numerical methods for time harmonic locally perturbed ...€¦ · julien coatleven : analyse...

Numerical methods for time harmonic locally perturbed periodic media

Introduction

Sonia Fliss(Ecole Nationale Supérieure de Techniques Avancées)

Intensiv Programme / Summer School „Periodic Structures in Applied Mathematics“Göttingen, August 18 – 31, 2013

This project has been funded with support from the European Commission. This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Grant Agreement Reference Number: D-2012-ERA/MOBIP-3-29749-1-6

Exact boundary conditions, Wave propagation and Periodic mediaSonia Fliss 1

Sonia Fliss

POEMS (UMR 7231 CNRS-INRIA-ENSTA)

Mainly based on joint works with Julien Coatleven, Patrick Joly and Jing-Rebecca Li

Numerical methods for time harmonic locally perturbed periodic media -

Introduction

1

Main objectives of the course

Sonia Fliss Exact boundary conditions, Wave propagation and Periodic media 2

Give some elementary introduction to the theory of source problems fortime harmonic wave propagation in locally perturbed periodic mediaEnlighten the link with the spectral theory of periodic differential operators

Sonia Fliss : Etude mathématique et numérique de la propagation des ondes dans des milieux périodiques localement perturbés (2009)

Julien Coatleven : Analyse mathématique et numérique de quelques problèmes d’ondes en milieux périodiques (2011)

Sonia Fliss, Eric Cassan and Damien Bernier: New approach to describe light refraction at the surface of a photonic crystal, JOSA B, vol. 27, pp. 1492-1503, 2010

Sonia Fliss, Patrick Joly and Jing Rebecca Li: Exact boundary conditions for periodic waveguides containing a local perturbation, Commun. Comput. Phys., vol.1(6), pp.945-973, 2006.

Sonia Fliss and Patrick Joly : Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media, APNUM, vol. 59(9), pp. 2155-2178, 2009

Sonia Fliss and Patrick Joly : Wave propagation in locally perturbed periodic media (case with absorption): Numerical aspects, JCP, vol. 231(4), pp. 1244-1271, 2012

Julien Coatleven and Patrick Joly : Operator Factorization for Multiple-Scattering Problems and an Application to Periodic Media, Commun. Comput. Phys., vol. 11(2), pp303-318, 2012

Main references of this work

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http://uma.ensta-paristech.fr/publication.php?id=1029




















Motivation

Our initial objective was to develop efficient numerical methods for the computation of the wave propagation in periodic structures with localised periodicity defects when

the period of the medium is small regarding to its size

the wavelength is comparable to the period of the medium

➡ The medium is supposed infinite

➡ The periodicity has to be considered as such

Sonia Fliss Exact boundary conditions, Wave propagation and Periodic media

Basic principle

Take advantage of the periodicity of the reference unperturbed medium to restrict the numerical computations to a neighborhood of the perturbation.

33

This issue for homogeneous medium...

4

This topic has been widely studied in the mathematical literature in the case where the exterior medium is homogeneous.

First approach : find a transparent or quasi transparent artificial boundary condition

�i

The exact DtN operator approach : Abboud (1993)

Integral representation technique : Jonhson & Nédélec (1980), Levillain (1990), Jami & Lenoir (1977), Hazard et Lenoir (1996)

Local radiation condition : Engquist & Majda (1979), Bayliss & Turkel (1980), Hagstrom (1999)

Exact boundary conditions, Wave propagation and Periodic mediaSonia Fliss4


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�i

Application to periodic media ?They are based on analytical techniques (separation of variables, Green’s functions, asymptotic expansions, ...) which use the homogeneous nature of the exterior medium.


The PML technique : Berenger (1994)

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Second approach : surround the computational domain by a “special” absorbing layer to absorb the outgoing waves




6




Application to periodic media ?They are based on the characterization of the outgoing waves in homogeneous medium.In a periodic medium, outgoing waves interact with the heterogeneities of the exterior medium up to infinity.




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The pole condition method : Hohage-Schmidt-Zschiedrich (2003a,b)



This issue for heterogeneous medium...

7

There are very few works in the same spirit in the general case.

Many work in the case of the diffraction of a plane wave by periodic gratings (the medium is periodic in only one direction and homogeneous at infinity in the other direction)

Incident field

The DtN approach : Abboud (1993), Bonnet & Tillequin (2000, 2001), Tausch & Butler (2000)

The integral equation approach : Meier, Arens, Chandler Wilde, Kirsch (2000)

The PML approach : Karpin Haijun Wu



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Exact boundary conditions, Wave propagation and Periodic mediaSonia Fliss

Few works in the case of bounded periodic media

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8



The DtN approach : Yuan&Lu (2006&2007), Ehrhardt&Zheng (2008), Ehrhardt et al. (2006)

Exact boundary conditions, Wave propagation and Periodic mediaSonia Fliss

Few works in the case of bounded periodic media

Question

How can restrict the computation around the perturbation when the surrounding media is periodic (and the periodicity is considered as such) and infinite?

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The basic mathematical model

Scalar wave equation with a varying coefficient


when the function is seen as a local perturbation of a periodic function ⇢(x)

There exists a periodic function ⇢per(x) such that supp (⇢� ⇢per) is small

⇢(x)@2U

@t2��U = F (x, t),

Given a frequency and assuming , it is natural to look for a solution of the form

F (x, t) = f(x, t) e�i!t

U(x, t) = u(x) e�i!t

! > 0

The limiting amplitude principle holds when

limt!+1

ei!t U(x, t) = u(x)

which leads to the Helmholtz equation

��u� ⇢(x)!2 u = f(x), (not always well-posed)

(unbounded domains, non uniform in space)

9

The basic mathematical model


We shall consider the related dissipative model ( )

⇢(x)⇣@2U"

@t2+ "

@U"

@t

⌘��U" = F (x, t)

It is well known that (time domain limiting absorption)

U"(·, t) �! U(·, t) (" �! 0)

" � 0

��u" � ⇢(x) (!2 + i!")u" = f(x),

This model leads to the dissipative Helmholtz equation

It is well known that (dissipative limiting amplitude)

limt!+1

ei!t U"(x, t) = u"(x)

The limiting absorption principle holds when

lim"!0

u"(x) = u(x) (unbounded domains, non uniform in space)

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delicate

easy

Limiting amplitude and limiting absorption


u(x) e�i!tu"(x) e�i!t

U"(x, t) U(x, t)

limiting absorption" �! 0

limiting amplitudet �! +1

In this lecture, we will study mainly the time harmonic wave equation (Helmholtz equation) in the dissipative case.For the 1D and the quasi-1D models, we will study the limiting absorption principle in order to define the physical solution of the Helmholtz equation in the case without absorption.

In the conclusion, I will give some ideas for the other problems.

with absorption

withoutabsorption

Time domain

Frequency domain

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�

Problem 1 : the 1D problem on the real lineScalar wave Problem


Time harmonic with absorption

�d

2u"

dx

2� ⇢(x)(!2 + ı"!)u" = f(x), in R

1D Case

�i = [a�, a+]

x 2 R

Supp(f) � �i

Supp(�� per) ⇥ �i

⇢per(x+ 1) = ⇢per(x)a� a+

⌦i

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Problem 2 : the 2D waveguideScalar wave Problem



�⇥u� � �(x) (⇥2 + ı⇤⇥)u� = f(x)

Waveguide case

x � � = R� [0, 1]

�i = [a�, a+]� [0, 1]

Supp(f) � �i


⌦i

13

Scalar wave Problem



�⇥u� � �(x) (⇥2 + ı⇤⇥)u� = f(x)

2D case - simple scattering

x � R2

�i = [a�, a+]2

Supp(f) � �i


Problem 2 : the 2D plane with a local defect

⌦i

14

2D case - Multiple scattering

Scalar wave Problem



�⇥u� � �(x) (⇥2 + ı⇤⇥)u� = f(x)

x � R2

Supp(f) � �i


�i =N�

j=1

Oj

Problem 4 : the 2D plane with several defects

⌦i

15

�

Presentation of the problems


�� +

�i

a� a+

1D Case Waveguide case 2D case Simple scattering

2D caseMultiple scattering

Four problems of increasing difficulty

The half-space case

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Principle of the DtN approachScalar wave Problem



�⇥u� � �(x) (⇥2 + ı⇤⇥)u� = f(x)

�� +

�i

a� a+

�i

Principle of the DtN approach

Restrict the numerical computation around the defect

➡ Find exact or transparent boundary conditions

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�� +

a� a+

T "1. Major difficulty : characterize and compute the transparent DtN operator :

ui" +B" @nu

i" = 0 on ⌃

i= @⌦i

@nui" + T " u

i" = 0 on ⌃

i= @⌦i

{ �@nu

i" + i!ui

"

�+ C"

�@nu

i" � i!ui

"

�= 0 on ⌃

i= @⌦i

Alternative formulations

C" =�T " + i!I

��1�T " � i!I

�B" = T�1

"

Simplerto present

Better adapted to limiting absorption

�i

(Pi�)

��ui" � ⇢(x)(!2 + ı"!)ui

" = f(x), in ⌦i

@nui" + T " u

i" = 0 on ⌃i = @⌦i

�i

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�� +

�i

a� a+

2. Requirement : have a post-processing algorithm to reconstruct the solution in the exterior domain

T "1. Major difficulty : characterize and compute the transparent DtN operator :

(Pi�)

��ui" � ⇢(x)(!2 + ı"!)ui

" = f(x), in ⌦i

@nui" + T " u

i" = 0 on ⌃i = @⌦i

�i

19

�i




�� +

�i

a� a+

(Pi�)

�⇥ui� � �(x)(⇥2 + ı⇤⇥)ui

� = f(x), in �i

@nui" + T " u

i" = 0 on ⌃

i= @⌦i

From a domain decomposition approach, it is easy to have a general abstract definition ofthe transparent DtN operator T "

How to solve cheaplythis exterior domain ?

Given ' : @⌦i �! C T " ' := � @nu"(')|@⌦i

�⇥ue� � �per(x)(⇥2 + ı⇤⇥)ue

� = 0, in �e

(Pe� )

ue� = �, on �i

H1 (Pe� )is the unique solution of the exterior problemue

✏(')where

�e�e

�e �e

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Sonia Fliss

POEMS (UMR 7231 CNRS-INRIA-ENSTA)

Mainly based on joint works with Julien Coatleven, Patrick Joly and Jing-Rebecca Li

Numerical methods for time harmonic locally perturbed periodic media -

Introduction

21

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