Abstract
In the present paper we combine an energy variational approach with shape optimization techniques to compute numerically free surfaces in electromagnetic shaping and levitation of liquid metals in three dimensions. Assuming the domains to be starshaped, the surfaces are represented via an ansatz by spherical harmonics, which generalizes the approximation by Fourier series in two dimensions. We will show that all ingredients of the shape optimization algorithm, particularly the shape gradient and the cost functional, 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 and its Neumann-to-Dirichlet map.
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