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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:High-frequency homogenization for imperfect interf
aces and dispersive media - Marie Touboul (Imperia
l College London)
DTSTART;TZID=Europe/London:20230324T113000
DTEND;TZID=Europe/London:20230324T120000
UID:TALK195742AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/195742
DESCRIPTION:Classically\, dynamic homogenization is understood
as a low-frequency approximation to wave propagat
ion in heterogeneous media. A particularly success
ful approach is the two-scale asymptotic expansion
method and the notion of slow and fast variable1.
The idea of high-frequency homogenization (HFH)\,
introduced in2\, is to use similar asymptotic met
hods to approximate how the dispersion relation an
d the media behave near a given point that satisfi
es the dispersion relation. This talk will be divi
ded in two parts : in the first one\, we extend th
e HFH method to the case of materials where imperf
ect contacts occur \; in the second one\, we exten
d HFH to the case of dispersive media where the pr
operties of the material depend on the frequency.\
nFirstly\, we consider an array of imperfect inter
faces of the spring-mass type. HFH is applied to f
ind an approximation of the wavefield and the disp
ersion diagram near a given point of the dispersio
n diagram. Especially\, we consider edges of the B
rillouin zone and three cases : when the eigenvalu
e solution of the dispersion relation is a single
eigenvalue\, when it is a double eigenvalue (the c
ase of Dirac points where two branches intersect)\
, or when eigenvalues are single but nearby. For t
hese three cases\, an approximation of the dispers
ion relation and an effective equation for the zer
oth order wavefield are found. Furthermore\, for t
he same example\, the nearby case is observed to g
ive much longer lived approximations than the sing
le one as we get further from the edge. \; Thi
s first part is a joint work with Raphael Assier\,
Bruno Lombard and Cé\;dric Bellis.\nSecondl
y\, we consider a doubly periodic structure on a s
quare lattice. The physical properties (permittivi
ty or permeability in electromagnetism\, or effect
ive elastic parameters arising from high-contrasts
in elasticity\, for example) may depend in some c
onstituents of the unit cell on the frequency\, fo
llowing a Lorentz (or Drude) model. Far from the a
ccumulation points that may occur at resonance in
these cases\, we get the approximation of the disp
ersion relation and an effective equation for the
wavefield. \;This on-going work is a joint wor
k with Benjamin Vial\, Raphael Assier\, Sé\;
bastien Guenneau and Richard Craster. \;\n1 A.
Bensoussan\, J.-L. Lions\, G. Papanicolaou\, Asym
ptotic Analysis for Periodic Structures\, AMS Chel
sea Publishing (2011).\n2 R. V. Craster\, J. Kaplu
nov\, A. V. Pichugin. Proceedings of the Royal Soc
iety A: Mathematical\, Physical and Engineering Sc
iences\, \;The Royal Society\, \;466\, 234
1-2362 (2010).
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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