## "Composites, Metamaterials, and Inverse Problems"## in honour of Graeme Milton's 60th birthday
Date: December 13 - 15, 2016 Venue: Rm. 1501, Natural Sciences B/D E6-1 KAIST, Daejeon |

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Speaker: Marc Briane (INSA de Rennes)

Title: Realizability of some fields in Electrophysics and Mechanics

Abstract: This work deals with the reconstruction of physical coefficients from some fields in Physics and Mechanics. On the one hand, in collaboration with G.W. Milton and A. Treibergs (University of Utah), we have explicitly reconstructed the conductivity from the electric field and the current field using dynamical systems. The realizability of the electric is based on the gradient flow of the associated potential. The one of the current field in dimension three is much more delicate and needs a triple dynamical system in the directions of the current filed, its curl and the cross product of the two fields. On the other hand, in two-dimensional incompressible elasticity we have reconstructed the shear modulus from the strain tensor using semilinear hyperbolic equations. Contrary to the electric or the current filed we have not succeeded to obtain a global realizability result. For all the periodic fields arising in composite materials we have proved that the realizability in the whole space does not imply the realizability in the torus.

Speaker: Elena Cherkaev (University of Utah)

Title: Inverse homogenization: An inverse problem for the structure of composites

Abstract: Inverse homogenization is a problem of deriving information about the microgeometry of a composite from known effective properties. I will discuss an approach to this problem based on the reconstruction of the matrix-valued spectral measure in the analytic representation of the composites effective properties. This Stieltjes analytic representation relates the n-point correlation functions of the microstructure to the moments of the spectralmeasure of an operator depending on the composites geometry. I will show that the spectral measure which contains all information about the microstructure, can be uniquely recovered from effective measurements known in an interval of frequency. In particular, the volume fractions of materials in the composite and the spectral gaps at the ends of the spectral interval, can be uniquely reconstructed. I will discuss identification of microstructural parameters from electromagnetic and viscoelastic effective measurements, and show that matrix Pade approximants provide an efficient way to construct spectrally closely matched microstructures.

Speaker: Yury Grabovsky (Temple University)

Title: A construction of isotropic rank-one convex, non-quasiconvex functions by using the theory of exact relations for composite materials

Abstract: The well-known relation between G-closed sets and quasiconvex functions and L-closed sets (sets containing effective tensors of all laninates made from materials taken from that set) and rank-one convex functions has been used by Graeme Milton to construct a (7-phase) elastic composite whose effective tensor cannot be mimicked by laminates. In this talk I will present a construction of a polycrystalline L-relation (a rotationally-invariant submanifold of positive codimension that is L-closed) that is not G-closed. The same link between G-closed sets and quasiconvex functions and L-closed sets and rank-one convex functions leads to a construction of rank-one convex, non-quasiconvex functions. The rotational invariance of the L-relation leads to the isotropy of constructed functions. Construction of a polycrystalline L-relation that is not G-closed is based on the general theory of exact relations, developed in collaboration with Graeme Milton.

Speaker: Hyeonbae Kang (Inha University)

Title: Fine analysis of stress concentration between two inclusions with extreme material properties

Abstract: In composites consisting of inclusions and a matrix, some inclusions are located very close to each other. If these inclusions have an extreme material property, such as high or low conductivity in the context of electro-statics, or high stiffness in elasticity, then it is known that high stress occurs in between these inclusions. In last 10 years or so there has been significant progress in quantitative understanding of the stress concentration through fine analysis.

Speaker: **Mikyoung Lim (KAIST)**

Title: Imaging and cloaking based on multipole expansions

Abstract: An inclusion embedded in a homogeneous background scatters incoming electromagnetic waves. The scattered waves can be decomposed into multipole elds with coefficients expressed in terms of integrals on the interface. The coefficients of the multipole expansion contain signi cant information on the shape of the inclusion and its material parameter. In this talk I will present application of multipole expansions to the problem of shape reconstruction of inclusion and invisibility cloaking.

Speaker: Ross McPhedran (University of Sydney)

Title: Getting the Most Out of Light Absorption with Structured Systems

Abstract: I will review some work done over a long period on the use of structure in optical systems, with the aim of getting as high an absorption as possible. Soon after I started work on this problem with Professor David McKenzie we recruited a young student to help us with the development of multipole methods. Not only did he do this, but he developed the method of bounds for constraining optimal performance. I will talk about some work from this period, and its various developments over the period up until the present. One current problem concerns the achievement of high absorption in systems much thinner than the wavelength of light.

Speaker: Alexander Movchan (University of Liverpool)

Title: Flexural waves and localisation around semi-infinite structured channels

Abstract: The lecture presents new results on the analysis of scattering and localisation for flexural elastic waves by a semi-infinite structure consisting of several parallel gratings of rigid pins embedded into an elastic thin plate. We consider a time-harmonic regime, with the flexural displacement governed by the fourth-order partial differential equation. The approach proposed in this study uses a combination of an analytical

procedure and numerical computations based on the analysis of “two-grating Green’s functions” for an infinite flexural plate. An application of the z-transform enables us to reduce the mathematical formulation of the problem to an elegant functional equation of the Wiener-Hopf type. Blockages and waveguide modes are observed and analysed in detail. A homogenisation procedure is applied to study the modulation of a trapped wave within the structure of gratings. The theoretical predictions are accompanied by numerical simulations and discussions of applications in the design of filters for flexural waves in structured plates. The lecture is based on the results of the joint work with I.S. Jones, N.V. Movchan and S.G. Haslinger.

Speaker: Francois Murat (Université Pierre et Marie Curie - Paris VI)

Title: Homogenization of the brush problem with a source term in $L^1$

Abstract: We consider a domain which has the form of a brush (in dimension N = 3) or of a comb (in dimension N = 2), i.e. an open set which is composed of cylindrical vertical teeth distributed over a fixed basis. All the teeth have a similar fixed height; their cross sections can vary from one teeth to another one and are not supposed to be smooth; moreover the teeth can be adjacent, i.e. they can share parts of their boundaries. The diameter of every tooth is supposed to be less than or equal to $varepsilon$, and the asymptotic volume fraction of the teeth is supposed to be bounded from below away from zero, but no periodicity is assumed on the distribution of the teeth.

In this domain we study the asymptotic behavior of the solution of a second order elliptic equation with a zeroth order term which is bounded from below away from zero, when the homogeneous Neumann boundary condition is imposed on the whole of the boundary.

We first revisit the problem where the source term belongs to $L^2$. This is a classical problem but our homogenization result takes place in a geometry which is more general that the ones which have been considered before. Moreover we prove a corrector result which is new.

We then consider the case where the source term belongs to $L^1$. Working in the framework of renormalized solutions, and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved (such a definition is new), we identify the limit problem and prove a corrector result.

This is joint work with Antonio Gaudiello (Naples, Italy) and Olivier Guib'e (Rouen, France).

Speaker: Pierre Seppecher (IMATH Université de Toulon and LMA Marseille)

Title: Homogenisation of linear elastic structures leading to complete second gradient models

Abstract: The classical homogenization formula describes the limit material as a classical elastic model (a Cauchy material). This formula is often used out of its scope : when the constast between different parts of the material is very high or worse when empty holes are present. On the countrary, we propose cylindrical structures based on a thickened periodic graph and we prove, using the tools of Gamma-convergence and double scale limits, that they have to be described, in the asymptotic limit, as second gradient materials. The choice of the geometry allows us to reduce to the study of discrete systems which correspond to trusses or welded frames. We show how the homogenization of these discrete systems reduces to very simple algebraic formula.

Speaker: Abdul Wahab (KAIST)

Title: A joint sparsity algorithm for inverse elastic medium scattering

Abstract: In this talk a compressed sensing based algorithm will be presented for efficient and accurate reconstruction of the spatial support of multiple inhomogeneous elastic inclusions in a bounded elastic formation and their material parameters. Only a few measurements of the displacement field over a very coarse grid (in the sense of Nyquist sampling rate) will be taken into account, on contrary to classical algorithms assuming continuous or dense grid measurments. The proposed algorithm is not only very accurate since it does not require any linearization but is also computationally efficient as it is direct. The breakthrough comes from a novel interpretation of Lippmann-Schwinger type integral representation of the displacement field in terms of unknown densities (that are linked to the internal displacement and strain fields) having common sparse support on the location of inclusions. First, the support identification problem is recast as a joint sparse recovery problem that renders such densities and the support of the inclusions simultaneously. Then, using the leverage of the learned densities and associated internal information, a linear inverse problem for quantitative evaluation of material parameters is formulated. The resulting problem is then converted to a noise robust constraint optimization problem. For numerical implementation, modified Multiple Sparse Bayesian Learning (M-SBL) algorithm and the Constrained Split Augmented Lagrangian Shrinkage Algorithm (C-SALSA) are invoked. The efficacy of the proposed framework will be manifested through a variety of numerical examples. The significance of this investigation is due to its pertinences for bio-medical imaging and non-destructive testing, wherein the real physical measurments are only available on a sub-sampled coarse grid. It is believed that the proposed algorithm is the first one tailored for parameter reconstruction problems in elastic media using highly under-sampled data.

Speaker: Martin Wegener (KIT)

Title: Metamaterials: Mathematics -- Design – Experiments

Abstract: 3D printing is a huge trend worldwide. 3D printing benefits from metamaterials and vice versa: In 2D graphical printing, thousands of colors can be mixed from three basic colors. By analogy, thousands of different effective metamaterial properties can be obtained by clever microstructures composed of only a few constituent-material cartridges. I will focus on mechanical metamaterials (pentamode, negative thermal expansion, beyond ordinary elasticity, and nonlinear buckling) and Hall-effect metamaterials. For all of these, mathematics and design have been crucial for our experiments.

Speaker: Sanghyeon Yu (ETH Zurich)

Title: Plasmonic interaction between nanospheres

Abstract: Recently, the plasmon resonances of a densely packed composite of nanospheres have received considerable attention due to its super-resolution imaging capability. However, the numerical computation of the electric field for the composites is very hard because extremely fine mesh is required in the narrow gap region. In 1998, Cheng and Greengard developed a hybrid numerical scheme for densely packed composites

of conducting cylinders or spheres by combining the method of image charges and the multipole expansion. However, their method is not directly applicable for plasmonic nanospheres since the image charges solution for two spheres does not converge when the dielectric constant have negative real part. In this talk, I will present how to extend their hybrid scheme for plasmonic nanospheres composites. This is a joint work with Habib Ammari (ETH Zurich, Switzerland).

Speaker: Hai Zhang (HKUST)

Title: Mathematics of super-resolution in resonant media

Abstract: We develop a mathematical theory to explain the mechanism of super-resolution in resonant media which consists of sub-wavelength resonators. Examples includes: Helmholtz resonators, plasmonic particles, and bubbles. For the media consists of small finite number of resonators, we show that super-resolution is due to sub-wavelength propagating modes; for the case of large number of resonators, we derive an effective media theory and show that super-resolution is due to the effective high contrast in the wave speed. This is a joint work with Habib Ammari at ETH, Switzerland.