engleski

Classical optimal design on annulus

We optimize a distribution of two isotropic materials that occupy an annulus in two or three dimensions, heated by a uniform heat source, aiming to maximize the total energy. In elasticity, the problem models the maximization of the torsional rigidity of a cylindrical rod with annular cross section made of two homogeneously distributed isotropic elastic materials. Commonly, optimal design problems do not have solutions (such solutions are called classical ), so one considers proper relaxation of the original problem. Relaxation by the homogenization method consists in introducing generalized materials, which are mixtures of original materials on the micro-scale. However, by analysing the optimality conditions, we are able to show that the solution is unique, classical and radial. Depending on the amounts of given materials, we find two possible optimal configurations. The precise solution can be determined by solving a system of nonlinear equations, which can be done only numerically. If Ω is a ball, in order to maximize the energy the better conductor should be placed inside a smaller (concentric) ball, whose radius can easily be calculated from the constraint on given amounts of materials. By analysing the optimality conditions, we are able to show that in the case of annul us, the solution is also unique, classical and radial. Depending on the amounts of given materials, we find two possible optimal configurations. If the amount of the first phase is less than some critical value, then the better conductor should be placed in an outer annulus. Otherwise, the optimal configuration consists of an annulus with the better conductor, surrounded by two annuli of the worse conductor.

stationary diffusion ; optimal design ; homogenization ; optimality conditions

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