Numer Anal 38: 401–413

    Article  MATH  MathSciNet  Google Scholar 

  40. Steinbach O (1998) Fast solution techniques for the symmetric boundary element method in linear elasticity. Comput Methods Appl Mech Eng 157: 185–191

    Article  MATH  MathSciNet  Google Scholar 

  41. Steinbach O (2008) Numerical approximation methods for elliptic boundary value problems. Springer, Heidelberg

    MATH  Google Scholar 

  42. Wheeler LT, Sternberg E (1968) Some theorems in classical elastodynamics. Arch Ration Mech Anal 31: 51–90

    MathSciNet  Google Scholar 

  43. Yan G, Lin F-B (1994) Treatment of corner node problems and its singularity. Eng Anal Bound Elem 13: 75–81

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Structural Analysis, Graz University of Technology, Lessingstr. 25, 8010, Graz, Austria

    Thomas Rüberg

  2. Institute of Applied Mechanics, Graz University of Technology, Technikerstr. 4, 8010, Graz, Austria

    Martin Schanz

Authors
  1. Thomas Rüberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Schanz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Schanz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rüberg, T., Schanz, M. An alternative collocation boundary element method for static and dynamic problems. Comput Mech 44, 247–261 (2009). https://doi.org/10.1007/s00466-009-0369-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-009-0369-4

Keywords

Search

Navigation