# Interval Analysis of Vibro-Acoustic Systems by the Enclosing Interval Finite-Element Method
###### Abstract  
Traditional interval analysis methods for interior vibro-acoustic system with uncertain-but-bounded parameters are based on interval perturbation theory. However, the solution sets by traditional interval finite-element methods are intrinsically not capable of reflecting the actual bounds of results, due to the non-conservative approximation for neglecting the high-order terms of both Taylor and Neumann series. In order to cope with this problem, this paper introduces the concept of unimodal components from structural mechanics to factorize the uncertainties, and a new enclosing interval-finite element method (enclosing-IFEM) is proposed to predict the uncertain vibro-acoustic response. In the enclosing-IFEM, the global matrix is assembled with the mixed-nodal-element strategy (MNE), which is different from the element-by-element assembly strategy. Thus, the vibro-acoustic coupling equation can be transformed into an iterative enclosure formula, and it avoids conflicts between the Lagrange multiplier matrix and the coupling sub-block matrix. The focus of this research is to reduce the overestimation caused by dependency phenomenon in the result of the enclosing-IFEM, therefore, both Rump's and Neumaier-Pownuk methods are analyzed in residual convergence. Furthermore, taking the results of the Monte Carlo approach and other interval finite-element methods as the cross-references, both the efficiency and accuracy of the enclosing-IFEM are examined through two numerical validation examples. enclosing interval finite-element method; Sherman-Morrison-Woodbury-series-based interval perturbation finite-element method; uncertain vibro-acoustic systems; mixed-nodal-element assemblage; iterative enclosure formula  
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strategy (GOS), where the dependency of uncertainties can be implicitly taken into account from the inside input-parameters; therefore, the output result of optimization is absolutely immune to the excessive conservativeness [6]. The most representative method for this methodology is the interval perturbation finite-element method (IPFEM) [8; 9], in which the interval matrix and vector are expanded by the finite-order of both Taylor series and Neumann series. However, the non-conservative approximation of neglecting higher-order terms of series expansion may result in overly-narrow results and unpredictable distortion. The second approach is the modified interval arithmetic strategy (MIAS), where the main body of modification is dedicated towards limiting the overestimation [10; 11] by tracing the parametric dependency throughout the whole solving process [3]. With the techniques of unimodal components' decomposition [12; 13] and iterative enclosure method [14], a standard procedure called element-by-element-based interval finite-element method (EBE-IFEM) was proposed to analyze the uncertain truss system [11; 15] and uncertain interior acoustic field [16]. As the theoretical derivation of EBE-IFEM is based on the variational formulation for an uncoupled analytical field [11], none of the functional expressions can describe the vibro-acoustic coupling field with the element-by-element assembling strategy. Thus, it fails to build the iterative enclosure formula for the vibro-acoustic coupling problem, due to the direct conflicts between Lagrange multipliers matrix and coupling sub-block matrix in the vibro-acoustic governing equation. As the focus of interval analysis has been shifted to uncertain vibro-acoustic analysis [6], researchers have proposed many interval vibro-acoustic modeling methods, such as perturbation statistical energy analysis [2; 17; 18], and Kriging-based response surface method [19]. However, the mainstream approaches for low-frequency prediction are still various forms of IPFEMs, with some improvements to overcome the shortcoming of the non-conservative approximation. The pioneering work was carried out by [PERSON] [9] to cope with the vibro-acoustic problem with large uncertain level, by employing the sub-interval technique [20].