##
**A meshless model for transient heat conduction in functionally graded materials.**
*(English)*
Zbl 1097.80001

Summary: A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter. First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem. Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively. Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined. Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients. The accuracy of the proposed method is assessed through two numerical examples.

### MSC:

80A20 | Heat and mass transfer, heat flow (MSC2010) |

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

### Keywords:

transient heat conduction; virtual boundary collocation method; fundamental solution; superposition principle; radial basis functions; analog equation method; functionally graded media
Full Text:
DOI

### References:

[1] | Ingber MS (1994) A triple reciprocity boundary element method for transient heat conduction analysis. In Boundary Element Technology, Brebbia CA, Kassab AJ (eds), vol. IX. Elsevier Applied Science Amsterdam, 41-49 |

[2] | Jirousek, J.; Qin, QH, Application of Hybrid-Trefftz element approach to transient heat conduction analysis, Compu Struc, 58, 195-201 (1996) · Zbl 0900.73802 |

[3] | Nardini D, Brebbia CA (1982) A new approach to free vibration analysis using boundary elements. In Boundary Element Methods in Engineering Proceedings. 4^th International Seminar, Southampton. ed. C. A. Brebbia, Springer-Verlag, Berlin and New York, 312-326 · Zbl 0541.73104 |

[4] | Bulgakov, Int J Numer Meth Eng, 43, 713 (1998) · Zbl 0948.76050 |

[5] | Blobner, Int J Numer Meth Eng, 49, 1865 (1999) · Zbl 0941.80007 |

[6] | Singh, Eng Anal Bound Elem, 23, 419 (1999) · Zbl 0955.74074 |

[7] | Bialecki, Eng Anal Bound Elem, 26, 227 (2002) · Zbl 1002.80019 |

[8] | Nowak, Eng Anal Bound Elem, 6, 164 (1989) |

[9] | Zhu, Comput Mech, 21, 223 (1998) · Zbl 0920.76054 |

[10] | Sladek, Comput Mater Sci, 28, 494 (2003) |

[11] | Chen, Eng Anal Bound Elem, 26, 571 (2002) · Zbl 1026.76035 |

[12] | Chen, Eng Anal Bound Elem, 26, 577 (2002) · Zbl 1013.65128 |

[13] | Kupradze VD (1965) Potential methods in the theory of elasticity. Israel Program for Scientific Translations. Jerusalem |

[14] | Sun HC, Zhang LZ, Xu Q, Zhang YM (1999) Nonsingularity Boundary Element Methods, Dalian: Dalian University of Technology Press (in Chinese) |

[15] | Golberg MA, Chen CS (1996) Discrete Projection Methods for Integral Equations. Computational Mechanics. Southampton |

[16] | Ingber, Int J Numer Meth Eng, 60, 2183 (2004) · Zbl 1178.76276 |

[17] | Schaback, Adv Comput Math, 3, 251 (1995) · Zbl 0861.65007 |

[18] | Katsikadelis JT (1994) The analog equation method - a powerful BEM - based solution technique for solving linear and nonlinear engineering problems. In: Brebbia CA, (ed) Boundary Element Method XVI, CLM Publications Southampton, p 167 · Zbl 0813.65124 |

[19] | Golberg MA, Chen CS (1999) The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary Integral Methods: Numerical and Mathematical Aspects, In: Golberg MA(ed) Computational Mechanics Publications Southampton, 103-176 · Zbl 0945.65130 |

[20] | Mitic, Eng Anal Bound Elem, 28, 143 (2004) · Zbl 1057.65091 |

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.