Homogenization of the vibro-acoustic transmission on perforated plates

TL;DR

This paper develops a homogenized model for vibro-acoustic transmission through perforated elastic plates, using asymptotic analysis and finite element validation, to simplify complex fluid-structure interactions in perforated layers.

Contribution

It introduces a novel homogenized interface model for vibro-acoustic transmission in perforated plates based on two-scale asymptotic analysis.

Findings

The homogenized model accurately predicts vibroacoustic transmission.

Finite element implementation validates the model against direct numerical simulations.

The approach simplifies complex fluid-structure interaction analysis.

Abstract

The paper deals with modelling of acoustic waves which propagate in inviscid fluids interacting with perforated elastic plates. The plate can be replaced by an interface on which transmission conditions are derived by homogenization of a problem describing vibroacoustic fluid-structure interactions in a transmission layer in which the plate is embedded. The Reissner-Mindlin theory of plates is adopted for periodic perforations designed by arbitrary cylindrical holes with axes orthogonal to the plate midplane. The homogenized model of the vibroacoustic transmission is obtained using the two-scale asymptotic analysis with respect to the layer thickness which is proportional to the plate thickness and to the perforation period. The nonlocal, implicit transmission conditions involve a jump in the acoustic potential and its normal one-side derivatives across the interface which represents the plate with a given thickness. The homogenized model was implemented using the finite element method and validated using direct numerical simulations of the non-homogenized problem. Numerical illustrations of the vibroacoustic transmission are presented.

Equations

$$\begin{split}c^{2}\nabla^{2}p+\omega^{2}p&=0\quad\mbox{ in }\Omega^{*}\;,\\ \nabla\cdot{\mbox{\boldmath$\sigma$\unboldmath}}({\textbf{{u}}})+\omega^{2}\rho{\textbf{{u}}}&=0\quad\mbox{ in }\Sigma\;,\\ \mbox{ acoustic transmission: }&\\ \left.\begin{array}[]{rcl}{\rm{i}}\omega{\textbf{{n}}}\cdot{\textbf{{u}}}&=&{\textbf{{n}}}\cdot\nabla p\\ {\textbf{{n}}}\cdot{\mbox{\boldmath$\sigma$\unboldmath}}({\textbf{{u}}})&=&{\textbf{{b}}}(p)={\rm{i}}\omega\rho_{0}p{\textbf{{n}}}\end{array}\right\}&\quad\mbox{ on }\partial_{*}\Sigma\;,\\ \mbox{ incident, or reflected acoustic waves in the fluid: }&\\ r{\rm{i}}\omega cp+c^{2}\frac{\partial{p}}{\partial{n}}&=s2{\rm{i}}\omega c\bar{p}\quad\mbox{ on }\partial_{\rm{ext}}\Omega^{G}\setminus\partial_{\textbf{{u}}}\Sigma\;,\\ \mbox{ clamped elastic structure: }&\\ {\textbf{{u}}}&=0\quad\mbox{ on }\partial_{\textbf{{u}}}\Sigma\;.\end{split}$$

$$\begin{array}{r}\partial {\Omega }_{\delta }={\Gamma }_{\delta }^{+}\cup {\Gamma }_{\delta }^{-}\cup {\partial }_{\mathrm{ext}}{\Omega }_{\delta },{\Gamma }_{\delta }^{\pm }={\Gamma }_0\pm \frac{\delta }{2}\overrightarrow{e_3},{\partial }_{\mathrm{ext}}{\Omega }_{\delta }=\partial {\Gamma }_0\times ]-\delta /2,\delta /2[,\end{array}$$

$$\begin{array}{rl}{\Sigma }^{\epsilon }& ={\Gamma }^{\epsilon }\times \epsilon \bar{h}]-1/2,+1/2[,\\ \partial {\Sigma }^{\epsilon }& ={\partial }_{\circ }{\Sigma }^{\epsilon }\cup {\partial }_{+}{\Sigma }^{\epsilon }\cup {\partial }_{-}{\Sigma }^{\epsilon }\cup {\partial }_{\text{u}}{\Sigma }^{\epsilon },\\ \text{mbox}where& \\ {\partial }_{\circ }{\Sigma }^{\epsilon }& ={\partial }_{\circ }{\Gamma }^{\epsilon }\times \epsilon \bar{h}]-1/2,+1/2[,\\ {\partial }_{\pm }{\Sigma }^{\epsilon }& ={\Gamma }^{\epsilon }\pm \epsilon \bar{h}/2,\end{array}$$

$$\begin{array}{rl}{\Gamma }^{\epsilon }& =\bigcup _{k\in {\mathbb{Z}}^2}\epsilon \left({\Xi }_K+\sum _{i=1,2}{k}_i{\ell }_i\overrightarrow{e_i}\right)\cap {\Gamma }_0,\end{array}$$

$$\begin{array}{rl}Y& =\Xi \times ]-\varkappa /2,+\varkappa /2],\\ S& ={\Xi }_S\times \bar{h}]-1/2,+1/2[,\end{array}$$

$$\begin{array}{rl}{\Omega }^{\ast \epsilon }& =\bigcup _{k\in {\mathbb{Z}}^2}\epsilon (Y^{\ast }+\sum _{i=1,2}{k}_i{\ell }_i\overrightarrow{e_i})\cap {\Omega }_{\delta },\end{array}$$

$$\begin{array}{rl}{\partial }_{\circ }S& ={\partial }_{\circ }{\Xi }_S\times \bar{h}]-1/2,+1/2[,\\ {\partial }_{\pm }S& ={\Xi }_S\pm \overrightarrow{e_3}\bar{h}/2,\\ {\partial }_{\#}S& =\partial \Xi \times \bar{h}]-1/2,+1/2[.\end{array}$$

$$\begin{array}{rl}{c}^2{\nabla }^2{P}^{\delta }+{\omega }^2{P}^{\delta }& =0\text{mbox}in{\Omega }_{\delta }^{+}\cup {\Omega }_{\delta }^{-},\\ \text{mbox}interfacecondition:& \\ P^{\delta }& ={\hat{p}}^{\epsilon }\text{mbox}on{\Gamma }_{\delta }^{\pm },\\ \text{mbox}otherboundaryconditions& \\ r\mathrm{i}\omega c{P}^{\delta }+c^2\frac{\partial P^{\delta }}{\partial n}& =s2\mathrm{i}\omega c\bar{p}\text{mbox}on{\partial }_{\mathrm{ext}}{\Omega }_{\delta }^G,\end{array}$$

$$\begin{split}c^{2}\nabla^{2}p^{\varepsilon}+\omega^{2}p^{\varepsilon}&=0\quad\mbox{ in }\Omega_{\delta}^{*\varepsilon}\;,\\ \mbox{ {interface conditions} }\quad\frac{\partial{p^{\varepsilon}}}{\partial{n}}&=-{\rm{i}}\omega g^{\varepsilon\pm}\quad\mbox{ on }\Gamma_{\delta}^{\pm}\;,\\ \nabla\cdot{\mbox{\boldmath$\sigma$\unboldmath}}({\textbf{{u}}}^{\varepsilon})+\omega^{2}\rho{\textbf{{u}}}^{\varepsilon}&=0\quad\mbox{ in }\Sigma_{\delta}^{\varepsilon}\;,\\ \mbox{ acoustic transmission: }&\\ \left.\begin{array}[]{rcl}{\rm{i}}\omega{\textbf{{n}}}\cdot{\textbf{{u}}}^{\varepsilon}&=&{\textbf{{n}}}\cdot\nabla p^{\varepsilon}\\ {\textbf{{n}}}\cdot{\mbox{\boldmath$\sigma$\unboldmath}}({\textbf{{u}}}^{\varepsilon})&=&{\textbf{{b}}}(p^{\varepsilon})={\rm{i}}\omega\rho_{0}p^{\varepsilon}{\textbf{{n}}}\end{array}\right\}&\quad\mbox{ on }\partial_{*}\Sigma_{\delta}^{\varepsilon}\;,\\ \mbox{ other boundary conditions }&\\ r{\rm{i}}\omega cp^{\varepsilon}+c^{2}\frac{\partial{p^{\varepsilon}}}{\partial{n}}&=s2{\rm{i}}\omega c\bar{p}\quad\mbox{ on }\partial_{\rm{ext}}\Omega^{G}\setminus\partial_{\textbf{{u}}}\Sigma_{\delta}^{\varepsilon}\;,\\ \mbox{ clamped elastic structure: }&\\ {\textbf{{u}}}^{\varepsilon}&=0\quad\mbox{ on }\partial_{\textbf{{u}}}\Sigma_{\delta}^{\varepsilon}\;.\end{split}$$

$$\begin{array}{rl}\mathrm{i}\omega g^{\epsilon \pm }& =\frac{\partial P^{\delta }}{\partial n^{\pm }}\text{mbox}on{\Gamma }_{\delta }^{\pm },\\ p^{\epsilon }& ={\hat{p}}^{\epsilon }=P^{\delta }\text{mbox}on{\Gamma }_{\delta }^{\pm },\end{array}$$

$$\begin{split}{W}_{ij}({\textbf{{v}}})&:=E_{ijkl}\partial_{l}v_{k}=[{{\rm I}\kern-1.99997pt{\rm E}}\overline{\nabla}^{S}{\textbf{{v}}}]_{ij},\\ Z_{i}(u_{3},{\mbox{\boldmath$\theta$\unboldmath}})&:=S_{ij}(\partial_{j}u_{3}-\theta_{j})=[{\textbf{{S}}}(\overline{\nabla}u_{3}-{\mbox{\boldmath$\theta$\unboldmath}})]_{i}.\end{split}$$

$$\begin{split}\overline{{\textbf{{w}}}}(x^{\prime},z)&=\overline{{\textbf{{u}}}}(x^{\prime})+hz{\mbox{\boldmath$\theta$\unboldmath}}(x^{\prime})\;,\quad w_{3}(x^{\prime},z)=u_{3}(x^{\prime})\;,z\in{[-1,1]}\;,x^{\prime}\in\Gamma^{\varepsilon}\;,\end{split}$$

$$\begin{split}\omega^{2}h\rho\;\overline{{\textbf{{u}}}}+h\overline{\nabla}\cdot{\textbf{{W}}}(\overline{{\textbf{{u}}}})&=-\overline{{\textbf{{f}}}}(p)\;,\quad\mbox{ in }\Gamma^{\varepsilon}\;,\\ \omega^{2}h\rho\;u_{3}+h\overline{\nabla}\cdot{\textbf{{Z}}}(u_{3},{\mbox{\boldmath$\theta$\unboldmath}})&=-f_{3}(p)\;,\quad\mbox{ in }\Gamma^{\varepsilon}\;,\\ \omega^{2}\frac{h^{3}}{12}\rho\;{\mbox{\boldmath$\theta$\unboldmath}}+\frac{h^{3}}{12}\overline{\nabla}\cdot{\textbf{{W}}}({\mbox{\boldmath$\theta$\unboldmath}})&=-\overline{{\textbf{{m}}}}\;\;,\quad\mbox{ in }\Gamma^{\varepsilon}\;,\\ h\overline{{\textbf{{n}}}}\cdot{\textbf{{W}}}(\overline{{\textbf{{u}}}})&=\overline{{\textbf{{f}}}}^{\partial}\;\quad\mbox{ on }\partial_{\circ}\Gamma^{\varepsilon}\;,\\ h\overline{{\textbf{{n}}}}\cdot{\textbf{{Z}}}(u_{3},{\mbox{\boldmath$\theta$\unboldmath}})&=f_{3}^{\partial}\;\quad\mbox{ on }\partial_{\circ}\Gamma^{\varepsilon}\;,\\ \frac{h^{3}}{12}\overline{{\textbf{{n}}}}\cdot{\textbf{{W}}}(\overline{{\textbf{{u}}}})&=\overline{{\textbf{{m}}}}^{\partial}\;\quad\mbox{ on }\partial_{\circ}\Gamma^{\varepsilon}\;,\\ {\textbf{{u}}}&=0\;\quad\mbox{ on }\partial_{\rm{ext}}\Gamma^{\varepsilon}\;,\\ {\mbox{\boldmath$\theta$\unboldmath}}&=0\;\quad\mbox{ on }\partial_{\rm{ext}}\Gamma^{\varepsilon}\;,\end{split}$$

$$\begin{array}{rl}{c}^2{\int }_{{\Omega }^{\ast \epsilon }}\nabla p^{\epsilon }\cdot \nabla q^{\epsilon }-{\omega }^2{\int }_{{\Omega }^{\ast \epsilon }}{p}^{\epsilon }q& =-\mathrm{i}\omega c^2\left({\int }_{{\Gamma }^{\pm \epsilon }}{g}^{\epsilon \pm }{q}^{\epsilon }d\Gamma +{\int }_{\partial {\Sigma }^{\epsilon }}\text{n}\cdot {\text{w}}^{\epsilon }{q}^{\epsilon }d\Gamma \right),\end{array}$$

$$\begin{split}&\omega^{2}h\int_{\Gamma^{\varepsilon}}\rho{\textbf{{u}}}^{\varepsilon}\cdot{\textbf{{v}}}^{\varepsilon}\omega^{2}\frac{h^{3}}{12}\int_{\Gamma^{\varepsilon}}\rho{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon}\cdot{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon}-h\int_{\Gamma^{\varepsilon}}[{{\rm I}\kern-1.99997pt{\rm E}}^{\varepsilon}\overline{\nabla}^{S}\overline{{\textbf{{u}}}}^{\varepsilon}]:\overline{\nabla}^{S}\overline{{\textbf{{v}}}}^{\varepsilon}\\ &-h\int_{\Gamma^{\varepsilon}}[{\textbf{{S}}}^{\varepsilon}(\overline{\nabla}u_{3}^{\varepsilon}-{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon})]\cdot(\overline{\nabla}v_{3}^{\varepsilon}-{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon})-\frac{h^{3}}{12}\int_{\Gamma^{\varepsilon}}[{{\rm I}\kern-1.99997pt{\rm E}}^{\varepsilon}\overline{\nabla}^{S}{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon}]:\overline{\nabla}^{S}{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon}\\ =&\int_{\Gamma^{\varepsilon}}{\textbf{{f}}}^{\varepsilon}(p^{\varepsilon})\cdot{\textbf{{v}}}^{\varepsilon}+\int_{\Gamma^{\varepsilon}}\overline{{\textbf{{m}}}}^{\varepsilon}(p^{\varepsilon})\cdot{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon}+\int_{\partial_{\circ}\Gamma^{\varepsilon}}\overline{{\textbf{{f}}}}^{\partial,\varepsilon}(p^{\varepsilon})\cdot\overline{{\textbf{{v}}}}^{\varepsilon}+\int_{\partial_{\circ}\Gamma^{\varepsilon}}\overline{{\textbf{{m}}}}^{\partial,\varepsilon}(p^{\varepsilon})\cdot{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon}\;,\end{split}$$

$$\begin{array}{rl}{\text{w}}^{\epsilon }(x^{\prime },x_3)& =(w_k^{\epsilon }(x^{\prime },x_3)),\\ w_{\alpha }^{\epsilon }(x^{\prime },x_3)& =u_{\alpha }^{\epsilon }(x^{\prime })-x_3{\theta }_{\alpha }^{\epsilon }(x^{\prime }),\alpha =1,2,\\ w_3^{\epsilon }(x^{\prime },x_3)& =u_3^{\epsilon }(x^{\prime }),\end{array}$$

$$\begin{array}{r}{\int }_{{\partial }_{\ast }{\Sigma }^{\epsilon }}{\text{b}}^{\epsilon }\cdot {\tilde{\text{w}}}^{\epsilon },\end{array}$$

$$\begin{array}{rl}{\overline{f}}_{\alpha }^{\epsilon }& =0,{\overline{m}}_{\alpha }^{\epsilon }=0,f_3^{\partial }=0,\\ {\overline{f}}_3^{\epsilon }& =\sum _{s=+,-}{b}_3(x^{\prime },s\epsilon \bar{h}/2)=\mathrm{i}\omega {\rho }_0(p^{\epsilon }(x^{\prime },\epsilon \bar{h}/2)-p^{\epsilon }(x^{\prime },-\epsilon \bar{h}/2)),\\ f_{\alpha }^{\partial ,\epsilon }& ={\int }_{-h^{\epsilon }/2}^{h^{\epsilon }/2}{b}_{\alpha }(x^{\prime },x_3)\mathrm{d}{x}_3=\mathrm{i}\omega {\rho }_0{\int }_{-h^{\epsilon }/2}^{h^{\epsilon }/2}{n}_{\alpha }(x^{\prime })p^{\epsilon }(x^{\prime },x_3)\mathrm{d}{x}_3,\\ m_{\alpha }^{\partial ,\epsilon }& =-{\int }_{-h^{\epsilon }/2}^{h^{\epsilon }/2}{x}_3{b}_{\alpha }(x^{\prime },x_3)\mathrm{d}{x}_3=-\mathrm{i}\omega {\rho }_0{\int }_{-h^{\epsilon }/2}^{h^{\epsilon }/2}{x}_3{n}_{\alpha }(x^{\prime })p^{\epsilon }(x^{\prime },x_3)\mathrm{d}{x}_3.\end{array}$$

$$\begin{split}\int_{\partial\Sigma^{\varepsilon}}{\textbf{{n}}}\cdot{\textbf{{w}}}^{\varepsilon}q^{\varepsilon}=\int_{\Gamma^{\varepsilon}}u_{3}^{\varepsilon}[\kern-1.69998pt{[}{q^{\varepsilon}(\cdot,x_{3})}]\kern-1.79993pt{]}_{{\varepsilon\bar{h}}}^{\pm}+\int_{\partial\Gamma^{\varepsilon}}\varepsilon\bar{h}\int_{-1/2}^{1/2}\bar{\textbf{{n}}}\cdot({\textbf{{u}}}^{\varepsilon}-\varepsilon\bar{h}\zeta{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon})q^{\varepsilon}(\cdot,\varepsilon\bar{h}\zeta)\mathrm{d}\zeta\;.\end{split}$$

$$\begin{split}\int_{\hat{\Omega}^{\varepsilon}}\hat{\nabla}p^{\varepsilon}\cdot\hat{\nabla}q^{\varepsilon}-\frac{\omega^{2}}{c^{2}}\int_{\hat{\Omega}^{\varepsilon}}p^{\varepsilon}q^{\varepsilon}=-\frac{{\rm{i}}\omega}{\varepsilon}\int_{\Gamma^{\pm}}\hat{g}^{\varepsilon\pm}q^{\varepsilon}\\ -\frac{{\rm{i}}\omega}{\varepsilon}\left[\int_{\Gamma^{\varepsilon}}u_{3}^{\varepsilon}[\kern-1.69998pt{[}{q^{\varepsilon}(\cdot,x_{3})}]\kern-1.79993pt{]}_{{\varepsilon\bar{h}}}^{\pm}+\int_{\partial_{\circ}\Gamma^{\varepsilon}}\varepsilon\bar{h}\int_{-1/2}^{1/2}\bar{\textbf{{n}}}\cdot(\overline{{\textbf{{u}}}}^{\varepsilon}-\varepsilon\bar{h}\zeta{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon})q^{\varepsilon}(\cdot,\varepsilon\bar{h}\zeta)\mathrm{d}\zeta\right]\;.\end{split}$$

$$\begin{split}&\omega^{2}\int_{\Gamma^{\varepsilon}}{\rho}{\textbf{{u}}}^{\varepsilon}\cdot{\textbf{{v}}}^{\varepsilon}+\omega^{2}\frac{h^{2}}{12}\int_{\Gamma^{\varepsilon}}{\rho}{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon}\cdot{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon}\\ &-\int_{\Gamma^{\varepsilon}}[{{\rm I}\kern-1.99997pt{\rm E}}^{\varepsilon}\overline{\nabla}^{S}\overline{{\textbf{{u}}}}^{\varepsilon}]:\overline{\nabla}^{S}\overline{{\textbf{{v}}}}^{\varepsilon}-\int_{\Gamma^{\varepsilon}}[{\textbf{{S}}}^{\varepsilon}(\overline{\nabla}u_{3}^{\varepsilon}-{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon})]\cdot(\overline{\nabla}v_{3}^{\varepsilon}-{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon})-\frac{h^{2}}{12}\int_{\Gamma^{\varepsilon}}[{{\rm I}\kern-1.99997pt{\rm E}}^{\varepsilon}\overline{\nabla}^{S}{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon}]:\overline{\nabla}^{S}{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon}\\ =&\frac{{\rm{i}}\omega\rho_{0}}{\varepsilon\bar{h}}\left[\int_{\Gamma^{\varepsilon}}v_{3}^{\varepsilon}[\kern-1.69998pt{[}{p^{\varepsilon}(\cdot,x_{3})}]\kern-1.79993pt{]}_{{\varepsilon\bar{h}}}^{\pm}+\varepsilon\bar{h}\int_{\partial_{\circ}\Gamma^{\varepsilon}}\int_{-1/2}^{1/2}p^{\varepsilon}(\cdot,\varepsilon\bar{h}\zeta)\bar{\textbf{{n}}}\cdot(\overline{{\textbf{{v}}}}^{\varepsilon}-\varepsilon\bar{h}\zeta{\mbox{\boldmath$\psi$\unboldmath}}^{\varepsilon})\mathrm{d}\zeta\right]\;.\end{split}$$

$$\begin{array}{rl}{\hat{g}}^{\epsilon +}(x^{\prime })& =g^0(x^{\prime })+\epsilon g^{1+}(x^{\prime },\frac{x^{\prime }}{\epsilon }),\\ {\hat{g}}^{\epsilon -}(x^{\prime })& =-g^0(x^{\prime })-\epsilon g^{1-}(x^{\prime },\frac{x^{\prime }}{\epsilon }).\end{array}$$

$$\begin{array}{r}\frac{1}{|\Xi |}{\int }_{\Xi }f=\sim {\int }_{\Xi }f\frac{1}{|\Xi |}{\int }_{D}f=:\sim {\int }_{D}f,\end{array}$$

$$\begin{array}{r}{\Vert p^{\epsilon }\Vert }_{L^2({\hat{\Omega }}^{\ast \epsilon })}\le C,{\Vert {\overline{\text{u}}}^{\epsilon }\Vert }_{[L^2({\Gamma }^{\epsilon }){]}^2}\le C,\\ {\Vert u_3^{\epsilon }\Vert }_{L^2({\Gamma }^{\epsilon })}\le C,{\Vert {\theta }^{\epsilon }\Vert }_{[L^2({\Gamma }^{\epsilon }){]}^2}\le C,\end{array}$$

$$\begin{array}{r}{\Vert \hat{\nabla }{p}^{\epsilon }\Vert }_{[L^2({\hat{\Omega }}^{\ast \epsilon }){]}^3}\le C,{\Vert \overline{\nabla }{\overline{\text{u}}}^{\epsilon }\Vert }_{[L^2({\Gamma }^{\epsilon }){]}^4}\le C,\\ {\Vert \overline{\nabla }{u}_3^{\epsilon }\Vert }_{[L^2({\Gamma }^{\epsilon }){]}^2}\le C,{\Vert \overline{\nabla }{\theta }^{\epsilon }\Vert }_{[L^2({\Gamma }^{\epsilon }){]}^4}\le C.\end{array}$$

$$\begin{array}{r}{\Vert \overline{\nabla }{p}^{\epsilon }\Vert }_{[L^2({\hat{\Omega }}^{\ast \epsilon }){]}^2}\le C,{\Vert {\partial }_z{p}^{\epsilon }\Vert }_{[L^2({\hat{\Omega }}^{\ast \epsilon }){]}^2}\le \epsilon C.\end{array}$$

$$\begin{array}{rl}{p}^{\epsilon }& \rightharpoonup p^0\text{mbox}w.in{L}^2(\hat{\Omega }),\\ {\partial }_z{p}^{\epsilon }& \rightharpoonup 0\text{mbox}w.in{L}^2(\hat{\Omega }),\end{array}$$

$$\begin{array}{rl}{\mathcal{T}}_{\epsilon }\left(p^{\epsilon }\right)& \rightharpoonup p^0\text{mbox}w.in{L}^2({\Gamma }_0\times Y^{\ast }),\\ {\mathcal{T}}_{\epsilon }\left(\overline{\nabla }{p}^{\epsilon }\right)& \rightharpoonup {\overline{\nabla }}_{x^{\prime }}{p}^0+{\overline{\nabla }}_{y^{\prime }}{p}^1\text{mbox}w.in{L}^2({\Gamma }_0\times Y^{\ast }),\\ \frac{1}{\epsilon }{\mathcal{T}}_{\epsilon }\left({\partial }_z{p}^{\epsilon }\right)& \rightharpoonup {\partial }_z{p}^1\text{mbox}w.in{L}^2({\Gamma }_0\times Y^{\ast }).\end{array}$$

$$\begin{array}{rl}{\text{u}}^{\epsilon }& \to {\text{u}}^0\text{mbox}s.in[L^2({\Gamma }_K){]}^2,\\ {\mathcal{T}}_{\epsilon }\left({\text{u}}^{\epsilon }\right)& \rightharpoonup {\text{u}}^0\text{mbox}w.in[L^2({\Gamma }_K\times {\Xi }_K){]}^3,\\ {\mathcal{T}}_{\epsilon }\left(\overline{\nabla }{\text{u}}^{\epsilon }\right)& \rightharpoonup {\overline{\nabla }}_{x^{\prime }}{\text{u}}^0+{\overline{\nabla }}_{y^{\prime }}{\text{u}}^1\text{mbox}w.in[L^2({\Gamma }_0\times {\Xi }_S){]}^6,\end{array}$$

$$\begin{split}{\mathcal{T}}_{\varepsilon}{\left({{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon}}\right)}&\rightharpoonup{\mbox{\boldmath$\theta$\unboldmath}}^{0}\quad\mbox{ w. in }[L^{2}(\Gamma_{0}\times\Xi_{S})]^{2}\;,\\ {\mathcal{T}}_{\varepsilon}{\left({\overline{\nabla}{\mbox{\boldmath$\theta$\unboldmath}}^{\varepsilon}}\right)}&\rightharpoonup\overline{\nabla}_{x^{\prime}}{\mbox{\boldmath$\theta$\unboldmath}}^{0}+\overline{\nabla}_{y^{\prime}}{\mbox{\boldmath$\theta$\unboldmath}}^{1}\quad\mbox{ w. in }[L^{2}(\Gamma_{0}\times\Xi)]^{4}\;,\end{split}$$

Full text

1 Introduction

The noise and vibration reduction belongs to important issues in design of structures used in the automotive industry, or civil engineering. The engine silencer used to reduce the noise emitted by the exhaust gas presents an important and well known example. However, there are many similar solid structures which can influence the acoustic wave propagation in fluid. Usually they involve porous, or perforated plates, or panels, such that they are permeable for the gas flow. The straightforward approach to modelling the acoustic wave propagation through vibrating perforated plates consists in solving directly the vibroacoustic problem with a 3D elastic structure describing the plate. However, its numerical treatment using the finite element method can lead to an intractable problem because of the prohibitive number of DOFs corresponding to the geometric complexity of the perforated structure. Therefore, it is reasonable to replace the elastic plate by an interface on which coupling transmission conditions are prescribed.

In this paper, we consider the acoustic wave propagation in an inviscid fluid interacting with elastic structures designed as periodically perforated plates. The aim is to derive non-local vibro-acoustic transmission conditions using the periodic homogenization method. Although similar problems have been treated in the literature, cf. [DELOURME201228], in this context, the plate elasticity has not been considered yet. As for the rigid structures, semi-empirical formulae for the acoustic impedance exist which were tuned by experiments, or developed using the electro-acoustic equivalent circuit theory [Jung-etal-2007-JKPS, Sakagami2010, Stremtan2012], or the Helmholtz-Kirchhoff integral theory [Zhou2013]. During the last decade, a number of works appeared which are based on a homogenization strategy. For a thin rigid perforated plate represented by interface $\Gamma_{0}$ and characterized by the thickness $\approx\delta$ it has been shown in [bb2005, DELOURME201228] that this interface is totally transparent for the acoustic field at the zero order $\delta^{0}$ terms of the model which describes the limit behaviour for $\delta\rightarrow 0$, cf. [Dorlemann2017]. For a higher order approximation, an approach based on the so-called inner and outer asymptotic expansions has been developed, such that two associated acoustic fields are coupled, one being relevant in the proximity of the perforations, the other at a distance from the limit interface, see e.g. [Clayes-Delourm-AA2013, Marigo-Maurel-JASA2016, Marigo-Maurel-PRSA2016]. In contrast with [bb2005] dealing with thin perforated interfaces only, in [rohan-lukes-waves07] we were concerned with homogenization of a fictitious layer in which rigid periodically distributed obstacles were placed. In particular, a rigid plate perforated by arbitrary shaped pores could be considered. Therein nonlocal transmission conditions were obtained as the two-scale homogenization limit of a standard acoustic problem imposed in the layer.

Here we follow the approach reported in [rohan-lukes-waves07] to develop vibroacoustic transmission conditions which substitute the vibroacoustic interaction on an elastic perforated plate immersed in the acoustic fluid. Up to our knowledge, despite some numerical studies, see *e.g. *. [Takahashi-Tanaka2002], a rigorous treatment of such a problem has not been treated using the homogenization method so far. As the result we obtain vibroacoustic transmission conditions in a form of an implicit Dirichlet-to-Neumann operator. Due to this operator, the elastic perforated plate can be replaced by an interface on which a jump of the global acoustic pressure is linked to the acoustic momenta associated with two faces of the homogenized plate. It allows us to obtain an efficient numerical model which takes into account geometrical details of the periodic perforation without need of discretizing the vibroacoustic problem at the global level. In other words, the homogenized interface provides a reduced model in which a complex 3D elastic structure is replaced by a 2D perforated plate model whose coefficients retain information about the perforation geometry. To do so, we rely on the homogenized Reissner-Mindlin plate tailor-made for the “simple” perforation represented by general cylindrical holes with axes orthogonal to the mid-plane of the plate. Elastic strongly heterogeneous plates were treated in [rohan-miara-CRAS2011, rohan-miara-ZAMM2015] where the framework of the Reissner-Mindlin theory was used to derive a model of phononic plates, cf. [Rohan2015bg-plates], but without the interaction with an exterior acoustic field.

The proposed modelling conception based on the problem decomposition and using the homogenization provides an alternative framework for modelling of microporous panels which are known for their capabilities of acoustic attenuation [Toyoda-JSV2005, Zhou2013, LIU2016149]. In [Maxit-JASA2012] the so-called patch transfer functions were developed for numerical modelling of compliant micro-perforated panels.

The plan of the paper is as follows. In Section 2 the vibroacoustic problem of the wave propagation in a waveguide containing the perforated plate is decomposed into the problem in a fictitious transmission layer (the “in-layer” problem) and the “outer” problem governing the acoustic field out of the layer. The “in-layer” vibroacoustic problem is treated using the homogenization method in Section 3, where the local problems imposed in the representative periodic cell are introduced and formulae for the homogenized coefficients are given. In Section LABEL:sec-global, as the main result of this paper, the global acoustic problem is established using the limit “in-layer” and the “outer” problems which are coupled using additional conditions derived by an additional integration and averaging procedure. The limit two-scale model of the homogenized layer is validated in Section LABEL:sec-valid using direct numerical simulations of the original problem. Finally, in Section LABEL:sec-simul, the proposed model is employed to simulate wave propagation in a waveguide equipped with the perforated plate. Some technical auxiliary derivations are presented in the Appendix.

Notation.

In the paper, the mathematical models are formulated in a Cartesian coordinate system $\mathcal{R}(\text{O};{\textbf{{e}}}_{1},{\textbf{{e}}}_{2},{\textbf{{e}}}_{3})$ where $O$ is the origin of the space and $({\textbf{{e}}}_{1},{\textbf{{e}}}_{2},{\textbf{{e}}}_{3})$ is a orthonormal basis for this space. The spatial position $x$ in the medium is specified through the coordinates $(x_{1},x_{2},x_{3})$ with respect to a Cartesian reference frame $\mathcal{R}$. The boldface notation for vectors, ${\textbf{{a}}}=(a_{i})$, and for tensors, ${\textbf{{b}}}=(b_{ij})$, is used. The gradient and divergence operators applied to a vector a are denoted by $\nabla{\textbf{{a}}}$ and $\nabla\cdot{\textbf{{a}}}$, respectively. By $\nabla^{S}{\textbf{{u}}}$ we denote the symmetrized gradient $\nabla{\textbf{{u}}}$, *i.e. *the strain tensor. When these operators have a subscript which is space variable, it is for indicating that the operator acts relatively at this space variable, for instance $\nabla_{x}=(\partial_{i}^{x})$. The symbol dot ‘$\cdot$’ denotes the scalar product between two vectors and the symbol colon ‘$:$’ stands for scalar (inner) product of two second-order tensors. Throughout the paper, $x$ denotes the global (“macroscopic”) coordinates, while the “local” coordinates $y$ describe positions within the representative unit cell $Y\subset{\mathbb{R}}^{3}$ where ${\mathbb{R}}$ is the set of real numbers. By latin subscripts $i,j,k,l\in\{1,2,3\}$ we refere to vectorial/tensorial components in ${\mathbb{R}}^{3}$, whereas subscripts $\alpha,\beta\in\{1,2\}$ are reserved for the tangential components with respect to the plate midsurface, *i.e. *coordinates $x_{\alpha}$ of vector represented by $x^{\prime}=(x_{1},x_{2})=(x_{\alpha})$ are associated with directions $({\textbf{{e}}}_{1},{\textbf{{e}}}_{2})$. Moreover, $\overline{\nabla}_{x}=(\partial_{\alpha})$ is the “in-plane” gradient. The gradient in the so-called dilated configuration with coordinates $(x^{\prime},z)$ is denoted by $\hat{\nabla}=(\overline{\nabla},\frac{1}{\varepsilon}\partial_{z})$. We also use the jump w.r.t. the transversal coordinate, $[\kern-1.69998pt{[}{q(\cdot,x_{3})}]\kern-1.79993pt{]}_{{r}}^{\pm}=q(\cdot,r/2)-q(\cdot,-r/2)$.

2 Formulation and decomposition of the vibroacoustic transmission problem

The aim of the paper is to find a representation of the vibro-acoustic interaction on a perforated plate. For this, homogenized vibroacoustic transmission conditions are derived using the asymptotisc analysis w.r.t. a scale parameter $\varepsilon$ which has a double role: on one hand it deals with the thickness of an elastic plate when considered as a 3D object, on the other hand it describes the size and spacing of holes periodically drilled in the plate structure.

The flowchart of deriving the transmission conditions for a limit global problem consists of the following steps:

The vibro-acoustic problem (later called the “global problem”) is formulated in a domain $\Omega^{G}\subset{\mathbb{R}}^{3}$ in which the perforated elastic plate is embedded, being represented by a planar surface – the plate midsurface.

• 2.

A transmission layer $\Omega_{\delta}$ of the thickness $\delta$ is introduced in terms of $\Gamma_{0}$ which constitutes its midsurface. This will allow to decompose the global problem into two subproblems: the vibroacoustic interaction in the layer $\Omega_{\delta}$ and the outer acoustic problems in $\Omega^{G}\setminus\Omega_{\delta}$. The two subproblems are coupled by natural transmission conditions on the “fictitious” interfaces $\Gamma_{\delta}^{\pm}$.

• 3.

We consider the layer thickness being proportional to the scale parameter, thus, $\delta=\varkappa\varepsilon$, where $\varkappa>0$ is fixed. The asymptotic analysis $\delta\approx\varepsilon\rightarrow 0$ considered for the problem in $\Omega_{\delta}$ with the Neumann type boundary conditions on $\Gamma_{\delta}^{\pm}$ leads to the homogenized vibroacoustic transmission problem defined on $\Gamma_{0}$. In this analysis, $\varepsilon$ has the double role announced above and the plate is described using its 2D representation in the framework of the Reissner-Mindlin plate theory. In Remark 2.4 we explain the dual interpretation of the plate thickness used in the asymptotic analysis of the vibro-acoustic problem.

• 4.

The final step is to derive the limit global problem for the acoustic waves in the fluid interacting with the homogenized perforated plate represented by $\Gamma_{0}$. For this, with a few modifications we follow the approach used in [rohan-lukes-waves07], where the rigid plate was considered; a given plate thickness $h$ corresponds to given finite thickness $\delta_{0}$ of the transmission layer. Then the continuity of the acoustic fields on interfaces $\Gamma_{\delta_{0}}^{\pm}$ yields the homogenized vibroacoustic transmission conditions which hold on $\Gamma_{0}$.

2.1 Global problem with transmission layer

In this section, we introduce the problem of acoustic waves in a domain $\Omega^{G}$ with embedded perforated elastic plate $\Sigma^{\varepsilon}$, see Fig. 2 and Fig. 3. The acoustic fluid occupies the domain $\Omega^{*\varepsilon}$. We consider a fictitious transimssion layer $\Omega_{\delta}$ with a thickness $\delta>0$, such that $\Sigma^{\varepsilon}\subset\Omega_{\delta}$. The plate thickness is $h^{\varepsilon}=\varepsilon\bar{h}$, while the layer thickness $\delta=\varkappa\varepsilon$ for a given fixed $\varkappa>0$.

For a fixed parameter $\varepsilon>0$, correspondingly to $\Omega^{*\varepsilon}$ and $\Sigma^{\varepsilon}$, we use a simplified notation $\Omega^{*}$ and $\Sigma$. The acoustic harmonic wave with the frequency $\omega$ is described by the acoustic potential $p:\Omega^{*}\ni x\mapsto{\mathbb{R}}^{3}$ in the fluid, the corresponding wave in the elastic body is described by the displacement field ${\textbf{{u}}}:\Sigma\ni x\mapsto{\mathbb{R}}^{3}$. Assuming the body is fixed to a rigid frame on the boundary $\partial_{\textbf{{u}}}\Sigma$ and interacting with the fluid on $\partial_{*}\Sigma=\partial\Sigma\setminus\partial_{\textbf{{u}}}\Sigma$, these fields satisfy the following equalities:

[TABLE]

Above, $c$ is the sound speed in the acoustic fluid, {\mbox{\boldmath\sigma\unboldmath}}({\textbf{{u}}}) is the stress in the lienar elastic solid, $\rho_{0}$ is reference fluid density, and by ${\textbf{{n}}}=(n_{i})$ we denote the normal vector. The constants $r,s\in\{0,1\}$ and $\bar{p}$ are defined to describe incident, reflected, or absorbed acoustic waves in the fluid, according to a selected part of the boundary.

2.2 Geometry of the perforated layer

Given a bounded 2D manifold $\Gamma_{0}\subset\{x\in\Omega^{G}|x_{3}=0\}$ representing the plate mid-plane, we introduce $\Omega_{\delta}=\Gamma_{0}\times]-\delta/2,\delta/2[\subset\Omega^{G}$, an open domain representing the transmission layer. This enables to decompose $\Omega^{G}$ into three nonoverlapping parts, as follows: $\Omega^{G}=\Omega_{\delta}\cup\Omega_{\delta}^{+}\cup\Omega_{\delta}^{-}$. Thus, the transmission layer is bounded by $\partial\Omega_{\delta}$ which splits into three parts:

[TABLE]

where $\delta>0$ is the layer thickness and $\vec{e_{3}}=(0,0,1)$, see Fig. 1. In the context of the transmission layer definition, we consider the plate as a 3D domain $\Sigma^{\varepsilon}$ defined in terms of the perforated midsurface $\Gamma^{\varepsilon}$; the following definitions are employed:

[TABLE]

where $\partial_{\textbf{{u}}}\Sigma^{\varepsilon}$ is the surface where the plate is clamped.

The midsurface $\Gamma^{\varepsilon}$ representing the perforated plate is generated using a representative cell $\Xi_{S}\subset{\mathbb{R}}^{2}$, as a periodic lattice. Let $\Xi={]0,\ell_{1}[}\times{]0,\ell_{2}[}$, where $\ell_{1},\ell_{2}>0$ are given (usually $\ell_{1}=\ell_{2}=1$) and consider the hole $\Xi^{*}\subset\Xi$, whereas its complement $\Xi_{S}=\Xi\setminus\overline{\Xi^{*}}$ defines the solid plate segment. Then

[TABLE]

Further we introduce the representative periodic cell $Y$ and define its solid part $S\subset Y$,

[TABLE]

so that $Y^{*}=Y\setminus\overline{S}$ is the fluid part. Obviously, in the transmission layer $\Omega_{\delta}$, the fluid occupies the part

[TABLE]

where $\vec{e_{1}}=(1,0,0)$ and $\vec{e_{2}}=(0,1,0)$.

For completeness, by virtue of (3) we can introduce the decomposition of boundary $\partial S=\partial_{\circ}S\cup\partial_{\pm}S\cup\partial_{\#}S$. For this we need the boundary $\partial\Xi_{S}=\partial_{\circ}\Xi_{S}\cup\partial_{\#}\Xi_{S}$, where $\partial_{\#}\Xi_{S}\equiv\partial\Xi$, so that the closed curve $\partial_{\circ}\Xi_{S}=\partial\Xi^{*}$ generates the cylindrical boundary $\partial_{\circ}S$:

[TABLE]

For the sake of simplicity, by $\partial\Xi_{S}$ we shall refere to $\partial_{\circ}\Xi_{S}$.

2.3 Problem decomposition

The domain split allows us to decompose problem (1) into three parts. By $P^{\delta}$ we denote the acoustic potential in $\Omega_{\delta}^{+}$ and $\Omega_{\delta}^{-}$, whereas $p^{\varepsilon}$ is the acoustic potential in the transmission layer $\Omega_{\delta}$, see Fig. 3. Further, by ${\rm{i}}\omega g^{\varepsilon\pm}$ we denote the acoustic fluid velocity projected into the normal of the interfaces $\Gamma_{\delta}^{\pm}$. The following subproblems are considered:

• 1,2: Given $\hat{p}^{\varepsilon}$ on $\Gamma_{\delta}^{\pm}$, find $P^{\delta}$ defined in $\Omega_{\delta}^{G}=\Omega_{\delta}^{+}\cup\Omega_{\delta}^{-}$, such that

[TABLE]

where $\partial_{\rm{ext}}\Omega_{\delta}^{G}=\partial\Omega^{G}\cap(\partial\Omega_{\delta}^{+}\cup\partial\Omega_{\delta}^{-})$ is the “external” boundary. As in problem (1), $r$ and $s$ are constants attaining values 0, or 1, whereas $\bar{p}$ is the amplitude of an incident wave. In the context of a waveguide, we consider $\partial_{\rm{ext}}\Omega_{\delta}^{G}$ to be decomposed into three parts, $\partial_{\rm in}\Omega_{\delta}^{G}$, $\partial_{\rm out}\Omega_{\delta}^{G}$, and $\partial_{w}\Omega_{\delta}^{G}$, denoting the input, the output and walls, respectively. By the constants $r,s$ in (8)3 different conditions on $\partial\Omega_{\delta}^{G}$ are respected: $r=s=0$ on the walls $\partial_{w}\Omega_{\delta}^{G}$, whereas $r=s=1$ on $\partial_{\rm in}\Omega_{\delta}^{G}$ and $r=1$, $s=0$ on $\partial_{\rm out}\Omega_{\delta}^{G}$, which accounts for the non-reflection condition.

• 3: Given $g^{\varepsilon\pm}$ on $\Gamma_{\delta}^{\pm}$, find $p^{\varepsilon}$ in $\Omega_{\delta}^{*\varepsilon}$ and ${\textbf{{u}}}^{\varepsilon}$ in $\Sigma^{\varepsilon}$, such that

[TABLE]

where $\partial_{*}\Sigma_{\delta}^{\varepsilon}=\partial\Sigma_{\delta}^{\varepsilon}\cap\partial\Omega_{\delta}^{*\varepsilon}$ is the surface of the elastic structure in contact with the fluid, thus, $\partial\Sigma_{\delta}^{\varepsilon}=\partial_{*}\Sigma_{\delta}^{\varepsilon}\cup\partial_{\textbf{{u}}}\Sigma_{\delta}^{\varepsilon}$.

• 4. For a fixed $\varepsilon$ and $\delta$, solutions to problems (8) and (9) are equivalent to the solution of (1), if the coupling conditions hold:

[TABLE]

where $n^{\pm}$ referes to normals ${\textbf{{n}}}^{\pm}$ outer to domains $\Omega_{\delta}^{\pm}$.

2.4 Plate model

The 3D model of an elastic plate involved in problem (1) can be replaced by a plate model which describes a thin structure. We assume a small $\varepsilon_{0}>0$ for which the limit model of acoustic transmission can be interpreted. In this paper we shall approximate behaviour of the thin elastic structure by the Reissner-Mindlin (R-M) plate model, which allows us to consider the effects related to shear stresses induced by rotations of the plate crossections w.r.t. the mid-plane.

The R-M plate model can be obtained by the asymptotic analysis of the corresponding 3D elastic structure while its thickness $\tilde{h}\rightarrow 0$. However, the obtained limit model is then interpreted in terms of a given thickness $h>0$. We shall discus this point in Remark 2.4.

The plate is represented by its perforated mean surface $\Gamma^{\varepsilon}$, therefore all involved variables depend on $\varepsilon$. However, for a while we drop the superscript ε related to these variables. The plate deflections are described by amplitude of the membrane elastic wave ${\textbf{{u}}}=(u_{1},u_{2})$, of the transverse wave $u_{3}$ and of the rotation wave {\mbox{\boldmath\theta\unboldmath}}=(\theta_{1},\theta_{2}). Two linear constitutive laws are involved, which depend upon the second order tensor ${\textbf{{S}}}=(S_{ij})=\varsigma\delta_{ij}$, where $\varsigma>0$ is the shear coefficient, and the fourth order elasticity tensor ${{\rm I}\kern-1.99997pt{\rm E}}=(E_{ijkl})$ which is given by the Hooke law adapted for the plane stress constraint; we define (all indices $i,j,k,l=1,2$)

[TABLE]

The Reissner-Mindlin plate model is derived using the following kinematic ansatz confining the displacement ${\textbf{{w}}}=(\overline{{\textbf{{w}}}},w_{3})$ in a plate with the actual thickness $h$,

[TABLE]

where $\overline{{\textbf{{u}}}}=(u_{1},u_{2})$ is “membrane-mode” displacements, i.e. vector ${\textbf{{u}}}=(\overline{{\textbf{{u}}}},u_{3})$ involves also the “transversal mode” (the deflection). The vector fields ({\textbf{{u}}},{\mbox{\boldmath\theta\unboldmath}}) satisfy the following equations in $\Gamma^{\varepsilon}$,

[TABLE]

where $\partial_{\circ}\Gamma^{\varepsilon}=\partial\Gamma^{\varepsilon}\setminus\partial_{\rm{ext}}\Gamma^{\varepsilon}$ describes the perforations. Above the applied forces ${\textbf{{f}}}=(\overline{{\textbf{{f}}}},f_{3})$, $\overline{{\textbf{{f}}}}^{\partial}$ and moments $\overline{{\textbf{{m}}}}$, $\overline{{\textbf{{m}}}}^{\partial}$ depend on the acoustic potential $p$. The crucial step in deriving the model of vibroacoustic transmission consists in describing these forces in terms of $p$ imposed on surface $\partial\Sigma^{\varepsilon}$ in the 3D plate representation.

**Remark

1. ** In our asymptotic analysis of the acoustic transmission layer, we shall use the plate thickness in two contexts:

The periodically perforated plate model defined in terms of the 2D domain $\Gamma^{\varepsilon}\subset\Gamma_{0}$ representing the mid-plane and the thickness $h=\varepsilon_{0}\bar{h}$ with $\varepsilon_{0}>0$ being fixed. In fact, for a given thickness $h$ and the perforation design (a given size of the holes yielding $\varepsilon_{0}$) we can obtain $\bar{h}$.

• 2.

To describe the interaction between the 3D elastic structure and the acoustic fluid, the thickness must be proportional to $\varepsilon$ which is also related to the transmission layer thickness $\delta=\varkappa\varepsilon$, thus, we consider $h^{\varepsilon}=\varepsilon\bar{h}$ and the elastic body occupying domain $\Sigma^{\varepsilon}$, see (3).

Thus, the homogenization of the periodically perforated plate is done by pursuing the asymptotic analysis $\varepsilon\rightarrow 0$ applied to the 2D plate model (13) divided by $h$. Whereas $h$ is fixed in the plate equation operator, beeing independent of $\varepsilon$, at the r.h.s. terms we get $1/(\varepsilon\bar{h})$ which is coherent with the dilation operation applied when dealing with fluid equation, see Section 2.7.

$\triangle$

2.5 Variational formulation of the vibroacoustic problem in the layer

In order to derive the homogenized model of the transmission layer, we shall need the variational formulation of problem (9) with the plate model (13).

Find $p^{\varepsilon}\in H^{1}(\Omega^{*\varepsilon})$ and ({\textbf{{u}}}^{\varepsilon},{\mbox{\boldmath\theta\unboldmath}}^{\varepsilon})\in(H^{1}_{0}(\Omega))^{5} such that

[TABLE]

for all $q\in H^{1}(\Omega^{*\varepsilon})$, where n is outward normal to domain $\Sigma^{\varepsilon}$, and

[TABLE]

for all test functions ({\textbf{{v}}}^{\varepsilon},{\mbox{\boldmath\psi\unboldmath}}^{\varepsilon})\in(H^{1}_{0}(\Omega))^{5}. In (14), the displacements ${\textbf{{w}}}^{\varepsilon}$ defined on the surface $\partial\Sigma^{\varepsilon}$ are expressed using the mid-plane kinematic fields. Due to (12), it holds that

[TABLE]

where $x^{\prime}\in\Gamma^{\varepsilon}$, $x_{3}\in\varepsilon\bar{h}]-1/2,1/2[$. In analogy, the test displacements $\tilde{w}_{k}^{\varepsilon}(x^{\prime},x_{3})$ $k=1,2,3$ can bentroduced in terms of the test functions ({\textbf{{v}}}^{\varepsilon},{\mbox{\boldmath\psi\unboldmath}}^{\varepsilon}) involved in (15); in this equation, the r.h.s. integrals express the virtual power

[TABLE]

where the traction stress ${\textbf{{b}}}^{\varepsilon}={\rm{i}}\omega\rho_{0}{\textbf{{n}}}p^{\varepsilon}$ is induced by the acoustic pressure in the fluid.

2.6 Fluid structure interaction on the plate surface $\partial\Sigma$

The forces and moments involved in the r.h.s. of (15) can be identified using the 3D representation of the plate surface $\partial_{*}\Sigma^{\varepsilon}$ decomposed according to (3). The actual surface traction $b_{i}^{\varepsilon}={\rm{i}}\omega\rho_{0}n_{i}p^{\varepsilon}$ is given by the acoustic potential and by the surface normal ${\textbf{{n}}}=(n_{i})$; note that $n_{\alpha}=0$, $\alpha=1,2$ on $\partial_{\pm}\Sigma^{\varepsilon}$, whereas $n_{3}=0$ on $\partial_{\circ}\Sigma^{\varepsilon}$. Hence, it can be shown that the following expressions hold:

[TABLE]

Then we consider the fluid equation. In (14), in the integral on $\partial\Sigma^{\varepsilon}$, the displacement field w must be expressed in terms of the mid-plane displacements and rotations ${\textbf{{u}}}^{\varepsilon}$ and {\mbox{\boldmath\theta\unboldmath}}^{\varepsilon}, as introduced in (16). This yields

[TABLE]

2.7 Dilated formulation

We can now state the vibro-acoustic problem in the dilated layer $\hat{\Omega}=\Gamma_{0}\times]-\varkappa/2,+\varkappa/2[$, where the fluid occupies domain $\hat{\Omega}^{*}=\{(x^{\prime},\varepsilon^{-1}x_{3})\in{\mathbb{R}}^{3}|x\in\Omega^{*\varepsilon}\}$, see (6). Using $z=\varepsilon^{-1}x_{3}$, while $x^{\prime}=(x_{\alpha})$, with new coordinates $(x^{\prime},z)$, the gradients are $\hat{\nabla}=(\partial_{\alpha},\varepsilon^{-1}\partial_{z})$, thus $\nabla p(x)=\hat{\nabla}p(x_{\alpha},z)$; to simplify the notation, we shall use the same notation for functions depending on $x_{3}$, but expressed in terms of $z$.

By virtue of the dilation and the periodic unfolding, the vibroacoustic problem can be transformed in the domain which does not change with $\varepsilon$. Consequently the standard means of convergence can be used to obtain the limit model.

Equation (14) with the substitution (19) can now be transformed by the dilatation (the same notation for all variables is adhered, but should be interpreted in this new context of this dilated formulation):

[TABLE]

Further we employ (18) to rewrite (15) which is divided by $h^{\varepsilon}$; by virtue of Remark 2.4, the plate thickness is given, i.e. $h=\varepsilon_{0}\bar{h}$, however, when dealing with the r.h.s. interaction terms, $h:=h^{\varepsilon}=\varepsilon\bar{h}$ in accordance with the dilation transformation. Thus we get

[TABLE]

It is worth noting that, in (20) and (21), the r.h.s. integrals provide a symmetry of the following formulation.

The problem formulation

The vibroacoustic interaction in the dilated layer $\hat{\Omega}_{\delta}$ is described by (p^{\varepsilon},{\textbf{{u}}}^{\varepsilon},{\mbox{\boldmath\theta\unboldmath}}^{\varepsilon})\in H^{1}(\hat{\Omega}^{*\varepsilon})\times(H_{0}^{1}(\Gamma^{\varepsilon}))^{5} which satisfy equations (20)-(21) for any test fields (q^{\varepsilon},{\textbf{{v}}}^{\varepsilon},{\mbox{\boldmath\psi\unboldmath}}^{\varepsilon})\in H^{1}(\hat{\Omega}^{*\varepsilon})\times(H_{0}^{1}(\Gamma^{\varepsilon}))^{5}.

Let $g^{0}\in L^{2}(\Gamma_{0})$ and $g^{1\pm}(x^{\prime},y^{\prime})\in L^{2}(\Gamma_{0}\times{\mathbb{R}}^{2})$, whereby $g^{1\pm}(x^{\prime},\cdot)$ being $\Xi$-periodic in the second variable; we define

[TABLE]

For any $\varepsilon>0$ and $\hat{g}^{\varepsilon\pm}$ defined according to (22), the vibroacoustic interaction problem constituted by equations (20)-(21) possesses a unique solution (p^{\varepsilon},{\textbf{{u}}}^{\varepsilon},{\mbox{\boldmath\theta\unboldmath}}^{\varepsilon}). As an essential step of the proof, the a priori estimates are derived in the Appendix LABEL:sec-appendixA.

3 Homogenization of the transmission layer

In this section, we introduce the convergence result which yields the limit acoustic pressure and the plate displacements and rotations. These are involved in the limit two-scale equations of the vibroacoustic problem imposed in the transmission layer. The asymptotic analysis is based on the unfolding method which was inaugurated in the seminal paper [Cioranescu-etal-2008] and elaborated further for thin structures in [Cioranescu2008-Neumann-sieve]. In our setting, the unfolding operator $\mathcal{T}_{\varepsilon}\,{}:L^{2}(\Omega_{\delta};{\mathbb{R}})\rightarrow L^{2}(\Gamma_{0}\times Y;{\mathbb{R}})$ transforms a function $f(x)$ defined in $\Omega_{\delta}$ into a function of two variables, $x^{\prime}\in\Gamma_{0}$ and $y\in Y$. For any $f\in L^{1}(Y)$, the cell average involved in all unfolding intergartion formulae will be abreviated by

[TABLE]

whatever the domain $D\subset\overline{Y}$ of the the integral is (*i.e. *volume, or surface).

3.1 The convergence results

Based on the a priori estimates derived in the Apendix A, the following theorem holds.

**Theorem 1.

---

Let us assume*

[TABLE]

then the folloving estimates can be obtained:

[TABLE]

Since $\hat{\nabla}p^{\varepsilon}=(\overline{\nabla}p^{\varepsilon},\varepsilon^{-1}\partial_{z}p^{\varepsilon})$, we have

[TABLE]

Due to Theorem 3.1 providing the estimates (24)-(26) we obtain the convergence of the unfolded functions(For the definition of the unfolding operator we refere *e.g. *to [Cioranescu-etal-2008]). First we observe (note (26)2):

[TABLE]

thus, $\partial_{z}p^{0}=0$. The classical results of the unfolding method of homogenization yield

[TABLE]

Above $p^{0}\in H^{1}(\Gamma_{0})$ and $p^{1}\in L^{2}(\Gamma_{0};H_{\underline{\#}}^{1}(Y^{*}))$, where $H_{\underline{\#}}^{1}(Y^{*})$ is the subspace of $H^{1}(Y)$ generated by $\Xi$-periodic functions (thus, the periodicity in $y_{\alpha}$ holds, but not in $y_{3}=z$), with vanishing average in $Y$.

For the plate responses we get

[TABLE]

where ${\textbf{{u}}}^{0}\in[H^{1}(\Gamma_{K})]^{3}$ and ${\textbf{{u}}}^{1}\in L^{2}(\Gamma_{K};[H_{\#}^{1}(\Xi_{K})]^{3})$. Here $H_{\#}^{1}(\Xi_{K})$ is subspace of$H^{1}(\Xi)$ involving only $\Xi$-periodic functions with vanishing average in $\Xi_{K}$. For the rotations we obtain

[TABLE]

where {\mbox{\boldmath\theta\unboldmath}}^{0}\in[H^{1}(\Gamma_{K})]^{2} and {\mbox{\boldmath\theta\unboldmath}}^{1}\in L^{2}(\Gamma_{0};[H_{\#}^{1}(\Xi_{S})]^{2}).

The limit vibro-acoustic problem can be derived by a formal approach which relies on the recovery sequences (w.r.t. $\varepsilon$) constructed in accordance with the convergence result. Neglecting the higher order terms in $\varepsilon$, the following approximate expansions for unfolded vibroacoustic fields (p^{\varepsilon},{\textbf{{u}}}^{\varepsilon},{\mbox{\boldmath\theta\unboldmath}}^{\varepsilon}) are considered:

[TABLE]

where $x^{\prime}\in\Gamma_{0}$, $y^{\prime}\in\Xi$ and $y=(y^{\prime},z)\in Y$; in (31), all the two-scale functions are $\Xi$-periodic in the second variable. Analogous expansions involving two-scale functions periodic in $y^{\prime}$ will be employed as the test functions involved in (20)-(21),

[TABLE]

It is worth to note that the use of the recovery sequences simplifies the derivation of limit equations of the vibro-acoustic model which, however, can be obtained more rigorously using the asymptotic analysis applied directley to equations (20)-(21). We shall substitute the ansatz (31)-(32) in unfolded equations (20)-(21) and explore the limit form for $\varepsilon\rightarrow 0$.

3.2 Limit fluid equation

The unfolded left hand side of (20) yields the following limit form:

[TABLE]

The unfolded right hand side integrals can be written, as follows:

[TABLE]

In the limit, the first integral related to the dilated fictitious interfaces $\Gamma^{\pm}$ yields

[TABLE]

The second integral in (34) can be rewritten, as follows (omitting the factor $-{\rm{i}}\omega c^{2}$)

[TABLE]

where only $q^{1}(x^{\prime},y^{\prime},\bar{h}\zeta)$ depends on $\zeta\in]-1/2,+1/2[$. Hence, since $\int_{-1/2}^{1/2}\zeta=0$, in the limit, the second integral in (34) yields

[TABLE]

Now the limit fluid equation constituted using (33)(35) and (37) attains the following form:

[TABLE]

where $\Delta g^{1}=g^{1+}-g^{1-}$.

3.3 Limit plate equation

The unfolded left hand side of (21) yields the following limit form:

[TABLE]

The unfolded right hand side integrals can be written in analogy with the ones involved in the fluid equation, see (36). Since the role of the solution and the test function switches, the unfolded form of (21) yields

[TABLE]

which, in the limit, yields an analogous expression as the one of (37). Thus, the limit of the plate equation (21) is constituted by (39) which equals to

[TABLE]

**Remark 2. ** Integrals over the plate surface involving $q^{1}$ and $p^{1}$ in (37) and (41), respectively, can be written in a more compact form; for any two-scale function $\varphi(x^{\prime},y^{\prime},\zeta)$ it holds that

[TABLE]