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Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

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Abstract

This paper presents a new strategy for the numerical evaluation of 2D nearly singular integrals that arise in the solution of 3D BEM using eight-node second-order quadrilateral surface elements. The strategy is an extension of the previous sinh transformation techniques, which is used to evaluate the 1D or 2D nearly singular integrals on simple geometry elements, such as linear or planar elements. The novel feature of the proposed method is that a new distance formula is introduced here, and based on this an ingenious combination of the distance formula and the sinh transformation is developed, and hence, the rapid variation of the distance formula on the integration interval can been smoothed out. Several numerical examples involving boundary layer effect and thin-body problems in 3D potential theory are given to verify the accuracy and efficiency of the presented method.

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References

  1. Cerrolaza M., Alarcon E.: A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods. Int. J. Numer. Meth. Eng. 28, 987–999 (1989)

    Article  MathSciNet  Google Scholar 

  2. Scuderi L.: A new smoothing strategy for computing nearly singular integrals in 3D Galerkin BEM. J. Comput. Appl. Math. 225, 406–427 (2009)

    Article  MathSciNet  Google Scholar 

  3. Ye W.J.: A new transformation technique for evaluating nearly singular integrals. Comput. Mech. 42, 457–466 (2008)

    Article  Google Scholar 

  4. Sladek V., Sladek J., Tanaka M.: Optimal transformations of the integration variables in computation of singular integrals in BEM. Int. J. Numer. Methods Eng. 47, 1263–1283 (2000)

    Article  Google Scholar 

  5. Johnston P.R.: Application of Sigmoidal transformations to weakly singular and near singular boundary element integrals. Int. J. Numer. Methods Eng. 45, 1333–1348 (1999)

    Article  MathSciNet  Google Scholar 

  6. Johnston Barbara M., Johnston P.R., Elliott D.: A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method. J. Comput. Appl. Math. 245, 148–161 (2013)

    Article  MathSciNet  Google Scholar 

  7. Johnston P.R., Elliott D.: A sinh transformation for evaluating nearly singular boundary element integrals. Int. J. Numer. Methods Eng. 62, 564–578 (2005)

    Article  MathSciNet  Google Scholar 

  8. Johnston B.M., Johnston P.R., Elliott D.: A sinh transformation for evaluating two- dimensional nearly singular boundary element integrals. Int. J. Numer. Methods Eng. 69, 1460–1479 (2007)

    Article  MathSciNet  Google Scholar 

  9. Elliott D., Johnston P.R.: The iterated sinh transformation. Int. J. Numer. Methods Eng. 75, 43–57 (2008)

    Article  MathSciNet  Google Scholar 

  10. Johnston P.R., Johnston B.M., Elliott D.: Using the iterated sinh transformation to evaluate two dimensional nearly singular boundary element integrals. Eng. Anal. Bound. Elem. 37, 708–718 (2013)

    Article  MathSciNet  Google Scholar 

  11. Lv J.H., Miao Y., Gong W.H. et al.: The sinh transformation for curved elements using the general distance function. Comput. Model. Eng. Sci. 93, 113–131 (2013)

    MathSciNet  Google Scholar 

  12. Lv J.H., Miao Y., Zhu H.P.: The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements. Comput. Mech. 53, 359–367 (2014)

    Article  MathSciNet  Google Scholar 

  13. Huang Q., Cruse T.A.: Some notes on singular integral techniques in boundary element analysis. Int. J. Numer. Methods Eng. 36, 2643–2659 (1993)

    Article  MathSciNet  Google Scholar 

  14. Ma H., Kamiya N.: Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. Eng. Anal. Bound. Elem. 26, 329–339 (2002)

    Article  Google Scholar 

  15. Ma H., Kamiya N.: A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two-and three-dimensional elasticity. Comput. Mech. 29, 277–288 (2002)

    Article  Google Scholar 

  16. Qin X.Y., Zhang J.M., Xie G.Z. et al.: A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. J. Comput. Appl. Math. 235, 4174–4186 (2011)

    Article  MathSciNet  Google Scholar 

  17. Miao Y., Li W., Lv J.H., Zhu H.P.: Distance transformation for the numerical evaluation of nearly singular integrals on triangular elements. Eng. Anal. Bound. Elem. 37, 1311–1317 (2013)

    Article  MathSciNet  Google Scholar 

  18. Zhang Y.M., Sun C.L.: A general algorithm for the numerical evaluation of nearly singular boundary integrals in the equivalent non-singular BIEs with indirect unknowns. J. Chin. Inst. Eng. 31, 437–447 (2008)

    Article  Google Scholar 

  19. Zhang Y.M., Gu Y., Chen J.T.: Boundary layer effect in BEM with high order geometry elements using transformation. Comput. Model. Eng. Sci. 45, 227–247 (2009)

    MathSciNet  Google Scholar 

  20. Zhang Y.M., Gu Y., Chen J.T.: Internal stress analysis for single and multilayered coating systems using the boundary element method. Eng. Anal. Bound. Elem. 35, 708–717 (2010)

    Article  Google Scholar 

  21. Zhang Y.M., Qu W.Z., Chen J.T.: BEM analysis of thin structures for thermoelastic problems. Eng. Anal. Bound. Elem. 37, 441–452 (2013)

    Article  MathSciNet  Google Scholar 

  22. Xie G.Z., Zhou F.L., Zhang J.M. et al.: New variable transformations for evaluating nearly singular integrals in 3D boundary element method. Eng. Anal. Bound. Elem. 37, 1169–1178 (2013)

    Article  MathSciNet  Google Scholar 

  23. Zhang Y.M., Li X.C., Sladek V., Sladek J., Gao X.W.: Computation of nearly singular integrals in 3D BEM. Eng. Anal. Bound. Elem. 48, 32–42 (2014)

    Article  MathSciNet  Google Scholar 

  24. Qin X.Y., Zhang J.M., Xie G.X. et al.: A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. J. Comput. Appl. Math. 235, 4174–4186 (2011)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    XiaoChao Li & Yu Su

  2. Institute of Applied Mathematics, Shandong University of Technology, Zibo, 255049, People’s Republic of China

    XiaoChao Li, Yaoming Zhang & Yanpeng Gong

  3. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024, Dalian, People’s Republic of China

    Yaoming Zhang & Xiaowei Gao

Authors
  1. XiaoChao Li
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  2. Yaoming Zhang
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  3. Yanpeng Gong
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  4. Yu Su
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  5. Xiaowei Gao
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Correspondence to Yaoming Zhang.

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Li, X., Zhang, Y., Gong, Y. et al. Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM. Acta Mech 226, 2873–2885 (2015). https://doi.org/10.1007/s00707-015-1361-z

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  • DOI: https://doi.org/10.1007/s00707-015-1361-z

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