KINETIC MODEL FOR ELASTIC WAVE PROPAGATION IN BOUNDED MEDIA AND APPLICATION TOHIGHFREQUENCY STRUCTURAL ACOUSTICSY. Le Guennec
†
, ´E. Savin
†
,
∗†
Onera – The French Aerospace Lab, F92322 Chˆatillon, France
∗
Email: Eric.Savin@onera.fr
Talk abstract
Semiclassical analysis of strongly oscillating solutionsof classical wave systems, such as the Navier equation of elastodynamics, shows that the associated elastic energydensity satisﬁes a Liouvilletype transport equation inphase space. Here this model is extended by consideringenergetic boundary and interface conditions consistentwith the boundary and interface conditions imposed tothe solutions of the underlying wave system in a boundedmedium. They are given in the form of power ﬂow reﬂection/transmission operators for the bicharacteristic curvesimpinging on a boundary or an interface. Specularliketransverse boundary reﬂections, diffuse reﬂections, orﬂuidstructure coupling may be treated as a particularcase of the proposed model. Nodal/spectral discontinuous ”Galerkin” ﬁnite element methods and MonteCarlomethods are implemented to integrate the transport equationssupplementedwiththeproposedboundaryandinterface conditions. An application to the transient dynamicsof a beam truss is proposed.
Kinetic model for elastic wave propagation
Let us consider the following Cauchy problem for elastic waves propagation in
O×
R
:
∂
2
t
u
ε
=
∇
x
·
(
C
:
∇
x
⊗
u
ε
)
,
u
ε
(
·
,
0) =
u
0
ε
, ∂
t
u
ε
(
·
,
0) =
v
0
ε
,
(1)depending on a small scaling parameter
ε
which quantiﬁes the rate of change of the initial data
u
0
ε
and
v
0
ε
with respect to the sizes of
O ⊂
R
d
or the propagation/observation distances. Here
u
ε
is the small displacement ﬁeld of the medium about a quasistatic equilibriumconsidered as the reference conﬁguration; and
C
(
x
)
isthe fourthorder elasticity tensor of the materials occupying
O
of which density is denoted by
(
x
)
. Sincehighfrequency waves are generated by an initial vibrational energy oscillating at wavelengths
ε
1
, the initial conditions
∇
x
⊗
u
0
ε
and
v
0
ε
shall be considered asstrongly
ε
oscillating functions, i.e. such that
(

ε
∇
x

ϕ
ε
)
2
remains at least locally integrable on
O
[1]. The planewaves
u
0
ε
(
x
) =
ε
A
(
x
)
e
i
k
·
x
/ε
and
v
0
ε
(
x
) =
B
(
x
)
e
i
k
·
x
/ε
,
k
∈
R
d
, typically fulﬁll this condition.
Ray method
The classical approach to solve this problem is the raymethod, consisting in seeking a solution of Eq. (1) in theform:
u
ε
(
x
,t
)
e
i
S
(
x
,t
)
/ε
∞
k
=0
ε
k
U
k
(
x
,t
)
,
(2)where the phase function
S
(
x
,t
)
is either real in theWKBJ method, or complex using Gaussian beams superposition [2]. Plugging the ansatz (2) into Eq. (1) it isshown, adopting an Eulerian point of view, that the phasefunction satisﬁes an eikonal equation and that the density

U
0

2
satisﬁes a linear transport equation of which coefﬁcients depend on the phase function. Let us introduce theacoustic tensor
Γ
(
x
,
k
)
and dispersion matrix
H
(
s
,
ξ
)
of the propagation medium as follows:
Γ
(
x
,
k
)
U
=
(
x
)
−
1
(
C
(
x
) :
U
⊗
k
)
k
,
H
(
s
,
ξ
) =
(
x
)(
Γ
(
x
,
k
)
−
ω
2
I
n
)
,
(3)where
s
= (
x
,t
)
,
ξ
= (
k
,ω
)
,
(
s
,
ξ
)
∈
T
∗
(
O ×
R
)
≡O×
R
t
×
R
d
k
×
R
ω
,
U
∈
R
n
, and
I
n
is the identity matrixof
R
n
.
H
does not depend on the scale
ε
, that is to sayhighfrequency wave propagation
ε
1
is considered ina slowly varying ”low frequency” medium. The eikonaland
0
th order transport equations read:
H
(
s
,
∇
s
S
)
U
0
=
0
,
∇
s
·
U
T
0
∇
ξ
H
(
s
,
∇
s
S
)
U
0
= 0
,
respectively. Adopting a Lagrangian point of view, thepair
(
s
,
∇
s
S
)
is the solution of the Hamiltonian systemassociated to the elastic wave equation, which consists insolving the eikonal equation by the method of characteristics. Thus introducing the Hamiltonian
H
= det
H
, theusual properties of the acoustic tensor
Γ
are such that:
H
(
s
,
ξ
) =
n
α
=1
H
α
(
s
,
ξ
)
,
H
α
(
s
,
ξ
) =
(
x
)
λ
2
α
(
x
,
k
)
−
ω
2
,
(4)where
λ
2
α
stands for the
α
th (positive) eigenvalue of
Γ
with
1
≤
α
≤
n
. As for isotropic elasticity for instance,
300
WAVES’2011
Γ
(
x
,
k
) =
λ
2P
(
x
,
k
)ˆ
k
⊗
ˆ
k
+
λ
2S
(
x
,
k
)(
I
d
−
ˆ
k
⊗
ˆ
k
)
with
ˆ
k
=
k
/

k

and
λ
α
(
x
,
k
) =
c
α
(
x
)

k

of multiplicity 1 if
α
= P
or 2 if
α
= S
,
c
P
and
c
S
being the elastic compressional and shear wave velocities, respectively, suchthat
c
S
< c
P
. The corresponding Hamiltonian equationsfor
1
≤
α
≤
n
are:
d
s
d
τ
=
∇
ξ
H
α
(
s
(
τ
)
,
ξ
(
τ
))
,
s
(0) =
s
0
,
d
ξ
d
τ
=
−
∇
s
H
α
(
s
(
τ
)
,
ξ
(
τ
))
,
ξ
(0) =
ξ
0
=
0
,
(5)in
T
∗
(
O×
R
)
\{
(
s
,
ξ
);
ξ
=
0
}
, with an initial condition
(
s
0
,
ξ
0
)
satisfying
H
α
(
s
0
,
ξ
0
) = 0
. The rays are deﬁnedas the projections on
O×
R
t
of the bicharacteristic curves
τ
→
(
s
α
(
τ
)
,
ξ
α
(
τ
))
solving (5), that is
τ
→
s
α
(
τ
)
. Theyare associated to polarizations, or modes
α
deﬁned bythe dispersion relations
H
α
(
s
,
ξ
) = 0
derived from theeikonal equation.
Kinetic model
Theraymethodwitharealphasehassomemajordrawbacks from either an Eulerian point of view or a Lagrangian point of view. The non linearity of the eikonalequation does not allow to superpose different phases, aswould be required with several monopoles for instance;one way to circumvent this difﬁculty is to consider viscosity solutions. From the Lagrangian point of view, raytracing is no longer possible on the caustics, where therays stack, because the amplitudes
U
k
rapidly increase intheir neighbourhood, and even blow up on the rays themselves; Gaussian beams may be considered to circumventthis shortcoming [2]. More generally the ansatz (2) of the ray method is only one
a priori
particular constructionof highfrequency solutions of wave equations. This approach also requires rather strong regularity assumptionsfortheinitialconditionsofthephase
S
andamplitude
U
0
.The more recent works of G´erard
et al.
[1], Papanicolaou& Ryzhik [3] or Akian [4], among others, on the semiclassical analysis of wave systems have generalized thistheory for weaker assumptions on their highfrequencysolutions and the initial conditions. They show that theenergy density associated to all oscillating solutions, resolved in the phase space position
×
wave vector, satisﬁesa Liouvilletype transport equation. The main mathematical tool for the derivation of a transport equation froma wave equation is the Wigner transform, of which highfrequency limit
ε
→
0
, the socalled Wigner measure,captures the vibrational energy density in phase space.The advantage of this representation is that it clears allclassical difﬁculties inherited from ray approaches, andit yields global propagation properties of the energy forweakened initial conditions. In return, the explicit knowledge of the phase is lost. The eikonal equation is replacedby the dependence of the Wigner measure vs.
ξ
, whichgives its propagation directions as obtained from the dispersion equation
H
(
s
,
ξ
) = 0
. Considering the solutions
τ
→
(
s
(
τ
)
,
ξ
(
τ
))
of the system (5) as the paths in phasespace of some energy ”particles” with a density denotedby
W
(
s
(
τ
)
,
ξ
(
τ
))
, the latter shall satisfy:
d
W
d
τ
=
{H
,W
}
= 0
(6)where
{
f,g
}
=
∇
ξ
f
·
∇
s
g
−
∇
s
f
·
∇
ξ
g
is the usualPoisson bracket. Eq. (6) is a Liouville equation, which isthe expression of the conservation of
W
in phase spacestarting from the initial data
W
(
s
0
,
ξ
0
)
. As the dispersionmatrix
H
is independent of time, Eq. (5) yields
d
ω
d
τ
= 0
and
W
has on
X
=
T
∗
(
O×
R
)
\{
k
=
0
}
the form:
W
=
n
α
=1
W
α
δ
0
(
H
α
)
.
(7)The Wigner transform of
u
ε
and its associated Wignermeasureareusedtolink
W
withtheelasticenergydensityassociated to the oscillating solutions
u
ε
of (1) for
ε
→
0
.
Boundary conditions
Boundary conditions for quadratic quantities such asthe vibrational energy density shall be constructed on thebasisoftheboundaryconditionsappliedtotheunderlyingdisplacement and stress ﬁelds, for example Dirichlet orNeumann boundary conditions, or interface jump conditions between substructures (assuming that the thicknessof that interface is much smaller than the wavelength).They basically translate into reﬂection/transmission operators for the energy rays. The consideration of boundaryconditions in kinetic models raises some signiﬁcant theoretical difﬁculties related to the polarization and the possible conversion of elastic waves, and to the critical angles of incidence arising for either transmission or reﬂection problems. Indeed, the ”energy” density
W
havingits support in the sets
X
α
=
{
(
s
,
ξ
)
∈X
;
H
α
(
s
,
ξ
) = 0
}
,the following condition has to be satisﬁed on the boundary
∂
O×
R
or an interface
Σ
⊂ O×
R
, oriented by itsnormal
(
n
,n
t
)
:
H
α
(
s
,
ξ
) =
a
α
(
s
)
k
2
n
+
b
α
(
s
,
k
)
k
n
+
H
α
(
s
,
k
,ω
) = 0
,
extracting the normal componant
k
n
=
k
·
ˆ
n
of the wavevector, with
k
= (
I
d
−
ˆ
n
⊗
ˆ
n
)
k
for its tangential component. Denoting by
∆
α
(
s
,
k
,ω
)
the discriminant of this
WAVES’2011
301secondorder equation with respect to
k
n
with real coefﬁcients, the cotangent bundle to the boundary
T
∗
∂
O×
R
splits as
T
∗
∂
O×
R
=
H
α
∪
E
α
∪
G
α
(
∀
α
), where thehyperbolic
H
α
, elliptic
E
α
and tangent
G
α
diffraction regions on
∂
O×
R
are deﬁned by (see for example [4,5]):
H
α
=
{
(
s
,
k
,ω
)
∈
T
∗
∂
O×
R
; ∆
α
(
s
,
k
,ω
)
>
0
}
,
E
α
=
{
(
s
,
k
,ω
)
∈
T
∗
∂
O×
R
; ∆
α
(
s
,
k
,ω
)
<
0
}
,
G
α
=
{
(
s
,
k
,ω
)
∈
T
∗
∂
O×
R
; ∆
α
(
s
,
k
,ω
) = 0
}
,
respectively. The ﬁrst one corresponds to the transverse rays for which
k
n
is real (below critical incidence),the second one corresponds to the totally reﬂected raysfor which
k
n
∈
C
\
R
(above critical incidence), and thethird one corresponds to the tangent rays for which
k
n
=0
in the local frame of the tangent plane to the boundary at
s
(critical or tangential incidence). As for an interface
Σ
, these deﬁnitions have to be extended on bothsides since the acoustic tensor, and thus its eigenvalues,are
a priori
different;
T
∗
Σ
is then the union of all theseregions [5], but the latter are not necessarily disjoint.Mode conversions
α
↔
β
in
H
α
∩
H
β
or
H
α
∩
E
β
aredriven by the condition
d
ω
d
τ
= 0
derived from Eq. (5)which enforces the conservation of the Hamiltonian,that is
λ
α
(
x
(
τ
−
0
)
,
k
(
τ
−
0
)) =
λ
β
(
x
(
τ
+0
)
,
k
(
τ
+0
))
onany discontinuity front
Σ
of the density
W
given by
Σ =
{
(
s
,
ξ
)
∈
T
∗
(
O×
R
);
S
(
s
,
ξ
) = 0
}
, where therefore
τ
0
>
0
is such that
S
(
s
(
τ
0
)
,
ξ
(
τ
0
)) = 0
. Besides, theRankineHugoniot condition on
Σ
∩
(
∪
nα
=1
X
α
)
for thetransport equation (6) reads:
{H
,
S}
W
= 0
,
(8)where
f
=
f
(
τ
+0
)
−
f
(
τ
−
0
)
stands for the jump of
f
on the front. This condition expresses the conservationof the normal energy ﬂux on the discontinuity, but
a priori
it does not describe how the energy density is distributed among the different modes by the reﬂections andtransmissions on an interface
Σ
for example. Howeversolving a Riemann problem at that interface gives a decomposition of the normal energy ﬂux into rays movingforward and backward from it, including the effects of reﬂection and transmission. Thus the power ﬂow reﬂection and transmission coefﬁcients for the transport problem may be derived and understood in terms of some particular Riemann solutions [6]. It may be observed in addition that since
W
β
is necessarily zero in a neighbourhoodof
E
β
(away from
T
∗
Σ
) because its support is in
X
β
, itstrace on
T
∗
Σ
is zero as well, thus
W
β
≡
0
by the modeconversion
α
→
β
within
H
α
∩
E
β
. For slender structures such as thick beams or plates for example, the powerﬂow reﬂection/transmission coefﬁcients at boundaries orinterfaces are then derived from the dispersion properties of the constitutive and balance equations for thesesystems [7]. For the MindlinReissnerUﬂyand’s thick plate kinematic model, the energy density is split intothree propagative modes
α
∈ {
T
,
S
,
P
}
with
λ
α
(
x
,
k
) =
c
α
(
x
)

k

such that
c
T
< c
S
< c
P
. Polarization
T
corresponds to transverse shear and the polarizations
S
,
P
correspond to bending and inplane vibrational energies inthe mean surface of the plate. The normal power ﬂow reﬂection/transmission coefﬁcients at the junction of platesand at a ﬁxed or a free boundary are derived for the hyperbolic regions and the hyperbolicelliptic region as functions of the angle between the plates.
Numerical simulations
Numerical methods with low numerical dispersion anddissipation errors are needed to perform longtime simulations of the transport equation in order to exhibit itsdiffusion limit if any. MonteCarlo methods [8] and discontinuous ﬁnite element methods [9] both have theseproperties. MonteCarlo methods are easy to implementsince they are based on a physical interpretation of thetransport equations. They also allow to compute localsolutions, a decisive advantage over energetic methods(such as ﬁnite elements) for largescale computations.Their main drawback is their lack of versatility for complex geometries as typically encountered in structural dynamics and acoustics. Finite element methods are, onthe contrary, much more ﬂexible and can be applied totruly complex geometries. Among them, the discontinuous ”Galerkin” (DG) ﬁnite element method has srcinallybeen introduced in order to compute neutronic transfers;it is therefore adapted to the integration of transport equations. In the DG method, boundary ﬂuxes at the edgesor faces of the elements maintain the consistency of thenumerical solution with the continuous, nondiscretizedtransport equations, which are otherwise discretized oneach element without any continuity relation between elements. The numerical ﬂuxes are constructed in orderto get to satisfy the reﬂection/transmission conditions derived from the analysis of Riemann solutions for boundaries and interfaces where the propagation media are discontinuous [10].As an illustration of the previous setting, consideran hexahedral beam truss constituted by
13
threedimensional thick beams, the 13
th
beam linking two oppositeverticesofthetrussthroughitsdiagonal. Allbeamshave the same crosssections and are made from the samehomogeneous, isotropic elastic material. The initial con
302
WAVES’2011
Beam #11387212341151069
Figure 1: An hexahedral beam truss and the initialcondition in beam
#1
.
0 10 20 30 40 5000.050.10.150.20.250.3
t
×
T
beam # 1 beam # 5 beam # 10beam #110102030405000.050.10.150.20.250.3
t
×
T
beam # 2beam # 4beam # 6beam #80102030405000.10.20.3
t
×
T
beam # 3beam # 7beam # 9beam #120102030405000.20.40.60.81
t
×
T
group #1group #2group #3beam #13total
Figure 2: Evolution of the total vibrational energy foreach beam of the truss.dition is a pure compressional wave applied to, say, beamlabelled
#1
and travels from the left to the right; see Figure 1. Figure 2 shows the evolution of the total vibrationalenergy for each beam computed by a DG method. Thetime scale
T
=
L/c
T
is the time needed by a transverseshear wave of velocity
c
T
to cross the edge of length
L
of the truss. The energetic behaviour of parallel beamsis similar at late times; hence these results are displayedfor each beam group, and then the sums of the energiesfor each group. The ﬁrst group is constituted by thebeams parallel to the initially loaded beam
#1
(beams
#1
,
5
,
10
,
11
), the second group is constituted by the vertical beams
#2
,
4
,
6
,
8
, and the third group is constitutedby the remaining beams
#3
,
7
,
9
,
12
whereas the diagonalbeam
#13
isconsideredapart. Thetotalenergycomputedfor the entire truss is virtually conserved; this is a requiredproperty of the numerical scheme retained. Moreover,the vibrational energy in each beam tends to a limit as
t
→
+
∞
depending on its mechanical parameters. Thisphenomenom charaterizes the diffusive limit of transportequations holding at late times.
References
[1] P. G´erard, P.A. Markowich, N.J. Mauser,F. Poupaud, “Homogenization limits and Wignertransforms”, Communications on Pure and AppliedMathematics, vol. L, pp. 323379, 1997.[2] S. Bougacha, J.L. Akian, R. Alexandre, “Gaussian beams summation for the wave equation in aconvex domain”, Communications in MathematicalSciences, vol. 7, pp. 9731008, 2009.[3] G.C. Papanicolaou, L.V. Ryzhik, “Waves and Transport”, in Hyperbolic Equations and Frequency Interactions (L. Caffarelli, W. E, eds.), IAS/Park CityMathematics Series, vol. 5, pp. 305382, AmericanMathematical Society, Providence RI, 1999.[4] J.L. Akian, “Spacetime semiclassical measures forthreedimensional elastodynamics: boundary conditions for the hyperbolic set”, Asymptotic Analysis,submitted, 2010.[5] L. Miller, “Refraction of highfrequency waves density by sharp interfaces and semiclassical measuresat the boundary”, Journal de Math´ematiques Pureset Appliqu´ees, vol. 79, pp. 227269, 2000.[6] R.J. LeVeque, FiniteVolume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2004.[7] ´E. Savin, “A transport model for highfrequencyvibrational power ﬂows in coupled heterogeneousstructures”, Interaction and Multiscale Mechanics,vol. 1, pp. 5381, 2007.[8] B. Lapeyre, ´E. Pardoux, R. Sentis, Introduction toMonteCarlo Methods for Transport and DiffusionEquations, Oxford University Press, Oxford, 2003.[9] J.S. Hesthaven, T. Warburton, Nodal DiscontinuousGalerkin Methods, Springer, New York NY, 2008.[10] ´E. Savin, “Numerical simulation of transient vibrational power ﬂows in slender heterogeneous structures”, in Proceedings of the 10th International Conference on Computational Structures Technology,Valencia, 1417 September 2010, paper #193, CivilComp Press, Stirlingshire, 2010.