درباره تجزیه و تحلیل حساسیت از مدل های مواد متخلخل

Porous materials are used in many vibroacoustic applications. Different available models describe their behaviors according to materials' intrinsic characteristics. For instance, in the case of porous material with rigid frame, and according to the Champoux–Allard model, five parameters are employed. In this paper, an investigation about this model sensitivity to parameters according to frequency is conducted. Sobol and FAST algorithms are used for sensitivity analysis. A strong parametric frequency dependent hierarchy is shown. Sensitivity investigations confirm that resistivity is the most influent parameter when acoustic absorption and surface impedance of porous materials with rigid frame are considered. The analysis is first performed on a wide category of porous materials, and then restricted to a polyurethane foam analysis in order to illustrate the impact of the reduction of the design space. In a second part, a sensitivity analysis is performed using the Biot–Allard model with nine parameters including mechanical effects of the frame and conclusions are drawn through numerical simulations.

Porous materials are used in a variety of acoustic applications. Prediction tools of acoustic characteristics for these materials are necessary. Zwikker and Kosten [1]and Biot [2] and [3] developed the first popular porous media models. A thorough review of these models and further developments was performed by Attenborough [4]. For the case of rigid frame porous media, Allard [5] and [6] developed a five parameters model, based on the idea of Johnson [7] and [8]. Measurement and identification of characteristics can be difficult and time consuming, and understanding the model sensitivity can make the optimization of the sound packages easier and facilitate new concept developments. Only very few papers in the open literature on porous media deal with sensitivity of models. Some elements about first-order estimation of impact of parameters are presented for instance in [9]. This gives a useful information, which remains limited to very small variations of parameters without considering any coupling effect between them. To the author's knowledge, one of the most advanced studies on the sensitivity analysis of porous materials models has been proposed by Bolton et al. [10] and [11], in which a singular value decomposition is performed on the so-called sensitivity matrix, which is build from first-order estimation of derivatives (finite differences) and concatenates effects on absorption coefficient or transmission loss factor for different frequencies. A Singular Value Decomposition is then performed to check the coupling effects between the parameters (the parameters being considered as independent) in order to reduce the size of the design space for identification purpose. The aim of this contribution is to go one step further, by applying rigorous sensitivity analysis techniques to porous material models. For illustration purpose, the main features of interest are the acoustic impedance and the absorption coefficient of a sample of porous material backed by an impervious rigid wall. The model used for the description of the acoustic performances is the Champoux–Allard one (depending on five parameters: porosity, flow resistivity, tortuosity, viscous and thermal characteristic lengths). It should be noticed that the methodology is general and can be applied to more complicated porous material models. For instance, the considered sensitivity approach is also applied in this paper using the Biot–Allard poroelastic model. In this paper we focus on global sensitivity analysis techniques. We classically distinguish two families of methods, namely the local and the global ones. Local sensitivity techniques are low cost, very easy to implement, but they are only able to capture the sensitivity of the model in a limited subset of the design space. On the other hand, global sensitivity analyses, which require a larger computational cost, give information about sensitivity which are valid for the whole design space and can deal with interactions effects between parameters. Sobol and FAST global sensitivity methods are considered here. The main issue is to clarify how the variability associated with the model inputs affects the model outputs [12]. Sensitivity analysis is also expected to (but not limited to) determine which input parameters contribute the most to output variabilities [13]; which parameters are insignificant; and estimate parameter interactions. A comprehensive review of the different sensitivity analysis methods, including their advantages and drawbacks, has been proposed by Helton et al. [14] and Frey et al. [15]. A comparison of these methods can be found in [16], [17], [18], [19] and [20]. Among the available sensitivity analysis methods, we propose in this paper to apply Sobol [21] and FAST [22]. The paper is structured as follows. Section 2 provides a brief survey of the sensitivity analysis methods considered in the paper. Section 3 recalls the porous material models used in this work. In Section 4, the Champoux–Allard model is first considered. Sensitivity results are presented and discussed. This analysis is performed considering a large design space whose parameters represent a wide variety of porous materials. Some remarks are drawn about agreements between Sobol and FAST results. Then, a comparison is performed between the first-order sensitivity and total sensitivity indexes, in order to evaluate interaction effects vs. total ones. In Section 5, another sensitivity analysis is performed when the design space is limited to a specific porous material, namely a polyurethane foam. In particular, a focus is made on the choice of the probability functions used in the sensitivity analysis. Some comments related to modelling and characterization of porous materials are given. 6 and 7 are dedicated to sensitivity analysis of materials that exhibit fluid–structure coupled behavior in the frequency range of interest. To that end, the Biot–Allard model is considered, first with a large porous materials data base, then a restriction to a given type of material. Finally, some recommendations and concluding remarks are given in Section 8.

Models considered for porous materials analysis are defined according to some parameters. The Champoux–Allard model (with five material parameters) and the Biot–Allard model (with nine parameters) are considered in this work for sensitivity purposes, as examples. The work presented in this paper quantifies the sensitivity of parameters for these models in the case of a 25 mm thick porous material sample with rigid frame, considering surface impedance and acoustic absorption indicators. The sensitivity analysis is frequency dependent and uses two methods: Sobol and extended FAST. As expected, both sensitivity methods are in good agreement when comparing first-order and total sensitivity indexes, including coupling effects. Both techniques lead to similar results concerning the first-order and total sensitivity indexes, even if the FAST technique is faster, but provides only mean first-order and total sensitivity indexes (which are often sufficient in practical cases). The results are presented together with the normalized standard deviation to improve readability and interpretation of results. The studies performed on porous materials illustrate the preponderant impact of flow resistivity on acoustic performances. Nevertheless, some other parameters can have a strong impact on the vibroacoustic behavior. The sensitivity of these parameters is strongly frequency dependent, since some of them can be irrelevant in a frequency band, and becoming very important for other frequency ranges. It appears that no general hierarchy of parameters for porous materials can be drawn. For the cases which have been studied, all acoustic parameters (i.e. those related to the fluid phase) have exhibited an important participation to the total sensitivity for one of the considered outputs (real, imaginary part of impedance, absorption coefficient) for a given frequency range. Mechanical parameters are generally less influent, except in the zones in which strong couplings between fluid and solid phases appear. In these zones, elastic parameters are very important, in particular the Poisson ratio. This can lead to drastic change in the material vibroacoustic behavior even for limited fluctuation of the parameter. One of the key aspects for the sensitivity analysis is the designer's knowledge concerning parameters variabilities: consequences in terms of hierarchy and quantification of coupling effects can strongly depend on the upper and lower bounds defined for the parameters, or by the probability density functions which are chosen. It has been shown that global sensitivity analysis can help the designer to choose a material, since it allows one to focus on the most influent parameters of the material for the desired vibroacoustic output. It is also very useful for parameter identification purposes, since it helps the analyst to mostly devote effort on observable and influent parameters only. The first drawback of the global sensitivity analysis in this context is the large number of model evaluations which are required, due to the number of considered parameters and of the model non-linearity. Practically, this is not a drastic limitation since outputs of interest are accessible through analytical expressions, as it is the case in this paper. For more general case studies where sensitivity evaluation is expected to be time consuming, model reduction can be a solution to render this kind of analysis feasible. The second drawback in the analysis proposed here is that all material parameters are considered as independent. When the analyst has some knowledge about the relationships between the parameters, they should be taken into account in the sensitivity analysis.