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For an educational purpose we develop the Python package AutoFreeFem which generates all ingredients for shape optimization with non-linear multi-physics in FreeFEM++ and also outputs the expressions for use in LaTex. As an input, the objective function and the weak form of the problem have to be specified only once. This ensures consistency between the simulation code and its documentation. In particular, AutoFreeFem provides the linearization of the state equation, the adjoint problem, the shape derivative, as well as a basic implementation of the level-set based mesh evolution method for shape optimization. For the computation of shape derivatives we utilize the mathematical Lagrangian approach for differentiating PDE-constrained shape functions. Differentiation is done symbolically using Sympy. In numerical experiments we verify the accuracy of the computed derivatives. Finally, we showcase the capabilities of AutoFreeFem by considering shape optimization of a non-linear diffusion problem, linear and non-linear elasticity problems, a thermo-elasticity problem and a fluid-structure interaction problem.

These are the lecture notes of a short course on the homogenization method for topology optimization of structures, given by Gr\'egoire Allaire, during the "GSIS International Summer School 2018" at Tohoku University (Sendai, Japan). The goal of this course is to review the necessary mathematical tools of homogenization theory and apply them to topology optimization of mechanical structures. The ultimate application, targeted in this course, is the topology optimization of structures built with lattice materials. Practical and numerical exercises are given, based on the finite element free software FreeFem++., Comment: 146 pages, based on the "GSIS International Summer School 2018" at Tohoku University (Sendai, Japan)

This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, $\epsilon$ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times $t$ of order $\epsilon^{-2+\delta}$ for any $\delta>0$. Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of $\epsilon$, yields an approximation of the exact solution with error $O(\epsilon^N)$ for times $\epsilon^{-N}$ with $N$ as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms.

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.

We study the homogenization of a Schr\"{o}dinger equation in a locally periodic medium. For the time and space scaling of semi-classical analysis we consider well-prepared initial data that are concentrated near a stationary point (with respect to both space and phase) of the energy, i.e. the Bloch cell eigenvalue. We show that there exists a localized solution which is asymptotically given as the product of a Bloch wave and of the solution of an homogenized Schr\"{o}dinger equation with quadratic potential.

We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection-diffusion equations, one in the bulk and one on the pore boundaries, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model [G. Allaire et al, Chemical Engineering Science, 65 (2010), pp.2292-2300], [G. Allaire et al, SIAM J. Math. Anal., 42 (2010), pp.125-144], [Allaire et al, IMA J Appl Math., 77 (2012), pp.788-815]. Under a technical assumption of equal drift velocities in the bulk and on the pore boundaries, we obtain a nonlinear monotone diffusion equation as the homogenized model. Our main technical tool is the method of two-scale convergence with drift [Marusic-Paloka et al, Journal of London Math. Soc., 72 (2005), pp.391-409]. We provide some numerical test cases in two space dimensions to support our theoretical analysis., Comment: 43 pages, 5 figures

This paper is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal eigenfunctions of an associated spectral problem and to cancel any exponential decay in time of the solution using the principal eigenvalue of the same spectral problem. We employ the notion of two-scale convergence with drift in the asymptotic analysis of the factorized model as the lengthscale of the oscillations tends to zero. This combination of the factorization method and the method of two-scale convergence is applied to upscale an adsorption model for multicomponent flow in an heterogeneous porous medium., Comment: 28 pages

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O'Brien's argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations., Comment: 46 pages

We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of $N$ chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behavior of its solution depending on a parameter $\beta$ which is the square of the ratio between a characteristic pore length and the Debye length. For small $\beta$ we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the {\it Donnan effect}, observing that the ions for which the charge is the one of the solid phase are expelled from the pores. For large $\beta$ we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid, so that the simplified additive theories, such as the one of the popular Derjaguin, Landau, Verwey and Overbeek (DLVO) approach, can be used. We show that the asymptotic behavior is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential).

This short note is a sequel to our previous papers on the asymptotic behavior of Bloch wave packet solutions of the wave equation in periodic media. The purpose is to prove that the group velocity for these Bloch wave packets is bounded by the maximal speed of propagation for the original wave equation.

We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schr\"odinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations.

We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.