Wavelet-based boundary element methods for exterior electromagnetic shaping

In the present paper we survey on the application of shape optimization for exterior electromagnetic shaping in two and three dimensions. An energy variational approach is used to compute numerically free surfaces either of columns of molten metal or of liquid bubbles, levitating in the electromagnetic field. The cost functional and its shape gradient can be computed by boundary integrals. A wavelet-based fast boundary element method of optimal complexity is employed for the computation of the exterior magnetic field. We sketch both scalar and vector valued boundary update methods for gaining satisfactory flexibility for the class of shapes under consideration. Numerical results are carried out to illustrate our algorithms for solving such problems.