# Interval Analysis of Vibro-Acoustic Systems by the Enclosing Interval Finite-Element Method
## 3 Enclosing Interval Finite-Element Method for Coupling Problem
### Neumaier-Pownuk Method in Vibro-Acoustic Coupling Problem  
Although the parametric uncertain truss systems were analyzed by [PERSON] [14] and [PERSON] [39] in their respective research work, the application of iterative procedure constituted by the Neumaier-Pownuk method has not been widely accepted in interval finite-element analysis, due to lack of the generic finite-element framework to transform the interval parametric system into the specific interval linear equation (called the iterative enclosure formula). To carry out the iteration of the Neumaier-Pownuk method, the following definitions should be given first. The radius \(\text{rad}\left(\mathbf{u}^{I}\right)\), the magnitude \(\text{mag}\left(\mathbf{u}^{I}\right)\) of an interval vector \(\mathbf{u}^{I}\), and the residual of the approximate solution \(\text{res}\left(\mathbf{x}^{*}\right)\) of the interval uncertain linear system \(\mathbf{A}^{I}\mathbf{x}^{*}=\mathbf{b}^{I}\) are described by these formulas:  
\[ \begin{split}\text{rad}\left(\mathbf{u}^{I}\right)& =\left|\left(\mathbf{u}^{I}-\mathbf{u}^{c}\right)\right|\\ \text{mag}\left(\mathbf{u}^{I}\right)&=\left| \mathbf{u}^{I}\right|=\text{max}\left(\underline{\mathbf{u}},\overline{ \mathbf{u}}\right)\\ \text{res}\left(\mathbf{x}^{*}\right)&=\mathbf{b}^{I }-\mathbf{A}^{I}\mathbf{x}^{*}\end{split}  \tag{33}\]  
The flow chart of the Neumaier-Pownuk method is shown in Figure 6. The Neumaier-Pownuk method does not require strong regularity of the coefficient matrix \(\mathbf{Z}_{cp}^{c}\), as long as the matrix is invertible, and \(\mathbf{G}=\left(\mathbf{Z}_{cp}^{c}\right)^{-1}\) and defines the diagonal matrix as \(\mathbf{D}\). Let the initial vector \(\mathbf{w}\) be a unit vector, which means all the elements of the unit vector are equal to 1; then the trial vector becomes  
\[ \begin{split}\mathbf{w}^{\prime}&=\mathbf{w}-\text{ rad}\left(\mathbf{D}\right)\text{mag}\left(\mathbf{B}_{cp}\mathbf{G}\mathbf{A}_{cp} \right)\\ \mathbf{w}^{\prime\prime}&=\mathbf{D}\,\text{mag} \left(\mathbf{B}_{cp}\mathbf{G}\mathbf{F}^{I}\right)\end{split}  \tag{34}\]  
If there exists an entry in the trial vector greater than zero, the initial value of the iteration can be written as  
\[\mathbf{H}_{0}=\mathbf{w}^{\prime\prime}/\mathbf{w}^{\prime},\mathbf{Y}_{0}= \mathbf{B}_{cp}\mathbf{G}\left(\mathbf{F}^{I}+\mathbf{A}_{cp}\mathbf{H}_{0}\right) \tag{35}\]  
Otherwise, by the eigenvalue calculation, the eigenvector with respect to the largest one of eigenvalue takes place of \(\mathbf{w}\), and participates in the initialization. Repeat the loop, until satisfying the stopping criteria; for the n steps' intermediate results of \(\mathbf{H}_{n}\), the iteration sometimes can be relaxed to \(\mathbf{h}_{tem}\). \[ \begin{split}\mathbf{h}_{\text{tem}}&=\mathbf{D} \mathbf{Y}_{n},\mathbf{H}_{n+1}=\mathbf{H}_{n}\cap\mathbf{h}_{\text{lem}}\\ \mathbf{y}_{\text{tem}}&=\mathbf{B}_{cp}\mathbf{G} \left(\mathbf{F}^{I}+\mathbf{A}_{cp}\mathbf{H}_{n+1}\right),\mathbf{Y}_{n+1}= \mathbf{Y}_{n}\cap\mathbf{y}_{\text{tem}}\end{split}  \tag{36}\]  
Figure 5: Flow chart of [PERSON]’s inclusion method. Distinguished from the [PERSON]'s inclusion method and original Neumaier-Pownuk method, the truncation technique of residual is introduced to accelerate the iteration. In detail, the algorithm will stop when there is no further improvement for residual,  
\[ \begin{split}&\left(\mathbf{P}^{I}\right)^{n}=\mathbf{G}\left( \mathbf{F}^{I}+\mathbf{A}_{cp}\mathbf{H}_{n}\right)\\ &\mathbf{d}^{n}=\text{mag}\left(\text{res}\left(\left(\mathbf{P} ^{I}\right)^{n}\right)\right)\\ &\left\|\left(\mathbf{d}\right)^{n+1}-\left(\mathbf{d}\right)^{n }\right\|_{\infty}\leq\left\|\eta\cdot(\mathbf{d})^{n}\right\|_{\infty}\end{split}  \tag{37}\]  
where the \(\left\|\cdot\right\|_{\infty}\) represents the maximum absolute value belonging to vector's component, which is widely used in the iterative method.