# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 2 Compressible Flow
### Discrete Flow Problem
The boundary state \(\mathbf{w}_{B}\) is defined on the basis of the Dirichlet boundary conditions (3) and extrapolation:  
\[\mathbf{w}_{B} =(\rho_{D},\rho_{D}\upsilon_{D1},\rho_{D}\upsilon_{D2},c_{\bar{ \
u}}\rho_{D}\theta_{\Gamma}^{(L)}+\frac{1}{2}\rho_{D}|\mathbf{\upsilon}_{D}|^{2}) \quad\text{ on }\Gamma_{I}, \tag{12}\] \[\mathbf{w}_{B} =\mathbf{w}_{\Gamma}^{(L)}\quad\text{ on }\Gamma_{O},\] (13) \[\mathbf{w}_{B} =(\rho_{\Gamma}^{(L)},\rho_{\Gamma}^{(L)}z_{D1},\rho_{\Gamma}^{(L )}z_{D2},c_{\bar{\
u}}\rho_{\Gamma}^{(L)}\theta_{\Gamma}^{(L)}+\frac{1}{2} \rho_{\Gamma}^{(L)}|\mathbf{z}_{D}|^{2})\quad\text{ on }\Gamma_{W}. \tag{14}\]  
In order to avoid spurious oscillations in the approximate solution in the vicinity of discontinuities or steep gradients, we apply artificial viscosity forms. They are based on the discontinuity indicator  
\[g_{t}(K)=\frac{1}{h_{K}|K|^{3/4}}\int_{\partial K}[\overline{\rho}_{h}]^{2}\ \ \mathrm{d}S,\quad K\in\mathcal{T}_{ht}. \tag{15}\]  
By \([\overline{\rho}_{h}]\), we denote the jump of the function \(\overline{\rho}_{h}\) on the boundary \(\partial K\), and \(|K|\) denotes the area of the element \(K\). Then, we define the discrete discontinuity indicator \(G_{t}(K)=0\quad\text{if }g_{t}(K)<1,\ G_{t}(K)=1\quad\text{if }g_{t}(K)\geq 1\), and the artificial viscosity forms (see [31])  
\[\hat{\beta}_{h}(\overline{w}_{h},\mathbf{w}_{h},\mathbf{\varphi}_{h},t)= \upsilon_{1}\sum\limits_{K\in\mathcal{T}_{ht}}h_{K}G_{t}(K)\int_{K}\
abla\mathbf{w}_ {h}\cdot\
abla\mathbf{\varphi}_{h}\ \ \mathrm{d}x, \tag{16}\] \[\hat{f}_{h}(\overline{w}_{h},\mathbf{w}_{h},\mathbf{\varphi}_{h},t)= \upsilon_{2}\sum\limits_{\Gamma\in\mathcal{F}_{h}^{1}}\frac{1}{2}\big{(}G_{t}(K_ {\Gamma}^{(L)})+G_{t}(K_{\Gamma}^{(R)})\big{)}\int_{\Gamma}[\mathbf{w}_{h}]\cdot[ \mathbf{\varphi}_{h}]\ \ \mathrm{d}S, \tag{17}\]  
with parameters \(\upsilon_{1}\), \(\upsilon_{2}=O(1)\). Because of the time discretization, we consider a partition \(0=t_{0}<t_{1}<\ldots<t_{M}=T\) of the time interval \([0,T]\) and denote \(I_{m}=(t_{m-1},t_{m})\), \(\overline{I}_{m}=[t_{m-1},t_{m}]\), \(\tau_{m}=t_{m}-t_{m-1}\), for \(m=1,\ldots,M\). We define the space \(\mathbb{S}_{h\tau}^{\eta}=(S_{h\tau}^{\eta})^{4}\), where  
\[ \begin{split} S_{h\tau}^{\eta}=\Big{\{}&\phi;\, \phi(x,t)=\sum_{i=0}^{q}\,t^{i}\phi_{i}(x),\,\phi_{i}\in S_{ht}^{r},\\ & t\in I_{m},\;x\in\Omega_{t},\;m=1,\ldots,M\Big{\}},\end{split}  \tag{18}\]  
with integers \(r,q\geq 1\). Here, \(P^{q}(I_{m})\) denotes the space of all polynomials in \(t\) on \(I_{m}\) of degree \(\leq q\) and the space \(S_{ht}^{r}\) is defined in (4). For \(\mathbf{\varphi}\in\mathbb{S}_{h\tau}^{\eta}\), we introduce the following notation:  
\[\mathbf{\varphi}_{m}^{\pm}=\mathbf{\varphi}(t_{m}^{\pm})=\lim_{t\to I_{m\pm}}\mathbf{ \varphi}(t),\quad\{\mathbf{\varphi}\}_{m}=\mathbf{\varphi}_{m}^{+}-\ \mathbf{\varphi}_{m}^{-}. \tag{19}\]  
In order to bind the solution on intervals \(I_{m-1}\) and \(I_{m}\), we augment the resulting identity by the penalty expression \(\big{(}\{\mathbf{w}_{h\tau}\}_{m-1},\mathbf{\varphi}_{h\tau}(t_{m-1}^{+})\big{)}_{t_{m -1}}\). The initial state \(\mathbf{w}_{h\tau}(0-)\in\mathbb{S}_{h0}^{r}\) is defined as the \(L^{2}(\Omega_{I0})\)-projection of \(\mathbf{w}^{0}\) on \(\mathbb{S}_{h0}^{r}\), i.e.,  
\[(\mathbf{w}_{h\tau}(0-),\mathbf{\varphi}_{h})_{\Omega_{t_{0}}}=\Big{(}\mathbf{w}^{0},\mathbf{ \varphi}_{h}\Big{)}_{\Omega_{t_{0}}}\quad\forall\mathbf{\varphi}_{h}\in\mathbb{S }_{h0}^{r}. \tag{20}\]  
Furthermore, we define the prolongation \(\overline{\mathbf{w}}_{h\tau}(t)\) of \(\mathbf{w}_{h\tau}|_{I_{m-1}}\) on the interval \(I_{m}\). In what follows, we introduce the notation  
\[(a,b)_{\omega}=\int_{\omega}ab\;\mathrm{d}x, \tag{21}\]  
for functions \(a,\;b\) defined in a set \(\omega\subset\mathbb{R}^{2}\). Now, the space-time DG approximate solution is defined as a function \(\mathbf{w}_{h\tau}\in\mathbb{S}_{h\tau}^{\eta}\) satisfying (20) and the following relation for \(m=1,\ldots,M\):  
\[