Neumann Homogenization via Integro-Differential Operators, Part 2: singular gradient dependence

We continue the program initiated in a previous work, of applying integro-differential methods to Neumann Homogenization problems. We target the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with \emph{non-co-normal} oscillatory Neumann conditions. Our analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. Also, we use homogenization results for regular Dirichlet problems to build barriers for the oscillatory Neumann problem with the singular gradient term. We note that our method allows to recast some existing results for fully nonlinear Neumann homogenization into this same framework. This version is the journal version.Comment: This is the revised version that appeared in SIM

An introduction to the qualitative and quantitative theory of homogenization

We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar's method of oscillating test functions. In partic- ular, we present a self-contained and elementary argument for the construction of the sublinear corrector of stochastic homogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In the second part we briefly discuss the representation of the homogenization error by means of a two- scale expansion. In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular, we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter in combi- nation with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and the homogenization error.Comment: Lecture notes of a minicourse given by the author during the GSIS International Winter School 2017 on "Stochastic Homogenization and its applications" at the Tohoku University, Sendai, Japan; This version contains a correction of Lemma 2.1

Multiscale computational first order homogenization of thick shells for the analysis of out-of-plane loaded masonry walls

This work presents a multiscale method based on computational homogenization for the analysis of general heterogeneous thick shell structures, with special focus on periodic brick-masonry walls. The proposed method is designed for the analysis of shells whose micro-structure is heterogeneous in the in-plane directions, but initially homogeneous in the shell-thickness direction, a structural topology that can be found in single-leaf brick masonry walls. Under this assumption, this work proposes an efficient homogenization scheme where both the macro-scale and the micro-scale are described by the same shell theory. The proposed method is then applied to the analysis of out-of-plane loaded brick-masonry walls, and compared to experimental and micro-modeling results.Peer ReviewedPostprint (author's final draft

Three scales asymptotic homogenization and its application to layered hierarchical hard tissues

In the present work a novel multiple scales asymptotic homogenization approach is proposed to study the effective properties of hierarchical composites with periodic structure at different length scales. The method is exemplified by solving a linear elastic problem for a composite material with layered hierarchical structure. We recover classical results of two-scale and reiterated homogenization as particular cases of our formulation. The analytical effective coefficients for two phase layered composites with two structural levels of hierarchy are also derived. The method is finally applied to investigate the effective mechanical properties of a single osteon, revealing its practical applicability in the context of biomechanical and engineering applications