Analysisofamultiscalemethodforhighlyhetero- geneous ... · DFG-project: Wave propagation in periodic structures and negative refraction mechanisms I Veselago 1968: negative refractive

$Page 1: Analysisofamultiscalemethodforhighlyhetero- geneous ... · DFG-project: Wave propagation in periodic structures and negative refraction mechanisms I Veselago 1968: negative refractive$
Analysis of a multiscale method for highly hetero-geneous Helmholtz problemsWorkshop “Waves in periodic media and metamaterials”, Cargèse 2016

M. Ohlberger B. Verfürth November 23-25, 2016living knowledgeWWU Münster

WESTFÄLISCHEWILHELMS-UNIVERSITÄTMÜNSTER

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WESTFÄLISCHEWILHELMS-UNIVERSITÄTMÜNSTER Multiscale method for Helmholtz problems 2 /28

Motivation/Goal

DFG-project: Wave propagation in periodic structures andnegative refraction mechanisms

I Veselago 1968: negativerefractive index possible inmaterials with negative " and µ

I Since 2000, experimentaldesign of such metamaterials,mostly relying on periodicstructures Figure: Negative refraction1

We study artificial magnetism, i.e. occurrence of µeff 6= 1 innon-magnetic composites

1from Wikimedia Commons

Barbara Verfürth ([email protected])

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OverviewHeterogeneous Problem

Two-scale equation Homogenized eq.

Homogenization

Discrete two-scale eq. HMM

Discretization

Two-scale LOD

Reduction of pollution effect


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Homogenization


Discretization

Two-scale LOD



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Homogenization


Discretization

Two-scale LOD



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Problem setting

Homogenization

Heterogeneous Multiscale Method

Numerical experiment

Localized Orthogonal Decomposition


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Heterogeneous ProblemI two-dimensional geometry, magnetic field H = (0,0, u) fulfills

Helmholtz equation

�r · ("�1r ru)� k2u = 0

I scattering problem: scatterer ⌦ surrounded by vacuum ("�1r = 1

outside) and illuminated by incoming wave uiI boundary condition: G �� ⌦ sufficiently large and

ru · n� iku = g := rui · n� ikuiI �-periodic structure with high contrast inside ⌦

⌦

G

D

�2"�1i

Y⇤

"�1e


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Helmholtz equation

�r · ("�1r ru)� k2u = 0


outside) and illuminated by incoming wave ui

I boundary condition: G �� ⌦ sufficiently large andru · n� iku = g := rui · n� ikui

I �-periodic structure with high contrast inside ⌦

⌦

G

D

�2"�1i

Y⇤

"�1e


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Helmholtz equation

�r · ("�1r ru)� k2u = 0



ru · n� iku = g := rui · n� ikui

I �-periodic structure with high contrast inside ⌦

⌦

G

D

�2"�1i

Y⇤

"�1e


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Helmholtz equation

�r · ("�1r ru)� k2u = 0



ru · n� iku = g := rui · n� ikuiI �-periodic structure with high contrast inside ⌦

⌦

G

D

�2"�1i

Y⇤

"�1e


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Heterogeneous ProblemFind u� 2 H1(G) such that

(a�ru�,r )G � k2(u�, )G � ik(u�, )@G = (g, )@G 8 2 H1(G)

periodic inclusions with high permittivity

I [Bouchitté, Felbacq 2004]

I [Bourel, Bouchitté, Felbacq 2009/2015], [Cherednichenko, Cooper 2015]

I Bouchitté, Lamacz, Schweizer 2010–2016

⌦

G

D

�2"�1i

Y⇤

"�1e


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Problem setting

Homogenization





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Two-scale equation

Theorem (cf. [Allaire 1992])

For � ! 0, (two-scale) limits of u� andp|a�|ru� characterized by triple

u := (u, u1, u2) 2 H := H1(G)⇥ L2(⌦;H1],0(Y

⇤))⇥ L2(⌦;H10(D)),

theunique solution to the two-scale equation

B(u, ) = (g, )@G 8 2 H

with two-scale sesquilinear form

B(v, ) := ("�1e (rv +ryv1),r +ry 1)⌦⇥Y⇤ + ("�1

i ryv2,ry 2)⌦⇥D

� k2(v + �Dv2, + �D 2)⌦⇥Y

+ (rv,r )G\⌦ � k2(v, )G\⌦ � ik(v, )@G


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Two-scale equation

Theorem (cf. [Allaire 1992])

For � ! 0, (two-scale) limits of u� andp|a�|ru� characterized by triple

u := (u, u1, u2) 2 H := H1(G)⇥ L2(⌦;H1],0(Y

⇤))⇥ L2(⌦;H10(D)), the

unique solution to the two-scale equation

B(u, ) = (g, )@G 8 2 H

with two-scale sesquilinear form

B(v, ) := ("�1e (rv +ryv1),r +ry 1)⌦⇥Y⇤ + ("�1

i ryv2,ry 2)⌦⇥D

� k2(v + �Dv2, + �D 2)⌦⇥Y

+ (rv,r )G\⌦ � k2(v, )G\⌦ � ik(v, )@G


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Homogenized equation

Theorem (cf. [Bouchitté, Felbacq 2004])

u from the two-scale limit is the unique solution of

Beff(u, ) = (g, )@G 8 2 H1(G),

Beff(u, ) := (aeffru,r )G � k2(µeff u, )G � ik(u, )@G

with effective parameters aeff and µeff computed from cell problems.

I aeff classical elliptic effective matrix (on Y⇤)I µeff :=

RY 1+ k2�Dw dy,

I where w 2 H10(D) solves

("�1i ryw,ry 2)D � k2(w, 2)D = (1, 2)D 8 2 2 H1

0(D)


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Homogenized equation

Theorem (cf. [Bouchitté, Felbacq 2004])

u from the two-scale limit is the unique solution of

Beff(u, ) = (g, )@G 8 2 H1(G),

Beff(u, ) := (aeffru,r )G � k2(µeff u, )G � ik(u, )@G

with effective parameters aeff and µeff computed from cell problems.

I aeff classical elliptic effective matrix (on Y⇤)I µeff :=

RY 1+ k2�Dw dy,

I where w 2 H10(D) solves

("�1i ryw,ry 2)D � k2(w, 2)D = (1, 2)D 8 2 2 H1

0(D)


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Effective permeability

µeff – variant I

µeff :=RY 1+ k2�Dw dy

µeff – variant II (cf. [Bouchitté, Felbacq])µeff = 1+

Pn

k2"i�n�k2"i

�RD �n

�2

with (�n,�n) eigenpair of Dirichlet Laplacian on D

I µeff k-dependent, real part can have positive or negative sign!I upper bound |µeff| CI lower bound Im(µeff) � Ck�2


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µeff – variant I

µeff :=RY 1+ k2�Dw dy

µeff – variant II (cf. [Bouchitté, Felbacq])µeff = 1+

Pn

k2"i�n�k2"i

�RD �n

�2

with (�n,�n) eigenpair of Dirichlet Laplacian on D



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20 30 40 50 60

�20

�10

0

10

20

Wavenumber k

Realandimaginarypartof

µeff

Re(µeff)Im(µeff)



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Stability of the effective equation

Theorem ([Ohlberger, V. 2016])

If G, ⌦ are star-shaped w.r.t. a ball and aeff|G\⌦ � aeff|⌦ neg. semi-definite,we have

krukG + kkukG . k3kfkG + k3/2kgkH1/2(@G).


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I with Im(µeff ) � C independent of k: stability

krukG + kkukG . kkfkG + k1/2kgkH1/2(@G).

I inf-sup constant for B and Beff like k�4

I higher regularity u 2 H1+spw (G) for s > 1/2 with

kukH1+spw

C(kkuk1,k + kfkG + kgkH1/2(@G))


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Idea of proofI stability for smooth a

I imaginary part of Beff(u, u): estimate for kuk⌦, kuk@GI real part of Beff(u, u): estimate for krukGI real part of Beff(u, x ·ru) (Rellich-type estimates): estimate for kukG\⌦

I smoothing of aeff: higher regularity of u needed


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Problem setting

Homogenization





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Numerical solution of multiscale problems

Direct solution infeasible) need for multiscale methods!

Examples:I Multiscale Finite Element Method (Efendiev, Hou)I Variational Multiscale Method (Hughes et.al., Larson, Målqvist)I Localized Orthogonal Decomposition (Målqvist, Peterseim)I Heterogeneous Multiscale Method


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Idea of the HMM

I FE method on coarse domain ⌦ withmacroscopic sesquilinear form

I local cell problems around eachquadrature point (localreconstructions)

I HMM also direct discretization withquadrature of the two-scale equation

TjY�l

xl

I related approaches: Engquist, E, Abdulle, Gloria, Grote, Stohrer, Ciarlet,Rungborg, Holst, Ming, Zhang, . . .


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Discretization of the two-scale equation

I triangulations TH of G and Th of YI discrete triple space VH,h ⇢ H consisting of linear Lagrange

elements: V1H ⇢ H1(G), eV1

h(Y⇤) ⇢ H1

],0(Y⇤) and V1

h(D) ⇢ H10(D)

I uH,h := (uH, uh,1, uh,2) 2 VH,h discrete two-scale solution to

B(uH,h, H,h) = (g, H)@G 8 H,h = ( H, h,1, h,2) 2 VH,h

I uh,1 = Kh,1(uH) and uh,2 = Kh,2(uH), where correctors Kh,1 and Kh,2solve discretized cell problems

I cell problems can be transformed back to �-scaled unit squaresaround macroscopic quadrature points) formulation in fashion ofthe Heterogeneous Multiscale Method


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h(Y⇤) ⇢ H1

],0(Y⇤) and V1

h(D) ⇢ H10(D)






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h(Y⇤) ⇢ H1

],0(Y⇤) and V1

h(D) ⇢ H10(D)






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Numerical analysis


Assuming sufficient regularity of the analytical two-scale solution, it holds:If k and H, h are coupled by

k5(H+ h) . 1,

then the discretization is stable

infvH,h2VH,h

sup H,h2VH,h

ReB(vH,h, H,h)

kvH,hke k H,hke& k�4

and quasi-optimal

ku� uH,hke . infvH,h2VH,h

ku� vH,hke . k4(H+ h)kgkH1/2(@G).


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Problem setting

Homogenization





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Setting for the numerical experiment

I scatterer ⌦ = (0.375,0.625)2

I artificial domain G = (0.25,0.75)2

I incoming wave exp(�ikx1) from theright

I inclusion D = (0.25,0.75)2 in theunit square

I "�1e = 10, "�1

i = 10� 0.01iI two situations: k = 38 “normal”

transmission; k = 29 “forbidden”frequency with Re(µeff) < 0

102 103 104

100

101

�0.64

k = 38k = 29

Figure: Energy error between HMMand (homogenized) referencesolution vs. number of grid entities


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k = 38

Figure: real part HMM solution Figure: line plot y = 0.545


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k = 38

Figure: real part “reconstruction” Figure: line plot y = 0.545


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k = 29

Figure: real part HMM solutionFigure: line plot y = 0.545


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k = 29

Figure: real part “reconstruction”Figure: line plot y = 0.545


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Problem setting

Homogenization





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Two-scale Localized Orthogonal DecompositionLOD for Helmholtz: Brown, Gallistl, Peterseim; LOD in general: Målqvist, Peterseim, Henning,Hellman, Morgenstern, Elfverson, Abdulle,. . .

I coarse discretization scales Hc > H and hc > h and stablequasi-interpolation operator (projection) IHc,hc : VH,h ! VHc,hc

I splitting VH,h = VHc,hc �WH,h withWH,h := ker IHc,hc

I subscale correction operator Q1 : VHc,hc ! WH,h defined via

B(w,Q1(vHc,hc)) = B(w, vHc,hc) 8w 2 WH,h.

I modified coarse scale space VHc,hc,1 := (1� Q1)VHc,hcI Find uHc,hc 2 VHc,hc such that

B(uHc,hc , Hc,hc) = (g, Hc)@G 8 Hc,hc 2 VHc,hc,1.

I exponential decay of correctors) truncation of corrector problemsto patches with sizemHc andmhc


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I splitting VH,h = VHc,hc �WH,h withWH,h := ker IHc,hcI subscale correction operator Q1 : VHc,hc ! WH,h defined via


I modified coarse scale space VHc,hc,1 := (1� Q1)VHc,hc

I Find uHc,hc 2 VHc,hc such that




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I splitting VH,h = VHc,hc �WH,h withWH,h := ker IHc,hcI subscale correction operator Q1 : VHc,hc ! WH,h defined via


I modified coarse scale space VHc,hc,1 := (1� Q1)VHc,hcI Find uHc,hc 2 VHc,hc such that




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Numerical Analysis


Under the assumption of a faithful reference solution uH,h 2 VH,h, it holds:If k and Hc, hc, and m are coupled by

k(Hc + hc) . 1, m ⇡ log(k),

the two-scale LOD is stable

infvHc,hc2VHc,hc

sup Hc,hc2VHc,hc,m

B(vHc,hc , Hc,hc)

kvHc,hcke k Hc,hcke& k�4

and quasi-optimal

kuH,h � uHc,hcke . infvHc,hc2VHc,hc

kuH,h � vHc,hcke.


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Summary and OutlookSummary

I homogenization for a Helmholtz problem with highly heterogeneousparameter

I stability of two-scale solution with factor k3

I Heterogeneous Multiscale Method is quasi-optimal if k5(H+ h) smallI Two-scale Localized Orthogonal Decomposition can be used to

reduce the pollution effect) k(Hc + hc) small andm ⇡ log(k)sufficient for quasi-optimality

OutlookI numerically investigate the resolution conditionI 3d case: highly heterogeneous bulk inclusions for a curl-curl-problemI highly heterogeneous parameter and more challenging topologies

(Helmholtz resonators, split rings, wires, etc.)


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ReferencesM. Ohlberger, B. VerfürthAnalysis of multiscale methods for the two-dimensional Helmholtzequation with highly heterogeneous coefficient. Part I.Homogenization and the Heterogeneous Multiscale MethodarXiv num. 1605.03400 arXiv Preprint

M. Ohlberger, B. VerfürthAnalysis of multiscale methods for the two-dimensional Helmholtzequation with highly heterogeneous coefficient. Part II. Two-scaleLocalized Orthogonal DecompositionarXiv num. 1605.03410 arXiv Preprint

Thank you for your attention!


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Analysisofamultiscalemethodforhighlyhetero- geneous ... · DFG-project: Wave propagation in periodic structures and negative refraction mechanisms I Veselago 1968: negative refractive