Inverse estimation of the boundary condition in three-dimensional heat conduction

Abstract

An inverse method based on the symbolic approach is proposed to determine the boundary condition in three-dimensional inverse heat conduction problems. This method uses symbols to represent the unknown boundary condition and then executes an implicit alternating-direction finite-difference method to calculate the temperature distribution. The calculated results are expressed explicitly as functions of the symbol of the undetermined boundary condition. Then, a set of linear equations is constructed from the comparison between the output symbolic temperature and the measured numerical temperature. Thus, the set of linear equations is solved through the linear least-squares error method and leads to the solution of the unknown boundary condition. Results from examples confirm that the proposed method is applicable in solving the three-dimensional inverse heat conduction problem. In the example problems, a temporally dependent and a spatially - temporally dependent boundary condition are used to demonstrate the procedure of the proposed method. In the first example, the temporally dependent boundary conduction, the result shows that only one-point measurement is needed to estimate the temporally dependent condition whether the measurement errors are considered or not. In the second problem, the four- and eight-point measurement methods are adopted to estimate the temporally - spatially dependent boundary condition. The result shows that only the four-point measurement is needed to estimate the surface temperature when measurement errors are neglected. When measurement errors are considered, the eight-point measurement is required in order to increase the congruence of the estimated results to the exact solutions.