Optimal design deals with the problem of finding a shape or structural arrangement of a single or several materials such that a certain optimality criterion is achieved. It is a very active research topic, that experienced a burst of new ideas during last decades, accompanied by numerical simulations and real world applications.

Homogenisation is one of the most successful mathematical methods when analysing mixtures of two or more materials. If the components involved in an optimal design problem have a low contrast, a small amplitude homogenisation method can be applied. Its main idea consists of taking a expansion of a solution to a problem under consideration with respect to a small parameter representing perturbation of the coefficient. Originally introduced by L. Tartar for a stationary diffusion problem, the approach has subsequently been elaborated and applied to more general homogenisation, optimal design and inverse problems. In all these papers H-measures are used as the main analytical tool, and the analysis is performed up to the second order expansion. Handling of terms appearing in higher order expansion, is, however, not achievable by a standard application of H-measures. Recently, an alternative approach has been introduced in resulting in explicit expressions for higher order correction terms. It has been applied to the stationary diffusion equation aiming to describe diffusion properties of the homogenised limit.

Within this project we aim to apply this approach to a non-stationary diffusion equation, as well as to the optimal design problem for two phase composites, and to obtain corresponding results up to the third (and higher) order expansion. Special, but important, case of the periodic setting will be used to compare the new approach with the classical periodic homogenisation techniques.

Literature:

- Lazar, M.

*Exploring limit behaviour of non-quadratic terms via H-measures. Application to small amplitude homogenisation.*Applicable analysis. 96 (2016) , 16; 2832-2845 - Allaire, G; Gutierrez, S.

Optimal Design in Small Amplitude Homogenization, Math. Model. Numer. Anal. 41 (2007) 543–574.