Radiative transfer in coupled plates

DÉPARTEMENT DYNAMIQUE DES STRUCTURES ET DES SYSTÈMES COUPLÉS Rapport Technique Radiative transfer in coupled plates. RT 2/11234 DDSS december 2006 RT 2/11234 DDSS –2– DECEMBER 2006 DNO ÉMETTEUR : Département Dynamique des Structures et des Systèmes Couplés DATE : december 2006 TITRE : Radiative transfer in coupled plates. AUTEURS : E. Savin NUMÉRO D’ORIGINE DU DOCUMENT : RT 2/11234 DDSS Pages : 30 Figures : 15 Réf. Biblio : 17 Version : 12th December 2006 RÉSUMÉ D’AUTEUR : The mathematical theory of microlocal analysis of hyperbolic partial differential equations shows that the energy density associated to their high-frequency solutions satisfies Liouville-type transport equations, or radiative transfer equations for randomly heterogeneous materials with correlation lengths comparable to the (small) wavelength. The main limitation to date to the existing theory is the consideration of boundary or interface conditions for the energy and power flow densities. This report deals with the radiative transfer regime in a randomly heterogeneous two-plate system. First, we propose an analytical model for the derivation of high-frequency reflection/transmission coefficients for the power flows at the plates junction. These results are used in subsequent computations to solve numerically the radiative transfer equations for this system, including the interface conditions. A transport model is finally proposed for the high-frequency guided waves which could possibly propagate along the junction line. NOTIONS D’INDEXAGE : Vibration, high frequency, transport, radiative transfer, wave propagation, guided wave. RT 2/11234 DDSS –4– DECEMBER 2006 DNO CONTENTS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. REFLECTION/TRANSMISSION COEFFICIENTS IN COUPLED PLATES 2.1. Notations and basic equations . . . . . . . . . . . . . . . . . . . . . . . 2.2. Low-frequency range analysis . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Wave components reflection/transmission coefficients . . . . . . 2.2.2. Power flow reflection/transmission coefficients . . . . . . . . . 2.3. High-frequency range analysis . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Wave components reflection/transmission coefficients . . . . . . 2.3.2. Power flow reflection/transmission coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 7 7 9 11 13 14 3. APPLICATION TO RADIATIVE TRANSFER IN COUPLED PLATES 3.1. Radiative transfer equations for a thick plate . . . . . . . . . . . 3.2. Radiative transfer for two coupled plates . . . . . . . . . . . . . . 3.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 20 22 23 4. RADIATIVE TRANSFER ALONG THE JUNCTION LINE . . . . . . . . . . . . . . . . . . 4.1. High-frequency guided wave problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Derivation of the transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 26 27 5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6. BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . . . 5 RT 2/11234 DDSS –5– DECEMBER 2006 DNO 1. INTRODUCTION The energy density associated to a high-frequency wave propagating in an isotropic, visco-elastic medium satisfies a Liouville-type transport equation in the phase space "positions × wave vectors", or a radiative transfer equation if the medium is randomly heterogeneous at length scales comparable to the (small) wavelength. This result also applies to acoustic waves, electromagnetic waves, or quantum mechanics for example [1]. It has been proved rigorously by several authors (see e.g. [2–4]) in the deterministic case, and in the random case for waves solution of a Schrödinger equation [5]. Applications dealing with beams, plates, shells or poro-visco-elastic media have been presented in [6, 7]. The objective of this research is to construct a general model which describes the energy density evolution at high-frequencies in complex structures, in order to predict, for example, the vibro-acoustic behavior of such systems in the steady-state high-frequency regimes, or their transient responses to impact loads such as pyrotechnic shocks. The main limitation of the proposed theory to date lies in the consideration of boundary and interface conditions for quadratic quantities (e.g., energy and power flow densities) consistent with the boundary/interface conditions imposed to the displacement and stress fields. Some preliminary results were given in [8] for the steady-state Navier equation in a bounded smooth domain with Dirichlet boundary conditions; their generalization to arbitrary boundary conditions is the subject of ongoing research. The energy density in bounded media may be characterized by the semi-classical measures (or Wigner measures) of the true solutions as done in [8], or the explicit Wigner measures of approximate solutions (which coincide with the Wigner measures of the true solutions) as initiated in [9]. Such approximate solutions are constructed as the superposition of Gaussian beams for arbitrary initial conditions, and they allow to account for the boundary conditions within convex domains. For non-convex domains, other classes of approximations need be introduced. In this report we focus on the formal development of boundary and interface conditions for a randomly heterogeneous two-plate system using thick plate kinematics. Our derivation of power flow reflection/transmission coefficients for the junction is based on a wave analysis, where the wave components are obtained from the high-frequency transport properties of the plates [6]. This approach also allows us to obtain the Dirichlet and Neumann boundary conditions for a semi-infinite thick plate. Radiative transfer equations can be solved numerically for the coupled system assuming that the radiative transfer model applies within each plate, and that power flows at their junction are reflected and/or transmitted (with possible mode conversions) according to the laws derived above. This analysis neglects the waves guided along the junction line, although a significant amount of energy may be transported by the latter. Thus a transport model for high-frequency guided waves need be developped as well. This report is organized as follows: the derivation of power flow reflection/transmission coefficients by a wave component approach is outlined in section 2. Then the radiative transfer regime for two coupled plates is presented in section 3. Some numerical simulations of this regime have also been performed. In section 4 we propose a transport model for the guided waves possibly propagated by the junction line. For that purpose, RT 2/11234 DDSS –6– DECEMBER 2006 DNO a general boundary impedance condition is introduced. Finally some conclusions are offered in section 5. 2. REFLECTION/TRANSMISSION COEFFICIENTS IN COUPLED PLATES In this section we show how to compute the power reflection/transmission coefficients, also called efficiencies in the literature, of wave components in two coupled, semi-infinite plates. Similar results have been reported recently in [10] for Euler-Bernoulli beams and Kirchhoff-Love plates, or in [11] for thick shells, among many other examples. Most studies are dedicated to the low-frequency range for thin plates (KirchhoffLove kinematics); a summary of the main results is presented in section 2.2 for consistency purposes and to introduce the theoretical setting. In the high-frequency range, where energy wave components have different group velocities and modes [6], an adapted analysis has to be developped. It is proposed in section 2.3 and extends to plates the approach initiated in [12] for beams. 2.1. Notations and basic equations The local coordinates on the plate mid-surfaces are denoted by (x, y) such that x ∈ ❘− for plate #1 and x ∈ ❘+ for plate #2. The junction Γ is the line {x = 0} and φ stands for the angle between plates, see FIG. 2.1. The in-plane motions of the plates are denoted by u = (u, v), the normal displacement (deflection) by w, 8 transmitted waves incident wave θ φ 8 reflected waves 0 Figure 2.1 – Two coupled thin plates. and the vector of changes of slope by θ. Their masses per unit surface are ̺j for either j = 1 or j = 2, Ej are the Young’s modulii, G′j = κj Gj are the reduced shear modulii where κj are the usual shear reduction factors Ej , and hj are the thicknesses. depending on cross-section geometries and Poisson’s coefficients νj , Gj = 2(1+ν j) The membrane forces tensor ◆, bending moments tensor ▼ and shear force T for both plates are given by (dropping subscripts for clarity purposes):   ◆ = νC(divu)■2 + (1 − ν)C∇ ⊗s u , ▼ = νD(divθ)■2 + (1 − ν)D∇ ⊗s θ , (2.1)  ′ T = G h(∇w − θ) , where: C= Eh Eh3 , D = 1 − ν2 12(1 − ν 2 ) RT 2/11234 DDSS –7– DECEMBER 2006 DNO are the plates membrane and bending stiffnesses, respectively, ■2 is the identity matrix of ❘2 , and ∇ = (∂x , ∂y ) is the gradient vector on the mid-surface. The dynamic equilibrium equations are:   ̺ü = Div◆ , 2 (2.2) ̺ h12 θ̈ = Div▼ + T ,  ̺ẅ = divT . 2.2. Low-frequency range analysis Let us assume that θ = −∇w (Kirchhoff-Love kinematics). Then the membrane forces tensor ◆, bending moments tensor ▼ and shear force T for both plates are given by:   ◆ = νC(divu)■2 + (1 − ν)C∇ ⊗s u , ▼ = −νD∆w ■2 − (1 − ν)D∇ ⊗ ∇w , (2.3)  T = Div▼ , and the dynamic equilibrium equations reduce to:  ̺ü = Div◆ , ̺ẅ = divDiv▼ . q q Let cp = C̺ and cs = Gh . Plates wavenumbers for plane waves at the frequency ω are denoted by: ̺ r 2 ω ω ω 4 ̺ω kp (ω) = , ks (ω) = , kE (ω) = = cp cs D cE (ω) (2.4) for, respectively, the in-plane longitudinal and transverse motions, and bending motion. They are such that kp2 = λ2p + k 2 , ks2 = λ2s + k 2 and kE2 = λ2E + k 2 = λ2e − k 2 , where k is the horizontal (parallel to the junction) wavenumber which is kept unchanged by the reflection/transmission processes at the junction. It is a real number for propagating waves, however λp , λs and λE may be either real (propagating, far-field waves) or purely imaginary (evanescent, near-field waves) depending on the value of k with respect to kp , ks and kE . By definition, λe is always real here. 2.2.1. Wave components reflection/transmission coefficients Let us consider an incident plane wave travelling in plate #1 in the direction of increasing x with an angle θ with respect to the normal to the junction. It may be either a bending wave: wi (x, y) = e−ikE1 (x cos θ+y sin θ) , k = kE1 sin θ , or an in-plane longitudinal wave: upi (x, y) =   cos θ −ikp1 (x cos θ+y sin θ) e , k = kp1 sin θ , sin θ or an in-plane transverse wave: usi (x, y) =   − sin θ −iks1 (x cos θ+y sin θ) e , k = ks1 sin θ , cos θ RT 2/11234 DDSS –8– DECEMBER 2006 DNO with i = √ −1. The reflected waves in plate #1 are: wr (x, y) = Aei(λE1 x−ky) + Be(λe1 x−iky) for the bending motion, and F ur (x, y) = kp1     G −λp1 i(λp1 x−ky) k e + ei(λs1 x−ky) k λ ks1 s1 for the in-plane motion. The transmitted waves in plate #2 are: wt (x, y) = He−i(λE2 x+ky) + Je−(λe2 x+iky) for the bending motion, and K ut (x, y) = kp2     L −k −i(λs2 x+ky) λp2 −i(λp2 x+ky) e + e k ks2 λs2 for the in-plane motion. Coefficients A, B, F , G, H, J, K and L are determined from the continuity conditions at the junction x = 0, that is, if n̂ is the unit normal to the junction, {u, w, ∂n̂ w} and {◆n̂, T · n̂, ▼ : n̂ ⊗ n̂} are continuous across the junction. These conditions yield:  ∂x wt = ∂x wi + ∂x wr ,     wt = (wi + wr ) cos φ − (ui + ur ) sin φ ,     ut = (wi + wr ) sin φ + (ui + ur ) cos φ ,    vt = vi + vr , 2 2 D2 (∂x wt + ν2 ∂y wt ) = D1 [∂x2 (wi + wr ) + ν1 ∂y2 (wi + wr )] ,     −D2 ∂x ∆wt = −D1 ∂x ∆(wi + wr ) cos φ − C1 [∂x (ui + ur ) + ν1 ∂y (ui + ur )] sin φ ,     C2 (∂x ut + ν2 ∂y ut ) = −D1 ∂x ∆(wi + wr ) sin φ + C1 [∂x (ui + ur ) + ν1 ∂y (ui + ur )] cos φ ,    (1 − ν2 )C2 (∂x vt + ∂y ut ) = (1 − ν1 )C1 [∂x (vi + vr ) + ∂y (ui + ur )] . They can be written as [T ]C = Ui with C = (A, B, F, G, H, J, K, L)T and:   T11 T12 [T ] = T21 T22 where: T11 λE1 kE1  λ e1 e2 E2 −i kλE1 0 0 −i kλE1 kE1 k − cos φ − cos φ − λp1 sin φ  1 sin φ  1    kp1 ks1 =  , T12 =  0 λp1 k 0   − sin φ − sin φ kp1 cos φ − ks1 cos φ s1 0 0 − λks1 0 0 − kkp1  T21 0 0 λp2 kp2 k kp2  0 0   , − kks2   λs2 ks2   ν1 k2 +λ2E1 λ2e1 −ν1 k2 −( k2 ) 0 0 2 kE1 E1    λE1 cos φ i λe1 cos φ µ ( ν1 k2 +λ2p1 ) sin φ −µ (1 − ν ) kλs1 sin φ  kE1  2 1 1 1 kE1 kp1 ks1 kp1 , = 2 2 ν1 k +λp1  λE1  λe1 kλs1 sin φ i sin φ −µ ( cos φ ) cos φ µ (1 − ν )  kE1  2 1 1 1 kp1 ks1 kE1 kp1   λ2s1 −k2 kλp1 0 0 2 k2 kp1 ks1 p1 (2.5) (2.6) RT 2/11234 DDSS –9– DECEMBER 2006 DNO and finally: T22 q    =   ̺2 D2 ν2 k2 +λ2E2 ( k2 ) ̺1 D1 E2 ̺2 kE1 λE2 ( k2 ) ̺1 E2 q ̺2 D2 ν2 k2 −λ2e2 ( k2 ) ̺1 D1 E2 ̺2 kE1 λe2 i ̺1 ( k 2 ) E2 0 0 0 0 introducing the frequency parameter µ defined by: µ= 0 0      2 2 C2 ν2 k +λp2 C2 kλs2  µ1 C1 ( kp1 kp2 ) −µ1 (1 − ν2 ) C1 ( kp1 ks2 ) (1−ν2 )C2 λ2s2 −k2 (1−ν2 )C2 kλp2 ( ) ( ) 2 (1−ν (1−ν1 )C1 kp1 ks2 1 )C1 kp1 kp2 0 0 kE kp C = 3 . kp kE D (2.7) (2.8) The loading vectors are:
T UEi = cos θ, cos φ, sin φ, 0, (cos2 θ + ν1 sin2 θ), cos θ cos φ, − cos θ sin φ, 0 for an incident bending wave, and: Upi = 0, − cos θ sin φ, cos θ cos φ, sin θ, 0, −µ1 (cos2 θ + ν1 sin2 θ) sin φ, µ1 (cos2 θ + ν1 sin2 θ) cos φ, sin 2θ T  kp1 ks1 sin 2θ sin φ, −µ sin 2θ cos φ, cos 2θ Usi = 0, sin θ sin φ, − sin θ cos φ, cos θ, 0, µ1 kkp1 1 ks1 kp1 s1 for incident in-plane waves. 2.2.2. Power flow reflection/transmission coefficients The transient power flow within a thin Kirchhoff-Love plate is [13]: Π = −ℜe{◆u̇ − ▼∇ẇ + Tẇ} and the time averaged power flow associated to travelling plane waves is: 1 < Π >= ℜe{iω 2 ◆u − ▼∇w + Tw
The incident power flow at the junction x = 0 for a bending wave is thus:   cos θ E 2 , < Πi >= ω ̺1 cE1 (ω) sin θ while for a longitudinal in-plane wave it is: < Πpi   1 2 cos θ >= ω ̺1 cp1 , sin θ 2 < Πsi   1 2 cos θ . >= ω ̺1 cs1 sin θ 2 and for a transverse in-plane wave it is: }.
T , RT 2/11234 DDSS – 10 – DECEMBER 2006 DNO The reflected power flow away from the junction is:        1 2 1 2 2 −ℜe{λp1 } 2 −ℜe{λs1 } 2 2 −ℜe{λE1 } + cs1 |G| , + cp1 |F | < Πr >= ω̺1 cE1 (ω)|A| k k k 2 2 and the transmitted power flow away from the junction is:        1 2 1 2 2 2 ℜe{λE2 } 2 ℜe{λp2 } 2 ℜe{λs2 } + cp2 |K| < Πt >= ω̺2 cE2 (ω)|H| + cs2 |L| . k k k 2 2 These results show that if either λp or λs is purely imaginary, the reflected/transmitted power flows are localized along the junction. Power reflection/transmission coefficients for coupled thin plates are computed by: ρ11 αβ = − < Παr · n̂1 > < Πβi · n̂1 > 12 , ταβ =− < Παt · n̂2 > < Πβi · n̂1 > , α, β ∈ {E, p, s} . They are plotted as functions of the dimensionless frequency number µ1 and the junction angle φ ∈ [0, π[ on k FIG. 2.2 and FIG. 2.3 for a bending incident wave in plate #1 with θ = 0◦ (k = 0) and θ = 30◦ (k = 2β ), respectively. Both plates have the same parameters E, h, ̺, and ν with ν1 = ν2 = 0.3. Note that the lowfrequency limit corresponds to µ1 → ∞ and the high-frequency limit corresponds to µ1 → 0; in the latter case Kirchhoff-Love’s model is irrelevant and a more adapted high-frequency model is desired. Figure 2.2 – Power reflection/transmission coefficients of two connected, semi-infinite thin plates for a bending incident wave in plate #1; incidence angle θ = 0◦ , E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3; (a) reflection coefficient for bending energy, (b) reflection coefficient for in-plane longitudinal energy, (c) reflection coefficient for in-plane transverse energy, (d) transmission coefficient for bending energy, (e) transmission coefficient for in-plane longitudinal energy, (f) transmission coefficient for in-plane transverse energy. RT 2/11234 DDSS – 11 – DECEMBER 2006 DNO Figure 2.3 – Power reflection/transmission coefficients of two connected, semi-infinite thin plates for a bending incident wave in plate #1; incidence angle θ = 30◦ , E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3; (a) reflection coefficient for bending energy, (b) reflection coefficient for in-plane longitudinal energy, (c) reflection coefficient for in-plane transverse energy, (d) transmission coefficient for bending energy, (e) transmission coefficient for in-plane longitudinal energy, (f) transmission coefficient for in-plane transverse energy. 2.3. High-frequency range analysis In the general case θ is not directly related to w and one has to consider the more general UflyandMindlin kinematics for thick plates. The wave components analysis of section 2.2 √ has to be modified accordingly, following for example the derivation of reference [14]. Introducing ct = κcs and kt (ω) = cωt , the plates wavenumbers for plane waves at the frequency ω solutions of (2.2) are: ω ω kp (ω) = , ks (ω) = cp cs for the in-plane motion, and: s ka (ω) = ± s
1 2 kp + kt2 + 2 r kE4 +
2 1 2 kt − kp2 , 4 r
2
1 2 1 2 2 if ω ≥ ωc , kp + kt − kE4 + kt − kp2 kb (ω) = ± 4 s2r
2 1
1 2 kE4 + kt − kp2 − kp2 + kt2 if ω < ωc , kb (ω) = ±i 4 2 ka kb ks kc (ω) = kp kt RT 2/11234 DDSS – 12 – DECEMBER 2006 DNO √ t . We observe that if kb is purely imaginary, so is kc . for the bending motion; the cutoff frequency is ωc = 12c h 2 2 As in section 2.2, these wavenumbers are written kp = λp + k 2 , ks2 = λ2s + k 2 , ka2 = λ2a + k 2 , kb2 = λ2b + k 2 , and kc2 = λ2c + k 2 , where k is the horizontal (parallel to the junction) wavenumber which is kept unchanged by the reflection/transmission processes. As above, it is a real number for propagating waves, however λp , λs , λa , λb , and λc may be either real (propagating, far-field waves) or purely imaginary (evanescent, near-field waves) depending on the value of k with respect to kp , ks , ka , kb , and kc , respectively. An incident wave travelling in plate #1 in the direction of increasing x with an angle θ with respect to the normal to the junction may be written either:     1  w(x, y)   cos θ −ika1 (x cos θ+y sin θ) 2 = e , k = ka1 sin θ , kt1 θ(x, y) i −ika1 1 − k2 a1 sin θ for bending motion, or:  cos θ −ikp1 (x cos θ+y sin θ) e , k = kp1 sin θ , upi (x, y) = sin θ   − sin θ −iks1 (x cos θ+y sin θ) usi (x, y) = e , k = ks1 sin θ cos θ  (2.9) (2.10) for in-plane motions. Then the reflected waves in plate #1 are:     1   w(x, y) 2 −λa1  ei(λa1 x−ky) = A kt1 θ(x, y) r −i(1 − k2 ) a1 k     0 1   i(λ x−ky) b1    k  ei(λc1 x−ky) −λb1 e + iF +B k2 −i(1 − k2t1 ) λc1 k b1 for the bending motion, and: H ur (x, y) = kp1     I −λp1 i(λp1 x−ky) k ei(λs1 x−ky) e + k ks1 λs1 (2.11) for in-plane motions. The transmitted waves in plate #2 are:     1   w(x, y) λ  e−i(λa2 x+ky) =J k2 θ(x, y) t −i(1 − k2t2 ) a2 a2 k     0 1   2 λb2  e−i(λb2 x+ky) − iL −k  e−i(λc2 x+ky) +K kt2 −i(1 − k2 ) λc2 k b2 for the bending motion, and: M ut (x, y) = kp2 for in-plane motions.     λp2 −i(λp2 x+ky) N −k −i(λs2 x+ky) e + e k ks2 λs2 (2.12) RT 2/11234 DDSS – 13 – DECEMBER 2006 DNO 2.3.1. Wave components reflection/transmission coefficients In the high-frequency limit µ = kkEp → 0 we have ka = kt , kb = kp and kc = ks . The analysis in [6] has shown that five propagating modes exist in this limit: two modes with wavenumber kp , two modes with wavenumber ks , and one mode with wavenumber kt . Introducing as above kt2 = λ2t + k 2 , where k ∈ ❘ and λt ∈ ❘ if k ≤ kt or λt ∈ i❘ if k > kt , the different types of waves travelling in the plates may be modified as follows. Incident waves travelling in plate #1 in the direction of increasing x with an angle θ with respect to the normal to the junction may be written either: wi (x, y) = e−ikt1 (x cos θ+y sin θ) , k = kt1 sin θ for quasi-shear motion, or:  cos θ −ikp1 (x cos θ+y sin θ) e , k = kp1 sin θ , θ pi (x, y) = −ikp1 sin θ   − sin θ −iks1 (x cos θ+y sin θ) e , k = ks1 sin θ θ si (x, y) = −iks1 cos θ  for quasi-bending motions, or as given by Eqs. (2.9)–(2.10) for in-plane motions. The reflected waves in plate #1 may be written: wr (x, y) = Aei(λt1 x−ky) for quasi-shear motion,     −λp1 i(λp1 x−ky) k θ r (x, y) = iB e + iF ei(λs1 x−ky) λs1 k for quasi-bending motions, and by Eq. (2.11) for in-plane motions. The transmitted waves in plate #2 may be written: wt (x, y) = Je−i(λt2 x+ky) for quasi-shear motion,    λp2 −i(λp2 x+ky) −k −i(λs2 x+ky) e θ t (x, y) = −iK e − iL λs2 k  for quasi-bending motions, and by Eq. (2.12) for in-plane motions. Coefficients A, B, F , H, I, J,K, L, M and N are obtained from the continuity conditions at the junction x = 0, that is, {u, θ, w} and {◆n̂, ▼n̂, T · n̂} are continuous across the junction. These conditions yield the equations:  θt = θi + θr ,  D2 (∂x θtx + ν2 ∂y θty ) = D1 [∂x (θix + θrx ) + ν1 ∂y (θiy + θry )] ,  (1 − ν2 )D2 (∂x θty + ∂y θtx ) = (1 − ν1 )D1 [∂x (θiy + θry ) + ∂y (θix + θrx )] , and:                wt ut vt G′2 h2 (∂x wt − θx ) C2 (∂x ut + ν2 ∂y ut ) (1 − ν2 )C2 (∂x vt + ∂y ut ) = (wi + wr ) cos φ − (ui + ur ) sin φ , = (wi + wr ) sin φ + (ui + ur ) cos φ , = vi + vr , = G′1 h1 [∂x (wi + wr ) − (θix + θrx )] cos φ − C1 [∂x (ui + ur ) + ν1 ∂y (ui + ur )] sin φ , = G′1 h1 [∂x (wi + wr ) − (θix + θrx )] sin φ + C1 [∂x (ui + ur ) + ν1 ∂y (ui + ur )] cos φ , = (1 − ν1 )C1 [∂x (vi + vr ) + ∂y (ui + ur )] . RT 2/11234 DDSS – 14 – DECEMBER 2006 DNO The first four equations are written [S]C = [U ] for C = (B, F, K, L), with: [S] = and: λp1 kp1  k  ks1  λ2p1 +ν1 k2  2  kp1 2 λkp12 k s1  p2 − λkp1 − kkp1 λs1 ks1 −2 λks12 k s1 λ2s1 −k2 2 ks1 k ks1 ̺2 h22 λ2p2 +ν2 k2 ( k2 ) ̺1 h21 p2 ̺ h2 −2 ̺12 h22 ( λkp22 k ) 1 s2 k kp1  λs2  ks1 ̺2 h22 λs2 k  −2 ̺1 h2 ( k2 ) 1 s2  ̺2 h22 k2 −λ2s2 ( k2 ) ̺1 h21 s2  , (2.13)  ks1 − cos θ sin θ kp1 kp1  sin θ cos θ    ks1 [U ] =  . 2 2 cos θ + ν1 sin θ − sin 2θ  k2 − kp1 − cos 2θ 2 sin 2θ  (2.14) s1 The last six equations are written [T ]D = [V ] for D = (A, H, I, J, M, N ), with:   λp2 k sin φ − sin φ −1 0 0 cos φ kp2 ks2   λp1 λp2 k k − − sin φ cos φ − cos φ  0 kp1 ks1 kp2 ks2   λ λ k k s2 s1  0 − 0 − kp1 ks1 kp2 ks2   2 2 , ̺2 cp2 λp2 +ν2 k [T ] =  ̺2 ct2 λt2 ̺2 cs2 λs2 k  λkt1 0 0 ( ) cos φ ̺1 ct1 ( k2 ) sin φ −2 ̺1 ct1 ( k2 ) sin φ  ̺1 ct1 kt2   t1 p2 s2 2 2   λ2p1 +ν1 k2 ̺2 cp2 λp2 +ν2 k ̺ c ̺ c c λ k λ λ k 2 t2 2 s2 s1 s1 t2 s2  0 −( 2 ) 2 c ( 2 ) − ̺ c ( k ) sin φ ̺ c ( 2 ) cos φ −2 ̺ c ( 2 ) cos φ k k k k p1 1 p1 t2 1 p1 1 p1   p1 2 kλp1p1kks1 0 s1 λ2s1 −k2 2 ks1 p2 0 s2 ( λp2 k ) 2 ̺̺21 ccs2 s1 kp2 ks2 ̺2 cs2 λ2s2 −k2 ( k2 ) ̺1 cs1 s2 (2.15) and:  1 0 0 0 0  0 0 0 cos θ − sin θ      0 0 0 sin θ cos θ   k [V ] = cos θ f (φ) − p1 cos θ f (φ) + ks1 sin θ . 0 0 1 1   kt1 kt1   f2 (φ) f2 (φ) cos θ2 + ν1 sin θ2 − kkp1 sin 2θ   0 s1 kp1 sin 2θ cos 2θ 0 0 0 ks1  f1 and f2 depend on the solution C to the previous system by:  λp2 ̺2 kt1 k f1 (φ) = − λkp1 B + F + K− ̺1 kt2 t1  kt2  kt1 λp2 K − kkt2 L sin φ . f2 (φ) = − ̺̺21kkp1 kt2 t2 2.3.2. Power flow reflection/transmission coefficients The transient power flow within a thick Mindlin plate is: Π = −ℜe{◆u̇ + ▼θ̇ + Tẇ} k L kt2  cos φ , (2.16) RT 2/11234 DDSS – 15 – DECEMBER 2006 DNO and the time averaged power flow associated to travelling plane waves is: 1 < Π >= ℜe{iω 2 ◆u + ▼θ + Tw The incident power at the junction x = 0 is thus: < Πti
}.   1 2 cos θ >= ω ̺1 ct1 sin θ 2 for the shear component, < 4 kp1 1 >= ω 2 ̺1 cp1 4 2 kE1 Πpb i  cos θ sin θ  , < Πsb i 2 2 kp1 ks1 1 >= ω 2 ̺1 cs1 4 2 kE1  cos θ sin θ  for the bending components, and: <     1 2 1 2 cos θ cos θ sn , < Πi >= ω ̺1 cs1 >= ω ̺1 cp1 sin θ sin θ 2 2 Πpn i for the in-plane components. The reflected power flows away from the junction are:        1 k −ℜe{λp1 } t 2 −ℜe{λt1 } 2 + AF + AB < Πr >= ω̺1 ct1 |A| ℜe{λs1 } k k 2 for the shear component, < Πbr     4  kp1 1 2 −ℜe{λs1 } 2 −ℜe{λp1 } 2 >= ω̺1 cp1 4 |B| + |F | k k 2 kE1 for the bending component, and: < Πnr      1 2 2 −ℜe{λs1 } 2 2 −ℜe{λp1 } + cs1 |I| >= ω̺1 cp1 |H| k k 2 for the membrane component. The transmitted power flows away from the junction are:        1 −k ℜe{λp2 } t 2 ℜe{λt2 } 2 < Πt >= ω̺2 ct1 |J| − JL − JK ℜe{λs2 } k k 2 for the shear component, < Πbt     4  kp2 1 2 ℜe{λs2 } 2 ℜe{λp2 } 2 + |L| >= ω̺2 cp2 4 |K| k k 2 kE2 for the bending component, and: < Πnt      1 2 2 ℜe{λs2 } 2 2 ℜe{λp2 } + cs2 |N | >= ω̺2 cp2 |M | k k 2 for the membrane component. RT 2/11234 DDSS – 16 – DECEMBER 2006 DNO These results are used to compute the various high-frequency power reflection/transmission coefficients for coupled Mindlin plates by: ρ11 αβ = − < Παr · n̂1 > < Πβi · n̂1 > 12 , ταβ =− < Παt · n̂2 > < Πβi · n̂1 > , α, β ∈ {t, pn, pb, sn, sb} . (2.17) They are frequency-independent and plotted as functions of φ (the junction angle in the range [0, π[) on FIG. 2.4 through FIG. 2.6 for various incidence angle θ and two identical plates: E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. FIG. 2.7 through FIG. 2.9 are the same for two different plates: 2E1 = E2 , all other parameters 2 → 0 (Neumann boundary condition) being unchanged. It can be checked in addition that the limiting cases E E1 E1 and E2 → 0 (Dirichlet boundary condition) yield the expected boundary reflection laws for plate #1 with due consideration of the critical incidence angle: r 1−ν θc = arcsin 2 and of course no transmission. The corresponding reflection coefficients as functions of the incidence angle θ are plotted on FIG. 2.10 and FIG. 2.11 for Dirichlet and Neumann boundary conditions, respectively, with ν = 0.3 so that θc ≃ 36◦ . One can observe that the quasi-shear flow is uncoupled from the in-plane and bending flows in both cases, independently of the incidence direction. Mode conversions occur only for p and s waves. In−plane P incident wave 1 0.5 0 1 0 30 In−plane S incident wave 60 90 φ 150 180 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave trans. quasi−shear wave trans. in−plane P wave trans. in−plane S wave 0.5 0 120 0 30 Quasi−shear incident wave 60 90 120 150 180 0 60 90 120 150 180 1 φ 0.5 0 30 φ Figure 2.4 – High-frequency power reflection/transmission coefficients for two connected, semi-infinite Mindlin plates; incidence angle θ = 0◦ , E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. RT 2/11234 DDSS – 17 – DECEMBER 2006 DNO In−plane P incident wave 1 0.5 0 0 30 In−plane S incident wave 60 90 120 150 180 0 30 Quasi−shear incident wave 60 90 120 150 180 1 φ 0.5 0 1 φ refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave trans. quasi−shear wave trans. in−plane P wave trans. in−plane S wave 0.5 0 0 30 60 90 φ 120 150 180 Figure 2.5 – High-frequency power reflection/transmission coefficients for two connected, semi-infinite Mindlin plates; incidence angle θ = 45◦ , E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. In−plane P incident wave 1 0.5 0 0 30 In−plane S incident wave 60 1 90 φ 150 180 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave trans. quasi−shear wave trans. in−plane P wave trans. in−plane S wave 0.5 0 120 0 30 Quasi−shear incident wave 60 90 120 150 180 0 60 90 120 150 180 1 φ 0.5 0 30 φ Figure 2.6 – High-frequency power reflection/transmission coefficients for two connected, semi-infinite Mindlin plates; incidence angle θ = 89◦ , E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. RT 2/11234 DDSS – 18 – DECEMBER 2006 DNO In−plane P incident wave 1 0.5 0 1 0 30 In−plane S incident wave 60 90 φ 150 180 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave trans. quasi−shear wave trans. in−plane P wave trans. in−plane S wave 0.5 0 120 0 30 Quasi−shear incident wave 60 90 120 150 180 0 60 90 120 150 180 1 φ 0.5 0 30 φ Figure 2.7 – High-frequency power reflection/transmission coefficients for two connected, semi-infinite Mindlin plates; incidence angle θ = 0◦ , 2E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. In−plane P incident wave 1 0.5 0 0 30 In−plane S incident wave 60 90 120 150 180 0 60 90 120 150 180 120 150 180 1.5 φ 1 0.5 0 30 Quasi−shear incident wave 6 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave trans. quasi−shear wave trans. in−plane P wave trans. in−plane S wave 4 2 0 φ 0 30 60 90 φ Figure 2.8 – High-frequency power reflection/transmission coefficients for two connected, semi-infinite Mindlin plates; incidence angle θ = 45◦ , 2E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. RT 2/11234 DDSS – 19 – DECEMBER 2006 DNO In−plane P incident wave 1 0.5 0 0 30 In−plane S incident wave 60 90 120 150 180 0 30 Quasi−shear incident wave 60 90 120 150 180 φ 1 0.5 0 φ 1 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave trans. quasi−shear wave trans. in−plane P wave trans. in−plane S wave 0.5 0 0 30 60 90 φ 120 150 180 Figure 2.9 – High-frequency power reflection/transmission coefficients for two connected, semi-infinite Mindlin plates; incidence angle θ = 89◦ , 2E1 = E2 , h1 = h2 , ̺1 = ̺2 , and ν1 = ν2 = 0.3. In−plane P incident wave 1 0.5 0 0 15 In−plane S incident wave 30 4 45 θ 60 75 90 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave 3 2 1 0 0 15 Quasi−shear incident wave 30 45 60 75 90 0 30 45 60 75 90 θ 1 0.5 0 15 θ Figure 2.10 – High-frequency power reflection coefficients for a semi-infinite Mindlin plate with Dirichlet boundary condition; ν = 0.3. RT 2/11234 DDSS – 20 – DECEMBER 2006 DNO In−plane P incident wave 1 0.5 0 0 15 In−plane S incident wave 30 45 60 75 90 0 15 Quasi−shear incident wave 30 45 60 75 90 θ 10 5 0 θ 1 refle. quasi−shear wave refle. in−plane P wave refle. in−plane S wave 0.5 0 0 15 30 45 θ 60 75 90 Figure 2.11 – High-frequency power reflection coefficients for a semi-infinite Mindlin plate with Neumann boundary condition; ν = 0.3. 3. APPLICATION TO RADIATIVE TRANSFER IN COUPLED PLATES The derivation of section 2 is used to simulate energy flows in coupled thick plates by the radiative transfer theory outlined in [6]. The latter is recalled in the following section 3.1 for a single plate, then it is applied to coupled plates in a subsequent section 3.2 including the interface conditions. Here one considers that the radiative transfer regime holds in each plate, and the power flows are reflected/transmitted along the junction line assuming that the reflection/transmission efficiencies are those derived in section 2.3. Some numerical simulations by the direct Monte-Carlo method are presented in section 3.3. 3.1. Radiative transfer equations for a thick plate One considers high-frequency wave propagation in an heterogeneous, visco-elastic thick plate with statistically isotropic random perturbations of its mechanical parameters, namely its density ̺, Young’s modulus E, and reduced shear modulus G′ . The plate is embedded in the domain O of ❘2 . The equations of motion are (2.2) together with the material behavior laws (2.1). The evolution of the associated energy density is RT 2/11234 DDSS – 21 – DECEMBER 2006 DNO described by the following multigroup radiative transfer equations [6]: Z σpp (x, k, p)[Wp (x, p, t) − Wp (x, k, t)]δ(cp |p| − cp |k|) dp ∂t Wp + {ωp , Wp } = ❘2 Z σps (x, k, p)[Ws (x, p, t) − Wp (x, k, t)]δ(cs |p| − cp |k|) dp , (3.1) + ❘2 ∂t Ws + {ωs , Ws } = Z ❘2 σss (x, k, p)[Ws (x, p, t) − Ws (x, k, t)]δ(cs |p| − cs |k|) dp Z σsp (x, k, p)[Wp (x, p, t) − Ws (x, k, t)]δ(cp |p| − cs |k|) dp , (3.2) + ❘2 and ∂t wt + {ωt , wt } = Z ❘2 σtt (x, k, p)[wt (x, p, t) − wt (x, k, t)]δ(ct |p| − ct |k|) dp , (3.3) which couple three different energy propagation modes (or rays) α = p, s or t. {f, g} = ∇k f · ∇x g − ∇x f · ∇k g stands for the usual Poisson’s bracket, ωα (x, k) = cα (x)|k| is the eigenfrequency associated to the mode α of which energy velocity is cα , and σαβ ≥ 0 is the scattering cross-section which gives the rate of conversion of an energy ray β in the direction p̂ to another ray α in the direction k̂, at position x and wavenumber |k|. The standard notation k = |k|k̂ is used throughout this section. In a deterministic medium all scattering crosssections are zero and right-hand-sides vanish in Eq. (3.1) through (3.3) which become Liouville, or transport equations. Then the space-time energy density of the plate is given by: Z XZ wt (x, k, t)dk + TrWα (x, k, t)dk , (3.4) E(x, t) = ❘2 α=s,p ❘2 while its power flow density vector is estimated by: Z Z X wt (x, k, t)k̂dk + TrWα (x, k, t)k̂dk . Π(x, t) = ct (x) cα (x) ❘2 α=s,p ❘2 (3.5) In the above, Wp is the so-called 2 × 2 coherence matrix of specific intensities for bending and in-plane longitudinal waves, and Ws is its counterpart for transverse waves. Both matrices are non-negative. wt ≥ 0 is the specific intensity, or phase space energy density, for quasi-shear waves. Wp (x, k, t), Ws (x, k, t) and wt (x, k, t) are derived as weak-* (in the sense of temperate distributions) high-frequency limit measures of the Wigner transform of the state vector v = (u̇, θ̇, ẇ, ◆, T, ▼) which is solution of Eqs. (2.1)–(2.2); see e.g. [1–4] and also section 4.2 below. Expressions of the scattering cross-sections have been derived in [6]. They are given by: i h π 2 2 2 σtt (x, k, p) = ct (x)|k| R̺̂ (k − p) + (k̂ · p̂) R̂G (k − p) + 2(k̂ · p̂)ℜeR̺̂G (k − p) , (3.6) 2 for the propagative quasi-shear mode labelled t, and: h π σpp (x, k, p) = c2p (x)|k|2 (k̂ · p̂)2 R̺̂ (k − p) + [(1 − ν)(k̂ · p̂)2 + ν]2 R̂E (k − p) 2 i +2(k̂ · p̂)[(1 − ν)(k̂ · p̂)2 + ν]ℜeR̺̂E (k − p) , (3.7) RT 2/11234 DDSS – 22 – DECEMBER 2006 DNO σss (x, k, p) = π 2 c (x)|k|2 (k̂ · p̂)2 R̺̂ (k − p) + [2(k̂ · p̂)2 − 1]2 R̂E (k − p) 2 s i +2(k̂ · k̂′ )[2(k̂ · p̂)2 − 1]ℜeR̺̂E (k − p) , (3.8) h h π 2 2 σps (x, k, p) = cs (x)(1 − (k̂ · p̂) ) |p|2 R̺̂ (k − p) + 4|k|2 (k̂ · p̂)2 R̂E (k − p) 2 i +4|k||p|(k̂ · p̂)ℜeR̺̂E (k − p) (3.9) for the propagative bending and in-plane modes labelled p and s, with σsp (x, k, p) = σps (x, p, k). Here k 7→ R̺̂ (k), k 7→ R̂G (k), k 7→ R̂E (k), and k 7→ R̺̂G (k), k 7→ R̺̂E (k) are the power spectral density functions and cross-spectral density functions of the three stochastic processes which represent the dimensionless random perturbations of parameters ̺, 1/G′ and 1/E, respectively. The latter are real-valued, mean-zero and statistically homogeneous (stationary) at the same length scale as the wavelength. 3.2. Radiative transfer for two coupled plates Now we consider two thick plates which occupy the domains O1 and O2 of ❘2 such that their junction ∂O1 ∩ ∂O2 = Γ is a smooth curve of which outward unit normal with respect to Or , r = 1, 2, is denoted by n̂r (x), x ∈ Γ, with n̂1 (x) = −n̂2 (x). We introduce the notation k′ = PΓ k where PΓ = ■2 − n̂1 ⊗ n̂1 = ■2 − n̂2 ⊗ n̂2 is the orthogonal projection on the tangent line to Γ at x. The radiative transfer theory outlined in the previous section holds in the interior of each plate. The interface conditions at the junction are those derived in section 2.3, provided that the background media are homogeneous (crα is independent of x). r r r r Let wαn = Wα,11 and wαb = Wα,22 for either α = p or α = s. The radiative transfer equations in Or are: XZ r r r r ∂t wα + cα k̂ · ∇x wα = σαβ (x, k, p)(wβr (x, p, t) − wαr (x, k, t))δ(crβ |p| − crα |k|) dp (3.10) β ❘2 r r with α, β ∈ {t, pn, pb, sn, sb} and σtα = σαt = 0 whenever α 6= t. For the energy transfer in O1 flowing away from the boundary after reflection and transmission one has:  X 1 1 1 ′ 1 1 1 11 ′ 1 ′ 1 2 2 12 ′ 2 ′ 2 cα λ̂α wα (x, k , −λα , t) = cβ λ̂β ραβ (k )wβ (x, k , λβ , t) + cβ λ̂β ταβ (k )wβ (x, k , λβ , t) , x ∈ Γ , (3.11) β and for the energy transfer in O2 flowing away from the boundary one has:  X ′ 2 ′ 2 1 1 21 ′ 1 ′ 1 c2α λ̂2α wα2 (x, k′ , −λ2α , t) = c2β λ̂2β ρ22 (k )w (x, k , λ , t) + c λ̂ τ (k )w (x, k , λ , t) , x ∈ Γ , (3.12) αβ β β β β αβ β β β with α, β ∈ {t, pn, pb, sn, sb}, and ραβ and ταβ are given by Eq. (2.17). Here the normal wavenumbers on the boundary from Or , r = 1, 2, are: s  2 ω ω r ′ r ′ − |k′ |2 , λα (k ) = r λ̂α (k ) = cα crα RT 2/11234 DDSS – 23 – DECEMBER 2006 DNO such that k = k′ + λrα n̂r for the energy wave vector of the mode α in Or . We consider only propagating waves because the energy in evanescent waves is exponentially small away from the boundary. Therefore we assume that the normal components λrα are in ❘+ and the support of wαr is uniformly inside the ball {|k′ | < cωr }. The α tangent wave vector k′ is kept unchanged by the reflection/transmission process on the boundary as a result of the Snell-Descartes law. 3.3. Numerical examples Equations (3.10) through (3.12) supplemented with initial conditions of the form wαr (x, k, 0) = gαr (x, k), are solved numerically by a direct Monte-Carlo method [15]. It consists in re-interpreting the radiative transfer equations as forward Kolmogorov, or Fokker-Planck equations for some underlying jump Markov processes. A population of trajectories is constructed as follows: between jumps particles evolve on the characteristic rays of the stream operators −crα k̂ · ∇x in (3.10), the jump times are exponentially distributed with parameters equal to the total scattering cross-sections: XZ r r σαβ (x, k, p)δ(crβ |p| − crα |k|)dp , α, β ∈ {t, p, s} , Σα (x, k) = β ❘2 and the scattered modes and directions are distributed according to the normalized scattering cross-sections within the domains, and the normalized reflection/transmission coefficients on the junction. Averaging over these paths finally yields the desired approximate solutions of (3.10)–(3.12). Two models of correlation are considered: either Gaussian correlation functions Rmn (x − y) = ξm ξn rmn e−γ such that R̂mn (k) = 2 |x−y|2 2 ξm ξn rmn − |k| 4γ 2 , e 4πγ 2 or exponential correlation functions Rmn (x − y) = ξm ξn rmn e−γ|x−y| such that ξm ξn rmn R̂mn (k) = 2πγ 2  |k|2 1+ 2 γ − 23 . γ −1 is a correlation lengthscale of the perturbations, ξm , ξn , m, n ∈ {̺, G, E}, stand for the standard deviations of random perturbations of ̺, 1/G′ , or 1/E, and the rmn ’s are coherency coefficients such that |rmn | ≤ 1 and rmm = 1. The dimensionless wave number k0 = γ −1 |k| is introduced. FIG. 3.1 and FIG. 3.2 display the evolution in time of the energy density (3.4) for an L junction and k0 = 0.1 or k0 = 1, respectively. The initial condition has the form gt1 (x, k) = δ(x − x0 ) and gα1 ≡ 0 for all other modes. Plate #2 is initially unloaded. The exponential correlation model is used with rmn = 1, ξ̺ = 0.4, and ξG = ξE = 1 since the mass is known to fluctuate much less than the stiffness. Both plates have identical constant wave velocities cα and Poisson’s coefficient ν = 0.2. FIG. 3.3 and FIG. 3.4 are basically the same for a junction with an angle φ = π3 between 1 plates and an initial condition of the form gpn (x, k) = δ(x − x0 ), gα1 ≡ 0 for all other modes. Here a Gaussian correlation model is used with the same parameters as before. For all computations one million (106 ) paths have been simulated and averaged out. RT 2/11234 DDSS – 24 – DECEMBER 2006 DNO Figure 3.1 – High-frequency energy transfer in a randomly heterogeneous L junction with exponential correlation of random perturbations; k0 = 0.1, ν = 0.2, ξ̺ = 0.4, ξG = ξE = 1, and cs = 1.0. Figure 3.2 – High-frequency energy transfer in a randomly heterogeneous L junction with exponential correlation of random perturbations; k0 = 1, ν = 0.2, ξ̺ = 0.4, ξG = ξE = 1, and cs = 1.0. RT 2/11234 DDSS – 25 – DECEMBER 2006 DNO Figure 3.3 – High-frequency energy transfer in a randomly heterogeneous junction at φ = π3 with Gaussian correlation of random perturbations; k0 = 0.1, ν = 0.2, ξ̺ = 0.4, ξG = ξE = 1, and cs = 1.0. Figure 3.4 – High-frequency energy transfer in a randomly heterogeneous junction at φ = π3 with Gaussian correlation of random perturbations; k0 = 1, ν = 0.2, ξ̺ = 0.4, ξG = ξE = 1, and cs = 1.0. RT 2/11234 DDSS – 26 – DECEMBER 2006 DNO 4. RADIATIVE TRANSFER ALONG THE JUNCTION LINE "Surface" (surfaces are lines in ❘2 ) waves propagating along the junction line have been neglected so far. This section intends to describe a possible model of transport regime for such guided waves. For that purpose one basically follows [16, 17]. We first outline our model of guided waves along the real line, before we derive the corresponding transport equation in the high-frequency limit. 4.1. High-frequency guided wave problem A high-frequency, scalar plane wave propagating in Ω = ❘∗+ × ❘ may be written: Z i 1 e− ǫ (λ(k)x+ky) Aǫ (k, t)dk , t > 0 , Ψǫ (x, y, t) = 2π ❘ (4.1) with the mixed boundary condition on the junction line Γ = {x = 0} (a paraxial-like approximation in the rescaled variables t → ǫ−1 t and y → ǫ−1 y): ǫ∂x Ψǫ (0, y, t) + b(y)Ψǫ (0, y, t) + iǫc(y)∂t Ψǫ (0, y, t) = 0 , b(y), c(y) > 0 , t > 0 , (4.2) and some given initial conditions in Ω. ǫ is a small frequency parameter which decreases toward 0 in the highfrequency limit, as for example µ of Eq. (2.8), Aǫ (k, t) is the wave amplitude, and the normal wavenumber λ(k) is given by:  p 2 2 pkα − k if k ≤ kα , (4.3) λ(k) = 2 2 −i k − kα if k > kα . kα is the bulk wavenumber within the medium and α may be either p, s or t as in section 2.3.1. Introducing the function ψǫ (y, t) = Ψǫ (0, y, t), Eqs. (4.1)–(4.2) yield: iǫ∂t ψǫ (y, t) + P (y, ǫD)ψǫ (y, t) = 0 on Γ × ❘+ (4.4) where P (y, D) is the pseudo-differential operator on S (the Schwartz space of all C ∞ functions which are also rapidly decreasing toward 0 at infinity as well as all their derivatives) with symbol P (y, k) = b(y)−iλ(k) ; we c(y) assume here that the latter is C ∞ on ❘ × ❘ and we shall use the same notation for an operator and its symbol. Eq. (4.4) above is a particular case of the more general setting: R(ǫDt )ψǫ (y, t) + P (y, ǫDy )ψǫ (y, t) = 0 (4.5) considered in [17], where R(D) in (4.4) is the pseudo-differential operator with symbol R(ω) = −ω. The above equation can model, for instance, the temporal discretization of ∂t and P (y, D) could be extended to model spatial discretization. For example, the finite difference approximation of ǫ∂t with a time step 2ǫ∆t ω∆t . corresponds to R(ω) = i sin∆t RT 2/11234 DDSS – 27 – DECEMBER 2006 DNO 4.2. Derivation of the transport equation We shall derive the transport equation for guided waves starting from the Wigner distribution of the guided wave function ψǫ . It is a function of position y on the junction line and wave number k and it is scaled by the small frequency parameter ǫ. The Wigner transform of φ, ϕ ∈ S ′ (❘) is defined by: Z  ǫu  1 ǫu   ϕ y+ du . (4.6) Wǫ [φ, ϕ](y, k) = eiku φ y − 2π ❘ 2 2 Provided that the sequence ψǫ lies in a bounded subset of L2 (❘), the real sequence Wǫ [ψǫ ] := Wǫ [ψǫ , ψǫ ] has (up to an extracted subsequence) a weak-* limit as ǫ → 0 which is also a non-negative measure, the so-called Wigner measure W of ψǫ . One also observes that the energy and power flow densities are given by: Z Z
iǫ 2 ψǫ ∇ψ ǫ − ψ ǫ ∇ψǫ = Wǫ [ψǫ ](y, k, t)kdk , |ψǫ (y, t)| = Wǫ [ψǫ ](y, k, t)dk , 2 ❘ ❘ respectively. Therefore Wǫ [ψǫ ] may be interpreted as an energy density in phase space in the high-frequency regime. The main property we shall use below to derive the desired results is the following (see [3], Proposition 1.8): ǫ Wǫ [P (y, ǫD)φ, ϕ] = P (y, k)Wǫ [φ, ϕ] + {P, Wǫ [φ, ϕ]} + ǫ2 Sǫ , 2i (4.7) ǫ Wǫ [φ, P (y, ǫD)ϕ] = P (y, k)Wǫ [φ, ϕ] + {Wǫ [φ, ϕ], P } + ǫ2 Tǫ , 2i ′ 2 where Sǫ and Tǫ are bounded in S (❘ ) as ǫ → 0. Note that if Eq. (4.5) is considered in lieu of Eq. (4.4), then a spatio-temporal Wigner transform shall be introduced as done in [17]. Now from (4.4) and (4.7) we deduce that: ∂t Wǫ [ψǫ ] =
1
1 iP (y, k)Wǫ [ψǫ ] + iP (y, k)Wǫ [ψǫ ] + {P, Wǫ [ψǫ ]} − {Wǫ [ψǫ ], P } + O(ǫ) . ǫ 2 (4.8) Passing to the limit ǫ → 0 in L∞ (❘+ , S ′ (❘2 )) we first see that the following dispersion equation holds: iP (y, k)W (y, k, t) = 0 . (4.9) One concludes from Eq. (4.9) that the Wigner measure W is singular with support on the set {y, k ∈ ❘2 ; iλ(k) = p b(y)} which corresponds to two points in k-space at the distance K(y) = kα2 + b2 (y) from the origin k = 0 at every point y on the junction line. Terms of order O(1) finally yield the transport equation for the guided waves energy density:   k c′ (y) ′ c(y)∂t W + ∂y W + b (y) − b(y) ∂k W = 0 (4.10) b(y) c(y) as P is real on the support of W . 5. CONCLUSIONS In this report we have first proposed an analytical model for the calculation of high-frequency power flow reflection/transmission coefficients for the junction of two thick plates, and reflection coefficients for a RT 2/11234 DDSS – 28 – DECEMBER 2006 DNO semi-infinite thick plate with Dirichlet or Neumann boundary conditions. It is based on the model developped in [12] for coupled Timoshenko beams and the results of [6] describing the high-frequency waves propagation features within thick shells. It also accounts for possible mode conversions on the interface. This model has been used to simulate the radiative transfer regime in a two-plate system assuming that the radiative transfer equations derived for a single plate hold in each plate, and the power flows at the junction between the plates are reflected and/or transmitted according to the laws derived by the analytical approach. Numerical examples have been presented solving the radiative transfer equations by a direct Monte-Carlo method. The analytical model is readily useable in the discontinuous Galerkin finite element scheme with weakly enforced generalized interface conditions presented in [12]. In a last section of the report we have developped a transport model for the high-frequency waves possibly guided along the junction line between the plates. The analysis is based on the derivation in [16] where the authors also considered random perturbations of the boundary impedance at a length scale comparable to the wavelength. The issue of characterizing surface waves in the high-frequency regime is of considerable interest for the many applications in electronics, solid-state physics, acoustics, or geophysics for example. From experimental measurements and practice it is recognized that a significant amount of energy is likely to be transported along junctions and interfaces. RT 2/11234 DDSS – 29 – DECEMBER 2006 DNO 6. BIBLIOGRAPHY [1] G. C. Papanicolaou and L. V. Ryzhik, Waves and transport, In Hyperbolic Equations and Frequency Interactions, edited by L. Caffarelli and W. E, vol. 5 of IAS/Park City Mathematics Series, pp. 305–382, Providence, RI, American Mathematical Society (1999), ISBN 0-8218-0592-4. [2] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Revista Matemática Iberoamericana, 9 (3) (1993), pp. 553–618. [3] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Communications on Pure and Applied Mathematics, L (4) (1997), pp. 323–379, Erratum LIII(2) (2000), pp. 280-281. [4] J.-L. Akian, Wigner measures for high-frequency energy propagation in visco-elastic media, Technical report ONERA no RT 2/07950 DDSS, Châtillon (december 2003). 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