Computational Mechanics

, Volume 51, Issue 4, pp 443–453 | Cite as

A fast elasto-plastic formulation with hierarchical matrices and the boundary element method

  • Jürgen Zechner
  • Gernot Beer
Original Paper


Boundary element methods offer some advantages for the simulation of tunnel excavation since the radiation condition is implicitly fulfilled and only the excavation and ground surfaces have to be discretized. Hence, large meshes and mesh truncation, as required in the finite element method, are avoided. Recently, capabilities for efficiently dealing with inelastic behavior and ground support have been developed, paving the way for the use of the method to simulate tunneling. However, for large scale three dimensional problems one drawback of the boundary element method becomes prominent: the computational effort increases quadratically with the problem size. To reduce the computational effort several fast methods have been proposed. Here a fast boundary element solution procedure for small strain elasto-plasticity based on a collocation scheme and hierarchical matrices is presented. The method allows the solution of problems with the computational effort and sparse storage increasing almost linearly with respect to the problem size.


Boundary element method Hierarchical matrices Elasto-plasticity Tunneling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bebendorf M (2008) Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. Springer, BerlinzbMATHGoogle Scholar
  2. 2.
    Bebendorf M, Grzhibovskis R (2006) Accelerating galerkin bem for linear elasticity using adaptive cross approximation. Math Methods Appl Sci 29: 1721–1747MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bebendorf M, Rjasanow S (2003) Adaptive low-rank approximation of collocation matrices. Computing 70(1): 1–24MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beer G, Smith IM, Dünser C (2008) The boundary element method with programming. Springer, WienzbMATHGoogle Scholar
  5. 5.
    Beer G, Dünser C, Riederer K, Glauber P, Thöni K, Zechner J, Stettner M (2010) Computer simulation of conventional construction. In: Beer G (ed.) Technology innovation in underground construction. CRC Press/Balkema, Leiden, pp 129–161Google Scholar
  6. 6.
    Benedetti I, Aliabadi M, Davi G (2008) A fast 3d dual boundary element method based on hierarchical matrices. Int J Solids Struct 45: 2355–2376zbMATHCrossRefGoogle Scholar
  7. 7.
    Beylkin G, Coifman R, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms. Commun Pure Appl Math 44(2): 141–183MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bonnet M (2004) Boundary integral equation methods for elastic and plastic problems. In: Stein E, de Borst R, Hughes T (eds) Encyclopedia of computational mechanics. Wiley, Chichester, pp 719–749Google Scholar
  9. 9.
    Bonnet M, Mukherjee S (1996) Implicit bem formulations for usual and sensitivity problems in elasto-plasticity using consistent tangent operator concept. Int J Solids Struct 33(30): 4461–4480zbMATHCrossRefGoogle Scholar
  10. 10.
    Börm S (2010) Efficient numerical methods for non-local operators. \({\mathcal{H}^2}\)-matrix compression, algorithms and analysis. European Mathematical Society, ZürichCrossRefGoogle Scholar
  11. 11.
    Börm S, Grasedyck L (1999) Hlib - a library for \({\mathcal{H}}\)- and \({\mathcal{H}^2}\)-matrices.
  12. 12.
    Börm S, Grasedyck L (2005) Hybrid cross approximation of integral operators. Numer Math 101: 221–249MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Carrer J, Telles J (1992) A boundary element formulation to solve transient dynamic elastoplastic problems. Comput Struct 45(4): 707–713zbMATHCrossRefGoogle Scholar
  14. 14.
    Ding J, Ye W, Gray LJ (2005) An accelerated surface discretization-based bem approach for non-homogeneous linear problems in 3-d complex domains. Int J Numer Meth Eng 63(12): 1775–1795zbMATHCrossRefGoogle Scholar
  15. 15.
    Elleithy W, Grzhibovskis R (2009) An adaptive domain decomposition coupled finite element–boundary element method for solving problems in elasto-plasticity. Int J Numer Meth Eng 79(8): 1019MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Frangi A, Maier G (1999) Dynamic elastic-plastic analysis by a symmetric galerkin boundary element method with time-independent kernels. Comput Meth Appl Mech Eng 171: 281–308zbMATHCrossRefGoogle Scholar
  17. 17.
    Gao X (2002) A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems. J Appl Mech 69(2): 154–160zbMATHCrossRefGoogle Scholar
  18. 18.
    Gao X, Davies T (2000) An effective boundary element algorithm for 2d and 3d elastoplastic problems. Int J Solids Struct 37: 4987–5008zbMATHCrossRefGoogle Scholar
  19. 19.
    Gens A, Carol I, Gonzales N, Caballero A, Garolera D (2007) Library of model subroutines: soil and rock models. TunConstruct Deliv. doi: 10.1061/(ASCE)GM.1943-5622.0000173
  20. 20.
    Grasedyck L (2005) Adaptive recompression of \({\mathcal{H}}\)-matrices for bem. Comput 74: 205–223MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hackbusch W (1999) A sparse matrix arithmetic based on \({\mathcal{H}}\)-matrices. Comput 62: 89–108MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hackbusch W (2009) Hierarchische Matrizen. Springer, BerlinzbMATHCrossRefGoogle Scholar
  23. 23.
    Hackbusch W, Börm S (2002) \({\mathcal{H}^2}\)-matrix approximation of integral operators by interpolation. Appl Numer Math 43(1-2): 129–143MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ingber M, Mammoli A, Brown M (2001) A comparison of domain integral evaluation techniques for boundary element methods. Int J Numer Meth Eng 52: 417–432zbMATHCrossRefGoogle Scholar
  25. 25.
    Maerten F (2010) Adaptive cross-approximation applied to the solution of system of equations and post-processing for 3d elastostatic problems using the boundary element method. Eng Anal Bound Elem 34(5): 483–491zbMATHCrossRefGoogle Scholar
  26. 26.
    Maier G, Polizzotto C (1987) A galerkin approach to boundary element elastoplastic analysis. Comput Meth Appl Mech Eng 60(2): 175–194zbMATHCrossRefGoogle Scholar
  27. 27.
    Mantic V (1993) A new formula for the c-matrix in the somigliana identity. J Elast 33: 191–201MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Margonari M, Bonnet M (2005) Fast multipole method applied to elastostatic bem–fem coupling. Comput Struct 83(10): 700–717MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nardini D, Brebbia C (1983) A new approach to free vibration analysis using boundary elements. Appl Math Model 7(3): 157–162zbMATHCrossRefGoogle Scholar
  30. 30.
    Of G, Steinbach O, Urthaler P (2010) Fast evaluation of volume potentials in boundary element methods. J Sci Comput 32(2): 585–602MathSciNetzbMATHGoogle Scholar
  31. 31.
    Phillips J, White J (1997) A precorrected-fft method for electrostatic analysis of complicated 3-d structures. Comput Des Integr Circuits Syst 16(10): 1059–1072CrossRefGoogle Scholar
  32. 32.
    Rjasanow S, Steinbach O (2007) The fast solution of boundary integral equations. Mathematical and analytical techniques with applications to engineering. Springer, New YorkGoogle Scholar
  33. 33.
    Rokhlin V (1985) Rapid solution of integral equations of classical potential theory. J Comput Phys 60(2): 187–207MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Rüberg T, Schanz M (2009) An alternative collocation boundary element method for static and dynamic problems. Comput Mech 44: 247–261MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Simo J, Taylor R (1985) Consistent tangent operators for rate- independent elastoplasticity. Comput Meth Appl Mech Eng 48(1): 101–118MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Steinbach O (1999) Fast evaluation of newton potentials in boundary element methods. E-W J Numer Math 7(3): 211–222MathSciNetzbMATHGoogle Scholar
  37. 37.
    Telles J, Brebbia C (1981) The boundary element method in plasticity. Appl Math Model 5: 275–281MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Telles J, Carrer J (1991) Implicit procedures for the solution of elastoplastic problems by the boundary element method. Math Comput Model 15: 303–311zbMATHCrossRefGoogle Scholar
  39. 39.
    Wang P, Yao Z (2007) Fast multipole boundary element analysis of two-dimensional elastoplastic problems. Commun Numer Meth Eng 23: 889–903MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Zhao Z, Lan S (1999) Boundary stress calculation–a comparison study. Comput Struct 71(1): 77–85CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute for Structural AnalysisGraz University of TechnologyGrazAustria

Personalised recommendations