Talk by François Willot
On July 2nd, 2024, François Willot (Université PSL) gave a talk about "Homogenized metrics in planar, multiscale random sets" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
Show Abstract
We derive new bounds for the length of minimal paths spanning a class of heterogeneous media in two dimensions, based on random set theory. These bounds provide insights into homogenization problems studied in theoretical mechanics, involving perfectly-plastic materials or related to fracture. Here we consider media containing randomly-distributed particles
embedded in a matrix. Two types of particles are examined: ”porous“ inclusions that can be crossed at no cost and ”rigid“ obstacles that can not be crossed. The metric in the embeddingmatrix is the Euclidean distance. Our main interest is on the effect of a multiscale random dispersion of particles, simulated by hierachical Boolean random sets and random sequential adsorption models. The bounds are derived in the case of a vanishingly small particle density, and are compared with numerical computations, performed by means of specialized, discretization-free algorithms. Comparisons are also made with other known bounds and exact results predicted by nonlinear homogenization theories.
