# Workshop on "Homogenization and Multi Scale Analysis"

## Time

Oct. 3-7, 2011

Room 601, Pao Yue-Kong Library

## Introduce

The workshop on “Homogenization and Multi Scale Analysis” will be held Oct. 3-7, 2011, in Room 601, Pao Yue-Kong Library(North Gate Entrance), Shanghai Jiao Tong University, Shanghai, China.

## Committee

• Doina Cioranescu (Paris 6 University)
• Alain Damlamian (Paris 12 University)
• Ta Tsien Li (Fudan University)

## Speakers

• Assyr Abdulle (EPFL Lausanne, Switzerland)
• Dominique Blanchard (Rouen University, France)
• Guy Bouchitté (Toulon and Var University, France)
• Doina Cioranescu (Paris 6 University, France)
• Alain Damlamian (Paris 12 University, France)
• Patrizia Donato (Rouen University, France)
• Horia Ené (IMAR, Bucarest, Romania)
• Georges Griso (Paris 6 University, France)
• Stéphane Labbé (Grenoble University, France)
• Tiejun Li (Peking University, China)
• Liping Liu (Houston University, USA)
• Maria Eugenia Perez (Cantabria University, Spain)
• Annie Raoult (Paris 5 University, France)
• Bogdan Vernescu (WPI, Worcester, Ma, USA)
• Zhijian Yang (Wuhan University, China)
• Xingye Yue (Suzhou University, China)
• Lei Zhang （Oxford University，England）

## Title&Abstract

### General methods in homogenization. Applications to perforated domains

#### Speaker

Doina Cioranescu, Paris 6 Univesity

#### Abstract

The aim of the mathematical theory of homogenization is to study the asymptotical behavior of problems with several parameters which are small compared to the global dimension of the domain where these problems are setted. The development of this theory is related in particular, to that of composite materials which have tremendous applications nowadays. The interest of these materials comes form the fact that in general, they have \better” properties than their components (signi cant examples being the concrete or the superconducting materials). One can describe a composite at a local microscopic scale, by taking into consideration the characteristics of each heterogeneity and that of the surrounding material. Such a procedure becomes too dicult (or even impossible) if the number of heterogeneities is too big. The idea of homogenization is to describe the material at a macroscopic scale, replacing it by a ctitious homogeneous one (of same volume), neglecting the uctuations due to the heterogeneities but having a behavior as close as possible to the original one. For more detailed Information, see

AbstractDoina.pdf

### The heterogeneous multiscale method with application to brittle crack dynamics

Zhijian Yang

#### Abstract

A coupled atomistic and continuum model for numerical simulations of dynamics of crystalline solids will be presented. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling - the heterogeneous multiscale method (HMM). An explicit coupling condition at the atomistic/continuum interface is derived. Application to the dynamics of brittle cracks will be presented. Results of the coupled model will be compared with the empirical continuum models. In particular, process zone, stress intensity factor, stress field, and loading curve will be discussed in details. Different types of loadings will be applied to study the interaction between elastic waves and crack tip behavior. The inertia effects of the crack tip will also be investigated.

### Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions

Trasp Shanghai Donato3_10_11_Patrizia Donato.pdf

Patrizia Donato

#### Abstract

We present the homogeneization of a quasilinear elliptic equation with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, depending on a real parameter. We suppose that the data satisfy some suitable assumptions which insure the existence and the uniqueness of a solution of the problem, in particular suitable growth conditions on the nonlinear boundary term. On the the quasilinear term, some assumptions on themodulus of continuity, which are weaker than a Lipschitz condition, are prescribed. We study the convergence to a limit problem, which is identified by using the periodic unfolding method. We also prove the well-posedness of the limit system. To do that, we show that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the homogenized quasilinear problem has a unique solution and all the sequence of solutions converges.

### Multiscale method for nonlinear non monotone problems: analysis and examples

#### Speaker

Assyr Abdulle, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland

#### Abstract

Following the framework of the heterogeneous multiscale method (HMM), we present a finite element (FE) method for the efficient solution of linear and nonlinear homogenization problems. We first review the numerical strategy, relying on an efficient coupling of macro and micro solvers, and present several examples including parabolic and hyperbolic problems. We then discuss a fully discrete analysis of the FE-HMM for nonlinear (nonmonotone) elliptic problems, and present optimal convergence rates in the $H^1$ and $L^2$ norms. Uniqueness of the method is shown on sufficiently fine macro and micro meshes. Error estimates in periodic homogenization with non-smooth coefficients Speaker:Bogdan Vernescu??? Abstract: Error estimates for the classical elliptic homogenization problem, with non-smooth coefficients and general data, will be discussed using first order and second order boundary layer correctors. The theory makes essential use of the periodic unfolding method developed by Cioranescu, Damlamian and Griso [C. R. Acad. Sci. Paris, Ser. I 335, 2002, 99-104] and extends classical error estimate results that assume either smooth enough coefficients or smooth data. These results extend, in the case of nonsmooth coefficients, the convergence proof for the finite element multiscale method proposed by T.Hou et al. [J. of Comp. Phys., 134, 1997, 169-189] and the first order corrector analysis for the first eigenvalue of a composite media obtained by Vogelius et al.[Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299].

### A multi-scale approach for metamaterials

expose--Guy.pdf

Guy Bouchite

#### Abstract

It is now commonly admitted that obstacles made of metallic or dielectric inclusions placed periodically in a suitable way can behave like homogeneous materials with negative refractive index (or other rather unexpected exotic properties). We will present several 3D situations where such behaviors can be recovered rigorously by using multi-scale methods: - High conductivity fibers with very small filling ratio (negative effective permittivity) - Pendry metallic split ring structures - High contrast dielectric inclusions (artificial magnetism and negative permeability) In all these cases, the key point relies on a spectral problem in the periodic cell which accounts internal resonances and allows to decribe small scale oscillations of the electromanetic field. The macroscopic behavior of the metamaterial is then deduced by classical homogenization techniques.

### Vibrations of high-contrast media: asymptotics for spectra and long-time approaches from quasimodes

E_PEREZ_SHANGHAI2011_web_send.pdf

#### Speaker

Maria Eugenia Perez

#### Abstract

We provide an overview [1-5] on results related to the asymptotic behaviour of the eigenelements of certain singularly perturbed spectral problems arising in models of vibrations of high-contrast structures/media. For certain vibrating systems we can construct standing waves, which concentrate asymptotically its support at points, along lines, or in certain regions and which approach certain solutions of the evolution problems for long time. time.

[1] E. Pérez: Long time approximations for solutions of wave equations via standing waves from quasimodes. J. Math. Pure Appl., 90, 387–411 (2008). [2] M.Lobo and E. Pérez: Long time approximations for solutions of wave equations associated with Steklov spectral homogenization problems. Math. Meth. Appl. Sci., 33, 1356–1371 (2010). [3] S.Nazarov and E. Pérez: New asymptotic effects for the spectrum of problems on concentrated masses near the boundary. C.R. Mecanique. 337, 585—590 , (2010) [4] E. Pérez .: Long time approximations for solutions of evolution equations from quasimodes: perturbation problems. Math. Balkanica (N.S.), 25 (No. 1–2), 95–130. (2011) [5] E. Pérez: On quasimodes for compact operators and associated evolution problems. In Integral Methods in Science and Engineering: Computational and Analytical Methods, Birkhauser, Boston, 313—324 (2011)

### Effective properties of periodic composites and their optimization

LipingLiuShanghai2011.pdf

Liping Liu

#### Abstract

In these talks we will begin with formal justification of the homogenization limit of periodic composites by the method of two-scale convergence. Then we proceed to the derivation of bounds for multiphase composites. Both the method of Hashin-Shtrikman and the method of compensated compactness or translation will be covered. The optimal conditions for some desired effective property gives rise to an overdetermined condition which motivates the definition of a class of new optimal microstructures: periodic E-inclusions. These simple and analytically solvable microstructures shed light to many problems concerning materials design and optimization in a variety of physical context including conductivity problem, elasticity problem, themo-elasticity problem, magneto-electro-elasticity problem, etc.

### Ferromgnetism : from micro to meso scale, the micrognetism model

Shanghai031011_Stephane Labbe.pdf

#### Speaker

Stéphane Labbé, Grenoble University

#### Abstract

In this exposé, we will present joint work with Q. Jouet about the justification of the micrognematism meso-scale model from a classical microsopic description. The fisrt part of the presentation will be devoted to the modelization of ferromagnetic materials. We will introduce in there the microscopic description of crystals and expose the specifities inducing the ferromagnetic behavior. The second part of the presentation will be devoted to the gamma convergence result driving the microscopic model to the micromagnetic meso-scale model introduced by W.-F. Brown in 60’.

### From discrete to continuum for lattices with angular terms and for complex lattices

Shanghai_Annie Raoult.pdf

Annie Raoult

#### Abstract

A long lasting issue has been to construct continuum equivalent models for lattices, see for instance the review paper [1] and references therein. In particular, whether the Cauchy-Born rule (insofar as it is clearly stated) is a handy postulate or a rigorous result is under discussion in several recent papers [2], [3]. The use of an asymptotic procedure (letting the mesh size go to 0) in order to properly derive a “limit” problem is a natural idea that has been given firm grounds and applied in an extensive array of cases by Braides and coauthors [6], [7]. The main tools are to extend node deformations in piecewise affine functions and Gamma-convergence. Formal derivations can be found in [4], [5]. In this talk, we will focus on two examples. First, we consider square planar lattices that can either deform into the three-dimensional space or are restricted to remain planar. Interactions are not restricted to be pairwise; on the contrary we consider three point interactions. Examples are mechanical trusses with torques between bars and atomic lattices. We associate a microscopic energy with each of the four angles of the elementary cell. Under some relationships – that are satisfied in classical examples – between the four energies, we show that the limit energy can be obtained by mere quasiconvexification and that the limit process does not involve any relaxation at the microscopic scale, see [8], [9]. Let us mention that since the energies take angles into account, admissible deformations cannot be allowed to send adjacent nodes on a single point. This natural requirement is often left aside in mathematical convergence analysis and induces here some technicalities. Examples of configurations with null limit energy will be given. Second, with application to carbon lattices (graphenes) in mind, we consider hexagonal lattices that are so-called complex lattices. We concentrate on pairwise interactions. The limit energy is obtained in a more intricate way than in the square case. In order to obtain a complete picture of the node deformations we propose [10], [11] to supplement the classical piecewise affine function with an increment, which is piecewise constant. The limit energy is obtained by first minimizing with respect to the increment, then homogenizing over a set of parallelograms, and by using a discrete version of the slicing method. Related results are found in [12] where Voronoi/Delaunay tesselations are used.

[1] J.L. Ericksen, On the Cauchy-Born rule, Math. Mech. Solids, vol. 13 (2008), 199–220. [2] FG. Friesecke, F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional massspring lattice, J. Nonlinear Sci., vol. 12 (2002), 445–478. [3] W. E, P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Rational Mech. Anal., 183 (2007), 241–297. [4] H. Tollenaere, D. Caillerie, Continuous modeling of lattice structures by homogenization. Adv. Eng. Softw. 29, (1998), 699–705. [5] D. Caillerie, A. Mourad, A. Raoult, Discrete homogenization in graphene sheet modeling, J. Elast., 84 (2006), 33–68. [6] A. Braides, M.S. Gelli, From Discrete to Continuum: a Variational Approach, Lecture Notes, SISSA, Trieste (2000). [7] A. Braides, M.S. Gelli. Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids, 7 (2002), 41–66.

### String method for the stochastic Cahn-Hilliard dynamics

TiejunLi.pdf

Tiejun Li

#### Abstract

We consider the nucleation of stochastic Cahn-Hilliard dynamics with the standard double well potential. We design the string method for computing the most probable transition path in the zero temperature limit based on large deviation theory. We derive the nucleation rate formula for the stochastic Cahn-Hilliard dynamics through finite dimensional discretization. We also discuss the algorithmic issues for calculating the nucleation rate, especially the high dimensional sampling for computing the determinant ratios.

### Junction of a plate and a rod in nonlinear elasticity

Shanghai2011_Dominique Blanchard.pdf

#### Speaker

Dominique Blanchard

#### Abstract

The junction problem between a plate and a rod is investigated as their thicknesses tend to zero. This multi-structure is made of elastic Saint-Venant- Kirchhoff’s materials (possibly different in the plate and in the rod). It is clamped on a part of the lateral boundary of the plate and it is free on the rest of its boundary. The goal is to characterize the limit of the rescaled infimum of the elastic energy as the minimum of a limit energy. This is achieved through two main arguments : the derivation of nonlinear Korn’s inequalities (in the plate and in the rod) and of sharp estimates in the 3d geometrical junction. Both rely on a split- ting technique of a large deformation (in each part) of the structure as a ”mean deformation” and a ”local” deformation (warping). The applied forces are scaled in such a way that the limit energy corresponds to a Von-K ́arm ́an’s plate and a nonlinear rod model coupled via the bending in the plate and the stretching in the rod.

### Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a 2D model problem

numhom-sjtu-oct062011_Lei Zhang1.pdf

ac-sjtu-oct062011_Lei Zhang2.pdf

Lei Zhang

#### Abstract

We present a new variant of the geometry reconstruction approach for the formulation of atomistic/continuum coupling methods (a/c methods). For multi-body nearest-neighbour interactions on the 2D triangular lattice, we show that patch test consistent a/c methods can be constructed for arbitrary interface geometries. Moreover, we prove that all methods within this class are first-order consistent at the atomistic/continuum interface and second-order consistent in the interior of the continuum region.

### Error estimates in periodic homogenization with non-smooth coefficients

Shanghai 2011_Bogdan Vernescu.pdf

Bogdan Vernescu

#### Abstract

Error estimates for the classical elliptic homogenization problem, with non-smooth coefficients and general data, will be discussed using first order and second order boundary layer correctors. The theory makes essential use of the periodic unfolding method developed by Cioranescu, Damlamian and Griso [C. R. Acad. Sci. Paris, Ser. I 335, 2002, 99-104] and extends classical error estimate results that assume either smooth enough coefficients or smooth data. These results extend, in the case of nonsmooth coefficients, the convergence proof for the finite element multiscale method proposed by T.Hou et al. [J. of Comp. Phys., 134, 1997, 169-189] and the first order corrector analysis for the first eigenvalue of a composite media obtained by Vogelius et al.[Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299].

### Relationships between the multiscale methods for elliptic homogenization problems

Shanghai2011_Xingye YUE.pdf

#### Speaker

Xingye Yue, Soochow University

#### Abstract

In this talk, we will give a short review on multiscale methods for elliptic homogenization problems. We will emphasize the intrin- sic links between some popular methods such as generalized finite element methods (GFEM), residual-free bubble methods (RFB), vari- ational multiscale methods (VMS), multiscale finite element methods (MsFEM) and heterogeneous multiscale methods (HMM).

### The homogenization of a Fredholm alternative in a domain which is periodically perforated

Fredholm_Shanghai_2011-09-28_Alain Damlamian.pdf

Alain Damlamian

#### Abstract

In the appendix of [1], the homogenization of a diffusion problem with homogeneous Neumann boundary conditions was presented in a periodically perforated domain $\Omega\epsilon^*$, when the equation has a zero order term. The domain is obtained by perforating a fixed domain $\Omega$ periodically by small holes of size $\epsilon$. This lecture will address the case without zero order term, which therefore involves a Fredholm alternative. However, there are multiple obstacles to solving this problem, if only because the perforated domain $\Omega\epsilon^*$ is not connected, a condition which is fundamental for the Fredholm alternative. The lecture will present a joint paper with Doina Cioranescu and Georges Griso, where an appropriate approximate problem (itself satisfying a Fredholm alternative) is used. The geometric considerations follow from the presentation of the periodic unfolding method for problems with holes (see [2]) and a result concerning the existence of a uniformly bounded Poincaré-Wirtinger constant for the approximating sequence of perforated domains under consideration. Non-homogeneous Neumann conditions will also be considered on the boundary of the perforations. Surprisingly, at the limit, they generate a non-homogeneous Neumann condition on the boundary of the domain $\Omega$. References: [1] G. Allaire-F. Murat:Homogenization of the homogeneous Neumann problem with nonisolated holes, including Appendix with A.K. Nandakumar, Asymptotic Analysis 7 (1993), 81-95. [2] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, submitted.

### A 3D approximation for nonlinear elastic thin beams

SlidesShanghaiGriso_Georges Griso.pdf

#### Speaker

Georges Griso, Laboratoire J.-L. Lions{CNRS, Bo^te courrier 187, Universite Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France, e-mail: griso@ann.jussieu.fr

to be announced.