Optimal Heating of a 2-D Solid Material with Thermal Dependent Properties

In this paper the optimal control of boundary heat flux in a 2-D solid body with an arbitrary shape is performed in order to achieve the desired temperature distribution at a given time interval. The boundary of the body is subdivided into a number of components. On each component a time-dependent heat flux is applied which is independent of the others. Since the thermophysical properties are temperature-dependent, the problem is treated as a nonlinear inverse heat conduction problem. Conjugate gradient method (CGM) along with adjoint problem is utilized in order to solve the inverse problem. Optimization process is employed for the heat flux imposed on each of the boundary component individually which was previously shown to be more efficient than optimizing the entire heat flux array simultaneously. Three versions of CGM; that is, the Fletcher-Reeves (FR), Polak-Ribiere (PR) and Powell-Beale are utilized for comparison. As a test case, heating of an Aluminum bar with a square cross section and temperature-dependent thermo-physical properties is considered. Results show that for large time-steps the Powell-Beale version with normalized search direction, and for small time-steps the Polak-Ribiere version are the most efficient method with the least error in the estimated temperature field. Moreover, for large time step size results show that addition of regularization term to the Error Function reduces the amplitude of oscillations in the estimated heat flux.