# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 3 Dynamic Elasticity Problem  
Following [32], we consider an elastic body represented by a bounded polygonal domain \(\Omega^{b}\subset\mathbb{R}^{2}\). By \(\partial\Omega^{b}\), we denote the boundary of the domain \(\Omega^{b}\) and assume that \(\partial\Omega^{b}=\Gamma^{b}_{D}\cup\Gamma^{b}_{N}\), where \(\Gamma^{b}_{D}\cap\Gamma^{b}_{N}=\emptyset\). The deformation of the domain \(\Omega^{b}\) is described by the displacement \(\mathbf{u}=(u_{1},u_{2}):\Omega^{b}\times[0,T]\to\mathbb{R}^{2}\) and the deformation mapping \(\mathbf{\psi}(\mathbf{X},t)=\mathbf{X}+\mathbf{u}(\mathbf{X},t),\;\mathbf{X}\in\Omega^{b},\;t\in[0,T]\). Further, we set  
\[\mathbf{F}=\
abla\mathbf{\psi},\quad J=\det\mathbf{F}>0,\quad\text{Cof}\mathbf{F}=J(\mathbf{F}^{-T}), \tag{23}\]  
where \(\mathbf{F}^{-T}=(\mathbf{F}^{-1})^{T}\). By \(\mathbf{P}\), we denote the first Piola-Kirchhoff stress tensor, which depends on the elasticity model. It is a function of \(\mathbf{u}\) via \(\mathbf{F}\): \(\mathbf{P}=\mathbf{P}(\mathbf{F}(\mathbf{u}))\) (we can refer  
\[u =u_{D} \text{in }\Gamma_{D}^{b}\times[0,T], \tag{36}\] \[P(F)n =\mathcal{g}_{N} \text{in }\Gamma_{N}^{b}\times[0,T],\] (37) \[u(\cdot,0)=u^{0}, y(\cdot,0) =\psi^{0} \text{in }\Omega^{b}. \tag{38}\]  
We construct a triangulation \(\mathcal{T}_{h}^{b}\) of \(\overline{\Omega}^{b}\) with standard properties. The approximate solution at every time instant \(t\in[0,T]\) will be sought in the finite-dimensional space  
\[S_{h}^{b,s}=\Big{\{}v\in L^{2}(\Omega^{b});v|_{K}\in P_{s}(K),K\in\mathcal{T}_ {h}^{b}\Big{\}}^{2}, \tag{39}\]  
where \(s>0\) is an integer. By \(\mathcal{F}_{h}^{b}\), we denote the system of all faces of all elements \(K\in\mathcal{T}_{h}^{b}\) and distinguish their sets of boundaries, \"Dirichlet\", \"Neumann\" and inner faces: \(\mathcal{F}_{h}^{b,B}=\Big{\{}\Gamma\in\mathcal{F}_{h}^{b};\Gamma\subset \partial\Omega^{b}\Big{\}}\), \(\mathcal{F}_{h}^{b,D}=\Big{\{}\Gamma\in\mathcal{F}_{h}^{b};\Gamma\subset \Gamma_{b}^{b}\Big{\}}\), \(\mathcal{F}_{h}^{b,N}=\Big{\{}\Gamma\in\mathcal{F}_{h}^{b};\Gamma\subset \Gamma_{N}^{b}\Big{\}}\) and \(\mathcal{F}_{h}^{b,I}=\mathcal{F}_{h}^{b}\backslash\mathcal{F}_{h}^{b,B}\). For \(\Gamma\in\mathcal{F}_{h}^{b}\), the symbols \(\mathbf{n}_{\Gamma},H_{\Gamma},K_{\Gamma}^{(L)},K_{\Gamma}^{(R)}\) and for \(\mathbf{\varphi}\in S_{h}^{b,s}\) symbols \(\mathbf{\varphi}_{\Gamma}^{(L)}\), \(\mathbf{\varphi}_{\Gamma}^{(R)}\), \(\mathbf{\varphi}_{\Gamma}\) and \([\mathbf{\varphi}]_{\Gamma}\) have the same meaning as in Section 2.2. If \(\mathbf{a}=(a_{ij})_{i,j=1}^{2},\mathbf{b}=(b_{ij})_{i,j=1}^{2}\) are tensors, then we define the tensor product by \(\mathbf{a}:\mathbf{b}=\sum_{i,j=1}^{2}a_{ij}b_{ij}\). The DG discretization in space is formulated with the use of the following forms. Linear elasticity form:  
\[ \begin{array}{l}a_{h}^{b}(\mathbf{u},\mathbf{\varphi})\\ =\sum\limits_{K\in\mathcal{T}_{h}^{b}}\int_{K}\mathbf{\sigma}(\mathbf{u}):\mathbf{e}(\mathbf{ \varphi})\,\mathrm{d}\mathbf{x}-\sum\limits_{\Gamma\in\mathcal{F}_{h}^{b,I}}\int _{\Gamma}(\langle\mathbf{\sigma}(\mathbf{u})\rangle\cdot\mathbf{n})\cdot[\mathbf{\varphi}]\, \mathrm{d}\mathbf{S}\\ -\sum\limits_{\Gamma\in\mathcal{F}_{h}^{b,D}}\int_{\Gamma}(\mathbf{\sigma}(\mathbf{u} )\cdot\mathbf{n})\cdot\mathbf{\varphi}\,\mathrm{d}\mathbf{S}\\ -\Theta\sum\limits_{\Gamma\in\mathcal{F}_{h}^{b,D}}\int_{\Gamma}(\langle\mathbf{ \sigma}(\mathbf{\varphi})\rangle\cdot\mathbf{n})\cdot[\mathbf{u}]\,\mathrm{d}\mathbf{S}\\ -\Theta\sum\limits_{\Gamma\in\mathcal{F}_{h}^{b,D}}\int_{\Gamma}(\mathbf{\sigma}( \mathbf{\varphi})\cdot\mathbf{n})\cdot\mathbf{u}\,\mathrm{d}\mathbf{S},\end{array}  \tag{40}\]  
where \(\mathbf{\sigma}(\mathbf{u})\) is defined by (29). Here, the parameter \(\Theta\) is chosen as \(1,0,-1\) for SIPG, IIPG, NIPG, respectively, version of the elasticity form. Nonlinear IIPG elasticity form (\(\Theta=0\)):  
\[