Analysis of Heat Conduction in a Quadratic Functionally Graded Plane by Boundary Element Method Based on the Variable Transmission Approach

Authors

1 yazd

2 kashan

Abstract

In this research, the governing differential equation of heat conduction problem in a non-homogenous, functionally graded material (FGM) is solved using the boundary elements method (BEM). Except for some limited cases, there is not known Green function or fundamental singular solution for this kind of problems, which is necessary to have the boundary elements analysis. In this paper, the thermal conductivity of the functionally graded plate is considered as a quadratic function of one direction and then an auxiliary variable is introduced into the governing differential equation in order to simplify the problem to a kind with known fundamental singular solution and therefore, heat transfer is solved in a 2D functionally graded material by simple boundary elements method which is not possible by common methods. Based on the proposed approach, a computer code is developed using the MATLAB. The validity of this developed code is verified by solving and analyzing a number of heat transfer problems.

Keywords

References

  1. Birman, V. and Byrd, L. W. “Modeling and Analysis of Functionally Graded Materials and Structures”, J. Appl. Mech. Rev. Vol. 60, No. 5, pp.195-216, 2007.##
    1. Ehsani, F., and Ehsani, F. “Transient Heat Conduction in Functionally Graded Thick Hollow  Cylinders under ‎Non-Uniform Heat Generation by  Homotopy Perturbation Method”, J. Basic Appl. Sci. Res. Vol. ‎2, No. 10‎, pp. ‎10676-‎10676, 2012.##
    2. Wu-Xiang, L. “Analysis of Steady Heat Conduction for 3D Axisymmetric Functionally Graded Circular Plate”, J. Central South Uni. Vol. 20, No. 6, pp. 1616-1622, 2013.##
    3. Wang, H., Qin, Q. H., and Kang, Y. L. “A Meshless Model for Transient Heat Conduction in Functionally Graded Materials”, Comput. Mech. Vol. 38, No. 1, pp. 51-60, 2005.##
  2. Sladek, J., Sladek, V., Tan, C. L., and Atluri‎, S. N. “Analysis of Transient Heat Conduction in 3D Anisotropic Functionally Graded ‎Solids by the MLPG Method”‎, Comput. Modell. Eng. Sci. Vol. 32, No. 3, pp. ‎161-174‎, 2008.##
  3. Khosravifard, A., Hematiyan, M. R., and Marin, L. “Nonlinear Transient Heat Conduction Analysis of Functionally Graded Materials in the Presence of Heat Sources Using an Improved Meshless Radial Point Interpolation Method”, Appl. Math. Modell. Vol. 35, No. 9, pp. 4157-4174, 2011.##
  4. Dai, B., Zhang B., Liang, Q., and Wang, L. “Numerical Solution of Transient Heat Conduction Problems Using Improved Meshless Local Petrov–Galerkin Method”, Appl. Math. Comput. Vol. 219, No. 19, pp. 10044-10052, 2013.##
  5. Brebbia, C. A., and Dominguez, J. “Boundary Elements: an Introductory Course”, WIT press, Southampton, UK, 1994.##
    1. Khodadad, M., and Dashti, M. “Boundary Elements Method”, Tehran: Mogestan, 2011. (in Persian)##
    2. 10.  Sutradhar, A., Reeder, J., and Gray, L. J. “Symmetric Galerkin Boundary Element Method”, Springer Science & Business Media, 2008.##
  6. 11.  Katsikadelis, J. T. “The BEM for Nonhomogeneous Bodies”, Arch. Appl. Mech. Vol. 74, No. 11-12, pp. 780-789, 2005.##
    1. 12.  Nerantzaki, M. S., and Kandilas, C. B. “A Boundary Element Method Solution for Anisotropic Nonhomogeneous Elasticity”, Acta Mech. Vol. 200, No. 3-4, pp. 199-211, 2008.##
    2. 13.  Kandilas, C. B. “Transient Elastodynamic Analysis of Nonhomogeneous Anisotropic Plane Bodies”, Acta Mech. Vol. 223, No. 4, pp. 861-878, 2012.##

14. Gray, L. J., Kaplan, T., Richardson, J. D., and Paulino, G. H. “Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction”, J. Appl. Mech. Vol. 70, No. 4, pp. 543-549, 2003.##

15. Sutradhar, A., and Paulino, G. H. “A Simple Boundary Element Method for Problems of Potential in Nonhomogeneous Media”, Int. J. Numer. Methods Eng. Vol. 60, No. 13, pp. 2203–2230, 2004.##

Aerospace Mechanics
Volume 15, Issue 2 - Serial Number 56
September 2020
Pages 77-89

Files

History

  • Receive Date: 10 December 2016
  • Revise Date: 20 February 2019
  • Accept Date: 19 September 2018
  • Publish Date: 22 June 2019

Share

How to cite

Statistics