A meshless singular boundary method for three-dimensional elasticity problems.

*(English)*Zbl 1352.74039Summary: This study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration-free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed.

##### MSC:

74B05 | Classical linear elasticity |

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

65N38 | Boundary element methods for boundary value problems involving PDEs |

##### Keywords:

meshless boundary collocation method; method of fundamental solutions; singular boundary method; boundary element method; three-dimensional elasticity problems
PDF
BibTeX
XML
Cite

\textit{Y. Gu} et al., Int. J. Numer. Methods Eng. 107, No. 2, 109--126 (2016; Zbl 1352.74039)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Tanaka, Analysis of cracked shear deformable plates by an effective meshfree plate formulation, Engineering Fracture Mechanics 144 pp 142– (2015) |

[2] | Cheng, Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements 29 (3) pp 268– (2005) · Zbl 1182.65005 |

[3] | Hematiyan, Efficient evaluation of weakly/strongly singular domain integrals in the BEM using a singular nodal integration method, Engineering Analysis with Boundary Elements 37 (4) pp 691– (2013) · Zbl 1297.65176 |

[4] | Bui, Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method, Engineering Structures 47 pp 90– (2013) |

[5] | Nguyen, Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method, Engineering Analysis with Boundary Elements 44 pp 87– (2014) · Zbl 1297.74108 |

[6] | Racz, Novel adaptive meshfree integration techniques in meshless methods, International Journal for Numerical Methods in Engineering 90 (11) pp 1414– (2012) · Zbl 1242.74217 |

[7] | Belytschko, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 (1-4) pp 3– (1996) · Zbl 0891.73075 |

[8] | Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics 9 (1) pp 69– (1998) · Zbl 0922.65074 |

[9] | Mukherjee, The boundary node method for potential problems, International Journal for Numerical Methods in Engineering 40 (5) pp 797– (1997) · Zbl 0885.65124 |

[10] | Chen, Some comments on the ill-conditioning of the method of fundamental solutions, Engineering Analysis with Boundary Elements 30 (5) pp 405– (2006) · Zbl 1187.65136 |

[11] | Nguyen, Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation 79 (3) pp 763– (2008) · Zbl 1152.74055 |

[12] | Chen, Meshfree particle methods, Computational Mechanics 25 (2-3) pp 100– (2000) |

[13] | Liu, A meshfree method: meshfree weak-strong (MWS) form method, for 2-D solids, Computational Mechanics 33 (1) pp 2– (2003) · Zbl 1063.74105 |

[14] | Marin, Treatment of singularities in the method of fundamental solutions for two-dimensional Helmholtz-type equations, Applied Mathematical Modelling 34 (6) pp 1615– (2010) · Zbl 1193.35223 |

[15] | Chen, A study on the method of fundamental solutions using an image concept, Applied Mathematical Modelling 34 (12) pp 4253– (2010) · Zbl 1201.35023 |

[16] | Fu, Boundary knot method for heat conduction in nonlinear functionally graded material, Engineering Analysis with Boundary Elements 35 (5) pp 729– (2011) · Zbl 1259.80025 |

[17] | Gu, Singular boundary method for solving plane strain elastostatic problems, International Journal of Solids and Structures 48 (18) pp 2549– (2011) |

[18] | Gu, Improved singular boundary method for elasticity problems, Computers & Structures 135 (0) pp 73– (2014) |

[19] | Gu, Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, International Journal of Heat and Mass Transfer 90 pp 291– (2015) |

[20] | Karageorghis, The method of fundamental solutions for the numerical solution of the biharmonic equation, Journal of Computational Physics 69 (2) pp 434– (1987) · Zbl 0618.65108 |

[21] | Young, Novel meshless method for solving the potential problems with arbitrary domain, Journal of Computational Physics 209 (1) pp 290– (2005) · Zbl 1073.65139 |

[22] | Sarler, Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Engineering Analysis with Boundary Elements 33 (12) pp 1374– (2009) · Zbl 1244.76084 |

[23] | Karageorghis, Stress intensity factor computation using the method of fundamental solutions, Computational Mechanics 37 (5) pp 445– (2006) · Zbl 1138.74336 |

[24] | Chen, MA mesh-free approach to solving the axisymmetric Poisson’s equation, Numerical Methods for Partial Differential Equations 21 (2) pp 349– (2005) · Zbl 1072.65154 |

[25] | Poullikkas, The method of fundamental solutions for three-dimensional elastostatics problems, Computers & Structures 80 (3-4) pp 365– (2002) · Zbl 1067.74023 |

[26] | Marin, An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation, Computational Mechanics 45 (6) pp 665– (2010) · Zbl 1398.65278 |

[27] | Guiggiani, A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, Journal of Applied Mechanics 57 (4) pp 906– (1990) · Zbl 0735.73084 |

[28] | Cheng, Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions, Engineering Analysis with Boundary Elements 24 (7-8) pp 531– (2000) · Zbl 0966.65088 |

[29] | Sladek, Numerical integration of logarithmic and nearly logarithmic singularity in BEMs, Applied Mathematical Modelling 25 (11) pp 901– (2001) · Zbl 0994.65131 |

[30] | Chen, The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, International Journal for Numerical Methods in Engineering 43 (8) pp 1421– (1998) · Zbl 0929.76098 |

[31] | Young, The method of fundamental solutions for 2D and 3D Stokes problems, Journal of Computational Physics 211 (1) pp 1– (2006) · Zbl 1160.76332 |

[32] | Marin, The MFS for the Cauchy problem in two-dimensional steady-state linear thermoelasticity, International Journal of Solids and Structures 50 (20-21) pp 3387– (2013) |

[33] | Gu, Infinite domain potential problems by a new formulation of singular boundary method, Applied Mathematical Modelling 37 (4) pp 1638– (2013) · Zbl 1349.65686 |

[34] | Guiggiani, Direct computation of Cauchy principal value integrals in advanced boundary elements, International Journal for Numerical Methods in Engineering 24 (9) pp 1711– (1987) · Zbl 0635.65020 |

[35] | Karageorghis, The method of fundamental solutions for axisymmetric elasticity problems, Computational Mechanics 25 (6) pp 524– (2000) · Zbl 1011.74005 |

[36] | Karageorghis, The method of fundamental solutions for an inverse boundary value problem in static thermo-elasticity, Computers & Structures 135 (0) pp 32– (2014) |

[37] | Johansson, A method of fundamental solutions for the radially symmetric inverse heat conduction problem, International Communications in Heat and Mass Transfer 39 (7) pp 887– (2012) |

[38] | Karageorghis, A survey of applications of the MFS to inverse problems, Inverse Problems in Science and Engineering 19 (3) pp 309– (2011) · Zbl 1220.65157 |

[39] | Chen, Multilevel compact radial functions based computational schemes for some elliptic problems, Computers & Mathematics with Applications 43 (3-5) pp 359– (2002) · Zbl 0999.65143 |

[40] | Beskos, Boundary element methods in dynamic analysis, Applied Mechanics Reviews 40 (1) pp 1– (1987) · Zbl 0645.73034 |

[41] | Beskos, Boundary element methods in dynamic analysis: part II (1986-1996), Applied Mechanics Reviews 50 (3) pp 149– (1997) |

[42] | Chati, The boundary node method for three-dimensional linear elasticity, International Journal for Numerical Methods in Engineering 46 (8) pp 1163– (1999) · Zbl 0951.74075 |

[43] | Chati, The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems, Engineering Analysis with Boundary Elements 25 (8) pp 639– (2001) · Zbl 1065.74626 |

[44] | Johnston, A sinh transformation for evaluating nearly singular boundary element integrals, International Journal for Numerical Methods in Engineering 62 (4) pp 564– (2005) · Zbl 1119.65318 |

[45] | Sladek, Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity, International Journal for Numerical Methods in Engineering 36 (10) pp 1609– (1993) · Zbl 0772.73091 |

[46] | Gu, Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method, Computational Mechanics 53 (6) pp 1223– (2014) · Zbl 1398.74047 |

[47] | Marin, A meshless method for solving the Cauchy problem in three-dimensional elastostatics, Computers & Mathematics with Applications 50 (1-2) pp 73– (2005) · Zbl 1127.74014 |

[48] | Gao, The radial integration method for evaluation of domain integrals with boundary-only discretization, Engineering Analysis with Boundary Elements 26 (10) pp 905– (2002) · Zbl 1130.74461 |

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.