engleski

Classical solutions in optimal design problems

In optimal design problems, one is trying to find the best arrangement of given materials, such that the obtained body has some optimal properties. The optimality of an arrangement is measured by a cost functional, which is usually an integral functional depending on the distribution of materials and the state function, obtained as a solution of the associated boundary value problem (the state equation). Commonly, optimal design problems do not have solutions (we refer to such solutions as classical), so one considers proper relaxations of original problems. Relaxation by the homogenization method consists in introducing generalized materials, which are mixtures of original materials on the micro-scale. We consider optimal design problems for stationary diffusion in the case of two isotropic phases, with several state equations, and a convex combination of compliances as the cost functional. If domain is spherically symmetric, and the right-hand sides of state equations are radial functions, we are able to show that there exists an optimal (relaxed) design which is a radial function. As a consequence, we can write down a simpler relaxation of the original optimal design problem, whose necessary conditions of optimality enable us to calculate a radial optimal design. By using this method, we shall present some problems having classical solutions.

homogenization ; optimal design ; saddle point

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