# 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
### Unimodal Components' Decomposition for Uncertain Structures  
Since [PERSON] introduced the Taylor series expansion in interval analysis [33; 34], the Taylor series expanding method has become the most widely used tool in perturbation-based IFEMs, but it suffers from the non-conservative approximation for neglecting high-order terms in Neumann series expansion [8]. An alternative tracing method originates from the unimodal components' decomposition [12; 13], which can be also regarded as the practice to the suggestion of uncertainty factorization proposed by [PERSON] [35]. With the Sherman-Morrison-Woodbury series expansion [13], the previous concept of unimodal components' decomposition is derived from the approximate explicit solutions of structural mechanics. Moreover, the basic and simple element expressions such as truss, space frame, and 2D constant strain triangle have been provided in literature [13; 36] with explicit formulas. In order to build the \"plate-cavity coupling model\", MITC4 (four-node mixed interpolation of tensorial components) for bending plate element [37] would be derived with approximate formulas by the proposed \"generalized unimodal components' decomposition\" method. Here, using \(E,\rho_{s},h,v\) as the [PERSON]'s modulus, density, thickness of plate, and [PERSON]'s ratio for material, the MITC4 element can be decomposed as follows with Gaussian integration. \[ \begin{split}\mathbf{K}_{e}^{l}=&\sum_{s=1}^{N}\sum_{t=1}^{N }\begin{pmatrix}\mathbf{B}_{b}^{\top}(\xi_{s},\eta_{t})\mathbf{D}_{b}\mathbf{B }_{b}(\xi_{s},\eta_{t})\\ +\mathbf{B}_{s}^{\top}(\xi_{s},\eta_{t})\mathbf{D}_{s}\mathbf{B}_{s}(\xi_{s}, \eta_{t})\end{pmatrix}w_{st}J_{st}\\ &=\mathbf{A}_{ke}\operatorname{diag}\Bigl{(}\Lambda_{ke}\boldsymbol{\alpha}_{ k}^{l}\Bigr{)}\mathbf{A}_{ke}^{\top}\\ &=\left\{\mathbf{A}_{ke}^{1}(\xi_{1},\eta_{1})\cdots\mathbf{A}_{ ke}^{N\times N}(\xi_{N},\eta_{N})\right\}\\ &\cdot\operatorname{diag}\Biggl{(}\left(\begin{array}{c}\Lambda_{ ke}^{1}(\xi_{1},\eta_{1})\\ \vdots\\ \Lambda_{ke}^{N\times N}(\xi_{N},\eta_{N})\end{array}\right)\boldsymbol{a}_{k} ^{l}\Biggr{)}\cdot\left\{\begin{array}{c}\mathbf{A}_{ke}^{1}(\xi_{1},\eta_{1 })^{\top}\\ \vdots\\ \mathbf{A}_{ke}^{N\times N}(\xi_{N},\eta_{N})^{\top}\end{array}\right\}\\ &\mathbf{A}_{ke}^{i}=\left\{\begin{array}{c}\mathbf{B}_{b}(\xi_{s},\eta_{t}) \\ \mathbf{B}_{s}(\xi_{s},\eta_{t})\end{array}\right\}^{\top}\left[\begin{array}[] {c}1,0,0,0\\ v,1,0,0\\ 0,0,1,0\\ 0,0,0,1\\ 0,0,0,0,1\end{array}\right]\sqrt{w_{st}J_{st}}\\ &\mathbf{A}_{ke}^{i}=\left\{\begin{array}{c}\frac{1}{12(1-v^{2})},\frac{1}{ 12},\frac{1}{24(1+v)},0,0\\ 0,0,0,\frac{\kappa}{2(1+v)},\frac{\kappa}{2(1+v)}\end{array}\right\}^{\top}, \boldsymbol{\alpha}_{k}^{l}=\left[Eh^{3},Elh\right]^{\top}\end{split}  \tag{13}\]  
\[ \begin{split}\mathbf{B}_{b}=\left[\begin{array}{ccc}0&N_{i,x} &0\\ \cdots&0&0&N_{i,y}\\ 0&N_{i,y}&N_{i,x}\end{array}\right],\quad\left\{\begin{array}{c}N_{i,x}\\ N_{i,y}\end{array}\right\}=\mathbf{J}\cdot\left\{\begin{array}{c}N_{i,\xi}\\ N_{i,\eta}\end{array}\right\}\quad i=1,2,3,4\end{split}  \tag{14}\]  
\[ \begin{split}\mathbf{B}_{s}=1/8\cdot\mathbf{J}^{-1}\mathbf{N}_{ \gamma}\mathbf{A}_{\gamma}\\ \mathbf{N}_{\gamma}=\left[\begin{array}{cccccc}\frac{1}{2}(1-\eta),0,\frac{1 }{2}(1+\eta),0\\ 0,\frac{1}{2}(1+\xi),0,\frac{1}{2}(1-\xi)\end{array}\right]\\ \mathbf{A}_{\gamma}=\left[\begin{array}{cccccc}-2&-x_{12}&-y_{12}&2&-x_{12}&-y _{12}&0&0&0&0&0\\ 0&0&0&-2&-x_{23}&-y_{23}&2&-x_{23}&-y_{23}&0&0&0\\ 0&0&0&0&0&2&x_{34}&y_{34}&-2&x_{34}&y_{34}\\ -2&x_{41}&y_{41}&0&0&0&0&0&2&x_{41}&y_{41}\end{array}\right]\\ \end{split}  \tag{15}\]  
\[ \begin{split}\mathbf{D}_{b}=\frac{El^{3}}{12(1-v^{2})}\left[ \begin{array}{ccc}1&v&0\\ v&1&0\\ 0&0&(1-v)/2\end{array}\right]\\ \mathbf{D}_{s}=\frac{\kappa Eh}{2(1+v)}\left[\begin{array}{ ccc}1&0\\ 0&1\end{array}\right]\\ \end{split}  \tag{16}\]  
\[