# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 5 Realization of the Discrete Nonlinear Elasticity Problem
### Important Ingredients of the Newton Method Implementation  
Let \(\psi_{i},\ i=1,\ldots,N=\text{dim}\mathcal{V}\), be a basis of \(\mathcal{V}\). The solution \(\mathbf{u}_{h\tau}\) can be expressed as  
\[\mathbf{u}_{h\tau}=\mathbf{u}_{h\tau}(\mathbf{\alpha})=\sum_{i=1}^{2N}\alpha_{i}\mathbf{\phi} _{i}, \tag{52}\]  
where \(\mathbf{\alpha}=(\alpha_{i})_{i=1}^{2N}\) are the FE coefficients and \(\mathbf{\phi}_{i}=(\psi_{i},0)\) for \(1\leq i\leq N\) and \(\mathbf{\phi}_{i}=(0,\psi_{i-N})\) for \(N<i\leq 2N\) form the basis of \(\left|\mathcal{V}\right|^{2}\). In order to apply the Newton method as defined in Section 5.1, it is necessary to differentiate the form \(a_{h}^{b}(\mathbf{u}_{h\tau}(\mathbf{\alpha}),\mathbf{\varphi})\) (and subsequently the tensor \(\mathbf{P}\)) with respect to the coefficients \(\mathbf{\alpha}\). For clarity, we shall denote the gradient with respect to \(\mathbf{\alpha}\) by \(\
abla_{\mathbf{\alpha}}\) and the gradient with respect to \(\mathbf{X}\) by \(\
abla_{\mathbf{X}}\). Obviously,  
\[ \begin{array}{ll}\frac{\partial}{\partial\mathbf{x}_{h\tau}}&=(\psi_{i},0),&1 \leq k\leq N,\ i=k,\\ \frac{\partial}{\partial\mathbf{x}_{h\tau}}\mathbf{u}_{h\tau}&=(0,\psi_{i}),&N<k\leq 2 N,\ i=k-N,\end{array}  \tag{53}\]and  
\[\
abla_{\mathbf{X}}\mathbf{u}_{h_{\text{IT}}} =\sum\limits_{i=1}^{N}\alpha_{i}\
abla_{\mathbf{X}}(\psi_{i},0)+\sum \limits_{i=1}^{N}\alpha_{i+N}\
abla_{\mathbf{X}}(0,\psi_{i}) \tag{54}\] \[= \begin{pmatrix}\sum\limits_{i=1}^{N}\alpha_{i}\frac{\partial\psi_ {i}}{\partial x_{1}},&\sum\limits_{i=1}^{N}\alpha_{i}\frac{\partial\psi_{i}}{ \partial x_{2}}\\ \sum\limits_{i=1}^{N}\alpha_{i+N}\frac{\partial\psi_{i}}{\partial x_{1}},&\sum \limits_{i=1}^{N}\alpha_{i+N}\frac{\partial\psi_{i}}{\partial x_{2}}\end{pmatrix} .\]  
By (23),  
\[\mathbf{P}(\mathbf{F})=\mathbf{P}(\
abla_{\mathbf{X}}\mathbf{X}+\
abla_{\mathbf{X}}\mathbf{u}) \tag{55}\]  
Since \(\
abla_{\mathbf{X}}(\mathbf{X})\) is the constant unit matrix \(\mathbb{I}\), we introduce the simplified notation  
\[\tilde{\mathbf{P}}(\
abla_{\mathbf{X}}\mathbf{u})=\mathbf{P}(\mathbb{I}+\
abla_{\mathbf{X}}\mathbf{u}). \tag{56}\]  
Now, the gradient of the form \(a_{h}^{b}\) can be expressed as  
\[ \begin{array}{l}\
abla_{\mathbf{A}}a_{h_{\text{IT}}}^{b}(\mathbf{u}_{h_{\text{IT}}} (\mathbf{\alpha}),\mathbf{\varphi})\\ =\sum\limits_{\mathbf{K}\in\mathcal{T}_{h}^{b}}\int_{\mathcal{K}}\
abla_{\mathbf{A}} \big{(}\tilde{\mathbf{P}}(\
abla_{\mathbf{X}}\mathbf{u}_{h_{\text{IT}}}(\mathbf{\alpha})): \
abla_{\mathbf{X}}\mathbf{\varphi}\big{)}\,\mathrm{d}\mathbf{x}\\ -\sum\limits_{\mathbf{\Gamma}\in\mathcal{F}_{h}^{b+1}\bigcup\mathcal{F}_{h}^{b \times D}}\int_{\Gamma}\
abla_{\mathbf{x}}\big{(}\big{\langle}\tilde{\mathbf{P}}( \
abla_{\mathbf{X}}\mathbf{u}_{h_{\text{IT}}}(\mathbf{\alpha}))\big{\rangle}\mathbf{n}\cdot[ \mathbf{\varphi}]\big{)}\,\mathrm{d}\mathbf{S}\\ +\sum\limits_{\mathbf{\Gamma}\in\mathcal{F}_{h}^{b+1}\bigcup\mathcal{F}_{h}^{b \times D}}\int_{\Gamma}\frac{C_{h}^{b}}{h_{\text{IT}}}\
abla_{\mathbf{x}}\big{(} [\mathbf{u}_{h_{\text{IT}}}(\mathbf{\alpha})]\cdot[\mathbf{\varphi}]\big{)}\,\mathrm{d} \mathbf{S}.\end{array}  \tag{57}\]  
Let \(\tilde{\mathbf{P}}(\
abla_{\mathbf{X}}\mathbf{u}_{h_{\text{IT}}}(\mathbf{\alpha}))=(P_{ij})_{i =1}^{2}\) (here, for simplicity, we do not explicitly write the dependence of \(P_{ij}\) on \(\
abla_{\mathbf{X}}\mathbf{u}_{h_{\text{IT}}}(\mathbf{\alpha})\)) and let \(\mathbf{\varphi}=(\varphi_{1},\varphi_{2})\).