Multiscale analysis of Prandtl-Ishlinskii operators

Abstract

Homogenization is a cost reducting mathematical method used to model composite materials. It replaces rapidly varying coefficients with constant ones, resulting in an idealized homogeneous material that exhibits similar macroscopic behavior, both qualitatively and quantitatively, to the actual material. The current paper focuses on the deterministic homogenization of the heat equation with hysteresis, which involves the Prandtl- Ishlinskii operator of play type. This equation serves as a model for heat conduction with phase transitions, accounting for undercooling and superheating effects. We consider a sequence of problems with spatially varying coefficients and utilize the concept of sigma-convergence to demonstrate the convergence of the corresponding solutions to the solution of the homogenized problem.