engleski

Analysis of optimality conditions by shape calculus for optimal designs in conductivity problems

We consider conductivity optimal design problems for two isotropic phases, possibly with several state equations. Our aim is to find a distribution of materials which maximizes the energy functional. By relaxing the problem via homogenization method, an application of classical methods of calculus of variations is enabled, leading to optimality conditions and various numerical methods. However, usually in spherically symmetric problems classical solutions occur, so it is reasonable to compare these methods to ones which are based on shape derivative analysis. Various numerical methods (both the first and the second order methods) show nice convergence properties, but we are here interested in theoretical analysis of optimality conditions obtained by shape derivatives. For problems on a ball, the first order optimality condition easily leads to few critical shapes. Thanks to symmetry assumptions, we are able to further analyse these critical shapes. Similar techniques are applied for classical isoperimetric problem, or similar questions of eigenfrequency optimization, where one is able, by the Fourier analysis, to check second order optimality condition for simple spherical candidates. Our problem is much more technically demanding, but we are able to express second order optimality conditions by Fourier analysis, not only in the case of simple spherical interface between two given materials, but also in the case where this interface is made of two spheres.

spherically symmetric problems

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