engleski

Multiple state optimal design problems and Hashin-Shtrikman bounds

In optimal design problems the goal is to determine the optimal distribution of given materials in a given domain. Optimality of a distribution is measured by an objective function, 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 for the corresponding partial differential equation. We are interested in thermal conductivity problem, where the state function is the temperature of considered body. Source term and boundary conditions are presumed fixed. Therefore, if we change them, optimal distribution changes. This situation refers to multiple state optimal design problems, where the optimality takes into account temperatures obtained for different right-hand sides. Relaxing the original problem by homogenization method, we are able to write optimality conditions. It was proved that these optimality conditions are satisfied by sequential laminates. Using the fact that we can calculate precise order of optimal laminates, we can simplify numerical algorithms for optimal distribution of materials. We show the connection of this problem to Hashin-Shtrikman energy bounds, which are known to be saturated by sequential laminates, but we precisely determine the order of these laminates.

Hashin-Shtrikman bounds; multiple state optimal design

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