A general class of design problems is examined in thermal transport systems. We mainly address inverse problems with overspecified coupled boundary conditions in one part of the boundary and unknown thermal conditions in another part of the boundary. The methods of choice for the solution of the above inverse problems are functional optimization methods using appropriately defined continuum sensitivity and adjoint problems. Conjugate gradient techniques, preconditioning and regularization are considered within an innovative object-oriented finite element framework.
Our particular interest for examining such inverse transport systems arises from the desire to address the design of directional solidification processes that lead to desired microstructures. As the main application of this paper, we will address the calculation of the mold/furnace heat flux conditions such that a desired solidification state (growth rate and temperature gradient) is achieved at the freezing interface. Changes in growth rate and thermal gradient at the freezing interface are known to alter the relative importance of thermal/mass transport and interfacial energy effects, and the magnitude of this partitioning of available driving forces dictates the formation of specific microstructures.