Implementation of sinh method in integration space for boundary integrals with near singularity in potential problems

Abstract

As a widely used numerical method, boundary element method (BEM) is efficient for computer aided engineering (CAE). However, boundary integrals with near singularity need to be calculated accurately and efficiently to implement BEM for CAE analysis on thin bodies successfully. In this paper, the distance in the denominator of the fundamental solution is first designed as an equivalent form using approximate expansion and the original sinh method can be revised into a new form considering the minimum distance and the approximate expansion. Second, the acquisition of the projection point by Newton-Raphson method is introduced. We acquire the nearest point between the source point and element edge by solving a cubic equation if the location of the projection point is outside the element, where boundary integrals with near singularity appear. Finally, the subtriangles of the local coordinate space are mapped into the integration space and the sinh method is applied in the integration space. The revised sinh method can be directly performed in the integration element. A verification test of our method is proposed. Results demonstrate that our method is effective for regularizing the boundary integrals with near singularity.

Key words： computer aided engineering (CAE); boundary element method (BEM); near singularity; sinh method; coordinate transformation; integration space

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11472102 and 51476149), the Research Found for the Doctoral Program of Zhengzhou University of Light Industry, Educational Commission of Henan Province (Grant Nos. 15A460002 and 15A460037), and Xi’an Jiaotong University State Key Laboratory for Manufacturing Systems Engineering (Grant No. sklms2015012).

References

1 Liu Y. Fast Multipole Boundary Element Method: Theory and Applications in Engineering. Cambridge: Cambridge University Press, 2009

2 Feng S, Cui X, Li A. Fast and efficient analysis of transient nonlinear heat conduction problems using combined approximations (CA) method. Transfer, 2016, 97: 638–644

3 Feng S, Cui X, Chen F, An edge/face-based smoothed radial point interpolation method for static analysis of structures. Engineering Analysis with Boundary Elements, 2016, 68: 1–10

4 Feng S, Cui X, Li G. Analysis of transient thermo-elastic problems using edge-based smoothed finite element method. International Journal of Thermal Sciences, 2013, 65: 127–135

5 Feng S, Cui X, Li G. Transient thermal mechanical analyses using a face-based smoothed finite element method (FS-FEM). International Journal of Thermal Sciences, 2013, 74: 95–103

6 Cui X, Liu G, Li G. A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids. Engineering Analysis with Boundary Elements, 2010, 34(2): 144–157

7 Cui X, Feng S, Li G. A cell-based smoothed radial point interpolation method (CS-RPIM) for heat transfer analysis. Engineering Analysis with Boundary Elements, 2014, 40: 147– 153

8 Cheng A H D, Cheng D T. Heritage and early history of the boundary element method. Engineering Analysis with Boundary Elements, 2005, 29(3): 268–302

9 Gao X. The radial integration method for evaluation of domain integrals with boundary-only discretization. Engineering Analysis with Boundary Elements, 2002, 26(10): 905–916

10 Gao X, Davies T G. Adaptive integration in elasto-plastic boundary element analysis. Journal of the Chinese Institute of Engineers, 2000, 23(3): 349–356

11 Zhang J, Qin X, Han X, A boundary face method for potential problems in three dimensions. International Journal for Numerical Methods in Engineering, 2009, 80(3): 320–337

12 Niu Z, Wendland W, Wang X, et al. A sim-analytic algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Computer Methods in Applied Mechanics and Engineering, 2005, 31: 949–964

13 Niu Z, Zhou H. The natural boundary integral equation in potential problems and regularization of the hypersingular integral. Computers & Structures, 2004, 82(2–3): 315–323

14 Zhou H, Niu Z, Cheng C, Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies. Engineering Analysis with Boundary Elements, 2007, 31(9): 739–748

15 Zhou H, Niu Z, Cheng C, Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems. Computers & Structures, 2008, 86(15–16): 1656–1671

16 Lv J, Miao Y, Zhu H. The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements. Computational Mechanics, 2014, 53(2): 359–367

17 Ma H, Kamiya N. A general algorithm for accurate computation of field variables and its derivatives near the boundary in BEM. Engineering Analysis with Boundary Elements, 2001, 25(10): 833–841

18 Ma H, Kamiya N. Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. Engineering Analysis with Boundary Elements, 2002, 26(4): 329–339

19 Ma H, Kamiya N. Nearly singular approximations of CPV integrals with end-and corner-singularities for the numerical solution of hypersingular boundary integral equations. Engineering Analysis with Boundary Elements, 2003, 27(6): 625–637

20 Hayami K, Matsumoto H. A numerical quadrature for nearly singular boundary element integrals. Engineering Analysis with Boundary Elements, 1994, 13(2): 143–154

21 Hayami K. Variable transformations for nearly singular integrals in the boundary element method. Publications of the Research Institute for Mathematical Sciences, 2005, 41(4): 821–842

22 Zhang Y, Gu Y, Chen J. Boundary layer effect in BEM with high order geometry elements using transformation. Computer Modeling in Engineering and Sciences, 2009, 45(3): 227–247

23 Zhang Y, Gu Y, Chen J. Boundary element analysis of the thermal behaviour in thin-coated cutting tools. Engineering Analysis with Boundary Elements, 2010, 34(9): 775–784

24 Xie G, Zhang J, Qin X, New variable transformations for evaluating nearly singular integrals in 2D boundary element method. Engineering Analysis with Boundary Elements, 2011, 35(6): 811–817

25 Xie G, Zhang J, Dong Y, An improved exponential transformation for nearly singular boundary element integrals in elasticity problems. International Journal of Solids and Structures, 2014, 51(6): 1322–1329

26 Johnston P R, Elliott D. A sinh transformation for evaluating nearly singular boundary element integrals. International Journal for Numerical Methods in Engineering, 2005, 62(4): 564–578

27 Johnston B M, Johnston P R, Elliott D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. International Journal for Numerical Methods in Engineering, 2007, 69(7): 1460–1479

28 Gu Y, Chen W, Zhang C. Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method. International Journal of Solids and Structures, 2013, 50(20–21): 3460–3471

29 Gu Y, Chen W, He X. Improved singular boundary method for elasticity problems. Computers & Structures, 2014, 135: 73–82

30 Lv J, Miao Y, Gong W, The sinh transformation for curved elements using the general distance function. Computer Modeling in Engineering & Sciences, 2013, 93(2): 113–131