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- 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.
- ItemFractal homogenization of a multiscale interface problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Heida, Martin; Kornhuber, Ralf; Podlesny, JoschaInspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergenceof the approximate problems is close to optimal.