on numerical methods for the computation of guided waves...

On numerical methods for the computation of guided wavesinvolving porous materialsA travel through the waveguide

B. Nennig*, Y. Aurégan+, W. Bi.+, R. Binois*,#,N. Dauchez#, J.-P. Groby+, E. Perrey-Debain#, L. Xiong+

*Institut supérieur de mécanique de Paris (SUPMECA), Laboratoire Quartz EA 7393,3 rue Fernand Hainaut, 93407 Saint-Ouen, France.

+Laboratoire d’Acoustique de l’Université du Maine, Unité Mixte de Recherche CNRS 6613,Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France

#Sorbonne universités, Université de Technologie de Compiègne, Laboratoire Roberval,UMR CNRS 7337, CS 60319, 60203 Compiègne cedex, France.

[email protected]

SAPEM 2017,December 4-6 2017, Le Mans, France

Introduction

IntroductionWaveguides with constant cross sectionWaveguides with periodic cross section

Conclusion and perspectives

Application fieldsOutlineAssumptions

Disclaimer !Focuses mainly on noise control and acoustic duct silencers

Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 2 / 45




Application fields

Guided waves in porous materials can be found in many noise control applications

(a) Muffler, perforated sheetand porous

(b) Engine intake, several stu-dies to put porous

(c) Splitter silencer, al-ternating layer of air andglasswool

I Provide ideal channels for noise transmission (buildings, aircraft, . . .)





Application fields

Guided waves can be useful to get information about poroelastic material skeleton.

I Direct measurement of quasi Lamb waves

(d) [Boecks, 2005] (e) [Geslain et al., 2017] SLaTCoW method





Outline

Wave decomposition - Analytical

Discretization - Numerical

Homogeneous

ZoomRIGIDWAVEGUIDE

Waveguides with constant cross section

Waveguides with periodic cross section

Homogeneousanisotropic

LayeredLayered

poroelastic

Biot

Periodic inclusions

Layered Periodic inclusions

EF

Pore network

Material param.effects

Exceptionalpoint

Winding Numberalgorithm

Bloch modes

Link withmacro-micro

Link withTortuosity

Link with DPPurely

geometric

Complex sym.orthogonality


Outline

1 IntroductionApplication fieldsOutlineAssumptions

2 Waveguides with constant cross sectionHomogeneous WaveguidesTransverse isotropics waveguidesLayered waveguidesLayered poroelastic waveguides

3 Waveguides with periodic cross sectionPeriodic pore modelingMetaporous

4 Conclusion and perspectivesConclusionsPersepectives in numerical methodes




Assumptions

Frequency domain e−iωt

Infinite duct, only few discussion on matching condition and so onNo flow here (see Brambley Wednesday !)Rigid waveguideWeak modal density, for high modal density [Bi, 2014 ; Chan, 2017] (seePerrey-Debain Wednesday)


Waveguides with constant cross section

Homogeneous Waveguides

Finite cross section

rigid wall

infiniteEF



Homogeneous WaveguidesTransverse isotropics waveguidesLayered waveguidesLayered poroelastic waveguides

Homogeneous Waveguides - Problem statement


rigid wall

infiniteEF

For a an infinite duct with invariant cross section, the pressure field can be written asp(x) = φ(x⊥)eiβx , where, β is the axial wavenumberIf the porous material fulfills Helmholtz eq., with the wavenumber kp

∆p + k2p p = 0, PDE inside the waveguide

∆⊥φ+ (k2p − β2)φ = 0,

∆⊥φ+ α2φ = 0, Where, α is the tranverse wavenumber∇⊥φ · n = 0, Boundary condition.

φ and α fulfill an eigenvalue problem. Here α are real and positive. The axialwavenumber βn =

√k2

p − α2n is complex, when kp ∈ C.

I The problem is purely geometricI Cut-off freq. in real case





Homogeneous Waveguides - Resolution

Wave decompositionIf the cross section is separable coordinate system for the wave equation, thewave propagation can solved explicitly. ex 1D : φ(y) = Aeiαy + Be−iαy .The BC (rigid) yield to a small size NL eigenvalue problem in α[

1 1eiαh e−iαh

]︸︷︷︸

M

(AB

)=(00

).

To solve it, generally we look for the root of the determinant

detM(α) = 0.

Newton-Raphson method, Muller’s method, the Secant method or simplexmethod : iterative algorithms requiering initial approximations for the zerosOther method based on Argument Principle (see later [Nennig,2010 ;Kravanja,2000]), without initial guess





Homogeneous Waveguides - Resolution (cont.)

Numerical methodsFinite element method, the discretization of the weak formulation yields to a realsymmetric generalized eigenvalue problem

(K− α2M)φ = 0

leading to orthogonal eigenvectors and positive eigenvalues.Finite difference, similar resultsWave finite element [Mace, 2005 ; Serra, 2015]Projection on orthogonal basis like Chebyshev spatial discretization technique[Chan, 2017], or the multimodal method [Bi, 2007] real symmetric matrix du toweak formulation.

I All first eigenvalue are obtained, without initial guess.I Big size problem, sparse matrix





Homogeneous Waveguides - orthogonality

The solution φn and αn are orthogonal. If we take 2 modes

∆⊥φn + α2nφn = 0

∆⊥φm + α2mφm = 0

+rigid BC

After multiplying by φ∗m and integrating by part twice and using the BC, we get

(α2n − α2

m)∫

Sφ∗mφn dS︸︷︷︸

L2−inner product

= 0.

This, the orthogonality relation reads∫Sφ∗mφn dS = Λnδmn.

I Changing boundary condition may change everything (ex : impedance)


Transverse isotropics waveguides


rigid walls

infiniteEF




Transverse isotropics waveguides


rigid walls

infiniteEF

plane of isotropy

stiff directionex

eyez

Transverse direction

isotropy axis

soft direction

I

T

Consider a 2D rigid-walled waveguide of height h filled with a TI fluid [Nennig et al.,submitted JASA].The wave propagation is described by [De Hoop, 1995 ; Torrent, 2008 ; Norris, 2009]

∇ · (τ∇p) +ω2

Keqp = 0,

where τ = ρ−1 is the inevrse of the density tensor.

We are looking for a guided mode propagating along the ex direction, having theexponential form p = φ(y)eiβx where

φ(y) = aeiα1y + beiα2y .





Transverse isotropics waveguides (cont.)

After insertion into the wave equation, transverse wavenumbers α1 and α2 must satisfy

τyyα2i + 2βτxyαi + τxxβ

2 −ω2

Keq= 0, with i = 1, 2.

Applying rigid conditions at the wall, i.e. uw · ey = 0, leads to a boundary condition ofmixed type and non-trivial solutions exist if

sin(

h√

∆2τyy

)= 0

where∆ = 4τyy

ω2

Keq+ 4β2(τ2xy − τxxτyy )

is the discriminant of the quadratic equation.





Transverse isotropics waveguides (cont.)

Propagation constant for guided modes are then found explicitly as

βn = ±τyy√τT τI

√ω2

τyy Keq−(nπ

h

)2

I Right and left propagating modes are identicalI The fundamental mode, n = 0, is also function of the transverse coordinate y

except when Θ = 0 or π/2I Solution depends on the fluid properties, not only on the WG shape


Layered waveguides


rigid wall

infiniteEF




Layered waveguides

HVAC Silencers are most often made of parallel baffles of porous material (rockwool)well described by equivalent fluid model

Airway

Porous material

Elementary cell

I Weak attenuation at low frequency [63 Hz - 500 Hz].I For this frequency range fundamental mode dominatesI Fundamental mode range depends on both the duct height h and of the number ofbaffle N : n = ninc + 2qN. [Mechel, 1990]

Refs : [Tam et al., 1991 ; Aurégan et al., 2001 ; Kirby et al., 2005 ; Binois et al., 2015 ;Nennig et al., 2015]





Layered waveguides


rigid wall

infiniteEF

In layered waveguide, each layer i = 1, I, satisfiesThe Hemlholtz equation, with different materials properties like density ρi ,celerity ci and wavenumber ki

∆p + k2i p = 0

Continuity of the pressure at each interfacesContinuity of the velocity ie

[1ρi∇⊥p · n

]= 0 at each interfaces

Applying wave decomposition and continuity condition yields

f (β) =I∑

i=1

αiρi

tanαi hi

where αi =√

k2i − β2. This equation must solved in β, the solution depends on

media properties.Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 16 / 45




Layered waveguidesThe associated weak formulation, is obtained by multiplying the wave equation (12)by a test function ψ and by integrating over the porous media occupying an arbitrarydomain Ω of boundary S

I∑i=1

1ρi

(−∫

Ω∇ψ · ∇φdΩ +

(k2

i − β2)∫

ΩψφdΩ

)= −

I∑i=1

1ρi

∫Sψ∂φ

∂ndS︸︷︷︸

=0

.

All boundary term vanished due rigid duct and transverse velocity continuity.This yields complex symetric eigenvalue problemK1 + k2

1M1 . . . 0...

⊕ ...0 . . . KI + k2

I MI

− β2M1 . . . 0

...⊕ ...

0 . . . MI

ψ1...ψI

= 0

I Complex symmetric eigenvalue problem ⊕stand for FEM assembly





Layered waveguides

Complex symmetric problems

Properties

Loss of orthogonality for L2 scalar product∫

S φ∗nφm dS = 0

Left and right eigenvectors are still orthogonal (adjoint), bi-orthogonality

I For complex symmetric problem (excepted for double root), a new orthogonalityrelation can be obtained without conjugate (not scalar product !)∫

Sφmφn dS = Λnδmn.

eg : Impedance lining, layered materials, . . .Refs : [Lawrie, 1999, 2006 ; Redon, 2011 ; Moiseyev, 2011]

I The EP/double root case will be discussed later





Layered waveguides - Example splitter silencer

With one layer of each material (air, porous), approximate solution can be obtained.αp

ρptanαpb +

αa

ρatanαaa = 0,

with αp =√

k2p − β2 and αa =

√k2

a − β2.

At low frequencies (viscous regime), the wavenumber order of magnitude in theporous material kp ∝

√ω, is larger than the axial wavenumber |β|.

When |km| |β|, the equation (19) is equivalent to a waveguide lined with alocally reacting material satisfying

ika

Z+αa

Zatanαaa = 0,

with Za = ρac0 and Z = iρpcp cot kpb.

This is equivalent to say that only plane wave propagate in the transverse directione±ikpy . Applying the continuity of the pressure between the micro-porous pm and theairway pa, yields to

pp(y) =pa

2 cos kpb(

eikpy + e−ikpy),∀ y ∈ [−b, b].





Layered waveguides - Example splitter silencer (cont.)

The pressure in the airway may considered as a constant in the transverse directionand is a forcing term for the pressure in the micro-porous media.

With these assumptions, the average pressure over the elementary cell reads,

〈pp〉pa

=12b

∫ b

−bpp(y) dy =

tan kpbkpb

.

is similar to the pressure diffusion function Fd of the double porosity model [Olny etal., 2003].





(a) f = fd3

(b) f = fd

(c) f = 3fd

Normalized pressure field (in absolute value) below, around and above ωd for a rockwool.





Layered waveguides - Optimal designThis closed form solution yields an optimal value (max Imβ) for the airflow resitivity

σ∗0 =

√3ρ0P0b2φ2

(1

γα∞+ φpφφpα∞

+ Υ)

depending on the waveguide height h and open

area ratio φp .

0 50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

30

f (Hz)

TL (

dB

)

(a) φp = 0.5, N = 1

0 500 1000 15000

10

20

30

40

50

60

70

80

90

f (Hz)

TL (

dB

)(b) φp = 0.5, N = 3

Transmission loss σ ∈ [0.1σ∗0 , . . . , 10σ∗0 ]. Obtained with a reference solution [Binois, 2013] on

realistic silencers. If σ < σ∗0 , (. . . ), If σ > σ∗0 ,( ), If σ = σ∗0 , ( ). Geometry H = 0.70

m and Ls = 1.20 m.

I What happens for absorption at normal incidence σ = σ∗0 ?


Layered poroelastic waveguides


rigid wall

infinite

Biot





Typical dissipative silencers (automotive industry, HVAC systems,...) consists of acylindrical expansion chamber filled with a sound absorbing material.

Air

Biot

z

rI II III

Prediction of the Transmission Loss (TL) is usually made using Mode MatchingMethods (Kirby & Denia, 2007 ; Lawrie & Kirby, 2005 ; and plenty more...)They all consider “fluid equivalent models” (Beraneck, 1947 ; Zwikker & Kosten,1949 ; Delany & Bazley, 1970)

I Why take into account the frame elasticity ? Can the skeleton improve the TL ?





Constitutive equations

For time-harmonic representation, the pressure p in the airway obeys to the(convected) wave equation

∆p −1c20

d2pdt2

= 0

In the poroelastic material [Biot, 1956]

∇ · σs + ω2(ρ11u + ρ12U) = 0,∇ · σf + ω2(ρ12u + ρ22U) = 0,

where U et u are respectively the fluid and the solid phase displacements and the Solidand fluid phase stress tensors are given by

σs = (A∇ · u + Q ∇ · U) I + 2N εs

σf = (Q ∇ · u + R ∇ · U) I

where εs = 1/2(∇u + (∇u)T) is the usual strain tensor.





Helmholtz decomposition

In the airway, we look for purely acoustic mode :

w = ∇ϕ0

In the porous media, both displacement fields are written as

u = ∇ϕ+∇∧ ψ and U = ∇χ+∇∧Θ.

After equations decoupling, we have

ϕ = ϕ1 + ϕ2 , χ = µ1ϕ1 + µ2ϕ2 , and Θ = µ3ψ

All potentials fulfill wave equation. Wave number ki and phase coupling coefficients µiare known (Allard, 1993)





Helmholtz decomposition > General potential form

We putX(r , θ, z, t) = X(r , θ) ei(βz−ωt)

Following (Gazis, 1951) we find for symetric modes

ϕ1 = [A1Jm(α1r) + B1Ym(α1r)] cosmθϕ2 = [A2Jm(α2r) + B2Ym(α2r)] cosmθψr = [A3Jm+1(α3r) + B3Ym+1(α3r)] sinmθψθ = − [A3Jm+1(α3r) + B3Ym+1(α3r)] cosmθ

ψz =[A′3Jm(α3r) + B′3Ym(α3r)

]sinmθ

Similarlyϕ0 = A0Jm(α0r) cosmθ





Continuity conditions

At the air-porous interface, the following coupling conditions must be verified

σtn = −p npp = p

φU · n + (1− φ)u · n︸︷︷︸Porous side

= w · n︸︷︷︸Airway

Here φ is the porosity and the pore pressure pp is obtained from the fluid phase tensoras −3φpp = trσf .

On the hard surface the poroelastic layer is assumed to be clamped

u = 0 and U · n = 0.





Continuity conditions > Eigenvalue equation

The modal vector containing the wave potential amplitudes

V =(

A0,A1,B1,A2,B2,A3,B3,A′3,B′3)T

,

must be a non-trivial solution of the 9× 9 algebraic system after boundary conditionsapplication

M(α0, α1, α2, α3, β)V = 0

Together with the dispersion equation (once inverted)

α0 =√

(k0 −Mβ)2 − β2 and αi =√

k2i − β2, i = 1, 2, 3

we haveM(β)V = 0






Instead of finding roots of f (β) = detM(β) it’s possible to find the zeros of theassociated polynomial Π with the same zeros [Kravanja & Van Barel, 2000 ; Nennig etal., 2010]

1 Residue theorem gives for analytical functions

Sn =12πi

∮C

f ′(β)f (β)

βn dβ =Nβ∑k=1

nkβnk

2 The coefficients of the polynom Π are related to Sn3 Get the zeros of the polynom Π by computing the eigenvalues of companion

matrix ie the root of f (β) in the interior of C

I Important to use integer number of sample on CI Works with all closed waveguides (always analytics function)I Variant with log, there is efficient way to generalized this to big matrix [Spence,2004 ; Güttel, 2017]. . .





Roots in the complex plane > frequency analysis

−800 −600 −400 −200 0 200 400 600 800

−600

−400

−200

0

200

400

600

PSfrag repla ements

Reβ

Imβ

(a) Integration path

−150 −100 −50 0 50 100 150−150

−100

−50

0

50

100

150

PSfrag repla ements

ReβImβ

(b) Frequency sweep

Roots in β complex plan for XFM foam in the silencer A and M = 0.

I Good numerical satiblity, because it never looks close to the root


Waveguides with periodic cross section



Periodic pore modelingMetaporous

As the governing equations, the boundary conditions, and the geometry are d-periodic,it follows from the Bloch theorem that the solutions are Bloch waves,

φ(x) = φ(x)eikB ·x, (1)

i.e. the fields can be split into a d-periodic field φ(x) modulated by a plane waveinvolving the Bloch wavevector kB = kBκ, where the unit vector κ = kB/kB stands forthe propagation direction.

The real part of kB measures the change in phase across the cellThe imaginary part of kB measures the attenuation.

Two way for solving this [Collet, 2011 ; Frazier, 2017]7 Choose ω as eigenvalue, suitable for free response, difficult with parameters f (ω)3 Choose kB as eigenvalue, suitable for forced response, i. e. most application in

waveguides

I Semi-analytical methods using plane wave expansion are also possible [Duclos,2009 ; Deymier, 2013]


Periodic pore modeling


rigid wall

infiniteZoom




Visco-thermal fluid

We use here linearized Navier and Stokes equations with Fourier flux q = −κ∇T , andwe assume a Newtonian fluid, ideal gas [Stinson, 1991]

−iωρ0v +∇p −∇ · τ = 0,−iωρ0CpT − κ∆T + iωp = Q,

∇ · v + iωTT0− iω

pP0

= 0.

Where v is the velocity, T the temperature and p the pressure fluctuation.

Taylor-Hood element are used to avoid looking problem in Stokes like equation. Thevelocity and the temperature is approximated with the quadratic lagrangian element,and the pressure with linear lagrangian element [Kampinga, 2010].

I The pressure is kept to ensure numerical stabilityI Possible to add the coupling with the skeletonI Possible to add other physics





ImplementationThe periodic formulation can be obtained thanks to a systematic approach, forinstance

∇ · u = [∇ · u + ikB · u] eikB ·u

∇u =[∇u + iu kt

B]

eikB ·x

We get a Quadratic eigenvalue problem in kB [Tisseur, 2001],

k2BK2 + kBK1 + K0 = 0

2N eigenvalues, only N independent eigenvectorsHere all parameters are real, then K2, K0 real symmetric, K1 Hermitian (emptydiag.) (similar gyroscopic pb when λ = ikb)kB and −k∗B are solutionThe left eigenvector for kBn is also the right eigenvector for −k∗Bn

I There is several possibilities for resolution (linearization be careful to respectsymetries, direct [TOAR, SLEPc])





Validation

0 2000 4000 6000 8000 100000

20

40

60

80

100

120

140

160

180

200

Re keq

Im keq

(a) keq

0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5

Re rhoeq

Im rhoeq

(b) ρeq

0 2000 4000 6000 8000 10000−2

0

2

4

6

8

10

12

14

16x 10

4

Re Keq

Im Keq

(c) Keq

Comparaison with FEM (dot) Zwicker and Kosten solution (solid) for circular pore r = 1 mm.

Using the 1st eigenvector, effective parameters ρeq and Keq , can be recover as definedin [Allard, 1993, Chap. 4]

−iωρeq vz = −∂p∂z. et Keq = −p/∇ ·

( v−iω

).

Here ¯ is the average over the cross section.





Examples

FEM computation on a staight pore with complex cross section





Periodic pore modeling

(a) géométrie

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

100

f (Hz)

Re keq

Im keq

(b) kB

Comparaison between FEM 3D model (dot) and homogenization [Boutin, 2015] (solid line) with arigid Helmholtz resonator in CC crystal (a = 6 mm, t = 0.3 mm, l = 1.5 mm, rc = 0.3 mm,rs = 2.75 mm)

I Strong connexion with homogenization theory


Metaporous


rigid wall

infinite




AimFew metaporous examples use to increase absorption, limit transmission through wallor through duct

(a) Yang et al., 2015 (b) Groby et al., 2014 (c) Lagarrigue al.,2015

(d) Aurégan et al.,2016

(e) Xiong et al., 2017

Focus here on duct parietal acoustic treatmentsI Bloch modes yields non hermitian quadratic eigenvalue problemI Geometric parameters in metamaterial are easy to tuneI Exploit easy tuning of metamaterial to find EP / force Bloch modes to coalesce





Configurations

Sample made of 5 mm metalic foam layers and blind cylinder inclusions with 2possible orientation [Xiong, 2017]

(f) Sample (g) Model

Pictures of (a) an open cylinder inclusion (filled with air), (b) zoom of the drilled metallic foamlayer of height 5 mm, and (c) a whole sample with cylinders (Fig. 7(a)) embedded in an alternatedway.





Experimental observation

p1

+

x

z

p1

-

p2

+

p2

-

u4 u3 u2u1 d1d2d3 d4hp

d

xc

Porous Air

ha

Unit cell

A 2D schematic view of the experimental setup. The measured sample, of length L = 200 mm,contains 8 unit cells. xc is the center of the rigid inclusion.

0 500 1000 1500 2000 2500 3000 35000

5

10

15

20

25

30

35

Frequency (Hz)

att.

(d

B)

(a)

hp

0 500 1000 1500 2000 2500 3000 35000

5

10

15

20

25

30

35

Frequency (Hz)

att.

(d

B)

(b)

hp

hp

’

0 500 1000 1500 2000 2500 3000 35000

5

10

15

20

25

30

35

Frequency (Hz)

att.

(d

B)

(c)

hp

hp

’

Comparison between the Bloch wave attenuation (·) and the measured transmission loss (line)with 8 unit cells for the configurations : (a) 5P, (b) P-t-P, and (c) P-u-P.





Parametric study

The inclusion is moved from δ from the centre of the cell is positive.I parametric (δ, f ) Quadratic eigenvalue problem

01000

20003000 −4.7

−4.61−4.5−4 −2

02

0

10

20

30

40

50

(mm)Frequency (Hz)

att.(dB)

freq. max.

Mode attenuations for the lower two Bloch modes as a function of frequency at different inclusionpositions δ.

I The attenuation reach a max. when 2 modes coalesce (eigenvector andeigenvalue)





Mode coalescence

050

100

02040600

500

1000

1500

2000

2500

3000

3500

Fre

quen

cy (

Hz)

0 20 40 60 80 1000

500

1000

1500

2000

2500

3000

3500

Fre

quen

cy (

Hz)

0

1

m1 m0

mS m1'

Coalescencem1

m0

m1'

mS

mL

mL

Dispersion curves for the Bloch wavenumber when ha = 135 mm and δ = −4.0 mm a) 3D view,b) projection in the frequency-Re kB plane. The inserted pictures show the (O, x , z)-cut of themodulus of the right eigenvectors at 100, 1375, 1385 and 2000 Hz.

I At the coalescence the double mode is localized in the liner, similar to impedancecase [Bi, 2016]

I (δ∗, f ∗) is an exceptional point EP [Kato, 1980 ; Heiss,1990 ; Berry,2004 ; Bi,2015]





Parametric study

Using a Taylor expansion in the vicinity of thedouble root k∗B [Tester,1973 ; Shenderov,2000],

kB − k∗B ≈ ±[(− 1

2∂2D∗∂k2B

)−1 ((δ − δ∗) ∂D∗

∂δ+ (f − f ∗) ∂D∗

∂f

)]1/2

I The EP is a branch point in the parameterplane

Riemann sheet of the real (a) and imaginary(b) part of the lower attenuated Blochwavenumber. The line indicates thecrossing of the different sheets.




ConclusionsPersepectives in numerical methodes


In a waveguide maximizing attenuation is different that for absorbing panelPartial filling / airwayHigher modes

Recently, we find new design rulesTune on EP with metaporous [Xiong et al., 2017]Use σ∗0 when porous remains in viscous regime [Nennig et al., 2015]Use transverse isotropic material

Other strategies have to be testedPlay with path in parameter space for asymmetric propagation [Doppler, 2016]

I Take advantage of complex problem to reveals new physics (EP, CPA, . . .)I Fine modeling of damping and radiation mechanism opens new routes (QM, EM)




ConclusionsPersepectives in numerical methodes


Numerical methods are now mature to tackle such problem, however, it remains somechallenges or some possible improvements

How to find efficiently EP (Jordan decomposition ?)How to speed-up ferquency sweepUse waveguide as an homogenization method. . .


The waveguide journey comes to an end,

Thankyouforyourattention!Thank you for your attention !

See you in porous session of CFA in Le Havre

This work was partly funded by the ANR Project METAUDIBLE No.ANR-13-BS09-0003-01 funded jointly by ANR and FRAE.

on numerical methods for the computation of guided waves...

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