Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions

ABSTRACT

The aim of this paper is to identify numerically the timewise thermal conductivity coefficients in the two-dimensional heat equation in a rectangular domain using initial and Dirichlet boundary conditions and the local heat flux as over-specification conditions. The measurement data represented by the local heat flux is shown to ensure the unique solvability of the inverse problem solution. The two-dimensional inverse problem is discretized using an alternating direction explicit method. The resulting constrained optimization problem is minimized iteratively by employing the MATLAB subroutine. Exact and noisy input data are inverted numerically. The  root mean square error values for various noise levels p for both smooth and non-smooth continuous timewise thermal conductivity coefficients Examples are compared. Numerical results are presented and discussed in order to illustrate the performance of the inversion for thermal conductivity components identification. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficient identification problems with applications in heat transfer and porous media.

Keywords:

• Inverse problem
• two-dimensional heat equation
• thermal conductivity
• local heat flux conditions
• nonlinear optimization

2010 Mathematics Subject Classifications:

• 65-XX
• 65Mxx
• 35R35

Acknowledgements

The comments and suggestions made by the editor and the referees are gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature

$a_1$	=	thermal conductivity (W m${}^{-1}$ K${}^{-1}$ )
$a_2$	=	thermal conductivity (W m${}^{-1}$ K${}^{-1}$ )
$c_1$	=	velocity (m s${}^{-1}$ )
$c_2$	=	velocity (m s${}^{-1}$ )
$c_3$	=	absorbance (AU)
f	=	heat source/force (J)
u	=	temperature (K or C)
${\nu }_1$	=	heat flux at x = 0 (W ${\mathrm{m}}^{-2}$ )
${\nu }_2$	=	heat flux at y = 0 (W ${\mathrm{m}}^{-2}$ )
t	=	time variable (s)
x, y	=	space variables (m)
$x_i,y_j$	=	space nodes (–)
$t_n$	=	time node (–)
$u_{i,j}^n$	=	values of u at the node $(i,j,n)$ (–)
l	=	end of space interval (–)
h	=	end of space interval (–)
T	=	end of time interval (–)
$M_1$	=	number of finite difference in x-coordinate (–)
$M_2$	=	number of finite difference in y-coordinate (–)
N	=	number of finite difference in t-coordinate (–)
F	=	nonlinear objective least-squares function (19) (–)
p	=	percentage of noise (–)
${\sigma }_1,{\sigma }_2$	=	standard deviations (–)
$Q_T$	=	fixed domain $(0,h)\times (0,l)\times (0,T)$ (–)
${\overline{Q}}_T$	=	closure of the solution domain $Q_T$ (–)