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

doi:10.1007/s11465-016-0396-8

Front. Mech. Eng.

2016, Vol. 11

Issue (4) : 412-422 https://doi.org/10.1007/s11465-016-0396-8

RESEARCH ARTICLE

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

Guizhong XIE¹,Dehai ZHANG²(

),Jianming ZHANG³,Fannian MENG¹,Wenliao DU¹(

),Xiaoyu WEN¹

¹. Mechanical and Electrical Engineering Institute, Zhengzhou University of Light Industry, Zhengzhou 450002, China
². Mechanical and Electrical Engineering Institute, Zhengzhou University of Light Industry, Zhengzhou 450002, China; State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China
³. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China

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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.

Keywords computer aided engineering (CAE) boundary element method (BEM) near singularity sinh method coordinate transformation integration space

Corresponding Author(s): Dehai ZHANG,Wenliao DU

Online First Date: 29 July 2016 Issue Date: 29 November 2016

Cite this article:

Guizhong XIE,Dehai ZHANG,Jianming ZHANG, et al. Implementation of sinh method in integration space for boundary integrals with near singularity in potential problems[J]. Front. Mech. Eng., 2016, 11(4): 412-422.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-016-0396-8
https://academic.hep.com.cn/fme/EN/Y2016/V11/I4/412

Fig.1 Vertical distance r₀, between the source point Q and the integration element S

Fig.2 Coordinate transformation from the ( ξ , η ) space to integration space ( α , β )

Fig.3 Meshes of the eighth hollow sphere. (a) External view; (b) internal view

Fig.4 Domain and boundary evaluation points of the eighth hollow sphere

Fig.5 Numerical solutions of domain sample points with linear solution

Fig.6 Numerical solutions of boundary sample points with linear solution

Fig.7 Numerical solutions of domain sample points with quadratic solution

Fig.8 Numerical solutions of boundary sample points with quadratic solution

Fig.9 Meshes of the hollow square

Fig.10 Domain and boundary evaluation points of the hollow square

Fig.11 Numerical solutions of domain sample points of cubic solution

Fig.12 Numerical solutions of boundary sample points of cubic solution

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 https://doi.org/10.1016/j.ijheatmasstransfer.2016.02.061
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 https://doi.org/10.1016/j.enganabound.2016.03.016
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 https://doi.org/10.1016/j.ijthermalsci.2012.10.007
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 https://doi.org/10.1016/j.ijthermalsci.2013.07.002
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 https://doi.org/10.1016/j.enganabound.2009.07.011
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 https://doi.org/10.1016/j.enganabound.2013.12.004
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 https://doi.org/10.1016/j.enganabound.2004.12.001
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 https://doi.org/10.1016/S0955-7997(02)00039-5
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 https://doi.org/10.1080/02533839.2000.9670555
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 https://doi.org/10.1002/nme.2633
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 https://doi.org/10.1016/j.compstruc.2003.06.002
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 https://doi.org/10.1016/j.enganabound.2007.01.007
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 https://doi.org/10.1016/j.compstruc.2007.10.002
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 https://doi.org/10.1007/s00466-013-0913-0
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 https://doi.org/10.1016/S0955-7997(01)00073-X
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 https://doi.org/10.1016/S0955-7997(02)00004-8
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 https://doi.org/10.1016/S0955-7997(02)00149-2
20	Hayami K, Matsumoto H. A numerical quadrature for nearly singular boundary element integrals. Engineering Analysis with Boundary Elements, 1994, 13(2): 143–154 https://doi.org/10.1016/0955-7997(94)90017-5
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 https://doi.org/10.2977/prims/1145474596
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 https://doi.org/10.1016/j.enganabound.2010.03.014
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 https://doi.org/10.1016/j.enganabound.2011.01.009
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 https://doi.org/10.1016/j.ijsolstr.2013.12.020
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 https://doi.org/10.1002/nme.1208
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 https://doi.org/10.1002/nme.1816
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 https://doi.org/10.1016/j.ijsolstr.2013.06.018
29	Gu Y, Chen W, He X. Improved singular boundary method for elasticity problems. Computers & Structures, 2014, 135: 73–82 https://doi.org/10.1016/j.compstruc.2014.01.012
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

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