engleski

Shape derivative method for optimal design in conductivity problem

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.

stationary diffusion, optimal design, homogenization, shape design, gradient method

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