A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation.

*(English)*Zbl 1163.74559Summary: A new fast multipole formulation for the hypersingular BIE (HBIE) for 2D elasticity is presented in this paper based on a complex-variable representation of the kernels, similar to the formulation developed earlier for the conventional BIE (CBIE). A dual BIE formulation using a linear combination of the developed CBIE and HBIE is applied to analyze multi-domain problems with thin inclusions or open cracks. Two pre-conditioners for the fast multipole boundary element method (BEM) are devised and their effectiveness and efficiencies in solving large-scale problems are discussed. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM using the dual BIE formulation. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale 2D multi-domain elasticity problems. The method can be applied to study composite materials, functionally-graded materials, and micro-electro-mechanical-systems with coupled fields, all of which often involve thin shapes or thin inclusions.

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

74B05 | Classical linear elasticity |

74E05 | Inhomogeneity in solid mechanics |

Full Text:
DOI

**OpenURL**

##### References:

[1] | Rokhlin V (1985). Rapid solution of integral equations of classical potential theory. J Comp Phys 60: 187–207 · Zbl 0629.65122 |

[2] | Greengard LF and Rokhlin V (1987). A fast algorithm for particle simulations. J Comput Phys 73(2): 325–348 · Zbl 0629.65005 |

[3] | Greengard LF (1988). The rapid evaluation of potential fields in particle systems. MIT Press, Cambridge · Zbl 1001.31500 |

[4] | Peirce AP and Napier JAL (1995). A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics. Int J Numer Meth Eng 38: 4009–4034 · Zbl 0852.73076 |

[5] | Gomez JE and Power H (1997). A multipole direct and indirect BEM for 2D cavity flow at low Reynolds number. Eng Anal Bound Elem 19: 17–31 |

[6] | Fu Y, Klimkowski KJ, Rodin GJ, Berger E, Browne JC, Singer JK, Geijn RAVD and Vemaganti KS (1998). A fast solution method for three-dimensional many-particle problems of linear elasticity. Int J Numer Meth Eng 42: 1215–1229 · Zbl 0904.73072 |

[7] | Nishimura N, Yoshida K and Kobayashi S (1999). A fast multipole boundary integral equation method for crack problems in 3D. Eng Anal Bound Elem 23: 97–105 · Zbl 0953.74074 |

[8] | Mammoli AA and Ingber MS (1999). Stokes flow around cylinders in a bounded two-dimensional domain using multipole-accelerated boundary element methods. Int J Numer Meth Eng 44: 897–917 · Zbl 0955.76061 |

[9] | Nishimura N (2002). Fast multipole accelerated boundary integral equation methods. Appl Mech Rev 55(4): 299–324 |

[10] | Greengard LF, Kropinski MC and Mayo A (1996). Integral equation methods for Stokes flow and isotropic elasticity in the plane. J Comput Phys 125: 403–414 · Zbl 0847.76066 |

[11] | Greengard LF and Helsing J (1998). On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites. J Mech Phys Solids 46(8): 1441–1462 · Zbl 0955.74054 |

[12] | Richardson JD, Gray LJ, Kaplan T and Napier JA (2001). Regularized spectral multipole BEM for plane elasticity. Eng Anal Bound Elem 25: 297–311 · Zbl 1007.74084 |

[13] | Fukui T (1998) Research on the boundary element method–development and applications of fast and accurate computations. Ph.D. dissertation (in Japanese), Department of Global Environment Engineering, Kyoto University |

[14] | Fukui T, Mochida T, Method K (1997) Crack extension analysis in system of growing cracks by fast multipole boundary element method (in Japanese), Seventh BEM technology conference (JASCOME, Tokyo, 1997) pp 25–30 |

[15] | Liu YJ and Nishimura N (2006). The fast multipole boundary element method for potential problems: a tutorial. Eng Anal Bound Elem 30(5): 371–381 · Zbl 1187.65134 |

[16] | Liu YJ (2005). A new fast multipole boundary element method for solving large-scale two-dimensional elastostatic problems. Int J Numer Meth Eng 65(6): 863–881 · Zbl 1121.74061 |

[17] | Wang P and Yao Z (2006). Fast multipole DBEM analysis of fatigue crack growth. Computat Mech 38: 223–233 · Zbl 1162.74047 |

[18] | Yamada Y, Hayami K (1995) A multipole boundary element method for two dimensional elastostatics. Report METR 95–07, Department of Mathematical Engineering and Information Physics, University of Tokyo · Zbl 0893.73074 |

[19] | Yao Z, Kong F, Wang H and Wang P (2004). 2D simulation of composite materials using BEM. Eng Anal Bound Elem 28(8): 927–935 · Zbl 1130.74476 |

[20] | Wang J, Crouch SL and Mogilevskaya SG (2005). A fast and accurate algorithm for a Galerkin boundary integral method. Comput Mech 37(1): 96 · Zbl 1158.65352 |

[21] | Rizzo FJ (1967). An integral equation approach to boundary value problems of classical elastostatics. Q Appl Math 25: 83–95 · Zbl 0158.43406 |

[22] | Mukherjeec S (1982). Boundary element methods in creep and fracture. Applied Science Publishers, New York |

[23] | Cruse TA (1988). Boundary element analysis in computational fracture mechanics. Kluwer, Dordrecht · Zbl 0648.73039 |

[24] | Brebbia CA and Dominguez J (1989). Boundary elements–an introductory course. McGraw-Hill, New York · Zbl 0691.73033 |

[25] | Banerjee PK (1994). The boundary element methods in engineering, 2nd edn. McGraw-Hill, New York |

[26] | Krishnasamy G, Rizzo FJ, Rudolphi TJ (1991) Hypersingular boundary integral equations: their occurrence, interpretation, regularization and computation. In: Banerjee PK et al (eds) Developments in boundary element methods, Chap 7. Elsevier, London |

[27] | Sladek V, Sladek J (eds) (1998) Singular integrals in boundary element methods. In: Brebbia CA, Aliabadi MH (eds) Advances in boundary element series, Computational Mechanics Publications, Boston · Zbl 0954.65082 |

[28] | Mukherjee S (2000). Finite parts of singular and hypersingular integrals with irregular boundary source points. Eng Anal Bound Elem 24: 767–776 · Zbl 0991.74079 |

[29] | Liu YJ and Rizzo FJ (1992). A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems. Comp Meth Appl Mech Eng 96: 271–287 · Zbl 0754.76072 |

[30] | Liu YJ and Rizzo FJ (1993). Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions. Comp Meth Appl Mech Eng 107: 131–144 · Zbl 0806.73073 |

[31] | Liu YJ, Rizzo FJ (1997) Scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation. J Acoust Soc Am 102(2), No. Pt.1, August, 926–932 |

[32] | Shen L and Liu YJ (2007). An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation. Comput Mech 40(3): 461–472 · Zbl 1176.76083 |

[33] | Liu YJ (2006). Dual BIE approaches for modeling electrostatic MEMS problems with thin beams and accelerated by the fast multipole method. Eng Anal Bound Elem 30(11): 940–948 · Zbl 1195.78068 |

[34] | Liu YJ and Shen L (2007). A dual BIE approach for large-scale modeling of 3-D electrostatic problems with the fast multipole boundary element method. Int J Numer Meth Eng 71(7): 837–855 · Zbl 1194.78058 |

[35] | Liu YJ (2008). A new fast multipole boundary element method for solving 2D Stokes flow problems based on a dual BIE formulation. Eng Anal Bound Elem 32(2): 139–151 · Zbl 1244.76052 |

[36] | Muskhelishvili NI (1958). Some basic problems of mathematical theory of elasticity. Noordhoff, Groningen |

[37] | Sokolnikoff IS (1956). Mathematical theory of elasticity. McGraw Hill, New York · Zbl 0070.41104 |

[38] | Mogilevskaya SG and Linkov AM (1998). Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 22(1): 88–92 · Zbl 0914.73073 |

[39] | Linkov AM and Mogilevskaya SG (1998). Complex hypersingular BEM in plane elasticity problems. In: Sladek, V and Sladek, J (eds) Singular integrals in boundary element method, pp 299–364. Computational Mechanics Publications, Southampton |

[40] | Liu YJ, Nishimura N, Otani Y, Takahashi T, Chen XL and Munakata H (2005). A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model. J Appl Mech 72(1): 115–128 · Zbl 1111.74528 |

[41] | Liu YJ, Nishimura N and Otani Y (2005). Large-scale modeling of carbon-nanotube composites by the boundary element method based on a rigid-inclusion model. Comput Mater Sci 34(2): 173– 187 |

[42] | Liu YJ, Xu N and Luo JF (2000). Modeling of interphases in fiber-reinforced composites under transverse loading using the boundary element method. J Appl Mech 67(1): 41–49 · Zbl 1110.74568 |

[43] | Krishnasamy G, Rizzo FJ and Liu YJ (1994). Boundary integral equations for thin bodies. Int J Numer Meth Eng 37: 107–121 · Zbl 0795.73076 |

[44] | Wang J, Mogilevskaya SG and Crouch SL (2003). Benchmarks results for the problem of interaction between a crack and a circular inclusion. J Appl Mech 70: 619–621 · Zbl 1110.74740 |

[45] | Mogilevskaya SG and Crouch SL (2004). A Galerkin boundary integral method for multiple circular elastic inclusions with uniform interphase layers. Int J Solids Struct 41: 1285–1311 · Zbl 1106.74420 |

[46] | Wang J, Mogilevskaya SG and Crouch SL (2005). An embedding method for modeling micromechanical behavior and macroscopic properties of composite materials. Int J Solids Struct 42: 4588–4612 · Zbl 1119.74536 |

[47] | Gross D and Seelig T (2006). Fracture mechanics with an introduction to micromechanics. Springer, The Netherlands · Zbl 1110.74001 |

[48] | Gomez JE and Power H (2000). A parallel multipolar indirect boundary element method for the Neumann interior Stokes flow problem. Int J Numer Meth Eng 48(4): 523–543 · Zbl 0974.76054 |

[49] | Yoshida K, Nishimura N and Kobayashi S (2001). Application of fast multipole Galerkin boundary integral equation method to crack problems in 3D. Int J Numer Meth Eng 50: 525–547 · Zbl 1004.74078 |

[50] | Shen L and Liu YJ (2007). An adaptive fast multipole boundary element method for three-dimensional potential problems. Comput Mech 39(6): 681–691 · Zbl 1198.74113 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.