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Kinetic model for elastic wave propagation in bounded media and application to high-frequency structural acoustics

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Semiclassical analysis of strongly oscillating solutions of classical wave systems, such as the Navier equation of elastodynamics, shows that the associated elastic energy density satisfies a Liouville-type transport equation in phase space. Here
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  KINETIC MODEL FOR ELASTIC WAVE PROPAGATION IN BOUNDED MEDIA AND APPLICATION TOHIGH-FREQUENCY STRUCTURAL ACOUSTICSY. Le Guennec † , ´E. Savin † , ∗† Onera – The French Aerospace Lab, F-92322 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 Liouville-type 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 reflec-tion/transmission operators for the bicharacteristic curvesimpinging on a boundary or an interface. Specular-liketransverse boundary reflections, diffuse reflections, orfluid-structure coupling may be treated as a particularcase of the proposed model. Nodal/spectral discontinu-ous ”Galerkin” finite element methods and Monte-Carlomethods are implemented to integrate the transport equa-tionssupplementedwiththeproposedboundaryandinter-face 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 elas-tic 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 quan-tifies 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 propaga-tion/observation distances. Here u ε  is the small displace-ment field of the medium about a quasi-static equilibriumconsidered as the reference configuration; and  C ( x )  isthe fourth-order elasticity tensor of the materials occu-pying  O  of which density is denoted by   ( x ) . Sincehigh-frequency waves are generated by an initial vibra-tional energy oscillating at wavelengths  ε    1 , the ini-tial 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 su-perposition [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 coeffi-cients 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 sayhigh-frequency 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 character-istics. 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 com-pressional 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 Theraymethodwitharealphasehassomemajordraw-backs from either an Eulerian point of view or a La-grangian 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 vis-cosity 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 them-selves; 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 high-frequency solutions of wave equations. This ap-proach 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 semi-classical analysis of wave systems have generalized thistheory for weaker assumptions on their high-frequencysolutions and the initial conditions. They show that theenergy density associated to all oscillating solutions, re-solved in the phase space position × wave vector, satisfiesa Liouville-type transport equation. The main mathemat-ical tool for the derivation of a transport equation froma wave equation is the Wigner transform, of which high-frequency limit  ε  →  0 , the so-called 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 knowl-edge 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 dis-persion 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 condi-tions between substructures (assuming that the thicknessof that interface is much smaller than the wavelength).They basically translate into reflection/transmission oper-ators for the energy rays. The consideration of boundaryconditions in kinetic models raises some significant theo-retical difficulties related to the polarization and the pos-sible conversion of elastic waves, and to the critical an-gles of incidence arising for either transmission or reflec-tion 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 bound-ary  ∂  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 com-ponent. Denoting by  ∆ α ( s , k  ,ω )  the discriminant of this  WAVES’2011  301second-order equation with respect to  k n  with real coeffi-cients, 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 re-gions 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 trans-verse 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 bound-ary at  s  (critical or tangential incidence). As for an in-terface  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 there-fore  τ  0  >  0  is such that S  ( s ( τ  0 ) , ξ ( τ  0 )) = 0 . Besides, theRankine-Hugoniot 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 pri-ori   it does not describe how the energy density is dis-tributed among the different modes by the reflections andtransmissions on an interface  Σ  for example. Howeversolving a Riemann problem at that interface gives a de-composition of the normal energy flux into rays movingforward and backward from it, including the effects of reflection and transmission. Thus the power flow reflec-tion and transmission coefficients for the transport prob-lem may be derived and understood in terms of some par-ticular Riemann solutions [6]. It may be observed in addi-tion 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 struc-tures such as thick beams or plates for example, the powerflow reflection/transmission coefficients at boundaries orinterfaces are then derived from the dispersion proper-ties of the constitutive and balance equations for thesesystems [7]. For the Mindlin-Reissner-Uflyand’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  corre-sponds to transverse shear and the polarizations  S ,  P  cor-respond to bending and in-plane vibrational energies inthe mean surface of the plate. The normal power flow re-flection/transmission coefficients at the junction of platesand at a fixed or a free boundary are derived for the hyper-bolic regions and the hyperbolic-elliptic region as func-tions of the angle between the plates. Numerical simulations Numerical methods with low numerical dispersion anddissipation errors are needed to perform long-time sim-ulations of the transport equation in order to exhibit itsdiffusion limit if any. Monte-Carlo methods [8] and dis-continuous finite element methods [9] both have theseproperties. Monte-Carlo 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 large-scale computations.Their main drawback is their lack of versatility for com-plex geometries as typically encountered in structural dy-namics and acoustics. Finite element methods are, onthe contrary, much more flexible and can be applied totruly complex geometries. Among them, the discontinu-ous ”Galerkin” (DG) finite element method has srcinallybeen introduced in order to compute neutronic transfers;it is therefore adapted to the integration of transport equa-tions. In the DG method, boundary fluxes at the edgesor faces of the elements maintain the consistency of thenumerical solution with the continuous, non-discretizedtransport equations, which are otherwise discretized oneach element without any continuity relation between el-ements. The numerical fluxes are constructed in orderto get to satisfy the reflection/transmission conditions de-rived from the analysis of Riemann solutions for bound-aries and interfaces where the propagation media are dis-continuous [10].As an illustration of the previous setting, consideran hexahedral beam truss constituted by  13  three-dimensional thick beams, the 13 th beam linking two op-positeverticesofthetrussthroughitsdiagonal. Allbeamshave the same cross-sections 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 Fig-ure 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 ver-tical 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. 323-379, 1997.[2] S. Bougacha, J.-L. Akian, R. Alexandre, “Gaus-sian beams summation for the wave equation in aconvex domain”, Communications in MathematicalSciences, vol. 7, pp. 973-1008, 2009.[3] G.C. Papanicolaou, L.V. Ryzhik, “Waves and Trans-port”, in Hyperbolic Equations and Frequency In-teractions (L. Caffarelli, W. E, eds.), IAS/Park CityMathematics Series, vol. 5, pp. 305-382, AmericanMathematical Society, Providence RI, 1999.[4] J.-L. Akian, “Space-time semiclassical measures forthree-dimensional elastodynamics: boundary condi-tions for the hyperbolic set”, Asymptotic Analysis,submitted, 2010.[5] L. Miller, “Refraction of high-frequency waves den-sity by sharp interfaces and semiclassical measuresat the boundary”, Journal de Math´ematiques Pureset Appliqu´ees, vol. 79, pp. 227-269, 2000.[6] R.J. LeVeque, Finite-Volume Methods for Hyper-bolic Problems, Cambridge University Press, Cam-bridge, 2004.[7] ´E. Savin, “A transport model for high-frequencyvibrational power flows in coupled heterogeneousstructures”, Interaction and Multiscale Mechanics,vol. 1, pp. 53-81, 2007.[8] B. Lapeyre, ´E. Pardoux, R. Sentis, Introduction toMonte-Carlo 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 vibra-tional power flows in slender heterogeneous struc-tures”, in Proceedings of the 10th International Con-ference on Computational Structures Technology,Valencia, 14-17 September 2010, paper #193, Civil-Comp Press, Stirlingshire, 2010.
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