# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 2 Compressible Flow
### Discrete Flow Problem  
We describe the discretization, which is used in our in-house solver. We assume that \(\Omega_{t}\) is a polygonal domain for every \(t\in[0,T]\). We denote by \(\mathcal{T}_{ht}\) a partition of the closure \(\overline{\Omega}_{t}\) into a finite number of closed triangles with disjoint interiors satisfying standard properties (see [29]). We suppose that \(\mathcal{T}_{ht}\) is an image of \(\mathcal{T}_{h0}\) under the regular mapping \"\(t\to\mathcal{A}\)\". Moreover, we assume that the ALE mapping \(\mathcal{A}_{t}\) is continuous and affine in \(\overline{\Omega}_{0}\). By \(\mathcal{F}\), we denote the system of all faces of all elements \(K\in\mathcal{T}_{ht}\). Moreover, we introduce the sets of boundary faces \(\mathcal{F}^{B}=\{\Gamma\in\mathcal{F};\,\Gamma\subset\partial\Omega_{t}\}\), \"Dirichlet\" boundary faces \(\mathcal{F}^{D}=\{\Gamma\in\mathcal{F}^{B};\,\text{a Dirichlet condition is prescribed on }\Gamma\}\) and inner faces \(\mathcal{F}^{I}=\mathcal{F}\setminus\mathcal{F}^{B}\). Each \(\Gamma\in\mathcal{F}\) is associated with a unit normal vector \(\mathbf{n}_{\Gamma}\) to \(\Gamma\). For \(\Gamma\in\mathcal{F}^  
\[ \begin{split}\hat{b}_{h}&(\overline{w}_{h},\mathbf{w}_{h}, \mathbf{\varphi}_{h},t)=\\ &-\sum\limits_{K\in\mathcal{T}_{h\!k_{\!1}}}\int_{K}\sum\limits_{ s=1}^{2}(\mathbb{A}_{\mathrm{s}}(\overline{w}_{h}(x))-z_{\mathrm{s}}(x))\mathbb{I})\mathbf{w}_{h}(x)) \cdot\frac{\partial\mathbf{\varphi}_{h}(x)}{\partial z_{s}}\ \ \mathrm{d}x\\ &+\sum\limits_{\Gamma\in\mathcal{F}_{ht}^{1}}\int_{\Gamma}\left( \mathbb{P}_{\bar{g}}^{+}\big{(}\big{\langle}\overline{w}_{h}\big{\rangle}_{ \Gamma},\mathbf{n}_{\Gamma}\big{)}\mathbf{w}_{h}^{(L)}+\mathbb{P}_{\bar{g}}^{-}\big{(} \big{\langle}\overline{w}_{h}\big{\rangle}_{\Gamma},\mathbf{n}_{\Gamma}\big{)}\bm {w}_{h}^{(R)}\big{)}\cdot[\mathbf{\varphi}_{h}]\ \ \mathrm{d}S\\ &+\sum_{\Gamma\in\mathcal{F}_{ht}^{8}}\int_{\Gamma}\left( \mathbb{P}_{\bar{g}}^{+}\big{(}\big{\langle}\overline{w}_{h}\big{\rangle}_{ \Gamma},\mathbf{n}_{\Gamma}\big{)}\mathbf{w}_{h}^{(L)}+\mathbb{P}_{\bar{g}}^{-}\big{(} \big{\langle}\overline{w}_{h}\big{\rangle}_{\Gamma},\mathbf{n}_{\Gamma}\big{)} \overline{w}_{h}^{(R)}\right)\cdot\mathbf{\varphi}_{h}\ \ \mathrm{d}S.\end{split}  \tag{9}\]  
We set \(\Theta=1\), \(\Theta=0\) or \(\Theta=-1\) and get the so-called symmetric (SIPG), incomplete (IIPG) or nonsymmetric (NIPG) version, respectively, of the discretization of viscous terms. In (7) and (8), \(C_{W}\) denotes a positive sufficiently large constant and \(\mathbb{K}^{T}\) is the transposed matrix to \(\mathbb{K}\). In the form (9), symbols \(\mathbb{P}_{\bar{g}}^{+}(\mathbf{w},\mathbf{n})\) and \(\mathbb{P}_{\bar{g}}^{-}(\mathbf{w},\mathbf{n})\) denote the \"positive\" and \"negative\" parts of the matrix \(\mathbb{P}_{\bar{g}}(\mathbf{w},\mathbf{n})=\sum_{s=1}^{2}(\mathbb{A}_{s}(\mathbf{w})-z_{ \mathrm{s}}\mathbb{I})n_{s}\) defined in the following way. By [30], this matrix is diagonalizable. It means that there exists a nonsingular matrix \(\mathbb{T}=\mathbb{T}(\mathbf{w},\mathbf{n})\) such that  
\[\mathbb{P}_{g}=\mathbb{T}\Lambda\mathbb{T}^{-1},\quad\Lambda=\text{diag}( \lambda_{1},\ldots,\lambda_{4}), \tag{10}\]  
where \(\lambda_{i}=\lambda_{i}(\mathbf{w},\mathbf{n})\) are eigenvalues of the matrix \(\mathbb{P}_{g}\). Now we define the \"positive\" and \"negative\" parts of the matrix \(\mathbb{P}_{g}\) by  
\[\mathbb{P}_{\bar{g}}^{\pm}=\mathbb{T}\Lambda^{\pm}\mathbb{T}^{-1},\quad\Lambda ^{\pm}=\text{diag}(\lambda_{1}^{\pm},\ldots,\lambda_{4}^{\pm}), \tag{11}\]  
where \(\lambda^{+}=\max(\lambda,0),\ \lambda^{-}=\min(\lambda,0)\).