engleski

On some properties of homogenised coefficients for stationary diffusion problem

We consider optimal design of stationary diffusion problems for two-phase materials. Speaking in the context of thermal conductivity, it consists in finding the best arrangement of two given materials in a fixed domain that maximises some functional, expressed in terms of temperature for some fixed source term on the right-hand side. Since problems of this kind usually have no solution, a relaxation (there are also some other approaches) consists in introducing the notion of composite materials, as fine mixtures of different phases, mathematically described by the homogenisation theory. Denoting the set of all possible composite materials with given local proportion $\theta$ of the first material by ${\cal K}(\theta)$ the problem can be written as an optimisation problem over this set. In their paper, Tartar and Murat (1985) described the set ${\cal K}(\theta)e$, for some vector $e$, and used this result to replace the optimisation over the complicated set ${\cal K}(\theta)$ by a much simpler one. Analogous characterisation even holds for the case of mixing more than two materials (possibly anisotropic), where the set ${\cal K}(\theta)$ is not effectively known (Tartar, 1995). We address the question of describing the set $\{(Ae, Af):A\in{\cal K}(\theta)\}$, for given vectors $e$ and $f$, which is important for optimal design problems with multiple state equations (different right-hand sides in stationary diffusion equation). In other words we are interested in describing two columns of matrices in ${\cal K}(\theta)$. In two dimensions we describe this set in appropriate coordinates and give some geometric interpretation. For the three dimensional case we consider the set $\{Af:A\in{\cal K}(\theta), Ae=t\}$, for a fixed $t$, and show how it can be reduced to a two dimensional one, although the solution involves tedious computations.

homogenisation; stationary diffusion equation

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