Compressible fluid

For a given a three-dimensional fluid domain $\Omega_\mathrm{F}$ and $(x,y,z)\in\Omega_\mathrm{F}$, we introduce the following quantities

• frequency $f=\omega/(2\pi)>0$ [Hz],
• mass density $\rho=\hat{\rho}(x,y,z,f)>0$ in $\Omega_\mathrm{F}$ [kg/m$^3$],
• speed of sound $c=\hat{c}(x,y,z,f)>0$ in $\Omega_\mathrm{F}$ [m/s],
• volumetric source $\boldsymbol{g}_{\mathrm{V}}(x,y,z,f)\in\mathbb{C}^3$ in $\Omega_\mathrm{F}$ [kg/m$^2$s$^2$]$^3$,
• surface pressure $p_{\mathrm{S}}(x,y,z,f)\in\mathbb{C}$ on $\Gamma_{\mathrm{N}}\cup\Gamma_{\mathrm{I}}$ [kg/ms$^2$],
• normal displacement $u_{\mathrm{D}}(x,y,z,f)\in\mathbb{C}$ on $\Gamma_{\mathrm{D}}$ [m],
• surface impedance $Z(x,y,z,f)\in\mathbb{C}$ on $\Gamma_{\mathrm{I}}$ [kg/m$^2$s].

Splitting the boundary $\Omega_\mathrm{F}$ into three disjoint parts, $\partial\Omega_\mathrm{F}=\Gamma_\mathrm{D}\cup\Gamma_\mathrm{N}\cup\Gamma_\mathrm{I}$, we seek the displacement field $\boldsymbol{u}(x,y,z)\in\mathbb{C}^3$ characterized by

\begin{equation*} \begin{array}{rcll} -\omega^2\rho\boldsymbol{u}-\mathrm{grad}(\rho c^2\mathrm{div}\boldsymbol{u}) & = & \boldsymbol{g}_{\mathrm{V}}&\mbox{ in }\Omega_\mathrm{F},\\ \boldsymbol{u}\cdot\boldsymbol{n} & = & u_{\mathrm{D}}&\mbox{ on }\Gamma_{\mathrm{D}}, & \mbox{(Dirichlet boundary)}\\ -\rho c^2\mathrm{div}\boldsymbol{u} & = & p_{\mathrm{S}}&\mbox{ on }\Gamma_{\mathrm{N}}, & \mbox{(Neumann boundary)}\\ -\rho c^2\mathrm{div}\boldsymbol{u}-i\omega Z(\boldsymbol{u}\cdot\boldsymbol{n}) & = & p_{\mathrm{S}}&\mbox{ on }\Gamma_{\mathrm{I}}, & \mbox{(Impedance boundary)} \end{array} \end{equation*}

being $\boldsymbol{n}$ the outward normal vector to $\partial\Omega_\mathrm{F}$.

The corresponding variational formulation of the problem reads as follows: find ${\boldsymbol{u}}\in H(\mbox{div},\Omega_\mathrm{F})$, such that ${\boldsymbol{u}} \cdot\boldsymbol{n} = u_\mathrm{D}$ on $\Gamma_\mathrm{D}$ and \begin{equation*} \left. \begin{array}{ccccccccccc} -\omega^2 \int_{\Omega_\mathrm{F}} \rho(\boldsymbol{u}\cdot \widetilde{\boldsymbol{u}}) ~\mbox{d}{x} + \int_{\Omega_\mathrm{F}} \rho c^2\mbox{div}\boldsymbol{u}~\mbox{div}\widetilde{\boldsymbol{u}} ~\mbox{d}{x} + i\omega \int_{\Gamma_\mathrm{I}} Z(\boldsymbol{u}\cdot\boldsymbol{n})(\widetilde{\boldsymbol{u}} \cdot\boldsymbol{n})~\mbox{d}{\gamma} = \int_{\Omega_\mathrm{F}} \boldsymbol{g}_\mathrm{V}\cdot\widetilde{\boldsymbol{u}} ~\mbox{d}{x} - \int_{\Gamma_\mathrm{I} \cup \Gamma_\mathrm{N}} p_\mathrm{S}(\widetilde{\boldsymbol{u}} \cdot\boldsymbol{n})~\mbox{d}{\gamma}, \end{array} \right. \end{equation*} for all $\widetilde{\boldsymbol{u}}\in H(\mbox{div},\Omega_\mathrm{F})$ (see, for instance, ), such that $\widetilde{\boldsymbol{u}} \cdot\boldsymbol{n} = 0$ on $\Gamma_\mathrm{D}$, where \begin{equation*} H(\mbox{div},\Omega_\mathrm{F})=\left\{\boldsymbol{v}\in\left[L^2(\Omega_\mathrm{F})\right]^3\;;\; \mbox{div}\boldsymbol{v}\in L^2(\Omega_\mathrm{F})\right\}. \end{equation*}

Given a tetrahedra mesh $\tau_h$ of the domain $\Omega_\mathrm{F}$, the approximate problem is defined using Raviart-Thomas finite elements : find ${\boldsymbol{u}_h}\in R_h(\Omega_\mathrm{F})$, such that ${\boldsymbol{u}_h} \cdot\boldsymbol{n} = u_\mathrm{D}$ on $\Gamma_\mathrm{D}$ and \begin{equation*} \left. \begin{array}{ccccccccccc} -\omega^2 \int_{\Omega_\mathrm{F}} \rho(\boldsymbol{u}_h\cdot \widetilde{\boldsymbol{u}}_h) ~\mbox{d}{x} + \int_{\Omega_\mathrm{F}} \rho c^2\mbox{div}\boldsymbol{u}_h~\mbox{div}\widetilde{\boldsymbol{u}}_h ~\mbox{d}{x} + i\omega \int_{\Gamma_\mathrm{I}} Z(\boldsymbol{u}_h\cdot\boldsymbol{n})(\widetilde{\boldsymbol{u}}_h \cdot\boldsymbol{n})~\mbox{d}{\gamma} = \int_{\Omega_\mathrm{F}} \boldsymbol{g}_\mathrm{V}\cdot\widetilde{\boldsymbol{u}}_h ~\mbox{d}{x} - \int_{\Gamma_\mathrm{I} \cup \Gamma_\mathrm{N}} p_\mathrm{S}(\widetilde{\boldsymbol{u}}_h \cdot\boldsymbol{n})~\mbox{d}{\gamma}, \end{array} \right. \end{equation*} for all $\widetilde{\boldsymbol{u}}_h\in R_h(\Omega_\mathrm{F})$, such that $\widetilde{\boldsymbol{u}}_h \cdot\boldsymbol{n} = 0$ on $\Gamma_\mathrm{D}$, where \begin{equation*} R_h(\Omega_\mathrm{F})=\left\{\boldsymbol{v}_h\in H(\mbox{div},\Omega_\mathrm{F})\;;\; \boldsymbol{v}_h|_{T}(x,y,z)=\left(a_{T}+d_{T}x,b_{T}+ d_{T}y,c_{T}+d_{T}z\right)\quad\forall\; {T}\in\tau_h\right\}, \end{equation*} being $a_T$, $b_T$, $c_T$ and $d_T$ complex constants, different for each tetrahedron $T\in\tau_h$.

Let $\boldsymbol{\phi}_j$ be the j-th basis function on the Raviart-Thomas space, defined by \begin{equation*} \boldsymbol{\phi}_j(\boldsymbol{a}_k)\cdot\boldsymbol{n}_k=A_k\delta_{jk}\quad\forall\;T\in\tau_h, \end{equation*} for all $1\leq j,k\leq N$, where $N$ is the global number of faces in $\tau_h$, $\boldsymbol{a}_k$ is the local vertex $k$ of the tetrahedron $T$, $\boldsymbol{n}_k$ is the outward normal vector to the face $k$ of $T$ and $A_k$ is the area of the $j$-th face of $T$.

Then, the approximate solution $\boldsymbol{u}_h\in R_h(\Omega_\mathrm{F})$ can be written in terms of the basis functions, \begin{equation*} \boldsymbol{u}_h(x,y,z)=\sum_{j=1}^N u_h^j\boldsymbol{\phi}_j(x,y,z), \end{equation*} and the discrete problem can be written in matrix formulation \begin{equation*} \left(K_\mathrm{F}-\omega^2 M_\mathrm{F}+i\omega C_\mathrm{F}\right)\boldsymbol{U}_h=G_\mathrm{F}, \end{equation*} being

\begin{equation*} \begin{array}{lcl} \displaystyle (K_\mathrm{F})_{jk} &=& \int_{\Omega_\mathrm{F}} \rho c^2\mbox{div}\boldsymbol{\phi}_k~\mbox{div}\boldsymbol{\phi}_j ~\mbox{d}{x},\\ \displaystyle(M_\mathrm{F})_{jk}&=&\int_{\Omega_\mathrm{F}} \rho(\boldsymbol{\phi}_k\cdot \boldsymbol{\phi}_j) ~\mbox{d}{x},\\ \displaystyle(C_\mathrm{F})_{jk}&=&\int_{\Gamma_\mathrm{I}} Z(\boldsymbol{\phi}_k\cdot\boldsymbol{n}) (\boldsymbol{\phi}_j\cdot\boldsymbol{n})~\mbox{d}{\gamma},\\ \displaystyle(G_\mathrm{F})_{j}&=&\int_{\Omega_\mathrm{F}} \boldsymbol{g}_\mathrm{V}\cdot\boldsymbol{\phi}_j ~\mbox{d}{x} -\int_{\Gamma_\mathrm{I} \cup \Gamma_\mathrm{N}} p_\mathrm{S}(\boldsymbol{\phi}_j \cdot\boldsymbol{n})~\mbox{d}{\gamma}, \end{array} \end{equation*}

for all $1\leq j,k\leq N$, where $\displaystyle\left(\boldsymbol{U}_h\right)_{k}= u_h^k$.

References

 F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991.

 Raviart P. A, Thomas J. M. A mixed finite element method for second order elliptic problems. In Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606. Springer, Berlin, 1977; 292–315.

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