Organizers: Thomas Eiter, Marita Thomas, Martin Heida

Abstract:

In many problems from material science, geophysics and fluid dynamics, certain material or geometrical properties vary strongly on a local scale. To study their effect on the macroscopic behavior and to obtain effective models, one can derive averaged equations by passage to a homogenized framework. Moreover, by introducing some randomness, one can take into account that the exact distribution of the small-scale fluctuations is usually unknown in applications. This minisymposium brings together experts on the mathematical investigation of such stochastic homogenization problems

Speakers:

14:30h: Peter Bella (TU Dortmund)

Stochastic homogenization of degenerate elliptic equations

I will discuss recent results on stochastic homogenization for linear elliptic equations with random coefficients. The main focus will be on the degenerate elliptic setting, where ellipticity is controlled by suitable (p,q)-moment conditions rather than uniform bounds. I will present several results, including large-scale regularity estimates. The talk is based on joint works with Mathias Schäffner.

15:00h: Stefan Neukamm (TU Dresden)

Quantitative stochastic homogenization of elastoplastic spring networks

We consider a discrete lattice of linear elastoplastic springs with hardening, modelled as an evolutionary rate-independent system with random, stationary and ergodic material parameters. Qualitative homogenization shows that in the large-scale limit the system converges to a continuum linear elastoplasticity model, whose stress-strain relation is described by a generalized Prandtl–Ishlinskii hysteresis operator.
We discuss the convergence of a periodic representative volume element (RVE) approximation of this hysteresis operator, and provide quantitative error estimates for both the systematic and random errors in the case of laminate coefficients with finite range of dependence. This yields the first quantitative homogenization result for a rate-independent system, and confirms numerically observed convergence rates. This is joint work with M. Abdel Wahab.

15:30h: Caterina Zeppieri (Universität Münster)

The random fractional obstacle problem

Nonlocal energies, such as fractional Sobolev seminorms, arise naturally in mathematical models involving long-range interactions. In this talk, we study minimizers of such energies that vanish on a collection of small balls with random centers and radii, leading to a bilateral (fractional) obstacle problem. I will present a homogenization result that holds under minimal assumptions on the distribution and size of the obstacles, which are generated by a stationary marked point process. In particular, the obstacles may overlap and form clusters with positive probability, giving rise to a complex microstructure. Our analysis identifies the limiting energy and shows how it reflects the underlying probability distribution of the obstacles.