Shape Optimization for Inverse Electromagnetic Casting Problems

In this paper we present an algorithm for inverse optimization problems concerning electromagnetic casting of molten metals. We are interested in locating suitable inductors around the molten metal so that the equilibrium shape be as near as possible to a desired target shape. A Simultaneous Analysis and Design (SAND) mathematical programming formulation is stated for the inverse problem. The resulting optimization problem is solved with FAIPA, a feasible directions interior-point algorithm. INTRODUCTION The industrial technique of electromagnetic casting allows for contactless heating, shaping and controlling of chemical aggressive, hot melts. The main advantage over the conventional crucible shape forming is that the liquid metal does not come into contact with the crucible wall, so there is no danger of contamination. This is very important in the preparation of very pure specimens in metallurgical experiments, as even small traces of impurities, such as carbon and sulphur, can affect the physical properties of the sample. Industrial applications are, for example, electromagnetic shaping of aluminum ingots using soft-contact confinement of the liquid metal, electromagnetic shaping of components of aeronautical engines made of superalloy materials (Ni,Ti, . . . ), control of the structure solidification, etc. The direct problem is to determine the resulting liquid metal shape for a given external current distribution. The model considered here concerns a vertically falling molten metal column shaped by an externally applied magnetic field created by a set of inductors. In general, the direct problem can be solved either directly studying the equilibrium equation at the interface, or minimizing an appropriate energy functional, the main advantage of this last method being that the resulting shapes are then mechanically stable [1, 2, 3]. The inverse problem consists in determining the exterior field, and therefore the external currents, for which the liquid metal takes on a given desired shape. In the two-dimensional case, the inverse shaping problem consists in finding a distribution of inductors in order that the generated exterior field makes the horizontal cross-section of the molten metal attain a prescribed shape. This is a very important problem that one needs to solve in order to define a process of electromagnetic liquid metal forming. In addition, from a practical point of view, the magnetic field has to be created by a simple configuration of inductors. The direct problem is inherently well posed, i.e. small variations in the applied currents cause small variations in the shape. The inverse problem is inherently ill posed: small variation of the liquid boundary may cause dramatic variations in the applied exterior field [4, 5]. In a previous work we studied the inverse electromagnetic shaping problem considering the case where the inductors are made of single solid-core wires with a negligible area of the cross-section [6]. Thus, the inductors were represented by points in the horizontal plane. In a second paper we considered the more realistic case where each inductor is a set of bundled insulated strands [7]. In the present paper we introduce a regularized cost function in order to treat the numerical instability. The most important effect of the regularization term is to control the magnetic pressure in the boundary of the computed shape. Thus, this term ensures that the boundary of the liquid metal is smooth. in ria -0 05 33 69 9, v er si on 1 8 N ov 2 01 0