# Interval Analysis of Vibro-Acoustic Systems by the Enclosing Interval Finite-Element Method
## 3 Enclosing Interval Finite-Element Method for Coupling Problem  
### [PERSON]'s Inclusion Method in Vibro-Acoustic Coupling Problem  
In order to confine the overestimation of results, [PERSON]'s inclusion method is developed by employing the fixed point theorem [7] to solve the interval uncertain system. Based on the fixed point theorem [15; 38], Equation (26) can be also rewritten as the fixed point form,  
\[\mathbf{Z}_{cp}^{c}\mathbf{P}^{I}=\mathbf{F}^{I}-\mathbf{A}_{cp}\operatorname{ diag}\Bigl{(}\mathbf{\Lambda}_{cp}\Delta\mathbf{\alpha}_{cp}^{I}\Bigr{)} \mathbf{B}_{cp}\mathbf{P}^{I} \tag{28}\]  
For practice, the [PERSON]'s inclusion method procedure is illustrated with a flow chart, given in Figure 5. Apparently, the matrix \(\mathbf{Z}_{cp}^{c}\) is non-singular and invertible by means of physical problem at the midpoint value of parametric interval. Therefore, with the special definition \(\mathbf{G}=\Bigl{(}\mathbf{Z}_{cp}^{c}\Bigr{)}^{-1}\), \(\mathbf{P}^{I}\) satisfies the fixed point form, as follows:  
\[\Bigl{(}\mathbf{P}^{I}\Bigr{)}^{n+1} =\mathbf{G}\mathbf{F}^{I}-\mathbf{G}\mathbf{A}_{cp}\cdot \operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{cp}\Delta\mathbf{\alpha}_{cp}^{ I}\Bigr{)}\mathbf{B}_{cp}\bigl{(}\mathbf{P}^{I}\bigr{)}^{n} \tag{29}\]  
Although the approximate solution set can be figured out by Equation (29), the accuracy of interval may be influenced by the redundant intermediate operation \(\mathbf{B}_{cp}\bigl{(}\mathbf{P}^{I}\bigr{)}\) so that, with the auxiliary variable \(\mathbf{v}^{I}=\mathbf{B}_{cp}\mathbf{P}^{I}\), the iterative form is transformed into  
\[\Bigl{(}\mathbf{v}^{I}\Bigr{)}^{n+1} =\mathbf{B}_{cp}\mathbf{G}\mathbf{F}^{I}-\mathbf{B}_{cp}\mathbf{ G}\mathbf{A}_{cp}\cdot\operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{cp}\Delta \mathbf{\alpha}_{cp}^{I}\Bigr{)}\bigl{(}\mathbf{v}^{I}\bigr{)}^{n} \tag{30}\]  
In addition, two basic aspects need be under consideration for the iteration: one is the initial value and another is the stopping criteria. It has been verified by [PERSON] [36] that \(\mathbf{v}^{I}=\mathbf{B}_{cp}\mathbf{G}\mathbf{F}^{I}\) is feasible to be the initialization of iteration. Additional stopping criteria by employing convergence condition is given by  
\[d =\operatorname{diag}\Bigl{(}\mathbf{\Lambda}_{cp}\Delta\mathbf{ \alpha}_{cp}^{I}\Bigr{)}\mathbf{v}^{I} \tag{31}\] \[\mathbf{d} =\{\bar{d}_{i}-\underline{d}_{i}\}=2\cdot\operatorname{rad}(d)\] \[\Bigl{\|}\left(\mathbf{d}\right)^{n+1}-\left(\mathbf{d}\right)^{ n}\Bigr{\|}_{1}\leq\left\|\varepsilon\cdot\left(\mathbf{d}\right)^{n}\right\|_{1}\]  
where \(\mathbf{d}\) is a vector to represent the width of the given interval vector \(d\). \(\left\|\cdot\right\|_{1}\) is the 1-norm of the vector, which represents the absolute sum of the elements of the vector. \(\varepsilon\) is the tolerance of the iteration. Assuming that after n times of steps, the output satisfies the stopping criteria, the final solution set is given by  
\[\mathbf{P}^{I}=\mathbf{G}\mathbf{F}^{I}-\mathbf{G}\mathbf{A}_{cp}\operatorname {diag}\Bigl{(}\mathbf{\Lambda}_{cp}\Delta\mathbf{\alpha}_{cp}^{I}\Bigr{)} \Bigl{(}\mathbf{v}^{I}\Bigr{)}^{n} \tag{32}\]  
Generally, the [PERSON]'s inclusion method is functional to approach the actual interval with high accuracy (less than 10 times the steps). However, it should be pointed out that it would fail in some cases with large-scale dimension (the curse of dimensionality), even if the uncertain level is small [14]. As a consequence, it calls for another algorithm to ensure the convergence of iteration.