Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

Abstract

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C ¹-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.

Keywords:

• stochastic homogenization
• heat transfer
• polycrystals
• contact conductivity
• two-scale

AMS Subject Classifications:

• 35B27
• 80M40
• 74Q10
• 60D05

Notes

1. This is a RACS consisting only of points.