# Uncertainty Analysis Based on Kriging Meta-Model for Acoustic-Structural Problems
## 1 Introduction  
The problem of interaction between flexible vibrating structures and an enclosed cavity is a complex domain that has been of great attentions for researchers in different fields such as automobile compartments, airplane cabins, etc. The vibration energy generated from various sources is transmitted into the compartment cavity through structural connections. Thus, vibration characteristics of the cavity and its boundary are very important factors which dominate acoustic response in a vehicle passenger compartment. The knowledge of acoustic resonances and their own modes (i.e., sound pressure distributions) is an essential task in the study of the interaction between the vibrating surface and the enclosed cavity, and ultimately in determining the level of the resulting interior noise. [PERSON] [1] presented a verification of finite element predictions of acoustic resonances in a vehicle cabin, while in [2], [PERSON] et al. presented an experimental verification of the finite element method to predict the noise reduction in the automobile. In [3], the author presented an active vibration control system for structural acoustic coupling of a 3D vehicle cabin model. The structural-acoustic coupling system is analyzed by combining the structural data from modal testing with the acoustic data from the finite element method. An excellent review was given by [4; 5] to investigate the vibro-acoustic behavior of such systems, such as modal interaction of coupled structural-acoustic systems. [PERSON] and [PERSON] [6] suggested an extended approach to numerically simulate complex noise problems. Several authors investigated in their studies the phenomenon of fluid-structure interaction applied to different applications [7; 8; 9], to determine the resonances of an irregularly shaped cavity. Thefinite element method (FEM) denotes a very powerful method, which can be used in order to obtain an approximate solution to the response of the coupled system. [PERSON] et al. [10] studied the vibro-acoustic response of a vehicle cabin in presence of uncertainties. In general, it is well known that the numerical study of the frequency response of the vibro-acoustic field without considering the uncertainty associated with different parameters has achieved great attention. However, there are many sources of uncertainty such as errors related to manufacturing, material characteristics, boundary conditions, (etc. ), that needed to be integrated in the study to ensure the robustness and reliability of results. Taking into account of uncertainties in systems analysis is a difficult domain including the following steps: quantification and identification of uncertainty, uncertainty propagation and post-processing to measure the influence of these uncertainties on the behaviour of the system. The main approaches for the inclusion of these uncertainties are probabilistic methods [11; 12] which the uncertain parameters can be modeled as random variables through the description of their probability density functions. Recently, the probabilistic modeling of vibro-acoustic problems has received the attention of some researchers [13; 14]. Among the probabilistic methods, the frequently used one is the statistical approach or sampling technique, like the Monte Carlo simulations (MCS) [15; 16]. This technique requires a large number of samples of the input variables to have reasonable accuracy. The problem is then solved for each realization. This method is widely used because it is simple to implement and very robust. However, the number of realisation should be large enough, i.e., \(10^{5}\) or \(10^{6}\) deterministic FEM need to be executed in order to obtain accurate results. In [17], the author used only MCS to account for uncertainties in vibro-acoustic problem which took a huge computational cost in its development process. Several authors used the non-parametric probabilistic approach to model the effect of uncertainties in their vibro-acoustic studies of car models [18; 19; 20].