The adjoint method for the design of directional binary alloy solidification processes in the presence of a strong magnetic field

An adjoint method formulation and numerical implementation is presented for a specific class of inverse design solidification problems. In particular, a method is developed to calculate the thermal boundary conditions on the mold walls for directional solidification processes under the influence of an external magnetic field such that a desired stable solid-liquid interface growth is achieved throughout solidification. Achieving a stable interface growth has profound implications on the obtained cast microstructures and is directly related to the quality of the solidified product.; The considered inverse solidification design problem belongs to a typical class of inverse heat transfer problems, in which, incomplete conditions are provided on one part of the boundary, whereas over-specified conditions are given on another part of the boundary. The design problem is mathematically posed as a functional optimization problem. The unknowns of the design problem are thermal boundary conditions on part of the mold walls. The cost functional is defined so as to represent the deviation of the freezing interface thermal conditions from local thermodynamic equilibrium. A continuum adjoint based method for gradient computations coupled with a conjugate-gradient optimization solver is employed to solve the inverse problem.; The design method is developed in two stages. First, the adjoint method is formulated for an inverse magneto-convection problem in a fixed domain with convection driven by buoyancy effects as well as a Lorentz force generated due to the applied magnetic field. The developed methods are demonstrated using various examples in which the exact solution to the inverse problem is known a priori. The examples demonstrate the accuracy and convergence behavior of the method in the framework of the conjugate gradient algorithm. The need for regularization is identified in one of the examples with uniformly distributed random errors in the input/measured temperature data where a H 1 regularized formulation is introduced in order to obtain stable solutions. The method is shown to be very robust and to work well for various problems including 3D applications. Secondly, the above developed adjoint technique is applied to the design of two typical solidification design problems. A directional binary alloy solidification process is examined in which melt convection is induced due to the combined action of buoyancy as well as Lorentz forces due to an external magnetic field. The goal of the inverse design problem is to identify the thermal conditions on the mold walls so that a desired stable flat-interface growth is realized throughout solidification. The above method is then extended to the design of a directional solidification process of a near-eutectic binary alloy in which convection is driven by the coupled action of buoyancy, thermocapillary and electromagnetic convection. Finally, the thesis concludes with a discussion of possible extensions to the proposed method.