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- ItemCorrector estimates for a thermo-diffusion model with weak thermal coupling(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Muntean, Adrian; Reichelt, SinaThe present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology weak thermal coupling refers here to the variable scaling in terms of the small homogenization parameter " of the heat conduction diffusion interaction terms, while the high-contrast is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with -independent estimates for the thermal and concentration fields and for their coupled fluxes.
- ItemPrecompact probability spaces in applied stochastic homogenization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, MartinWe provide precompactness and metrizability of the probability space Ω for random measures and random coefficients such as they widely appear in stochastic homogenization and are typically given from data. We show that these properties are enough to implement the convenient two-scale formalism by Zhikov and Piatnitsky (2006). To further demonstrate the benefits of our approach we provide some useful trace and extension operators for Sobolev functions on Ω, which seem not known in literature. On the way we close some minor gaps in the Sobolev theory on Ω which seemingly have not been proven up to date.
- 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.
- ItemError estimates for elliptic equations with not exactly periodic coefficients(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Reichelt, SinaThis note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.
- ItemThe dielectric constant of liquid electrolytes obtained from periodic homogenization theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Landstorfer, ManuelThe dielectric constant of an electrolytic solution is known to decrease with increasing salt concentration. This effect, frequently called dielectric decrement, is experimentally found for many salts and solvents and shows an almost linear decrease up to a certain salt concentration. However, the actual origin of this concentration dependence is yet unclear, and many different theoretical approaches investigate this effect. Here I present an investigation based on microscopic Maxwell equations and periodic homogenization theory. The microscopic perception of anions and cations forming a pseudo lattice in the liquid solution is exploited by multi-scale asymptotic expansions, where the inverse Avogadro number arises as small scaling parameter. This leads to a homogenized Poisson equation on the continuum scale with an effective or homogenized dielectric constant that accounts for microscopic field effects in the pseudo lattice. Incomplete dissociation is further considered at higher salt concentrations due to solvation effects. The numerically computed homogenized dielectric constant is then compared to experimental data of NaCl and shows a remarkable qualitative and quantitative agreement in the concentration range of (0 5)mol L.
- ItemStochastic homogenization on perforated domains II -- Application to nonlinear elasticity models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, MartinBased on a recent work that exposed the lack of uniformly bounded W1,p → W1,p extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear elasticity on such structures using instead the extension operators constructed in [11]. We thereby introduce two-scale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space Ω, e.g. abstract Gauss theorems.
- ItemMultiscale modeling of weakly compressible elastic materials in harmonic regime and applications to microscale structure estimation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Caiazzo, Alfonso; Mura, JoaquínThis article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive.We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.
- ItemCorrector estimates for a class of imperfect transmission problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Reichelt, SinaBased on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order , where > 0 describes the periodicity of the microstructure and 2 (0; 1/2 ] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.
- 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").