Numerical method for multi-phase inverse stefan design problems

The inverse problems for differential equations consist of stating the initial conditions, boundary conditions or thermophysical properties of the body. But the insufficiency of input information is compensated by some additional information on the effects of the input conditions. Generally, for the inverse Stefan problem, it is assumed that this additional information is the position of the freezing front, its velocity in normal direction or temperature in selected points of the domain. We may consider the usage of the demanded position of the moving front as the constraint for the cost functional. This kind of problem becomes an inverse design problem. In the paper, the multi-phase inverse Stefan design problems are formulated and described by means of the optimization method. These problems consist of the reconstruction of the function which describes the heat-transfer coefficient, when the positions of the moving interfaces of the phase change are well-known. The method consists of the minimization of a functional, the value of which is the norm of a difference between given position of the moving interface of the phase change and a position reconstructed from the selected function describing the heat-transfer coefficient In numerical calculations the Nelder-Mead optimization method and the generalized alternating phase truncation method were used.