Optimal control of induction heating for semi-solid alloy forming

Thixoforming technique requires reheating the feedstock to a semi-solid state in relatively short time with a uniform temperature distribution as well as an optimum liquid fraction to achieve a satisfactory globular microstructure. The skin effect of induction heating results in an exponential profile of power density (heat source) distribution along the radius of the cylindrical materials and an uneven temperature profile from surface to core. Our main objective of this study is to seek an optimal induction heating strategy by utilizing optimization methods to obtain the required temperature value without thermal gradient within the materials at the end of the process. This optimization problem is essentially an inverse heat transfer problem.; We start our study by investigating the inverse problem of determining the time-dependent heat flux imposed on the boundary of a solid slab from the final temperature distribution by using the conjugate gradient (CG) and the truncated singular value decomposition (TSVD) methods separately. The feasibility of recovering the boundary flux is found to depend on the total time of the heating/cooling process: the exact heat flux can be reconstructed only in a non-dimensional time range [tf-0.1, tf], beyond which we obtain only the time-averaged values. However, by using a modified conjugate gradient method, we may reconstruct the boundary heat flux for much larger times if its initial value is known. This kind of inverse problem is then extended to a two-dimensional system of natural convection in porous medium. The time and position dependent heat flux can be recovered with success in a non-dimensional time interval of the order of 0.1 for a Rayleigh number up to 1000. The sensitivity of this inverse problem is significantly reduced as the final time or Rayleigh number increases. In addition, the numerical solutions obtained from the noisy measurement data are also presented to show the regularization power of the conjugate gradient method.; Based on the previous study, the conjugate gradient method in conjunction with the adjoint method is implanted into an optimal control problem of estimating the timewise-varying strength of heat source (power density) with exponential profile along the radius of a cylinder to achieve a uniform temperature distribution at the final time. The physical and mathematical models take into account the temperature-dependent thermophysical properties and the formulation is performed in cylindrical coordinates system. Radiative and convective heat losses are also considered. The phase change process in the mushy region is simulated by applying the apparent heat capacity method. Such a model of induction heating system is proved to be reasonable by comparing the numerical solutions with experimental data. The optimal heating strategies estimated by regular CGM may achieve more uniform temperature distribution at the final time than the empirical methods obtained from experiments. A modified CGM may be a better approach to solve this optimal control problem because less iterations are needed and more perfect temperature uniformity could be achieved.; One might conclude that a general numerical algorithm has been successfully applied to the optimal heating control of semi-solid metal forming. This optimization method may provide a reasonable heating strategy in a few minutes for any kind of alloy materials of any size under any operating frequency of an induction heater.