The main topics of this project are different problems in homogenization, dimension reduction and structural optimization with the special emphasis on the applications in mechanics of solids. The physical idea of homogenization, i.e., averaging a highly heterogeneous media to derive its effective properties has a rather long history. Mathematically, it emerged from difficulties faced with (numerical) analysis of governing partial differential equations in cases when the material oscillates on a small scale or is significantly thinner in one or two directions. A passage to the limit (with respect to some material parameters, i.e., in appropriate topology) usually leads to models that are expected to be simpler, both from the analytical and numerical point of view. The nature of the oscillations of the material can be assumed to be ergodic, periodic or completely general and thus analysed by (stochastic) two-scale convergence method, compensated compactness approach (H-convergence, H-measures) or calculus of variations techniques (Gamma-convergence). Described procedure is well suited for modelling of composite materials, which are fine mixtures of two or more constituent materials. The special interest arises in high-contrast composites, where one obtains a metamaterial with a band gap structure of the spectrum. Optimizing design of devices made of such responsive material and structural material is of great interest. The research proposed by the project encompasses modelling of such materials, and constitutes of challenging and contemporary problems, including following topics:
- stochastic homogenization of high contrast media;
- homogenization and dimension reduction in elasto-plasticity;
- crystallization;
- quantitative homogenization for evolution for linear elastic thin structures in moderate contrast;
- homogenization of Kirchhoff-Love plate equation;
- structural optimization in 3D linearized elasticity;
- structural optimization for Kirchoff-Love plate equation;
- homogenization of Reissner-Mindlin plate equation;
- homogenization of Friedrichs systems…