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SUMMARY:High-frequency homogenization for imperfect interfaces and dispers
 ive media - Marie Touboul (Imperial College London)
DTSTART:20230324T113000Z
DTEND:20230324T120000Z
UID:TALK195742@talks.cam.ac.uk
DESCRIPTION:Classically\, dynamic homogenization is understood as a low-fr
 equency approximation to wave propagation in heterogeneous media. A partic
 ularly successful approach is the two-scale asymptotic expansion method an
 d the notion of slow and fast variable1. The idea of high-frequency homoge
 nization (HFH)\, introduced in2\, is to use similar asymptotic methods to 
 approximate how the dispersion relation and the media behave near a given 
 point that satisfies the dispersion relation. This talk will be divided in
  two parts : in the first one\, we extend the HFH method to the case of ma
 terials where imperfect contacts occur \; in the second one\, we extend HF
 H to the case of dispersive media where the properties of the material dep
 end on the frequency.\nFirstly\, we consider an array of imperfect interfa
 ces of the spring-mass type. HFH is applied to find an approximation of th
 e wavefield and the dispersion diagram near a given point of the dispersio
 n diagram. Especially\, we consider edges of the Brillouin zone and three 
 cases : when the eigenvalue solution of the dispersion relation is a singl
 e eigenvalue\, when it is a double eigenvalue (the case of Dirac points wh
 ere two branches intersect)\, or when eigenvalues are single but nearby. F
 or these three cases\, an approximation of the dispersion relation and an 
 effective equation for the zeroth order wavefield are found. Furthermore\,
  for the same example\, the nearby case is observed to give much longer li
 ved approximations than the single one as we get further from the edge.&nb
 sp\; This first part is a joint work with Raphael Assier\, Bruno Lombard a
 nd C&eacute\;dric Bellis.\nSecondly\, we consider a doubly periodic struct
 ure on a square lattice. The physical properties (permittivity or permeabi
 lity in electromagnetism\, or effective elastic parameters arising from hi
 gh-contrasts in elasticity\, for example) may depend in some constituents 
 of the unit cell on the frequency\, following a Lorentz (or Drude) model. 
 Far from the accumulation points that may occur at resonance in these case
 s\, we get the approximation of the dispersion relation and an effective e
 quation for the wavefield.&nbsp\;This on-going work is a joint work with B
 enjamin Vial\, Raphael Assier\, S&eacute\;bastien Guenneau and Richard Cra
 ster.&nbsp\;\n1 A. Bensoussan\, J.-L. Lions\, G. Papanicolaou\, Asymptotic
  Analysis for Periodic Structures\, AMS Chelsea Publishing (2011).\n2 R. V
 . Craster\, J. Kaplunov\, A. V. Pichugin. Proceedings of the Royal Society
  A: Mathematical\, Physical and Engineering Sciences\,&nbsp\;The Royal Soc
 iety\,&nbsp\;466\, 2341-2362 (2010).
LOCATION:Seminar Room 1\, Newton Institute
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