Gradient methods for multiple state optimal design problems |
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Authors: | Nenad Antonić Marko Vrdoljak |
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Institution: | (1) Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10 000 Zagreb, Croatia |
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Abstract: | We optimise a distribution of two isotropic materials α
I and β
I (α < β) occupying the given body in R
d
. The optimality is described by an integral functional (cost) depending on temperatures u
1, . . . , u
m
of the body obtained for different source terms f
1, . . . ,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 Gateaux 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 α
I almost everywhere on Ω. 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.
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Keywords: | Gradient methods Design problems |
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