Naslov
Gradient methods for multiple state optimal design problems

Autori
Antonić, Nenad ; Vrdoljak, Marko

Izvornik
Annali dell'Università di Ferrara. Sezione 7: Scienze matematiche (0430-3202) 53 (2007), 2; 177-187

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
optimal design; multiple state equations; stationary diffusion; gradient method

Sažetak
We optimise a distribution of two isotropic materials $\alpha\mI$ and $\beta\mI$ ($\alpha<\beta)$ occupying the given body in $\R^d$. The optimality is described by an integral functional (cost) depending on temperatures $u_1, \ldots u_m$ of the body obtained for different source terms $f_1, \ldots, f_m$ with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by the homogenisation theory. The necessary conditions of optimality are derived via the G\^{; ; ; a}; ; ; teaux derivative of the cost functional. Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal design. In the case $m<d$ we show that there exists an optimal design which is a rank-$m$ sequential laminate with matrix material $\alpha\mI$ almost everywhere on $\Omega$ . In contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems (although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient methods for minimising the cost functional.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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