engleski

Shape gradient method for optimal design on annulus

We consider multiple state optimal design problem for stationary diffusion equation with two isotropic phases, a better conductor with conductivity β and worse with conductivity α. Aim is to maximize a conic sum of energies obtained for each state problem. Commonly, optimal design problems do not have solutions (such solutions are called classical), so one considers proper relaxation of the original problem. From there the necessary and sufficient conditions of optimality are obtained. With spherically symmetric domain Ω and radial fi one can construct classical solutions. Analytical solutions give opportunity to test different numerical methods i.e. compare rates of convergence, stability or check for possible errors. For demonstration, shape gradient method was implemented in freefem++. Based on shape derivative, method creates vector field which moves interface between phases in order to increase value of object function. We have observed stable convergence under tests created from optimal problems with analytical solutions.

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

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