engleski

Optimality criteria method for optimal design problems

In this thesis, we study numerical solutions for optimal design problems. In such problems, the goal is to find an arrangement of given materials within the domain which minimizes (or maximizes) a particular integral functional, under constraints on the amount of materials and PDE constraints that underlay involved physics. We consider such problems in the frame of the stationary diffusion equation and linearized elasticity system for domains occupied by two isotropic materials. In Chapter 1 we review the basic facts about homogenization theory. The definition of H-convergence and composite materials is presented as well as some of their main properties. Chapter 2 focuses on the multiple state optimal design problems for the stationary diffusion equation. We give necessary condition of optimality for the relaxed formulation of optimal design problems, where the relaxation was obtained by the homogenization method. Moreover, we present a new variant of the optimality criteria method suitable for some minimization problems. The method is tested on various examples, and convergence is proved in the spherically symmetric case and the case when the number of states is less then the space dimension. The single state optimal design problem in linearized elasticity is addressed in Chapter 3. We give an explicit calculation of the lower Hashin-Shtrikman bound on the complementary energy in two and three space dimensions, and derive the optimality criteria method for two-dimensional compliance minimization problems.

Optimality criteria method, multiple state optimal design problems, homogenization, stationary diffusion, linearized elasticity, convergence, Hashin-Shtrikman bounds

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