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- ItemHomogenization of the nonlinear bending theory for plates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Neukamm, Stefan; Olbermann, HeinerWe carry out the spatially periodic homogenization of Kirchhoff's plate theory. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in Kirchhoff's plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions of class W2,2, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.
- ItemHomogenization of a porous intercalation electrode with phase separation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin; Landstorfer, Manuel; Liero, MatthiasIn this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.
- ItemHomogenization of Cahn-Hilliard-type equations via evolutionary Gamma-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Reichelt, SinaIn this paper we discuss two approaches to evolutionary Gamma-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Gamma-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savare 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Gamma-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
- ItemHomogenization in gradient plasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Hanke, HaukeThis paper yields a two-scale homogenization result for a rate-independent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by e > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor e? (? = 0) . We derive different limiting models for e ? 0 in dependence of &gamma ;. For ? > 1 the limiting model is the two-scale model derived in [MielkeTimofte07], where no gradient term was present. For ? = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for ? ? [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part
- ItemStochastic two-scale convergence and Young measures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin; Neukamm, Stefan; Varga, MarioIn this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.
- ItemStochastic unfolding and homogenization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Heida, Martin; Neukamm, Stefan; Varga, MarioThe notion of periodic two-scale convergence and the method of periodic un- folding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coecients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coecients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in dierent ways to the stochastic case. In this work we introduce a stochastic unfolding method that fea- tures many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogeniza- tion result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method descibed in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.
- ItemStochastic homogenization of Lambda-convex gradient flows(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heida, Martin; Neukamm, Stefan; Varga, MarioIn this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.
- ItemPulses in FitzHugh-Nagumo systems with rapidly oscillating coefficients(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Gurevich, Pavel; Reichelt, SinaThis paper is devoted to pulse solutions in FitzHughNagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHughNagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.
- ItemQuantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Gloria, Antoine; Neukamm, Stefan; Otto, FelixWe study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice Zd with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w. r. t. a Glauber dynamics on coefficient fields |as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i. e. of the generator of the process called "random environment as seen from the particle").