Marko Vrdoljak

Sequential laminates in multiple state optimal design problems

In the study of optimal design related to stationary diffusion problems with multiple-state equations, the description of the set H={(Aa1,...,Aam):A∈K(θ)} for given vectors a1,...,am∈ℝd (m<d) is crucial. K(θ) denotes all composite materials (in the sense of homogenisation theory) with given local proportion θ of the first material. We prove that the boundary of H is attained by sequential laminates of rank at most m with matrix phase αI and core βI (α<β). Examples showing that the information on the rank of optimal laminate cannot be improved, as well as the fact that sequential laminates with matrix phase αI are preferred to those with matrix phase βI, are presented. This result can significantly reduce the complexity of optimality conditions, with obvious impact on numerical treatment, as demonstrated in a simple numerical example.

Parabolic H-convergence and small-amplitude homogenization

H-convergence and small-amplitude homogenization is studied for linear parabolic problems with coefficients, which may depend on time. The small-amplitude homogenization consists of taking a sequence of coefficients, whose difference is proportional to a small parameter, and then computing the first correction in the limit. We recall the definition and main results on H-convergence for non-stationary diffusion equation, and prove that the smoothness (with respect to a parameter) is preserved in the process of taking the H-limit, which is essential for our purposes. The explicit expression for the correction is obtained by using a recently introduced parabolic variant of H-mesures.

Connecting Classical and Abstract Theory of Friedrichs Systems via Trace Operator

Based on recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern et al. (2007), as well as Antonic and Burazin (2010), we continue our efforts to relate these results to the classical Friedrichs theory. Following the approach via the trace operator, we extend the results of Antonic and Burazin (2011) to situations where the important boundary field does not consist only of projections, allowing the treatment of hyperbolic equations, besides the elliptic ones.

Classical Optimal Design in Two-Phase Conductivity Problems

We consider multiple state optimal design problems for stationary diffusion in the case of two isotropic phases, aiming to maximize a conic combination of energies. It is well known that for problems with one state equation, there are relaxed solutions corresponding to simple laminates at each point of the domain. As a consequence, one can write down a simpler relaxation, written only in terms of the local proportion of given materials. For multiple state optimal design problems we prove an analogous result in the spherically symmetric case. This simpler relaxation problem is represented by a convex-concave minimax problem, and its solution can be characterized by necessary and sufficient optimality conditions. The optimality conditions are further analyzed on some examples. If the domain is a ball, the presented method enables explicit calculation of an optimal design which is, in most cases, unique and classical. However, an example with a nonunique solution is presented as well.

On Some Properties of Homogenised Coefficients for Stationary Diffusion Problem

We consider optimal design of stationary diffusion problems for two-phase materials. Such problems usually have no solution. A relaxation consists in introducing the notion of composite materials, as fine mixtures of different phases, mathematically described by the homogenisation theory. The problem can be written as an optimisation problem over К(θ), the set of all possible composite materials with given local proportion θ. Tartar and Murat (1985) described the set К(θ)e, for some vector e, and used this result to replace the optimisation over the complicated set К(θ) by a much simpler one. Analogous characterisation holds even for the case of mixing more than two materials (possibly anisotropic), where the set К(θ) is not effectively known (Tartar, 1995).

On Principal Eigenvalue of Stationary Diffusion Problem with Nonsymmetric Coefficients

We consider the eigenvalue problem

Second-order equations as Friedrichs systems

\left\{ \begin{gathered} - div (A\nabla u) = \lambda \rho u \hfill \\ u \in H_0^1\left( \Omega \right)\hfill \\ \end{gathered} \right.

Gradient methods for multiple state optimal design problems

where Ω ∈ R d is open and bounded, ρ ∈ L∞(Ω) and A ∈ L∞(ΩM d×d ) satisfying