Wave transport in anisotropic media

Wave transport in anisotropic media

Éric Savineric.savin@{onera,centralesupelec}.fr

(+ Jean-Luc Akian, Ibrahim Baydoun, Didier Clouteau, Régis Cottereau, Johann Guilleminot)

Onera – Dépt. Mécanique des Fluides Numérique29, avenue de la Division Leclerc

F-92322 Châtillon cedex

CentraleSupélec – Département Mécanique et Génie CivilGrande Voie des Vignes

F-92295 Châtenay-Malabry

Elastic waves and anisotropyUltrasonic monitoring of polycrystals

Example: recrystallization of inco c© 718 superalloy (NiCr19Fe19Nb5Mo3) in hot compression.

Estimate of the grain size ` from the attenuation Σ = 2πλQ

such that Iout ≡ Iine−Σh:

Σ ∝ `3f 4 λ� ` (Rayleigh regime) ,Σ ∝ `f 2 λ ' ` (stochastic regime) ,Σ ∝ `−1 λ� ` (diffusive regime) .

Stanke-Kino JASA 75(3), 665 (1984)Goebbels Materials Characterization for Process Control and Product Conformity, CRC Press, Boca Raton FL (1994)

É. Savin Atelier Imagerie LMA 8/06/2015 2 / 30

http://dx.doi.org/10.1121/1.390577

http://www.crcpress.com/product/isbn/9780849389573

Elastic waves and anisotropyGlobal seismology

Example: texture and dynamics of the inner core from observations of PKiKP and PKIKPwaves (PKJKP are seldom observed and the reported evidences are highly controversial).

Cao-Romanowicz-Takeuchi Science 308, 1453 (2005)Wookey-Helffrich Nature 454, 873 (2008)

Monnereau-Calvet-Margerin-Souriau Science 328, 1014 (2010)


http://dx.doi.org/10.1126/science.1109134

http://dx.doi.org/10.1038/nature07131

http://dx.doi.org/10.1126/science.1186212

Elastic waves and anisotropyTriclinic random medium

Example: Velocity field in a half-space constituted by heterogeneous anisotropic materialswith an homogeneous isotropic background, impacted by a monopole on its surface;cP = 2000 m/s, cS = 1000 m/s, % = 2000 kg/m3.

Spectral FEM with LGL + PML7 · 107 dofs, O = 5× 5× 2.5 km3

Ta-Clouteau-Cottereau Eur. J. Comp. Mech. 19(1-3), 241 (2010)


http://dx.doi.org/10.3166/ejcm.19.241-253

Elastic waves and anisotropyAnisotropic diffusion

Example: "localization". Privileged directions for diffusion corresponding to the principalaxes of diffusion → localization when λ ∼ `sc?

f = 664± 40 kHz f = 195± 40 kHz

Lobkis-Weaver JASA 124(6), 3528 (2008)


http://dx.doi.org/10.1121/1.2999345

Outline

1 The acoustic tensor

2 The Wigner measure

3 Random media


Outline



3 Random media


Elastic waves in heterogeneous media

Hypotheses: linear, visco-elastic materials with density %(x) and relaxation tensor C(x, t),x ∈ O ⊆ R3, t ∈ R+.

Local balance of momentum for the displacement field u in O × R:

%∂2t u = Divσ .

Constitutive equation for the stress field σ in O × R+ as a function of the strain fieldε(u) = ∇x ⊗s u:

σ(x, t) = C(x, 0)ε(u) + ∂tC(x, ·) ∗t ε(u)︸︷︷︸ineffective in the high-frequency limit

.


Ultrasonic waves 1/2

High frequencies correspond to ε→ 0 for strongly "ε–oscillatory" initial conditions:

uε(x, 0) = u0ε(x) , ‖|ε∇x|u0ε‖L2loc<∞ ,

and:∂tuε(x, 0) = v0ε(x) , ‖|ε∇x|v0ε‖L2loc

<∞ .

Example: plane waves, ε ≡ (|k|L)−1, i =√−1,

u0ε(x) = εA(x)eiεk·x , v0ε(x) = B(x)e

iεk·x .


Ultrasonic waves 2/2

Introduce the acoustic (Christoffel) tensor Γ of the medium:

[Γ(x, k)]jk = %−1(x)(ej ⊗ k) : C(x) : (ek ⊗ k) , k ∈ R3 ,

where (ej , ek ) = δjk , 1 ≤ j , k ≤ 3.

The local balance of momentum for the displacement field uε in O × Rt is the elastic waveequation:

%(x)(Γ(x, iε∇x)− (iε∂t)2

)uε = O(ε) ,

supplemented with (e.g. Neumann or Dirichlet) boundary conditions on ∂O × Rt .

In Dirac formalism the acoustic tensor is %(x)[Γ(x, k)]jk = 〈k, j |C|k, k〉 and the waveoperator is %(x)[Γ(x, iε∇x)]jk = 〈i x

ε, j |C| k, i x

′

ε〉, for 〈j |k〉 = 〈ej |ek 〉 = δjk .


Some properties of Γ

Γ is symmetric, real, positive definite (in O × R3 \ {k = 0}): it has 3 (possibly rα-multiple)positive eigenvalues ω2α such that:

ωα(x, k) = cα(x, k)|k| , 1 ≤ α ≤ 3 ,

where k := k/|k|, and the real eigenvectors pα(x, k) = pα(x, k) can be orthonormalized,

Γ =3∑α=1

ω2αpα ⊗ pα , I =3∑α=1

pα ⊗ pα .

Example: three-dimensional isotropic medium C = λI⊗ I + 2µI � I, rP = 1, rS = 2, and

Γ(x, k) = c2P(x)k⊗ k + c2S(x)(I− k⊗ k) , ∀k ∈ S2 ,

with cP =√λ+2µ%

and cS =√µ%.


Elastic waves and anisotropyThree propagation speeds

Transverse isotropic medium

Orthotropic medium


Energy and power flow densities

The energy density (real, positive) and the power flow density (vector) are:

Eε(x, t) =12

(%|∂tuε|2 + σε : εε) = Tε + Uε ,

πε(x, t) = −σε∂tuε .

They satisfy for all fixed ε ∈]0, ε0] a conservation equation:

∂tEε + divπε = 0 .

What happens when ε→ 0???

For constant coefficients the usual parametrix (where Γ12 =

∑α ωαpα ⊗ pα):

uε(x, t) = F−1k→x

[cos(tΓ

12 (k)

)]? u0ε(x) + F−1k→x

[Γ−

12 (k) sin

(tΓ−

12 (k)

)]? v0ε(x)

propagates oscillations of wavelength ε which inhibit (uε) from converging strongly in asuitable sense since Γ is positive.


Outline



3 Random media


Why quadratic observables?

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

u

ε(x)

m(x)w[u

ε]0.5

Let:uε(x) = m(x) + σ(x) sin

x

ε,

then (uε) ⇀ m weakly in L2(R) as ε→ 0, but (uε) has no strong limit.



0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

u

ε(x)

m(x)w[u

ε]0.5

Now for any observable ϕ ∈ C0(R):

limε→0

(ϕ(x)uε, uε)L2 =

∫Rϕ(x)

((m(x))2 +

12

(σ(x))2)

dx .



0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

u

ε(x)

m(x)w[u

ε]0.5

Take an observable of the form:

ϕ(x , ∂x )uε(x) =12π

∫R

eik·xϕ(x , ik)uε(k)dk .



0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

u

ε(x)

m(x)w[u

ε]0.5

Take an observable of the form:

ϕ(x , ε∂x )uε(u) =12π

∫R

eik·xϕ(x , iεk)uε(k)dk .



0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

u

ε(x)

m(x)w[u

ε]0.5

Then:limε→0

(ϕ(x , ε∂x )uε, uε)L2 =

∫∫R2ϕ(x , ik)W[uε](dx , dk) ,

where W[uε] is the (positive) Wigner measure of (uε).


Wigner measure

Let ϕ ∈ C∞0 (R3x × R3

k) (a smooth observable), and for u ∈ L2(R3):

Opε[ϕ]u(x) = ϕ(x, ε∇x)u(x) = ϕ(x, ε∇x)

(∫eix·ku(k) dk

):=

∫eix·kϕ(x, iεk)u(k) dk .

Then if (uε) is bounded, ∀ϕ a smooth observable:

limε→0

(ϕ(x, ε∇x)uε, uε)L2 = Tr∫∫

ϕ(x, ik)W[uε](dx, dk) ,

and the positive, Hermitian measure W[uε] of the sequence (uε) always exists—up toextracting a sub-sequence if need be.

Lions-Paul Rev. Mat. Iberoamericana 9(3), 553 (1993)Gérard-Markowich-Mauser-Poupaud Commun. Pure Appl. Math. L(4), 323 (1997)

Martinez An Introduction to Semiclassical and Microlocal Analysis, Springer, Berlin (2002)Zworski Semiclassical Analysis, American Mathematical Society, Providence RI (2012)


http://dx.doi.org/10.4171/RMI/143

http://dx.doi.org/10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C

http://dx.doi.org/10.1007/978-1-4757-4495-8

http://dx.doi.org/

Wigner measureKinetic and strain energies

Example: the overall kinetic and strain energy densities are in the high-frequency limit ε→ 0

limε→0Tε(x, t) =

12%(x)

∫R3

TrW[ε∂tuε(·, t)](x, dk) ,

limε→0Uε(x, t) =

12%(x)

∫R3

Γ(x, k) : W[uε(·, t)](x, dk) ,

(eventually one can prove that they are equal in this very limit), or

limε→0Eε(x, t) =

3∑α=1

∫R3

Wα(x, t; dk) .


Wigner transform

The Wigner transform of u, v ∈ S′(R3):

Wε[u, v](x, k) =1

(2π)3

∫R3

eik·yu(x− εy)⊗ v(x) dy .

Then (T∗R3 ≡ R3x × R3

k):

(ϕ(x, ε∇x)u, v)L2 = Tr∫T∗R3

ϕ(x, ik)Wε[u, v](dx, dk) , ∀ϕ .

k allows to keep track of the direction of propagation of the oscillations of u, v.Let s = (x, t) ∈ O × R and ξ = (k, ω) ∈ Rd × R, the space-time Wigner transform reads:

Wε[u, v](s, ξ) =1

(2π)4

∫R4

ei(k·y+ωτ)u(x− εy, t − ετ)⊗ v(x, t) dydτ .


Wigner measure of the elastic wave equationDispersion properties

The dispersion matrix H of the medium, or inverse Green function in wave vector space:

H(s, ξ) = %(x)(Γ(x, k)− ω2I

)= %(x)

3∑α=1

(ω2α(x, k)− ω2)pα(x, k)⊗ pα(x, k) .

Then the wave equation reads:

H(s, ε∇s)uε(s) = O(ε) .

Let ϕ be a smooth observable on T∗(O × R):

O(ε) = (ϕ(s, ε∇s) ◦H(s, ε∇s)uε, uε)L2

= ((ϕH)(s, ε∇s)uε, uε)L2 ,

by the rule of composition Opε[ϕ1 ◦ϕ2] = Opε[ϕ1ϕ2 − iε∇ξϕ1 ·∇sϕ2 + O(ε2)].


Wigner measure of the elastic wave equationDispersion properties

Taking the limit ε→ 0:

0 = Tr∫T∗(O×R)

(ϕH)(s, ξ)W[uε](ds, dξ) , ∀ϕ ,

and thus:

0 = HW[uε] = %

( 3∑α=1

(ω2α − ω2)pα ⊗ pα

)W[uε] .

Since Hα = ω2α − ω2 6= Hβ whenever α 6= β, one finally has:

W[uε] =3∑α=1

Wαδ(Hα) pα ⊗ pα , Wα = p∗αW[uε]pα .


Wigner measure of the elastic wave equationEvolution properties

Again from the wave equation:

O(ε) = (ϕ(s, ε∇s) ◦H(s, ε∇s)uε, uε)L2 − (ϕ(s, ε∇s)uε,H(s, ε∇s)uε)L2

=1ε

([ϕ,H](s, ε∇s)uε, uε)L2 − i ({ϕ,H}(s, ε∇s)uε, uε)L2

by the rules of composition and transposition Opε[ϕ]∗ = Opε[ϕ∗ − iε∇ξ ·∇sϕ∗ + O(ε2)],where:

[A,B] = AB− BA (Lie bracket)

{A,B} = ∇ξA ·∇sB−∇sA ·∇ξB (Poisson bracket) .


0 = Tr∫T∗(O×R)

{ϕ,H}W[uε](ds, dξ) , ∀ϕ s.t. [ϕ,H] = 0 .

Choosing for instance ϕ = ϕI yields the Liouville transport equation:

dWα

dt= {Hα,Wα} = 0 .


Outline



3 Random media


Waves in random elastic mediaRandom elasticity tensor

Assume now that C depends on ε:

C(x) = C0(x) + εsC1

(xε

),

where (C1(y), y ∈ R3) is a second order, mean-zero, homogeneous random (fourth-order)tensor.

The correlation length of the random perturbations is the same as the (small) wavelength ε,ensuring maximum interactions between waves and the underlying medium.

Heterogeneities are small and their size is s = 12 , which is the unique scaling that allows

them to significantly modify the energy in the transport regime.


Waves in random elastic mediaRandom wave equation

Let ϕ be a smooth observable on T∗(O × R), then the wave equation reads:

O(ε32 ) = (ϕ ◦H0(x, t, ε∇x+∇y, ε∂t)uε, uε)L2 +

√ε (ϕ ◦H1(y, ε∇x+∇y)uε, uε)L2 ,

where y ≡ xε, ∇x ≡ ∇x + 1

ε∇y, and:

[H1(y, k)]jk = (ej ⊗ k) : C1(y) : (ek ⊗ k) , k ∈ R3 .


0 = Tr∫T∗(O×R)

(ϕH0)(s, ξ)W[uε](ds, dξ) , ∀ϕ ,

and the spectral expansion of the Wigner measure W[uε] is unchanged from thedeterministic case.


Waves in random elastic mediaEvolution properties of the Wigner measure

Again from the wave equation:

O(√ε) =

1ε

(ϕ ◦H0(x, t, ε∇x+∇y, ε∂t)uε, uε

)L2

+1√ε

(ϕ ◦H1(y, ε∇x+∇y)uε, uε

)L2

−1ε

(ϕ(s, ε∇s)uε,H0(x, t, ε∇x+∇y, ε∂t)uε

)L2−

1√ε

(ϕ(s, ε∇s)uε,H1(y, ε∇x+∇y)uε

)L2

=1ε

([ϕ,H0](s, ε∇s)uε, uε)L2 − i ({ϕ,H0}(s, ε∇s)uε, uε)L2

+i√ε

(L1(y, ε∇x+∇y)uε, uε)L2 .

It can be shown that:

limε→0

1√εE{(L1(y, ε∇x + ∇y)uε, uε)L2} =

∫T∗(O×R)

(ϕQ)(s, ξ)E{W[uε]}(ds, dξ) , ∀ϕ ,

in return for a mixing lemma–like assumption.

Then:

{Hα,E{Wα}} =3∑β=1

QαβE{Wβ} ,

where Qαβ are collision operators.


Radiative transfer in anisotropic elastic media

The radiative transfer (linear Boltzmann) equations for the specific intensitiesWα = p∗αE{W[uε]}pα of the three pseudo-modes 1 ≤ α ≤ 3:

∂tWα + {ωα,Wα} =3∑β=1

∫R3σαβ(x, k, k′)Wβ(t, x, k′)dk′ − Σα(x, k)Wα(t, x, k) ,

where by energy conservation Σα(x, k) =∑3β=1

∫R3 σαβ(x, k, k′)dk′: the attenuations (or

scattering cross-sections).

σαβ(x, k, k′): the differential scattering cross-sections.

��

��

σαβ ( k k’),

k ’

kW

W

β

α

Ryzhik-Papanicolaou-Keller Wave Motion 24(4), 327 (1996)Bal Wave Motion 43(2), 132 (2005)

Bal-Komorowski-Ryzhik Kinet. Relat. Models 3(4), 529 (2010)


http://dx.doi.org/10.1016/S0165-2125(96)00021-2

http://dx.doi.org/10.1016/j.wavemoti.2005.08.002

http://dx.doi.org/10.3934/krm.2010.3.529

Scattering cross-sections

The random tensor C1 has 21 different coefficients {cj (y), 1 ≤ j ≤ 21} in the generaltriclinic case.

Correlations of the random tensor:

E{cj (k)ck (q)} := (2π)3δ(k + q)Rjk (k) ,

where Rjk (y1 − y2) = E{cj (y1)ck (y2)} by the (statistical) homogeneity assumption.

The differential scattering cross-sections:

σαβ(x, k, q) =π

2Rαβ(x, k, k− q, q)

(2π)3ωα(x, k)ωβ(x, q)δ(ωα(x, k)− ωβ(x, q))

where:(2π)3δ0(p)Rαβ(x, k, p, q) := E{[p∗α(x, k)H1(x, k, p, q)pβ(x, q)]2} ,

and:[H1(x, k, p, q)]jk = %−1(x)(ej ⊗ k) : C1(p) : (ek ⊗ q) , k, p, q ∈ R3 .

In Dirac formalism Rαβ(x, k, ·, q) ∝ E{〈k, α|C1|β, q〉〈q, β|C1|α, k〉} i.e. the "square of aT-matrix", while the "on-shell" condition ωα(x, k) = ωβ(x, q) holds.

Baydoun-Savin-Clouteau-Cottereau-Guilleminot Wave Motion 51(8), 1325 (2014)



Scattering cross-sectionsEarth inner core

Example: (normalized) attenuation coefficients Σ#αβ of hcp-Fe (T = 5500◦K, P = 360GPa).

Vočadlo EPSL 254(1-2), 227 (2007)


http://dx.doi.org/10.1016/j.epsl.2006.09.046

Waves in heterogeneous mediaScattering mean free path

Scales: λ – wave length, ` – correlation length, L – propagation/observation distance...

...and a mesoscopic scale `sc ∝ Σ−1 – the mean free path.

Ryzhik HFWP’05 (2005)


http://www.cscamm.umd.edu/programs/hfw05/ryzhik_hfw05.pdf


Example: heterogeneous medium.

��

��

��

��

��

��

��

��

��

��

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��

k ’β k

WW α

k

W γ

’’

lα



Example: bounded medium, SEA method.

Phase distribution of the radial velocity

Herdic et al. JASA 117(6) 3667 (2005)


http://dx.doi.org/10.1121/1.1887125

Hot topics

Time-dependent acoustic tensor;

Diffusion limit(s);

Mode crossings, boundary conditions (very difficult).

Further reading...I Akian J.-L. Asymp. Anal. 78(1-2):37-83 (2012).I Alshits V.I., Lothe J. Wave Motion 40(4):297-313 (2004).I Bal G. Wave Motion 43(2):132-57 (2005).I Bal G., Komorowski T., Ryzhik L. Kinet. Relat. Models 3(4):529-644 (2010).I Brassart M. Limite semi-classique de transformées de Wigner dans des milieux périodiques ou

aléatoires. PhD Thesis, University of Nice Sophia Antipolis (2002).I Calvet M., Margerin L. J. Acoust. Soc. Am. 131(3):1843-61 (2012).I Gérard P., Markowich P.A., Mauser N.J., Poupaud F. Commun. Pure Appl. Math. L(4):323-79

(1997).I Margerin L., Sato H. J. Acoust. Soc. Am. 130(6):3674-90 (2011).I Papanicolaou G.C., Ryzhik L.V. Waves and transport. In: Caffarelli L, E W, editors. Hyperbolic

equations and frequency interactions. IAS/Park City Mathematics Series, vol. 5. AmericanMathematical Society, Providence RI (1999); pp. 305-82.


http://dx.doi.org/10.3233/ASY-2011-1084



http://dx.doi.org/10.3934/krm.2010.3.529

https://tel.archives-ouvertes.fr/tel-00002512/

https://tel.archives-ouvertes.fr/tel-00002512/

http://dx.doi.org/10.1121/1.3682048

http://dx.doi.org/10.1121/1.3652856

Funding: Fédération Francilienne de Mécanique (F2M) CNRS-FR2609.

Collaborators:


Wave transport in anisotropic media

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