Classical Shape Optimization : The Electromagnetic-Casting Problem
Résumé
The industrial technique of electromagnetic casting allows contactless heating, shaping and controlling of chemical aggressive, hot melts. The simplified model of the 3-dimensional electromagnetic shaping problem considered here concerns a bubble of liquid metal levitating in the electromagnetic field created by given conductors. Under suitable assumptions, the equilibrium configurations of the liquid metal are described by a set of equations involving a relationship between the electromagnetic, superficial and gravity forces at the boundary. In the two-dimensional case the model concerns a vertically falling molten metal column shaped by an externally applied magnetic field created by a set of inductors. In numerical simulation of electromagnetic casting one approach is to consider models where the computation of the free boundary amounts for solving a shape optimization problem. We consider direct and inverse problem. In the direct problem we compute a local minimum w.r.t. the shape of the total energy of the phenomenon under consideration. Our interest is to use the Newton method to obtain an optimal solution, which satisfies the Kuhn-Tucker conditions. In the inverse problem we are interested to locate suitable inductors around a molten metal so that the equilibrated shape is as near as possible to a desired one. Numerically, we construct a sequence of domains determinate by their boundaries. We consider piecewise linear surfaces with n nodes. The integral equation is solved by a Galerkin finite element method. After these numerical approximations, we obtain a Newton-like algorithm. A Simultaneous Analysis and Design (SAND), mathematical programming method is stated for the inverse problem. The resulting optimization problem is solved with FAIPA, a feasible interior point algorithm.