Structural Engineering and Mechanics

  • 제30권3호
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  • Pages.279-296
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  • 2008
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Exact integration for the hypersingular boundary integral equation of two-dimensional elastostatics

  • Zhang, Xiaosong (Department of Engineering Mechanics, Shijiazhuang Railway Institute) ;
  • Zhang, Xiaoxian (Department of Engineering, The University of Liverpool)
  • 투고 : 2007.01.02
  • 심사 : 2008.08.11
  • 발행 : 2008.10.20
⟨ 이전 논문 다음 논문 ⟩

초록

This paper presents an exact integration for the hypersingular boundary integral equation of two-dimensional elastostatics. The boundary is discretized by straight segments and the physical variables are approximated by discontinuous quadratic elements. The integral for the hypersingular boundary integral equation analysis is given in a closed form. It is proven that using the exact integration for discontinuous boundary element, the singular integral in the Cauchy Principal Value and the hypersingular integral in the Hadamard Finite Part can be obtained straightforward without special treatment. Two numerical examples are implemented to verify the correctness of the derived exact integration.

키워드

참고문헌

  1. Brebbia, C.A. (1978), The Boundary Element Method for Engineers. Pentech Press, London
  2. Brebbia, C.A. and Dominguez, J. (1989), Boundary Elements: An Introductory Course. Computational Mechanics Publications; Southampton
  3. Chen, J.T. and Hong, H.K. (1992), Boundary Element Methods, 2nd Edition, New World Press, Taipei
  4. Chen, J.T. and Hong, H.K. (1999), "Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series", Appl. Mech. Review, 52(1), 17-33 https://doi.org/10.1115/1.3098922
  5. Chen, J.T., Chen, K.H., Yeih, W. and Shieh, N.C. (1998), "Dual boundary element analysis for cracked bars under torsion", Eng. Comput., 15, 732-749 https://doi.org/10.1108/02644409810231871
  6. Chen, J.T., Hsiao, C.C., Chiu, P.Y. and Lee, Y.T. (2007), "Study of free-surface seepage problems using hypersingular equations", Comm. Numer. Meth. Eng., DOI:10.1002/cnm.925 (in Press)
  7. Chen, J.T., Kuo, S.R. and Lin, J.H. (2003), "Analytical study and numerical experiments for degenerate scale problems in boundary element method for two-dimensional elasticity", Int. J. Numer. Meth. Eng., 54, 1669- 1681 https://doi.org/10.1002/nme.476
  8. Chen, X.L. and Liu, Y.J. (2001), "Thermal stress analysis of multi-layer thin film and coatings by an advanced boundary element method", CMES: Comput. Modeling Eng. Sci., 2(3), 337-349
  9. Chien, C.C., You, Z.Y. and Chung, Y.L. (1997), "Internal stress calculation using an iterative subdivision scheme in the boundary element method", Eng. Anal. Bound. Elem., 20, 145-153 https://doi.org/10.1016/S0955-7997(97)00074-X
  10. Florez, W.F. and Power, H. (2001), "Comparison between continuous and discontinuous boundary elements in multidomain dual reciprocity method for the solution of two-dimensional Navier-Stokes equations", Eng. Anal. Bound. Elem., 25, 57-69 https://doi.org/10.1016/S0955-7997(00)00051-5
  11. Fratantonio, M. and Rencis, J.J. (2000), "Exact boundary element integrations for two-dimensional Laplace equation", Eng. Anal. Bound. Elem., 24, 325-342 https://doi.org/10.1016/S0955-7997(00)00005-9
  12. Friedrich, F. (2002), "A linear analytical boundary element method for 2D homogeneous potential problems", Comput. Geosci., 28(5), 679-692 https://doi.org/10.1016/S0098-3004(01)00093-0
  13. Gray, L.J., Chinta, B. and Kane, J.H. (1995), "Symmetric Galerkin fracture analysis", Eng. Anal. Bound. Elem., 15, 103-109 https://doi.org/10.1016/0955-7997(95)00023-H
  14. Guiggiani, M. (1995), "Hypersingular boundary integral equations have an additional term", Comput. Mech., 16, 245-248 https://doi.org/10.1007/BF00369869
  15. Hong, H.K. and Chen, J.T. (1988), "Derivation of integral equations of Elasticity", J. Eng. Mech., 114(6), 1028- 1044 https://doi.org/10.1061/(ASCE)0733-9399(1988)114:6(1028)
  16. Liang, M.T., Chen, J.T. and Yang, S.S. (1999), "Error estimation for boundary element method", Eng. Anal. Bound. Elem., 23, 257-265 https://doi.org/10.1016/S0955-7997(98)00086-1
  17. Liu, Y.J. (1998), "Analysis of shell-like structures by boundary element method based on 3-D elasticity: Formulation and verification", Int. J. Numer. Meth. Eng., 41, 541-558 https://doi.org/10.1002/(SICI)1097-0207(19980215)41:3<541::AID-NME298>3.0.CO;2-K
  18. Lu, Y.Y., Belytschko, T. and Liu, W.K. (1991), "A variationally coupled FE-BE method for elasticity and fracture mechanics", Comput. Meth. Appl. Mech. Eng., 85, 21-37 https://doi.org/10.1016/0045-7825(91)90120-U
  19. Luo, J.F., Liu, Y.J. and Berger, E.J. (1998), "Analysis of two-dimensional thin structures (from micro- to nanoscales) using the boundary element method", Comput. Mech., 22, 404-412 https://doi.org/10.1007/s004660050372
  20. Mera, N.S., Elliott, L., Ingham, D.B. and Lesnic, D. (2001), "A comparison of boundary element method formulations for anisotropic heat conduction problems", Eng. Anal. Bound. Elem., 25, 115-128 https://doi.org/10.1016/S0955-7997(00)00050-3
  21. Padhi, G.S., Shenoi, R.A., Moy, S.S.J. and McCarthy, M.A. (2001), "Analytic integration of kernel shape function product integrals in boundary element method", Comput. Struct., 79, 1325-1333 https://doi.org/10.1016/S0045-7949(01)00020-7
  22. Patterson, C. and Sheikh, M.A. (1989), "Interelement continuity in boundary element method", In Topics in Boundary Element Research. Springer-Verlag, 1, 123-141
  23. Paulino, G.H., Menon, G. and Mukherjee, S. (2001), "Error estimation using hypersingular integrals in boundary element methods for linear elasticity", Eng. Anal. Bound. Elem., 25, 523-534 https://doi.org/10.1016/S0955-7997(01)00019-4
  24. Singh, K.M. and Tanaka, M. (2001), "On non-linear transformation for accurate numerical evaluation of weakly singular boundary integrals", Int. J. Numer. Meth. Eng., 50, 2007-2030 https://doi.org/10.1002/nme.117
  25. Tadeu, A. and Antonio, J. (2000), "Use of constant, linear and quadratic boundary elements in 3D wave diffraction analysis", Eng. Anal. Bound. Elem., 25, 131-144
  26. Telles, J.C.F. (1987), "A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary integrals", Int. J. Numer. Meth. Eng., 24, 959-973 https://doi.org/10.1002/nme.1620240509
  27. Timoshenko, S.P. and Goodier, J.N. (1987), Theory of Elasticity. 3rd edition. McGraw-Hill, New York
  28. Trevelyan, J. (1992), "Use of discontinuous boundary elements for fracture mechanics analysis", Eng. Anal. Bound. Elem., 10, 353-358 https://doi.org/10.1016/0955-7997(92)90150-6
  29. Xu, J.M. and Brebbia, C.A. (1986), "Optimum positions for nodes in discontinuous boundary element", Proceeding of 8th International Conference on BEM, Springer-Verlag, 761-767
  30. Yoon, S.S. and Heister, S.D. (2000), "Analytic solution for fluxes at interior points for 2D Laplace equation", Eng. Anal. Bound. Elem., 24, 155-160 https://doi.org/10.1016/S0955-7997(99)00059-4
  31. Zhang, X.S. and Zhang X.X. (2004), "Exact integrations of two-dimensional high-order discontinuous boundary elements of elastostatics problems", Eng. Anal. Bound. Elem., 28, 725-732 https://doi.org/10.1016/j.enganabound.2004.02.003
  32. Zhang, X.S. and Zhang, X.X. (2002), "Coupling FEM and discontinuous BEM for elastostatics and fluidstructure interaction", Eng. Anal. Bound. Elem., 26, 719-725 https://doi.org/10.1016/S0955-7997(02)00031-0
  33. Zhang, X.S. and Zhang, X.X. (2003), "Exact integration in the boundary element method for two-dimensional elastostatics problems", Eng. Anal. Bound. Elem., 27, 987-997 https://doi.org/10.1016/S0955-7997(03)00091-2
  34. Zhang, X.S. and Zhang, X.X. (2004), "Exact integration for stress evaluation in boundary element analysis of two-dimensional elastostatics", Eng. Anal. Bound. Elem., 28, 997-1004 https://doi.org/10.1016/j.enganabound.2004.03.003
  35. Zhang, X.S., Ye, T.Q. and Wang, R.P. (1993), "Multi-domain boundary element method and its parallelization", In Boundary Element Method, Elsevier Science Publishers, 67-78

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