On numerical methods for the computation of guided waves involving porous materials

Transcription

1 On numerical methods for the computation of guided waves involving porous materials A travel through the waveguide B. Nennig *, Y. Aurégan +, W. Bi. +, R. Binois *,#, N. Dauchez #, J.-P. Groby +, E. Perrey-Debain #, L. Xiong + * Institut supérieur de mécanique de Paris (SUPMECA), Laboratoire Quartz EA 7393, 3 rue Fernand Hainaut, Saint-Ouen, France. + Laboratoire d Acoustique de l Université du Maine, Unité Mixte de Recherche CNRS 6613, Avenue Olivier Messiaen, Le Mans Cedex 9, France # Sorbonne universités, Université de Technologie de Compiègne, Laboratoire Roberval, UMR CNRS 7337, CS 60319, Compiègne cedex, France. benoit.nennig@supmeca.fr SAPEM 2017, December , Le Mans, France

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3 Application fields Outline Assumptions Disclaimer! Focuses mainly on noise control and acoustic duct silencers Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 2 / 45

4 Application fields Introduction Application fields Outline Assumptions Guided waves in porous materials can be found in many noise control applications (a) Muffler, perforated sheet and porous (b) Engine intake, several studies to put porous (c) Splitter silencer, alternating layer of air and glasswool Provide ideal channels for noise transmission (buildings, aircraft,...) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 3 / 45

5 Application fields Introduction Application fields Outline Assumptions Guided waves can be useful to get information about poroelastic material skeleton. Direct measurement of quasi Lamb waves (d) [Boecks, 2005] (e) [Geslain et al., 2017] SLaTCoW method Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 4 / 45

6 Application fields Outline Assumptions Outline Wave decomposition - Analytical Purely geometric Material param. effects Link with DP Winding Number algorithm RIGID WAVE GUIDE Homogeneous Homogeneous anisotropic Layered EF Layered poroelastic Biot Zoom Pore network Periodic inclusions Layered Periodic inclusions Bloch modes Complex sym. orthogonality Link with macro-micro Link with Tortuosity Exceptional point Discretization - Numerical Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 5 / 45

7 Outline 1 Introduction Application fields Outline Assumptions 2 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides 3 Periodic pore modeling Metaporous 4 Conclusions Persepectives in numerical methodes

8 Application fields Outline Assumptions Assumptions Frequency domain e iωt Infinite duct, only few discussion on matching condition and so on No flow here (see Brambley Wednesday!) Rigid waveguide Weak modal density, for high modal density [Bi, 2014 ; Chan, 2017] (see Perrey-Debain Wednesday) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 7 / 45

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10 Homogeneous Waveguides EF rigid wall infinite Finite cross section

11 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Homogeneous Waveguides - Problem statement EF rigid wall infinite Finite cross section For a an infinite duct with invariant cross section, the pressure field can be written as p(x) = φ(x )e iβx, where, β is the axial wavenumber If the porous material fulfills Helmholtz eq., with the wavenumber k p p + kp 2 p = 0, φ + (kp 2 β 2 )φ = 0, φ + α 2 φ = 0, φ n = 0, PDE inside the waveguide Where, α is the tranverse wavenumber Boundary condition. φ and α fulfill an eigenvalue problem. Here α are real and positive. The axial wavenumber β n = k 2 p α 2 n is complex, when k p C. The problem is purely geometric Cut-off freq. in real case Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 8 / 45

12 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Homogeneous Waveguides - Resolution Wave decomposition If the cross section is separable coordinate system for the wave equation, the wave propagation can solved explicitly. ex 1D : φ(y) = Ae iαy + Be iαy. The BC (rigid) yield to a small size NL eigenvalue problem in α [ ] ( ( 1 1 A 0 e iαh e B) }{{ iαh =. 0) } M To solve it, generally we look for the root of the determinant det M(α) = 0. Newton-Raphson method, Muller s method, the Secant method or simplex method : iterative algorithms requiering initial approximations for the zeros Other method based on Argument Principle (see later [Nennig,2010 ; Kravanja,2000]), without initial guess Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 9 / 45

13 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Homogeneous Waveguides - Resolution (cont.) Numerical methods Finite element method, the discretization of the weak formulation yields to a real symmetric generalized eigenvalue problem (K α 2 M)φ = 0 leading to orthogonal eigenvectors and positive eigenvalues. Finite difference, similar results Wave finite element [Mace, 2005 ; Serra, 2015] Projection on orthogonal basis like Chebyshev spatial discretization technique [Chan, 2017], or the multimodal method [Bi, 2007] real symmetric matrix du to weak formulation. All first eigenvalue are obtained, without initial guess. Big size problem, sparse matrix Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 10 / 45

14 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Homogeneous Waveguides - orthogonality The solution φ n and α n are orthogonal. If we take 2 modes φ n + α 2 nφ n = 0 φ m + α 2 mφ m = 0 +rigid BC After multiplying by φ m and integrating by part twice and using the BC, we get This, the orthogonality relation reads (α 2 n α 2 m) φ mφ n ds = 0. S }{{} S L 2 inner product φ mφn ds = Λnδmn. Changing boundary condition may change everything (ex : impedance) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 11 / 45

15 Transverse isotropics waveguides EF rigid walls infinite Finite cross section

16 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Transverse isotropics waveguides EF rigid walls infinite plane of isotropy stiff direction T e z e y e x Transverse direction isotropy axis soft direction I Finite cross section Consider a 2D rigid-walled waveguide of height h filled with a TI fluid [Nennig et al., submitted JASA]. The wave propagation is described by [De Hoop, 1995 ; Torrent, 2008 ; Norris, 2009] (τ p) + ω2 K eq p = 0, where τ = ρ 1 is the inevrse of the density tensor. We are looking for a guided mode propagating along the e x direction, having the exponential form p = φ(y)e iβx where φ(y) = ae iα 1y + be iα 2y. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 12 / 45

17 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Transverse isotropics waveguides (cont.) After insertion into the wave equation, transverse wavenumbers α 1 and α 2 must satisfy τ yy α 2 i + 2βτ xy α i + τ xx β 2 ω2 K eq = 0, with i = 1, 2. Applying rigid conditions at the wall, i.e. u w e y = 0, leads to a boundary condition of mixed type and non-trivial solutions exist if ( h ) sin = 0 2τ yy where = 4τ yy ω 2 is the discriminant of the quadratic equation. K eq + 4β 2 (τ 2 xy τ xx τ yy ) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 13 / 45

18 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Transverse isotropics waveguides (cont.) Propagation constant for guided modes are then found explicitly as ( ) β n = ± τyy ω 2 nπ 2 τt τ I τ yy K eq h Right and left propagating modes are identical The fundamental mode, n = 0, is also function of the transverse coordinate y except when Θ = 0 or π/2 Solution depends on the fluid properties, not only on the WG shape Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 14 / 45

19 EF rigid wall infinite Finite cross section

20 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides HVAC Silencers are most often made of parallel baffles of porous material (rockwool) well described by equivalent fluid model Elementary cell Airway Porous material Weak attenuation at low frequency [63 Hz Hz]. For this frequency range fundamental mode dominates Fundamental mode range depends on both the duct height h and of the number of baffle N : n = n inc + 2qN. [Mechel, 1990] Refs : [Tam et al., 1991 ; Aurégan et al., 2001 ; Kirby et al., 2005 ; Binois et al., 2015 ; Nennig et al., 2015] Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 15 / 45

21 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides EF rigid wall infinite Finite cross section In layered waveguide, each layer i = 1, I, satisfies The Hemlholtz equation, with different materials properties like density ρ i, celerity c i and wavenumber k i p + k 2 i p = 0 Continuity of the pressure at each interfaces Continuity of the velocity ie [ 1 ρ i p n ] = 0 at each interfaces Applying wave decomposition and continuity condition yields where α i = ki 2 media properties. f (β) = I i=1 α i ρ i tan α i h i β 2. This equation must solved in β, the solution depends on Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 16 / 45

22 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides The associated weak formulation, is obtained by multiplying the wave equation (12) by a test function ψ and by integrating over the porous media occupying an arbitrary domain Ω of boundary S I ( ) 1 I 1 ψφ dω = ψ φ ρ i n ds. i=1 Ω ψ φ dω + ( k 2 i β 2) Ω i=1 ρ i S } {{ } =0 All boundary term vanished due rigid duct and transverse velocity continuity. This yields complex symetric eigenvalue problem K 1 + k1 2M M β K I + ki 2M I 0... M I ψ 1.. = 0 ψ I Complex symmetric eigenvalue problem stand for FEM assembly Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 17 / 45

23 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Complex symmetric problems Properties Loss of orthogonality for L 2 scalar product S φ n φ m ds = 0 Left and right eigenvectors are still orthogonal (adjoint), bi-orthogonality For complex symmetric problem (excepted for double root), a new orthogonality relation can be obtained without conjugate (not scalar product!) S φ mφ n ds = Λ nδ mn. eg : Impedance lining, layered materials,... Refs : [Lawrie, 1999, 2006 ; Redon, 2011 ; Moiseyev, 2011] The EP/double root case will be discussed later Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 18 / 45

24 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides - Example splitter silencer With one layer of each material (air, porous), approximate solution can be obtained. α p tan α pb + αa tan α aa = 0, ρ p ρ a with α p = kp 2 β2 and α a = ka 2 β2. At low frequencies (viscous regime), the wavenumber order of magnitude in the porous material k p ω, is larger than the axial wavenumber β. When k m β, the equation (19) is equivalent to a waveguide lined with a locally reacting material satisfying ik a Z with Z a = ρ ac 0 and Z = iρ pc p cot k pb. + αa Z a tan α aa = 0, This is equivalent to say that only plane wave propagate in the transverse direction e ±ikpy. Applying the continuity of the pressure between the micro-porous p m and the airway p a, yields to p a ( p p(y) = e ik py + e ikpy), y [ b, b]. 2 cos k pb Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 19 / 45

25 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides - Example splitter silencer (cont.) The pressure in the airway may considered as a constant in the transverse direction and is a forcing term for the pressure in the micro-porous media. With these assumptions, the average pressure over the elementary cell reads, p p = 1 b tan kpb p p(y) dy = p a 2b k. b pb is similar to the pressure diffusion function F d of the double porosity model [Olny et al., 2003]. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 20 / 45

26 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides (a) f = f d 3 (b) f = f d (c) f = 3f d Normalized pressure field (in absolute value) below, around and above ω d for a rockwool. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 21 / 45

27 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides - Optimal design This closed form solution yields an optimal value (max Im β) for the airflow resitivity ) σ0 = 3ρ 0 P φ pφ + Υ depending on the waveguide height h and open γα φ pα b 2 φ 2 ( area ratio φ p TL (db) 15 TL (db) f (Hz) f (Hz) (a) φ p = 0.5, N = 1 (b) φ p = 0.5, N = 3 Transmission loss σ [0.1σ 0,..., 10σ 0 ]. Obtained with a reference solution [Binois, 2013] on realistic silencers. If σ < σ 0, (... ), If σ > σ 0,( ), If σ = σ 0, ( ). Geometry H = 0.70 m and L s = 1.20 m. What happens for absorption at normal incidence σ = σ 0? Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 22 / 45

28 Layered poroelastic waveguides Biot Finite cross section rigid wall infinite

29 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Layered poroelastic waveguides Typical dissipative silencers (automotive industry, HVAC systems,...) consists of a cylindrical expansion chamber filled with a sound absorbing material. I II Air III r z Biot Prediction of the Transmission Loss (TL) is usually made using Mode Matching Methods (Kirby & Denia, 2007 ; Lawrie & Kirby, 2005 ; and plenty more...) They all consider fluid equivalent models (Beraneck, 1947 ; Zwikker & Kosten, 1949 ; Delany & Bazley, 1970) Why take into account the frame elasticity? Can the skeleton improve the TL? Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 23 / 45

30 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Constitutive equations For time-harmonic representation, the pressure p in the airway obeys to the (convected) wave equation p 1 d 2 p c0 2 dt 2 = 0 In the poroelastic material [Biot, 1956] σ s + ω 2 (ρ 11 u + ρ 12 U) = 0, σ f + ω 2 (ρ 12 u + ρ 22 U) = 0, where U et u are respectively the fluid and the solid phase displacements and the Solid and fluid phase stress tensors are given by σ s = (A u + Q U) I + 2N ε s σ f = (Q u + R U) I where ε s = 1/2( u + ( u) T ) is the usual strain tensor. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 24 / 45

31 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Helmholtz decomposition In the airway, we look for purely acoustic mode : w = ϕ 0 In the porous media, both displacement fields are written as After equations decoupling, we have u = ϕ + ψ and U = χ + Θ. ϕ = ϕ 1 + ϕ 2, χ = µ 1 ϕ 1 + µ 2 ϕ 2, and Θ = µ 3 ψ All potentials fulfill wave equation. Wave number k i and phase coupling coefficients µ i are known (Allard, 1993) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 25 / 45

32 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Helmholtz decomposition > General potential form We put X(r, θ, z, t) = X(r, θ) e i(βz ωt) Following (Gazis, 1951) we find for symetric modes ϕ 1 = [A 1 J m(α 1 r) + B 1 Y m(α 1 r)] cos mθ ϕ 2 = [A 2 J m(α 2 r) + B 2 Y m(α 2 r)] cos mθ ψ r = [A 3 J m+1 (α 3 r) + B 3 Y m+1 (α 3 r)] sin mθ ψ θ = [A 3 J m+1 (α 3 r) + B 3 Y m+1 (α 3 r)] cos mθ ψ z = [ A 3 Jm(α 3r) + B 3 Ym(α 3r) ] sin mθ Similarly ϕ 0 = A 0 J m(α 0 r) cos mθ Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 26 / 45

33 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Continuity conditions At the air-porous interface, the following coupling conditions must be verified φu n + (1 φ)u n }{{} Porous side σ t n = p n p p = p = w n }{{} Airway Here φ is the porosity and the pore pressure p p is obtained from the fluid phase tensor as 3φp p = trσ f. On the hard surface the poroelastic layer is assumed to be clamped u = 0 and U n = 0. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 27 / 45

34 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Continuity conditions > Eigenvalue equation The modal vector containing the wave potential amplitudes V = ( A 0, A 1, B 1, A 2, B 2, A 3, B 3, A 3, B 3) T, must be a non-trivial solution of the 9 9 algebraic system after boundary conditions application M(α 0, α 1, α 2, α 3, β)v = 0 Together with the dispersion equation (once inverted) α 0 = (k 0 Mβ) 2 β 2 and α i = ki 2 β 2, i = 1, 2, 3 we have M(β)V = 0 Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 28 / 45

35 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Layered poroelastic waveguides Instead of finding roots of f (β) = det M(β) it s possible to find the zeros of the associated polynomial Π with the same zeros [Kravanja & Van Barel, 2000 ; Nennig et al., 2010] 1 Residue theorem gives for analytical functions S n = 1 f (β) 2πi f (β) βn dβ = C 2 The coefficients of the polynom Π are related to S n N β k=1 n k β n k 3 Get the zeros of the polynom Π by computing the eigenvalues of companion matrix ie the root of f (β) in the interior of C Important to use integer number of sample on C Works with all closed waveguides (always analytics function) Variant with log, there is efficient way to generalized this to big matrix [Spence, 2004 ; Güttel, 2017]... Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 29 / 45

36 Homogeneous Waveguides Transverse isotropics waveguides Layered poroelastic waveguides Roots in the complex plane > frequency analysis Imβ 0 Imβ Ñ ÒØ 600 ÈË Ö Ö ÔÐ Ñ ÒØ Reβ Reβ (a) Integration path (b) Frequency sweep Roots in β complex plan for XFM foam in the silencer A and M = 0. Good numerical satiblity, because it never looks close to the root Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 30 / 45

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38 Introduction Periodic pore modeling Metaporous As the governing equations, the boundary conditions, and the geometry are d-periodic, it follows from the Bloch theorem that the solutions are Bloch waves, φ(x) = φ (x)eikb x, (1) i.e. the fields can be split into a d-periodic field φ (x) modulated by a plane wave involving the Bloch wavevector kb = kb κ, where the unit vector κ = kb /kb stands for the propagation direction. The real part of kb measures the change in phase across the cell The imaginary part of kb measures the attenuation. Two way for solving this [Collet, 2011 ; Frazier, 2017] 7 Choose ω as eigenvalue, suitable for free response, difficult with parameters f (ω) 3 Choose kb as eigenvalue, suitable for forced response, i. e. most application in waveguides I Semi-analytical methods using plane wave expansion are also possible [Duclos, 2009 ; Deymier, 2013] Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 31 / 45

39 Periodic pore modeling Zoom rigid wall infinite Finite cross section

40 Periodic pore modeling Metaporous Visco-thermal fluid We use here linearized Navier and Stokes equations with Fourier flux q = κ T, and we assume a Newtonian fluid, ideal gas [Stinson, 1991] iωρ 0 v + p τ = 0, iωρ 0 C pt κ T + iωp = Q, v + iω T T 0 iω p P 0 = 0. Where v is the velocity, T the temperature and p the pressure fluctuation. Taylor-Hood element are used to avoid looking problem in Stokes like equation. The velocity and the temperature is approximated with the quadratic lagrangian element, and the pressure with linear lagrangian element [Kampinga, 2010]. The pressure is kept to ensure numerical stability Possible to add the coupling with the skeleton Possible to add other physics Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 32 / 45

41 Implementation Introduction Periodic pore modeling Metaporous The periodic formulation can be obtained thanks to a systematic approach, for instance u = [ û + ik B û] e ik B û u = [ û + iû k t B] e ik B x We get a Quadratic eigenvalue problem in k B [Tisseur, 2001], k 2 B K 2 + k B K 1 + K 0 = 0 2N eigenvalues, only N independent eigenvectors Here all parameters are real, then K 2, K 0 real symmetric, K 1 Hermitian (empty diag.) (similar gyroscopic pb when λ = ik b ) k B and kb are solution The left eigenvector for k Bn is also the right eigenvector for k Bn There is several possibilities for resolution (linearization be careful to respect symetries, direct [TOAR, SLEPc]) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 33 / 45

42 Validation Introduction Periodic pore modeling Metaporous Re keq Im keq 2.5 Re rhoeq Im rhoeq 16 x 104 Re Keq 14 Im Keq (a) k eq (b) ρ eq (c) K eq Comparaison with FEM (dot) Zwicker and Kosten solution (solid) for circular pore r = 1 mm. Using the 1st eigenvector, effective parameters ρ eq and K eq, can be recover as defined in [Allard, 1993, Chap. 4] iωρ eq v z = p ( ) v z. et Keq = p/. iω Here is the average over the cross section. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 34 / 45

43 Introduction Periodic pore modeling Metaporous Examples FEM computation on a staight pore with complex cross section Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 35 / 45

44 Periodic pore modeling Introduction Periodic pore modeling Metaporous Re keq Im keq f (Hz) (a) géométrie (b) k B Comparaison between FEM 3D model (dot) and homogenization [Boutin, 2015] (solid line) with a rigid Helmholtz resonator in CC crystal (a = 6 mm, t = 0.3 mm, l = 1.5 mm, r c = 0.3 mm, r s = 2.75 mm) Strong connexion with homogenization theory Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 36 / 45

45 Metaporous rigid wall infinite Finite cross section

46 Introduction Periodic pore modeling Metaporous Aim Few metaporous examples use to increase absorption, limit transmission through wall or through duct (a) Yang et al., 2015 (d) Aurégan 2016 (b) Groby et al., 2014 et al., (c) Lagarrigue 2015 al., (e) Xiong et al., 2017 Focus here on duct parietal acoustic treatments I Bloch modes yields non hermitian quadratic eigenvalue problem I Geometric parameters in metamaterial are easy to tune I Exploit easy tuning of metamaterial to find EP / force Bloch modes to coalesce Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 37 / 45

47 Configurations Introduction Periodic pore modeling Metaporous Sample made of 5 mm metalic foam layers and blind cylinder inclusions with 2 possible orientation [Xiong, 2017] (f) Sample (g) Model Pictures of (a) an open cylinder inclusion (filled with air), (b) zoom of the drilled metallic foam layer of height 5 mm, and (c) a whole sample with cylinders (Fig. 7(a)) embedded in an alternated way. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 38 / 45

48 Experimental observation Introduction Periodic pore modeling Metaporous z Unit cell + p 1 - p 1 x d Porous Air p 2 + p 2 - ha u 4 u 3u 2u 1 xc d 1d 2d 3d 4 hp A 2D schematic view of the experimental setup. The measured sample, of length L = 200 mm, contains 8 unit cells. x c is the center of the rigid inclusion. 35 (a) (b) (c) 30 att. (db) h p att. (db) 25 h p h p att. (db) 25 h 20 p 15 h p Frequency (Hz) Frequency (Hz) Frequency (Hz) Comparison between the Bloch wave attenuation ( ) and the measured transmission loss (line) with 8 unit cells for the configurations : (a) 5P, (b) P- -P, and (c) P- -P. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 39 / 45

49 Parametric study Introduction Periodic pore modeling Metaporous The inclusion is moved from δ from the centre of the cell is positive. parametric (δ, f ) Quadratic eigenvalue problem att.(db) freq. max Frequency(Hz) (mm) 2 Mode attenuations for the lower two Bloch modes as a function of frequency at different inclusion positions δ. The attenuation reach a max. when 2 modes coalesce (eigenvector and eigenvalue) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 40 / 45

50 Introduction Periodic pore modeling Metaporous Mode coalescence m1' ms ml Frequency (Hz) Frequency (Hz) 2500 m1' ml ms m0 m m1 Coalescence m Dispersion curves for the Bloch wavenumber when ha = 135 mm and δ = 4.0 mm a) 3D view, b) projection in the frequency-re kb plane. The inserted pictures show the (O, x, z)-cut of the modulus of the right eigenvectors at 100, 1375, 1385 and 2000 Hz. I At the coalescence the double mode is localized in the liner, similar to impedance case [Bi, 2016] I (δ, f ) is an exceptional point EP [Kato, 1980 ; Heiss,1990 ; Berry,2004 ; Bi, 2015] Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 41 / 45

51 Parametric study Introduction Periodic pore modeling Metaporous Using a Taylor expansion in the vicinity of the double root kb [Tester,1973 ; Shenderov,2000], k B k B ± [ ( 1 2 ) 1 ( 2 D k (δ δ 2 ) D δ B ) ] 1/2 + (f f ) D f The EP is a branch point in the parameter plane Riemann sheet of the real (a) and imaginary (b) part of the lower attenuated Bloch wavenumber. The line indicates the crossing of the different sheets. Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 42 / 45

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53 Conclusions Persepectives in numerical methodes In a waveguide maximizing attenuation is different that for absorbing panel Partial filling / airway Higher modes Recently, we find new design rules Tune on EP with metaporous [Xiong et al., 2017] Use σ0 when porous remains in viscous regime [Nennig et al., 2015] Use transverse isotropic material Other strategies have to be tested Play with path in parameter space for asymmetric propagation [Doppler, 2016] Take advantage of complex problem to reveals new physics (EP, CPA,...) Fine modeling of damping and radiation mechanism opens new routes (QM, EM) Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 43 / 45

54 Conclusions Persepectives in numerical methodes Numerical methods are now mature to tackle such problem, however, it remains some challenges or some possible improvements How to find efficiently EP (Jordan decomposition?) How to speed-up ferquency sweep Use waveguide as an homogenization method... Nennig, Aurégan, Bi, Binois, Dauchez, Groby, Perrey-Debain, Xiong Guided waves with porous materials 44 / 45

55 Thank you for your attention! The waveguide journey comes to an end, Thank you for your attention! See you in porous session of CFA in Le Havre This work was partly funded by the ANR Project METAUDIBLE No. ANR-13-BS funded jointly by ANR and FRAE.