Radial point interpolation based finite difference method for mechanics problems.

*(English)*Zbl 1129.74049Summary: We propose a radial point interpolation based finite difference method (RFDM). In this novel method, radial point interpolation using local irregular nodes is employed together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in collocation methods. A least-square technique is adopted, which leads to a system matrix with good properties such as symmetry and positive definiteness. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. The results are examined in detail in comparison with other numerical approaches such as the radial point collocation method that uses local nodes, conventional finite difference and finite element methods.

##### MSC:

74S20 | Finite difference methods applied to problems in solid mechanics |

PDF
BibTeX
XML
Cite

\textit{G. R. Liu} et al., Int. J. Numer. Methods Eng. 68, No. 7, 728--754 (2006; Zbl 1129.74049)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Meshfree Method: Moving beyond the Finite Element Method. CRC Press: Boca Raton, 2003. |

[2] | . An Introduction to Meshfree Methods and Their Programming. Springer: Dordrecht, 2005. |

[3] | Belytschko, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) |

[4] | Belytschko, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) |

[5] | Liu, International Journal for Numerical Methods in Fluids 20 pp 1081– (1995) |

[6] | Atluri, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067 |

[7] | Wang, International Journal for Numerical Methods in Engineering 54 pp 1623– (2002) |

[8] | Liu, Journal of Sound and Vibration 246 pp 29– (2001) |

[9] | Li, Computational Mechanics (2006) |

[10] | Liu, International Journal of Computational Methods (2006) |

[11] | Liu, International Journal of Computational Methods 2 pp 1– (2005) |

[12] | . Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific: Singapore, 2003. · Zbl 1046.76001 |

[13] | Monaghan, Computer Physics Communications 48 pp 89– (1988) |

[14] | Jensen, Computers and Structures 2 pp 17– (1972) |

[15] | Perrone, Computers and Structures 5 pp 45– (1975) |

[16] | Liszka, Computers and Structures 11 pp 83– (1980) |

[17] | Liszka, International Journal for Numerical Methods in Engineering 20 pp 1599– (1984) · Zbl 0544.65006 |

[18] | Meshless finite difference method I. Basic approach. Proceedings of the IACM-Fourth World Congress in Computational Mechanics Argentina, 1998. |

[19] | , , . Radial basis point interpolation collocation method for 2D solid problem. In Advances in Meshless and X-FEM Methods, Proceedings of the 1st Asian Workshop on Meshfree Methods, (ed.). World Scientific: Singapore, 2002; 35–40. |

[20] | , , . Collocation-based meshless method for the solution of transient convection-diffusion equation. First EMS-SMAI-SMF Joint Conference Applied Mathematics and Applications of Mathematics (AMAM 2003), Nice, France, 10–13 February 2003; List of Posters, Posters Session 2 (48). |

[21] | Liu, Computational Mechanics 36 pp 298– (2005) |

[22] | . Computational Inverse Techniques in Nondestructive Evaluation. CRC Press: Boca Raton, 2003. · Zbl 1067.74002 |

[23] | Onate, Computers and Structures 79 pp 2151– (2001) |

[24] | Liu, Computer Methods in Applied Mechanics and Engineering (2006) |

[25] | Liu, Computational Mechanics 33 pp 2– (2003) |

[26] | Liu, International Journal for Numerical Methods in Fluids 46 pp 1025– (2004) |

[27] | The theory of radial basis function approximation in 1990. In Advances in Numerical Analysis, (ed.). Oxford University Press: Oxford, 1992; 303–322. |

[28] | (ed.). In Handbook of Computational Solid Mechanics: Survey and Comparison of Contemporary Methods. Springer: Berlin, 1998. |

[29] | The Boundary Element Method for Engineers. Pentech Press: London, 1978. |

[30] | Gu, Computational Mechanics 28 pp 47– (2002) |

[31] | . Theory of Elasticity (3rd edn). McGraw-Hill: New York, 1970. |

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.