Use of a hybrid adaptive finite element/modal approach to assess the sound absorption of porous materials with meso-heterogeneities
Research highlights
► Numerical method to predict acoustic behavior of meso-perforated porous materials. ► Design of experiments for optimizing absorption of meso-perforated porous materials. ► Charts to evaluate influence of air gaps on average absorption of porous materials. ► Useful for characterization of porous material based on SWT measurements. ► Useful for quantifying the impact of air gaps and for designing efficient absorbers.
Introduction
Porous materials are classically used to absorb sound. However a large thickness is usually needed to attenuate sound at low frequencies. Multilayered configurations and special design such as resonators and resistive screens are classical alternatives. This is usually bulky and expensive. Recently, heterogeneous porous materials have been shown to exhibit interesting properties for noise control applications. Several studies have investigated the sound absorption performances of these materials [1], [2], [3], [4], [5], [6]. The potential of heterogeneous porous materials has also been considered in the scope of vibration damping and sound radiation of a panel inside an enclosure [7], [10] or outside an enclosure [8], [9]. The effect of such materials on the sound transmission of panels has been studied more recently [11]. Heterogeneous porous materials can either be made of patches of different materials [12], incorporate perforations [1], [3], [4], [5], [6], fluid inclusions [2], [4], [8], [9], [10] or embedded masses [7], [11]. Properly designed meso-perforations in highly resistive materials have been shown to lead to an important increase in a large frequency band of the normal incidence absorption coefficient especially at low frequencies [3]. This is a consequence of the pressure diffusion effect which takes place in the material near the perforation boundary. Olny et al. [5] described theoretically this attenuation mechanism. He showed that for highly resistive porous materials, perforations produce pressure diffusion around a characteristic frequency. The pressure in the perforations depends on the excitation frequency, the dimensions of the perforation and the physical properties of the porous material. Around the diffusion frequency, the pressure in the microporous domain varies at the same scale as the pressure in the perforation. The gradient of pressure between the two media induces a diffusion phenomenon which tends to equilibrate the pressure. The pressure varies strongly within a thin layer called diffusion layer thus resulting in extra dissipation due to the viscous forces during the diffusion. The amount of this extra dissipation is related to the pressure gradient between the perforations and the material and the size of the diffusion volume (diffusion layer in 2-D) which is by definition the volume inside which the dissipation occur. Atalla et al. [13] and Sgard et al. [6] discussed examples of this phenomenon together with its modeling using 3-D finite elements. Moreover, Olny et al. [5] and Sgard et al. [6] proposed rules to select good candidate materials favoring this phenomenon. Besides considering fluid meso-heterogeneities as a way of designing absorbing material with enhanced acoustic performances, fluid meso-heterogeneities such as air gaps are important to study in the scope of absorbing materials characterization methods based on Standing Wave Tube (SWT). Indeed, the presence of air gaps due to poor mounting conditions of a sample in a SWT affects the measured impedance and absorption coefficient and in consequence affects the accuracy of identification methods based on these measured indicators; the latter assume an ideally mounted sample [21].
The impact of air gaps on the sound absorption of porous materials has been widely investigated in the context of material acoustical characterization. Indeed, several materials such as felts and fibers and even foams may be difficult to cut and fit properly into an impedance tube thereby inducing air gaps which may perturb the characterization process. This is an important issue since an impedance tube test is a standard widely used to compare material performances. More importantly, it is nowadays a common method for determining Biot parameters using inverse methods ([14] and see Proceedings of SAPEM 2008 conference [15] which contains several papers on inverse characterization of porous materials based on SWT). These inverse methods are based on sound absorption models of an infinite rigid frame sample and thus the control of finite size effects (via mounting conditions) on the measured absorption is crucial. Several experimental, theoretical and numerical works have been done in the past on the effect of peripheral air gaps on the acoustic absorption of porous materials measured in SWT [16], [17], [18], [19], [20]. The main conclusion of these studies was that the main effect of air gaps is to decrease the real part of the surface impedance and the measured absorption coefficient. Pilon et al. [21] investigated simple layer materials with lateral air gaps in an impedance wave guide using an axi-symmetric finite element model. They pointed out that a lateral air gap may increase the sound absorption in particular for highly resistive materials. This can be physically explained by the pressure diffusion phenomenon described above [5]. They proposed in consequence a criterion to identify a priori materials which may be affected by air gaps in SWT absorption measurements based on the use of a simple permeability ratio and a set of pre-calculated charts.
This paper is an extension of [6] and is based on the work of Castel [22]. Its objective is to discuss the sound absorptive performance of porous materials with meso-perforations inserted in a rectangular waveguide using a numerical FEM based method. More specifically the paper concentrates on two applications (i) the improvement of porous materials noise absorption coefficient using meso-perforations (ii) the effects of peripheral air gaps on the normal incidence sound absorption of mono-layer and two-layer porous materials. The model is based on a 3D hybrid adaptive finite element-modal method. It consists of a finite element model of the porous materials and the meso-heterogeneities coupled to a modal approach for the sound field in the waveguide. The direct adaptive meshing procedure is based on a hybrid grid/plastering technique. The error on the solution is estimated from the second derivative of the pressure gradient along each finite element side. A convergence criterion sets the maximum variation of the gradient of the solution allowed over a finite element. This controls locally the interpolation error and thus the acoustic indicator of interest (absorption coefficient, dissipated powers, etc.)
The numerical model is then used to illustrate graphically the pressure diffusion phenomenon in porous materials with peripheral air gaps. For the first application, a numerical design of experiment is used to optimize the sound performance of a porous material with meso-perforations with a reduced number of numerical simulations. An example in which the optimization process is carried out on the thickness and size of the perforation is presented to illustrate the relevance of the approach. For the second application, a set of twenty fibrous materials spanning a large flow resistivity range is used to derive practical charts to help evaluate the influence of air gaps on the average sound absorption performance of porous materials. This could be helpful to both the experimenter regarding characterization of porous material based on SWT measurements and to the engineer by quantifying the impact of air gaps and providing guidelines to design efficient absorbers.
Section 2 proposes an efficient 3D hybrid adaptive finite element-modal method to predict the vibroacoustic behavior of multilayered heterogeneous porous materials in waveguides. Section 3 shows how this numerical model combined with design of experiments can be used to optimize the sound absorption performance of a porous material using the concept of meso-perforations. Finally Section 4 presents an example to illustrate visually the physics of the acoustical behavior of a porous medium surrounded by an air gap and presents charts quantifying the impact of air gaps on the expected sound absorption coefficient as a function of important parameters (size of the gaps, flow resistivity and thickness of the material). The latter are helpful in the scope of SWT measurements and design purposes.
Section snippets
Numerical model
The approach is based on a numerical model which consists of a finite element discretization of the different domains (porous and fluid). A modal approach is used to account for the coupling between the heterogeneous porous domain and the waveguide [13]. In particular, it accounts for evanescent modes caused by impedance discontinuities on the excitation surface. The proposed model solves the entire problem accounting for all the coupling conditions. The nodal solutions are used to compute
Optimization of sound absorption of meso-perforated porous materials
The sound absorption of meso-perforated porous material depends on the flow resistivity of the porous substrate, the meso-porosity ϕp (ratio of volume of perforation to total volume of material), the size of the perforation and their shape (see [6]). In this section, an example is given to illustrate what should be the thickness e and the perforation meso-porosity ϕp of a material, that is a good candidate to the phenomenon of pressure diffusion, to achieve the highest absorption.
In order to
Description of the configurations
The geometry of the problem is depicted in Fig. 3. It consists of two layers of porous materials inserted into a rectangular wave guide. No air gap is present in configuration 1. In configurations 2 and 3, the samples present lateral air gaps on either one of the layer. In the following, the presence of these air gaps is quantified using a leakage ratio defined as the ratio of air gaps cross section area to the SWT cross section area.
The pressure diffusion effect for a single layer material
This subsection presents a practical visual illustration of
Conclusion
This paper discussed the sound absorptive performance of a multi-layered porous material with meso-perforations inserted in a rectangular waveguide using a numerical hybrid adaptive finite element-modal method. Two specific applications have been investigated: (i) the improvement of porous materials noise reduction coefficient using meso-perforation (ii) the effects of peripheral air gaps on the normal incidence sound absorption of mono-layer and two-layer porous materials. The main
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