# 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
### Different Assembly Strategies for Interval Coupling System  
As the abovementioned in Section 1, the EBE-IFEM cannot be applied in vibro-acoustic analysis, and the direct explanation for the problem is shown in Figure 2. As the Lagrange multiplier in element-by-element assemblage is incompatible with the coupling sub-block matrix, the application of the EBE-IFEM is restricted to uncoupled problem [11]. To deal with this dilemma, the assemblage of element-by-element should be replaced by another one with the attributes of nodal indexes, so that the modified interval arithmetic approach would be applicable in the coupling problem. As expected, the answer is found in the unimodal components' analysis as well [12; 13]. Subsequently, in order to distinguish the referred three methods of elemental assembly, apart from the traditional nodal assemblage and element-by-element type, the new one is named with the mixed-nodal-element assemblage (MNE) [12] because this new assembly method has adopted their technical advantages, not only satisfying the coupling equation but also applying to the factorization of uncertainties. In addition, it may save the computational cost with the reduced model, in contrast to EBE-IFEM, when solving the uncoupled field.  
As per the above analysis, the effective assembly methods for coupling problem are the nodal assembly method and the mixed-nodal-element assemblage. The detail of the mixed-nodal-element assembling is depicted in Figure 3 for the general case of \(\mathbf{A}_{k}^{e,i}\) to \(\mathbf{A}_{k}\) assembling. In brief, each line-vector of the deterministic matrix \(\mathbf{A}_{k}^{e,i}\) should be assembled by the nodal index similar to the traditional nodal assemblage, but each column-vector of  
Figure 2: The failure reason for EBE-IFEM to analyze the vibro-acoustic coupling problem.  
\(\mathbf{A}_{k}^{c,i}\) should be assembled by element index similar to the EBE-assemblage. For the other deterministic matrix \(\mathbf{\Lambda}_{k}^{i}\), it is the same as the element-by-element type [38]; the system matrix \(\mathbf{\Lambda}_{k}\) with \(N\) sub-matrices is assembled as  
\[\mathbf{\Lambda}_{k}=\left[\begin{array}{cccc}\mathbf{\Lambda}_{k}^{1};&\cdots&;\mathbf{ \Lambda}_{k}^{N}\end{array}\right] \tag{20}\]  
Observing the different methodologies in the preprocessing phase, it is instructive to clarify the logic and relationship among the different IFEMs applied to the vibro-acoustic coupling problem. As shown in Figure 4, the existing methods for the coupling problem can be sorted into two classes: the perturbation-based method and the modified interval arithmetic strategy. With the two types of uncertain tracing means and assembly ways, the next step is to give an approximation of the inverse corresponding to the interval matrix. Referring to Neumann series approximation, SMW series approximation, and iterative enclosure solution, three kinds of IFEMs are finally established theoretically. The point to emphasize here is that the TN-MIPFEM is inspired by the SMW-IPFEM with two reasons [8; 21; 22]: on one hand, the SMW-IPFEM creatively provides a special Neumann series expansion to retain more high-order terms of the non-conservative approximation [21]; on the other hand, the interval-value SMW approximation is confined to the specific dyadic-product formula [8; 21; 22], which hinders the dynamic application of SMW-IPFEM.