Abstract : Composites are inhomogeneous mixtures of their component materials. Metamaterials are composites with properties that go beyond those of the constituent phases and possibly beyond naturally occurring materials. These have attracted increasing attention in the past twenty years. Many important mathematical questions have been addressed, yet many remain. For example, composites and metamaterials can guide fields and waves in desired ways, e.g., cloaking, but the limitations of this are not so clear. Also, what unusual effective behaviors are possible? Incorporating dispersion, dissipation, anisotropy, extreme moduli, etc. adds to the challenge. This minisymposium will present exciting new developments in the field.
Organizer(s) : Maxence Cassier (CNRS, Institut Fresnel), Graeme W. Milton (University of Utah), Anthony Stefan (Florida Institute of Technology), Aaron Welters (Florida Institute of Technology)
Abstract : Wave propagating in a medium with two constitutive parameters vanishing (double-zero index media) does not accumulate any phase retardation. Such a medium is not mathematically interesting but also bears unusual functionalities, such as wave front engineering, cloaking of objects and wave tunnelling. I will report our progresses on realizing double-zero index materials in two and three dimensions for electromagnetic and acoustic waves and discuss their special characteristics.
[04938] Rigidity and Elasticity of Kirigami and Origami Metamaterials
Format : Talk at Waseda University
Author(s) :
Ian Tobasco (University of Illinois Chicago)
Abstract : Kirigami metamaterials combine elasticity and geometry to create unusual bulk deformations. We derive a partial differential equation (PDE) for periodic kirigami, along with a strain-gradient like homogenized energy. Minimizing this energy amongst PDE solutions predicts the kirigami’s deformation, as we demonstrate via experiments and simulations. Time permitting, we present analogous results for origami. A key step in our analysis is a rigidity inequality showing that the metamaterial’s deformation is approximated by local mechanism motions.
[03863] The macroscopic behavior of the Kagome lattice metamaterial
Format : Talk at Waseda University
Author(s) :
Xuenan Li (New York University)
Abstract : Mechanism-based metamaterials are synthetic materials that exhibit microscale buckling in response to mechanical deformation. These artificial materials are like elastic composites, but more degenerate, since they can deform with zero elastic energy. We call such deformations with zero elastic energy mechanisms. My research focuses mainly on a rich example, the Kagome lattice metamaterial. This particular material has a huge variety of mechanisms, which might seem incompatible with having a meaningful macroscopic energy at first sight. In this talk, I will discuss the large-scale behavior of the kagome lattice metamaterial as a nonlinear homogenization problem and present our analysis of the well-defined macroscopic energy on this highly degenerate metamaterial. Our macroscopic theory reveals that compressive conformal maps achieve zero effective energy. I will also discuss the adequacy of our macroscopic theory with various numerical experiments. The theory is joint work with Robert Kohn, and the numerical results are joint work with Katia Bertoldi and Bolei Deng.
[04196] Rayleigh waves in 2D extremal materials
Format : Online Talk on Zoom
Author(s) :
Gengkai Hu (School of Aerospace Engineering, Beijing Institute of Technology)
Yu Wei (School of Aerospace Engineering, Beijing Institute of Technology)
Abstract : Rayleigh waves are guaranteed in Cauchy materials with positive definite elasticity tensors. However it's proved that 2D extremal materials with Rank-deficient elasticity tensors cannot support Rayleigh waves, they can appear if the second gradient effect is considered. Microstructural models corresponding to the examined continuum models of Cauchy and second gradient elasticity are constructed, both continuum and discrete models agree very well in terms of the velocity and ellipticity for the predicted Rayleigh waves.
[04081] Large-scale metasurface design with fast direct solvers
Format : Talk at Waseda University
Author(s) :
Owen Miller (Yale University)
Abstract : Metasurfaces offer nanophotonic performance for centimeter-scale optics applications. Yet simulating such large structures is beyond current simulation capabilities. We demonstrate a 2D “fast direct” integral-equation solver that can simulate and design a high-efficiency, high-numerical-aperture metalens that is 20,000 wavelengths in diameter. For a visible wavelength of 500nm, this corresponds to a design diameter of 1cm, achieved with full simulations of Maxwell’s equations.
[04889] Relaxation of variational principles for bounding the effective operators of composites
Format : Talk at Waseda University
Author(s) :
Aaron Welters (Florida Institute of Technology)
Abstract : An approach to the theory of composites is presented that allows a relaxation of the direct and dual minimization principles used to bound effective operators. This is based on representing the effective operator as the Schur complement of a positive semidefinite operator on a Hilbert space having a Hodge decomposition. We show the theory also applies in electric circuit theory for the Dirichlet-to-Neumann map and for the classical effective conductivity on a finite linear graph.
[04890] Bessmertnyĭ realizations of effective tensors for metamaterial synthesis: conjectures and counterexamples
Format : Talk at Waseda University
Author(s) :
Anthony Dean Stefan (Florida Institute of Technology)
Abstract : Effective tensors of isotropic n-phase composites are known to be homogeneous multivariate Herglotz functions. Recently, M. Bessmertnyĭ claimed to characterize any such rational function as being in the Bessmertnyĭ class because each partial Wronskian associated with it has a polynomial sum-of-squares representation. We disprove this claim by providing a counterexample derived from the basis generating polynomial for the Vámos matroid and give a conjecture on the realizability of effective tensors. Joint work with Aaron Welters.
[05015] Time domain analysis of resonant plasmonic nano-particles
Format : Talk at Waseda University
Author(s) :
Pierre Millien (CNRS)
Alice L. Vanel (CERN)
Lorenzo Baldassari (RICE)
Habib Ammari (ETHZ)
Abstract : We study the possible expansion of the electromagnetic field scattered by a strictly convex metallic nanoparticle with dispersive material parameters placed in a homogeneous medium in a low-frequency regime as a sum of modes oscillating at complex frequencies (diverging at infinity), known in the physics literature as the quasi-normal modes expansion. We show that such an expansion is valid in the static regime and that we can approximate the electric field with a finite number of modes. We then use perturbative spectral theory to show the existence, in a certain regime, of plasmonic resonances as poles of the resolvent for Maxwell's equations with non-zero frequency. We show that, in the time domain, the electric field can be written as a sum of modes oscillating at complex frequencies. We introduce renormalised quantities that do not diverge exponentially at infinity.
[05234] Active exterior thermal cloaking
Format : Talk at Waseda University
Author(s) :
Trent DeGiovanni (University of Utah)
Fernando Guevara Vasquez (University of Utah)
Maxence Cassier (CNRS, Institut Fresnel)
Sébastien Guenneau (Imperial College London)
Abstract : We consider the problem of concealing an object, in the presence of a known probing fielding, from the perspective of thermal measurements. This is achieved using specially designed sources. Such a cloak can be constructed by using the Green identities; however, this requires a continuous strip of sources that encloses the object. In this talk, we demonstrate an alternative approach to this cloaking problem that uses only a few sources.
[04994] Imaging conductivity with thermal noise induced currents
Format : Online Talk on Zoom
Author(s) :
Fernando Guevara Vasquez (University of Utah)
Trent DeGiovanni (University of Utah)
Abstract : Thermal fluctuations of charge carriers in a conductive body create small but detectable currents. We show that the variance of such currents can be used to image the conductivity of a body, including for complex conductivities. This is done by relating the stochastic problem to a deterministic inverse problem that is close to one arising in acousto-electric tomography.
