Discrete Models and Continuum Limits

Discrete energies with degenerate growth

In [NSS], we study stochastic homogenization in absence of uniform ellipticity. More precisely, we consider discrete lattice energies with random (stationary & ergodic) pair potentials satisfying a weighted version of the standard -growth condition. In a scalar setting, which include as a special case the well-studied random conductance model, we provide optimal moment conditions on the weight and its inverse, ensuring that the discrete energy -converges (as the lattice spacing tends to zero) almost surely to a deterministic integral functional satisfying standard -growth. Moreover, we provide related results for vector-valued problems and show how under stronger assumptions on the probability distribution, namely iid coefficients, the corresponding moment conditions on the weights can be relaxed.

Stochastic Unfolding for Discrete Systems

Two scale-convergence methods play a prominent role in homogenization. The merit in using two-scale convergence is the elegant and straightforward procedure leading to homogenization results in a broad range of problems. Periodic unfolding is a well-established method based on an unfolding operator, which reduces the notion of two-scale convergence to mere weak convergence in L^p-spaces. This method has been applied to a large number of homogenization problems featuring rapidly oscillating periodic coefficients.

In our work [NV] we extend the idea of unfolding from the periodic to the stochastic, possibly discrete, setting. Namely, we introduce an operator (a linear isometry) defined on random fields – the stochastic unfolding operator. It admits similar properties as its periodic counterpart and thus it affords a simple (operator theory flavored) homogenization procedure for problems involving random coefficients.

Homogenization of the evolution of spring networks

Spring network models are often used in practical applications in materials science and engineering.In such models, deformable solids are represented by particles in a lattice, connected by springs with certain material laws. Network models serve as meso-scale approximations of some continuum theories, as well as direct phenomenological representations of microscopic phenomena in materials. If the size of the springs is significantly smaller than the macroscopic material sample, the numerical treatment is costly and there is a need for upscaling/homogenization.

In [NV] we analyse a spring network model where the springs are modeled by one-dimensional elasto-plasticity with hardening. The material parameters are described by a rapidly oscillating random field. Using the stochastic unfolding method, we derive an effective continuum model in the limit as the characteristic size of the springs tends to zero.

Publications (and works in preperation)

• [NSS] Stefan Neukamm, Mathias Schäffner and Anja Schlömerkemper, Stochastic homogenization of nonconvex discrete energies with degenerate growth, arXiv:1606.06533. SIAM SIMA (to appear)
• [NV] Stefan Neukamm and Mario Varga. Stochastic homogenization and discrete-to-continuum limits via stochastic unfolding. SIAM MMS (accepted)
• [HNV] Martin Heida, Stefan Neukamm and Mario Varga. Stochastic homogenization via Stochastic Unfolding. WIAS Preprint No. 2460, 2017