Continuum
Models Discrete
Systems 13CMDS Investigators Workshop:Snowbird, Utah, USA, May 16, 2015 The
Workshop is sponsored by NSF

The purpose of CMDS13 Investigators Workshop: "At the Frontiers of Computation and Materials" is to bring together local researchers working in mathematics for complex materials, to give the floor to young researchers in the area, and to discuss current trends of the field. The informal and friendly atmosphere made this meeting fruitful for exchange of ideas, methods and results.
Program:
Paricipants9:00  9:50 – Kirill Cherednichenko. Homogenisation of the system of highcontrast Maxwell equations
(guest lecture)9:50  10:10 – Coffee break
10:10  10:40  Patrick Bardsley. Imaging with correlated sourse pairs.
10:40  11:10  Kyle Steffen. Models of porous media flow and their numerical solution.
11:10  11:40  Ornella Mattei. Bounds on the transient response of twocomponent viscoelastic composites
11:40  12:10  Michal Kordy. Hexahedral edge element approximation of the electromagnetic field. Distortion of the elements12:10  12:50 – Discussion: "New trends in Math Methods for materials" (Graeme Milton, Elena Cherkaev, Kenneth Golden, Kirill Cherednichenko, Andrej Cherkaev).
1:00  2:30  Lunch
PhotosPatrick Bardsley
Maxence Cassier Kirill Cherednichenko
Andrej Cherkaev Elena Cherkaev Yekaterina Epshteyn Kenneth Golden Fernando GuevaraVasquez
Davit Harutyunyan Michal Kordy
Ornella Mattei
Graeme Milton Braxton Osting
Kyle Steffen
Andrejs Treibergs
Qing Xia
Abstract: I shall discuss the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities $\epsilon$ have a high degree of contrast between each other. I assume that the ratio between the permittivities of the components with low and high values of $\epsilon$ are of the order $\eta^2,$ where $\eta>0$ is the period of the medium. I determine the asymptotic behaviour of the electromagnetic response of such a medium in the ``homogenisation limit", as $\eta\to 0,$ and derive the limit system of Maxwell equations in ${\mathbb R}^3.$ The results that I shall present extend a number of conclusions of the paper [Zhikov ,V. V., 2004. On the bandgap structure of the spectrum of some divergentform elliptic operators with periodic coefficients, {\it St.\,Petersburg Math.\,J.}] to the case of the full system of Maxwell equations. This is joint work with Shane Cooper (Bath).
Patrick Bardsley: Imaging with correlated sourse pairs.
Abstract: We image scatterers in a homogeneous medium by sending correlated wave signals from two different locations and measuring the intensity of the echoes at a single receiver location. By altering the positions of the source pairs we form a linear system which we solve in the least squares sense to recover full waveform data. We can image with this data using classic techniques such as Kirchhoff migration which gives known resolution estimates. The same source pair strategy can be used when we probe the medium with correlated sources of noise (Gaussian processes) and measure autocorrelations at a single location.
Kyle Steffen: Models
of porous media flow and their numerical solution
Abstract: In this presentation I will discuss ongoing work on two models of porous media flow and their numerical solution. In the first half I will give a brief introduction to fluid flow through sea ice; discuss a random network model for effective fluid transport properties in sea ice from Zhu, et al. (2006); discuss current work on an extension of the model to sea ice with entrained algae; and give some preliminary results. In the second half I will introduce the StokesDarcy problem, a multiphysics model coupling free flow with flow in porous media; discuss the Difference Potentials Method, a framework for the development of highorder numerical methods for wellposed boundary value problems, including interface problems; and discuss progress towards a Difference Potentials Method for the StokesDarcy problem.
Ornella Mattei: Bounds on the transient response of twocomponent viscoelastic composites
Abstract: In the literature several results concerning bounds on the antiplane response of twophase viscoelastic materials have been proposed for time harmonic applied fields but very few results regarding bounds in the time domain for generic applied fields have been derived. In the latter case, the aim is to provide a bound on the averaged stress field for each moment of time, given the time varying averaged strain field, or viceversa. We obtain such bounds using the analytic method, first proposed by Milton (1981) and Bergman (1978) independently, which exploits the analytic properties of the effective viscoelastic parameters of the composite with respect to the viscoelastic parameters of the constituents. The resulting bounds become tighter the more information is incorporated about the composite geometry, such as the volume fractions of the constituents and whether it is isotropic or not.
Michal Kordy: Hexahedral
edge element approximation of the electromagnetic field.
Distortion of the elements
We consider the lowest order edge elements on the hexahedral mesh for approximation of the electric field in the frequency domain. It is known that if the elements are not parallelepipeds, then the convergence of the approximation is lost. Falk Gatto and Monk (2009) suggested adding a family of edge functions to recover the convergence. In the case when the element distortion is caused by dislocation of the vertices in one direction, we propose a slightly smaller family of functions that also recover the convergence. We propose a measure of the element distortion, that allow to automatically select the elements that are distorted significantly and add the shape functions only for those elements in the domain. We present numerical tests, which show that the additional functions indeed allow to recover the convergence and that the measure of element distortion successfully allows to select the elements, where the distortion of the approximation of the field is significantBased on a joint work with Elena Cherkaev and Phil Wannamaker.
Department of Mathematics, University of Utah, 155 S 1400 E ROOM 233, SALT LAKE CITY, UT 841120090 T:+1 801 581 6851, F:+1 801 581 4148 