Symbolic computation to estimate two-sided boundary conditions in two-dimensional conduction problems

A method using symbolic computation is proposed to determine the boundary conditions in two-dimensional inverse heat conduction problems, The method uses symbols to represent the unknown boundaries and then executes the finite difference method to calculate the temperature field, The calculated results are expressed explicitly as functions of the undetermined boundaries, Then, a direct comparison between the output symbolic temperature and the measurement temperature are made to construct a set of Linear equations, Thus, the linear least-squares method is adopted to solve the linear equations, Finally, the unknown boundary conditions are determined, Results from the examples confirm that the proposed method is applicable in solving the multidimensional inverse heat conduction problems, In the example problems, three kinds of measuring methods are adopted to estimate the surface temperature, The result shows that only three measuring points are needed to estimate the surface temperature when measurement errors are neglected, When measurement errors are considered, seven measuring points are required to increase the congruence of the estimated results to the exact solutions.