Martin Heida

An extension of the stochastic two-scale convergence method and application

The stochastic two-scale convergence method, recently developed in an article by Zhikov and Piatnitskii (Izv. Math. 70(1) (2006), 19-67) is extended to arbitrary probability spaces and is now based on the theory of stochastic geometry instead of random measures. It will be shown that former results on stochastic and periodic two-scale convergence fit into the new approach in a natural way. These results will be applied to functions with jumps on (n − 1)-dimensional manifolds, in particular to a homogenization problem of heat transfer through a composite material or polycrystal.

On gradient flows of nonconvex functionals in Hilbert spaces with Riemannian metric and application to Cahn-Hilliard equations

In Hilbert spaces with a densely defined Riemannian metric, we study gradient flows (curves of maximal slope) of the form ∂tu+∇lS(u) 3 f where S is a nonconvex functional, ∇lS(u) is the strong-weak closure of the subgradient of S and f is a time dependent right hand side. The article generalizes the results by Rossi and Savaré to this setting and provides some examples from multiphase systems. In particular, we treat Allen-Cahnand Cahn-Hilliard equations with mobility depending nonlinear on the concentration and its gradient. We also study systems of multiple phases derived by Heida, Málek and Rajagopal [20, 19] in a simplified form. In particular, we will show that a certain class of reaction-diffusion equations coming from a modeling approach by Rajagopal and Srinivasa [27] are automatically subject to the theory of curves of maximal slope.

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

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.

Stochastic homogenization of plasticity equations

In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter e > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit e → 0. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution induces a stress evolution . Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by −∇· Σ (∇ s u ) = f .