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Gradient methods for multiple state optimal design problems (CROSBI ID 136933)

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Antonić, Nenad ; Vrdoljak, Marko Gradient methods for multiple state optimal design problems // Annali dellNULLUniversità di Ferrara. Sezione 7: Scienze matematiche, 53 (2007), 2; 177-187

Podaci o odgovornosti

Antonić, Nenad ; Vrdoljak, Marko

engleski

Gradient methods for multiple state optimal design problems

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.

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

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Podaci o izdanju

53 (2)

2007.

177-187

objavljeno

0430-3202

1827-1510

Povezanost rada

Matematika

Indeksiranost