# Numerical Analysis of Vibroacoustic Response of Timber Floor Panels Damped with Porous Materials
## 3 Finite Element Formulation of Sound Radiation of an Elastic Structure with a Poroelastic Layer
### Variational Formulation  
The variational formulation of the problem is obtained from the local equations of the previous paragraph by applying the test function method. It is written for the different domains as follows [26; 27]:  
* Elastic structure: \[\int_{\Omega_{E}}\stackrel{{=E}}{{\sigma}}\left(\stackrel{{ =E}}{{u}}\right):\stackrel{{=E}}{{\varepsilon}}\left(\delta \stackrel{{=E}}{{u}}\right)dV-\ \omega^{2}\int_{\Omega_{E}}\rho_{E}\stackrel{{ \rightarrow^{E}}}{{u}}.\delta\stackrel{{=E}}{{u}}dV-\int_{ \sum_{EP}}\stackrel{{=E}}{{\sigma}}\left(\stackrel{{ -\varepsilon}}{{u}}\right)\stackrel{{\rightarrow}}{{n_{E}}} \delta\stackrel{{\rightarrow^{S}}}{{u}}dS=\int_{T_{t}}\stackrel{{ -\varepsilon}}{{F}}d\delta\stackrel{{\rightarrow^{E}}}{{u}}dS\] (5)
* Poroelastic material (solid phase): \[\int_{\Omega_{P}}\stackrel{{=S}}{{\sigma}}\left( \stackrel{{-S}}{{u}}\right):\stackrel{{=S}}{{ \varepsilon}}\left(\delta\stackrel{{-S}}{{u}}\right)dV-\omega^{2} \int_{\Omega_{P}}\stackrel{{\sim}}{{\rho}}_{s}\stackrel{{ -S}}{{u}}\delta\stackrel{{\rightarrow^{S}}}{{u}}dV-\int_{ \Omega_{P}}\stackrel{{\sim}}{{\gamma}}\stackrel{{ \rightarrow}}{{\gamma}}\stackrel{{\rightarrow}}{{\gamma}} \stackrel{{\rightarrow}}{{p}}^{F}.\delta\stackrel{{ \rightarrow^{S}}}{{u}}dV-\int_{\sum_{EP}}\stackrel{{=T}}{{\sigma}} \stackrel{{\rightarrow}}{{n_{P}}}.\delta\stackrel{{ \rightarrow^{S}}}{{u}}dS\] (6) \[-\int_{\sum_{EP}}\phi\left(1+\stackrel{{<}}{{R}} \right)p^{F}\delta\stackrel{{\rightarrow^{S}}}{{u}}\stackrel{{ \rightarrow}}{{n_{P}}}dS=0\] \[\bullet\] Poroelastic material (fluid phase): \[\int_{\Omega_{P}}\stackrel{{\phi^{2}}}{{\phi_{22}}} \stackrel{{\rightarrow}}{{\gamma}}\stackrel{{ \rightarrow}}{{p}}^{F}.\stackrel{{\rightarrow}}{{\
abla}} \delta p^{F}dV-\omega^{2}\int_{\Omega_{P}}\stackrel{{\phi^{2}}}{{ \gamma}}\stackrel{{\rightarrow}}{{u}}\stackrel{{ \rightarrow}}{{\gamma}}\stackrel{{\rightarrow}}{{\delta}} \stackrel{{\rightarrow}}{{\gamma}}\stackrel{{ \rightarrow}}{{\delta}}\stackrel{{\rightarrow}}{{p}}^{F}-\omega^{2} \int_{\sum_{EP}}\phi\left(1+\stackrel{{<}}{{R}}\right)\stackrel{{ \rightarrow}}{{u}}\stackrel{{\rightarrow}}{{n_{P}}}\delta p^{F}dS\] (7) \[-\omega^{2}\int_{\sum_{EP}}\phi\left(\stackrel{{ \rightarrow^{F}}}{{u}}\stackrel{{\rightarrow^{S}}}{{u}}\stackrel{{ \rightarrow}}{{u}}\stackrel{{\rightarrow}}{{u}}\right)\stackrel{{ \rightarrow}}{{n_{P}}}dS=0\] where \[\delta\stackrel{{\rightarrow^{E}}}{{u}}\stackrel{{ \rightarrow^{S}}}{{,}}\delta\stackrel{{\rightarrow^{S}}}{{u}}\], and \[\delta p^{F}\] are virtual quantities (displacements and pressures). By summing the coupling terms defined at the structure-poroelastic interface \[\sum_{EP}\] in the integral Equations (5)-(7), we find the following expression: \[I_{EP=}-\int_{\sum_{EP}}\phi\left(1+\frac{\widetilde{Q}}{R}\right)p^{F}\delta \stackrel{{\rightarrow^{S}}}{{u}}.\stackrel{{ \rightarrow}}{{n_{P}}}dS-\omega^{2}\int_{\sum_{EP}}\phi\left(1+\frac{ \widetilde{Q}}{R}\right)\stackrel{{\rightarrow^{S}}}{{u}}. \stackrel{{\rightarrow}}{{n_{P}}}\delta p^{F}dS\] (8)  
Equation (8) shows that the porous material is coupled to the elastic domain through a symmetrical coupling term. For most of porous materials used in acoustic, the bulk modulus of the poroelastic material is negligible compared with the bulk modulus of the material from which the skeleton is made; thus, it is assumed the coefficient \(\phi\left(1+\widetilde{Q}/\widetilde{R}\right)\approx 1\)[25]. In this case, the integrals in Equation (8) become with the same form as the usual elastic structure-acoustic coupling condition. Thus, at the surface \(\sum_{EP}\), the coupling is explicitly imposed with only the kinematic condition \(\stackrel{{\rightarrow^{S}}}{{u}}=\stackrel{{ \rightarrow^{E}}}{{u}}\).