Skip to main content
SpringerLink
Log in

An adaptive cell-based domain integration method for treatment of domain integrals in 3D boundary element method for potential and elasticity problems

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

An adaptive cell-based domain integration method (CDIM) is proposed for the treatment of domain integrals in 3D boundary element method (BEM). The domain integrals are computed in background cells rather than volume elements. The cells are created from the boundary elements based on an adaptive oct-tree structure and no other discretization is needed. Cells containing the boundary elements are subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements; and the sub-cells outside the domain are deleted to obtain the desired accuracy. The method is applied in the 3D potential and elasticity problems in this paper.

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. M. Jaswon, Integral equation methods in potential theory. I, in: Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 275, 1963, pp. 23–32.

    Article  MathSciNet  MATH  Google Scholar 

  2. M.A. Jaswon, G.T. Symm, Integral Equation Methods in Potential Theory and Elastostatics, Oxford University Press, United Kingdom, 1977.

    MATH  Google Scholar 

  3. M.K. Chati, S. Mukherjee, Y.X. Mukherjee, The boundary node method for three-dimensional linear elasticity, Int. J. Numer. Methods Eng. 46 (8) (1999) 1163–1184.

    Article  MATH  Google Scholar 

  4. J. Lv, Y. Miao, H. Zhu, Boundary node method based on parametric space for 2D elasticity, Eng. Anal. Boundary Elem. 37 (4) (2013) 659–665.

    Article  MathSciNet  MATH  Google Scholar 

  5. X. Li, J. Zhu, A Galerkin boundary node method and its convergence analysis, J. Comput. Appl. Math. 230 (1) (2009) 314–328.

    Article  MathSciNet  MATH  Google Scholar 

  6. Y. Miao, Y.-H. Wang, An improved hybrid boundary node method in two-dimensional solids, Acta Mech. Solida Sin. 18 (4) (2005) 307–315.

    Google Scholar 

  7. Y. Miao, Y.-h. Wang, Meshless analysis for three-dimensional elasticity with singular hybrid boundary node method, Appl. Math. Mech. 27 (5) (2006) 673–681.

    Article  MATH  Google Scholar 

  8. Y. Miao, T. He, Q. Yang, J. Zheng, Multi-domain hybrid boundary node method for evaluating top-down crack in Asphalt pavements, Eng. Anal. Boundary Elem. 34 (9) (2010) 755–760.

    Article  MATH  Google Scholar 

  9. Y. Miao, T. He, H. Luo, H. Zhu, Dual hybrid boundary node method for solving transient dynamic fracture problems, Comput. Model. Eng. Sci. (CMES) 85 (6) (2012) 481–498.

    MathSciNet  MATH  Google Scholar 

  10. J. Zhang, X. Qin, X. Han, G. Li, A boundary face method for potential problems in three dimensions, Int. J. Numer. Methods Eng. 80 (3) (2009) 320–337.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Zhou, J. Zhang, X. Sheng, G. Li, Shape variable radial basis function and its application in dual reciprocity boundary face method, Eng. Anal. Boundary Elem. 35 (2) (2011) 244–252.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Lv, Y. Miao, H. Zhu, Y. Li, A Kriging interpolation-based boundary face method for 3D potential problems, Eng. Anal. Boundary Elem. 37 (5) (2013) 812–817.

    Article  MathSciNet  MATH  Google Scholar 

  13. S.-C. Hsiao, A. Mammoli, M. Ingber, The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods, Comput. Mech. 32 (4-6) (2003) 226–233.

    Article  MATH  Google Scholar 

  14. D. Nardini, C. Brebbia, A new approach to free vibration analysis using boundary elements, Appl. Math. Model. 7 (3) (1983) 157–162.

    Article  MATH  Google Scholar 

  15. R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (3) (1995) 251–264.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Neves, C. Brebbia, The multiple reciprocity boundary element method in elasticity: a new approach for transforming domain integrals to the boundary, Int. J. Numer. Methods Eng. 31 (4) (1991) 709–727.

    Article  MATH  Google Scholar 

  17. S. Ahmad, P.K. Banerjee, Free vibration analysis by BEM using particular integrals, J. Eng. Mech. 112 (7) (1986) 682–695.

    Article  Google Scholar 

  18. E. Pan, A. Chengz, Treatment of body forces in single-domain boundary integral equation method for anisotropic elasticity, Transformation of Domain Effects to the Boundary, 14, 2003, p. 95.

  19. E. Pan, B. Amadei, A 3-D boundary element formulation of anisotropic elasticity with gravity, Appl. Math. Model. 20 (2) (1996) 114–120.

    Article  Google Scholar 

  20. X.-W. Gao, The radial integration method for evaluation of domain integrals with boundary-only discretization, Eng. Anal. Boundary Elem. 26 (10) (2002) 905–916.

    Article  MATH  Google Scholar 

  21. X.-W. Gao, Evaluation of regular and singular domain integrals with boundary-only discretization—theory and Fortran code, J. Comput. Appl. Math. 175 (2) (2005) 265–290.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Nintcheu Fata, Treatment of domain integrals in boundary element methods, Appl. Numer. Math. 62 (6) (2012) 720–735.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Hematiyan, A general method for evaluation of 2D and 3D domain integrals without domain discretization and its application in BEM, Comput. Mech. 39 (4) (2007) 509–520.

    Article  MATH  Google Scholar 

  24. M. Hematiyan, A. Khosravifard, T. Bui, Efficient evaluation of weakly/strongly singular domain integrals in the BEM using a singular nodal integration method, Eng. Anal. Boundary Elem. 37 (4) (2013) 691–698.

    Article  MathSciNet  MATH  Google Scholar 

  25. Z. Sedaghatjoo, H. Adibi, Calculation of domain integrals of two dimensional boundary element method, Eng. Anal. Boundary Elem. 36 (12) (2012) 1917–1922.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Dallner, G. Kuhn, Efficient evaluation of volume integrals in the boundary element method, Comput. Methods Appl. Mech. Eng. 109 (1) (1993) 95–109.

    Article  MathSciNet  MATH  Google Scholar 

  27. M.S. Ingber, A.A. Mammoli, M.J. Brown, A comparison of domain integral evaluation techniques for boundary element methods, Int. J. Numer. Methods Eng. 52 (4) (2001) 417–432.

    Article  MATH  Google Scholar 

  28. Y. Dong, C. Lu, Y. Li, J. Zhang, G. Xie, Y. Zhong, Accurate numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction problem, Eng. Anal. Boundary Elem. 60 (2015) 89–94.

    Article  MathSciNet  Google Scholar 

  29. Y. Dong, J. Zhang, G. Xie, C. Lu, L. Han, P. Wang, A general algorithm for the numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction, Eng. Anal. Boundary Elem. 51 (2015) 30–36.

    Article  MathSciNet  Google Scholar 

  30. H. Cheng, L. Greengard, V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys. 155 (2) (1999) 468–498.

    Article  MathSciNet  MATH  Google Scholar 

  31. Y. Fu, K.J. Klimkowski, G.J. Rodin, E. Berger, J.C. Browne, J.K. Singer, R.A. Van De Geijn, K.S. Vemaganti, A fast solution method for three-dimensional many-particle problems of linear elasticity, Int. J. Numer. Methods Eng. 42 (7) (1998) 1215–1229.

    Article  MATH  Google Scholar 

  32. Q. Wang, Y. Miao, H. Zhu, A fast multipole hybrid boundary node method for composite materials, Comput. Mech. 51 (6) (2013) 885–897.

    Article  MathSciNet  MATH  Google Scholar 

  33. Q. Wang, Y. Miao, J. Zheng, The hybrid boundary node method accelerated by fast multipole expansion technique for 3D elasticity, Comput. Model. Eng. Sci. 70 (2) (2010) 123–151.

    MathSciNet  MATH  Google Scholar 

  34. Q. Wang, Y. Miao, H. Zhu, C. Zhang, An O (N) fast multipole hybrid boundary node method for 3D elasticity, Comput. Mater. Continua 28 (1) (2012) 1–25.

    Google Scholar 

  35. G. Of, O. Steinbach, P. Urthaler, Fast evaluation of volume potentials in boundary element methods, SIAM J. Sci. Comput. 32 (2) (2010) 585–602.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Ding, W. Ye, L. Gray, An accelerated surface discretization-based BEM approach for non-homogeneous linear problems in 3-D complex domains, Int. J. Numer. Methods Eng. 63 (12) (2005) 1775–1795.

    Article  MATH  Google Scholar 

  37. O. Steinbach, L. Tchoualag, Fast Fourier transform for efficient evaluation of Newton potential in BEM, Appl. Numer. Math. 81 (2014) 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  38. Y. Shiah, Y.-C. Chaing, T. Matsumoto, Analytical transformation of volume integral for the time-stepping BEM analysis of 2D transient heat conduction in anisotropic media, Eng. Anal. Boundary Elem. 64 (2016) 101–110.

    Article  MathSciNet  Google Scholar 

  39. W. Qu, W. Chen, Z. Fu, Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method, Eng. Anal. Boundary Elem. 60 (2015) 2–9.

    Article  MathSciNet  Google Scholar 

  40. J. Telles, A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int. J. Numer. Methods Eng. 24 (5) (1987) 959–973.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 430072, Wuhan, China

    Qiao Wang, Wei Zhou, Yonggang Cheng, Gang Ma, Xiaolin Chang & Qiang Huang

Authors
  1. Qiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yonggang Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiaolin Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Qiang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Qiao Wang or Wei Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Q., Zhou, W., Cheng, Y. et al. An adaptive cell-based domain integration method for treatment of domain integrals in 3D boundary element method for potential and elasticity problems. Acta Mech. Solida Sin. 30, 99–111 (2017). https://doi.org/10.1016/j.camss.2016.08.002

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1016/j.camss.2016.08.002

Keywords

Search

Navigation