# Interval Analysis of Vibro-Acoustic Systems by the Enclosing Interval Finite-Element Method
## 2 Formulation of Vibro-Acoustic Coupling System with Uncertain-But-Bounded Parameters
### The Construction of the Interval Coupling Governing Equations  
#### 2.4.1 The Iterative Enclosure Formula of Interval Coupling Governing Equations  
In Section 2.2, the decompositions of stiffness and mass matrices were given. By the mixed-nodal-element assemblage (MNE) referred to in Section 2.3, the decomposition after assembling becomes  
\[ \begin{split}\mathbf{K}_{\text{MNE}}^{I}&=\mathbf{A}_{k} \operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{k}\mathbf{\alpha}_{k}^{I}\Bigr{)}\mathbf{ A}_{k}^{\top}=\mathbf{K}^{I}\\ \mathbf{M}_{\text{MNE}}^{I}&=\mathbf{A}_{m} \operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{m}\mathbf{\alpha}_{m}^{I}\Bigr{)}\mathbf{ A}_{m}^{\top}=\mathbf{M}^{I}\end{split}  \tag{21}\]  
Figure 4: Relationship between three different IPFEMs. Figure 3: The assembly method for the mixed-nodal-element assemblage. where \(\mathbf{K}_{\text{MNE}}^{I}\) and \(\mathbf{M}_{\text{MNE}}^{I}\) with mixed-nodal-element assemblage are equal to the \(\mathbf{K}^{I}\) and \(\mathbf{M}^{I}\) with traditional nodal assemblage, respectively. Supposing that the uncertainty is just restricted in structures and no damping is taken into account, by the same method as EBE-IFEM, the interval dynamic stiffness matrix is expressed with the formula similar to the generalized unimodal components' decomposition,  
\[ \begin{split}\mathbf{D}_{s}^{I}&=\mathbf{A}_{s} \operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{s}\boldsymbol{\alpha}_{s}^{I} \Bigr{)}\mathbf{B}_{s}\\ \mathbf{A}_{s}&=\Bigl{\{}\mathbf{A}_{k^{\prime}}- \omega^{2}\mathbf{A}_{m}\Bigr{\}}\\ \mathbf{\Lambda}_{s}&=\Bigl{\{}\begin{array}{cc} \boldsymbol{\Lambda}_{k}&0\\ 0&\boldsymbol{\Lambda}_{m}\end{array}\Bigr{\}}\\ \boldsymbol{\alpha}_{s}^{I}&=\Bigl{\{}\boldsymbol{\alpha}_{k}^{I}; \boldsymbol{\alpha}_{m}^{I}\Bigr{\}},\quad\mathbf{B}_{s}=\{\mathbf{A}_{k}, \mathbf{A}_{m}\}^{\top}\end{split}  \tag{22}\]  
where \(\mathbf{A}_{s},\boldsymbol{\Lambda}_{s},\mathbf{B}_{s}\) are all deterministic matrices for the coupling system, and the uncertainties are only isolated in \(\boldsymbol{\alpha}_{s}^{I}\). Then, substituting \(\mathbf{D}_{s}^{I}\) to Equation (10), the coupling equation is equivalent to  
\[ \begin{split}\left[\begin{array}{cc}\mathbf{D}_{s}^{I}&- \mathbf{R}_{cp}\\ -\rho_{a}\omega^{2}\mathbf{R}_{cp}^{\top}&\mathbf{K}_{a}-\omega^{2}\mathbf{M} _{a}\end{array}\right]\mathbf{P}^{I}=\mathbf{F}^{I}\end{split}  \tag{23}\]  
Since only \(\mathbf{D}_{s}^{I}\) is uncertain in the dynamic matrix \(\mathbf{Z}^{I}\), by introducing the centered-form from Equation (11), \(\boldsymbol{\alpha}_{s}^{I}=\boldsymbol{\alpha}_{s}^{c}+\Delta\mathbf{a}_{s}^ {I}\), the coupling Equation (23) can be transformed into  
\[ \begin{split}\left\{\begin{array}{cc}\mathbf{D}_{s}^{c}&- \mathbf{R}_{cp}\\ -\rho_{a}\omega^{2}\mathbf{R}_{cp}^{\top}&\mathbf{K}_{a}-\omega^{2}\mathbf{M} _{a}\end{array}\right.\right]+\mathbf{A}_{cp}\operatorname{diag}\Bigl{(} \mathbf{\Lambda}_{cp}\Delta\boldsymbol{\alpha}_{cp}^{I}\Bigr{)}\mathbf{B}_{cp }\Bigr{\}}\mathbf{P}^{I}=\mathbf{F}^{I}\end{split}  \tag{24}\]  
\[ \begin{split}\mathbf{A}_{cp}&=\Bigl{\{}\begin{array}{cc} \mathbf{A}_{s}\\ 0\end{array}\Bigr{\}};\quad\mathbf{B}_{cp}=\bigl{\{}\begin{array}{cc} \mathbf{B}_{s}&0\end{array}\bigr{\}}\\ \boldsymbol{\Lambda}_{cp}&=\boldsymbol{\Lambda}_{s};\quad\Delta \boldsymbol{\alpha}_{cp}^{I}=\Delta\boldsymbol{\alpha}_{s}^{I}\end{split}  \tag{25}\]  
Further defining the deterministic matrix in Equation (24) as \(\mathbf{Z}_{cp}^{c}\), it can be rewritten with simplified formula, which is called iterative enclosure formula [14]:  
\[ \begin{split}\Bigl{(}\mathbf{Z}_{cp}^{c}+\mathbf{A}_{cp} \operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{cp}\Delta\boldsymbol{\alpha}_{ cp}^{I}\Bigr{)}\mathbf{B}_{cp}\Bigr{)}\mathbf{P}^{I}=\mathbf{F}^{I}\end{split}