##
**A new boundary meshfree method with distributed sources.**
*(English)*
Zbl 1244.65189

Summary: A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |

### Keywords:

meshfree method; area distributed sources; method of fundamental solutions; boundary node method; potential problem
PDF
BibTeX
XML
Cite

\textit{Y. J. Liu}, Eng. Anal. Bound. Elem. 34, No. 11, 914--919 (2010; Zbl 1244.65189)

Full Text:
DOI

### References:

[1] | Mukherjee, S.; Mukherjee, Y. X., Boundary methods: elements, contours, and nodes. (2005), CRC: CRC Boca Raton · Zbl 1110.65002 |

[2] | Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics, 9, 1/2, 69-95 (1998) · Zbl 0922.65074 |

[3] | Golberg, M. A.; Chen, C. S., The method of fundamental solutions for potential, Helmhotz and diffusion problems, (Golberg, M. A., Boundary integral methods: numerical and mathematical aspects (1998), Computational Mechanics Publications: Computational Mechanics Publications Boston), 103-176 · Zbl 0945.65130 |

[4] | Fairweather, G.; Karageorghis, A.; Martin, P. A., The method of fundamental solutions for scattering and radiation problems, Engineering Analysis with Boundary Elements, 27, 7, 759-769 (2003) · Zbl 1060.76649 |

[5] | Berger, J. R.; Karageorghis, A., The method of fundamental solutions for heat conduction in layered materials, International Journal for Numerical Methods in Engineering, 45, 11, 1681-1694 (1999) · Zbl 0972.80014 |

[6] | Ramachandran, P. A., Method of fundamental solutions: singular value decomposition analysis, Communications in Numerical Methods in Engineering, 18, 11, 789-801 (2002) · Zbl 1016.65095 |

[7] | Smyrlis, Y.-S.; Karageorghis, A., A matrix decomposition MFS algorithm for axisymmetric potential problems, Engineering Analysis with Boundary Elements, 28, 5, 463-474 (2004) · Zbl 1074.65134 |

[8] | Mitic, P.; Rashed, Y. F., Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources, Engineering Analysis with Boundary Elements, 28, 2, 143-153 (2004) · Zbl 1057.65091 |

[9] | Poullikkas, A.; Karageorghis, A.; Georgiou, G., The method of fundamental solutions for three-dimensional elastostatics problems, Computers and Structures, 80, 3-4, 365-370 (2002) |

[10] | Liu, Y. J.; Nishimura, N.; Yao, Z. H., A fast multipole accelerated method of fundamental solutions for potential problems, Engineering Analysis with Boundary Elements, 29, 11, 1016-1024 (2005) · Zbl 1182.74256 |

[11] | Young, D. L.; Chen, K. H.; Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domain, Journal of Computational Physics, 209, 290-321 (2005) · Zbl 1073.65139 |

[12] | Chen, K. H.; Kao, J. H.; Chen, J. T.; Young, D. L.; Lu, M. C., Regularized meshless method for multiply-connected-domain Laplace problems, Engineering Analysis with Boundary Elements, 30, 882-896 (2006) · Zbl 1195.65200 |

[13] | Young, D. L.; Chen, K. H.; Chen, J. T.; Kao, J. H., A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains, CMES: Computer Modeling in Engineering & Sciences, 19, 3, 197-221 (2007) · Zbl 1184.65116 |

[14] | Šarler, B., 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, 1374-1382 (2009) · Zbl 1244.76084 |

[15] | Liu, Y. J., Fast multipole boundary element method—theory and applications in engineering (2009), Cambridge University Press: Cambridge University Press Cambridge |

[17] | Mukherjee, S., Boundary element methods in creep and fracture (1982), Applied Science Publishers: Applied Science Publishers New York · Zbl 0534.73070 |

[18] | Liu, Y. J.; Rudolphi, T. J., Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations, Engineering Analysis with Boundary Elements, 8, 6, 301-311 (1991) |

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.