Raymond Panneton

A mixed displacement-pressure formulation for poroelastic materials

Recently, finite element models based on Biot’s displacement (u¯,U¯) formulation for poroelastic materials have been extensively used to predict the acoustical and structural behavior of multilayer structures. These models while accurate lead to large frequency dependent matrices for three-dimensional problems necessitating important setup time, computer storage and solution time. In this paper, a novel exact mixed displacement pressure (u¯,p) formulation is presented. The formulation derives directly from Biot’s poroelasticity equations. It has the form of a classical coupled fluid-structure problem involving the dynamic equations of the skeleton in vacuo and the equivalent fluid in the rigid skeleton limit. The governing (u¯,p) equations and their weak integral form are given together with the coupling conditions with acoustic media. The numerical implementation of the presented approach in a finite element code is discussed. Examples are presented to show the accuracy and effectiveness of the presented...

NUMERICAL PREDICTION OF SOUND TRANSMISSION THROUGH FINITE MULTILAYER SYSTEMS WITH POROELASTIC MATERIALS

The sound transmission performance of finite multilayer systems containing poroelastic materials is of utmost importance for noise control in automobiles, aircrafts, buildings, and several other engineering applications. Currently, the need for tools predicting the acoustical and structural behaviors of such structures is considerably increasing. In this paper, such a tool is presented. It is applied to the sound transmission loss through multilayer structures made from a combination of elastic, air, and poroelastic materials. The presented approach is based on a three‐dimensional finite element model. It uses classical elastic and fluid elements to model the elastic and fluid media. For the poroelastic material, it uses a two‐field displacement formulation derived from the Biot theory. Furthermore, it couples with a boundary element approach to account, when important, for fluid–structure coupling and to calculate the transmission loss through the multilayer structure. Numerical predictions of the transm...

An efficient finite element scheme for solving the three-dimensional poroelasticity problem in acoustics

In this paper, the finite element method (FEM) is used to solve the three-dimensional poroelasticity problem in acoustics based on the isotropic Biot–Allard theory. A displacement finite element model is derived using the Lagrangian approach together with an analogy with solid elements. From this model, it is seen that the “damping” and “stiffness” matrices of the poroelastic media are complex and frequency dependent. This leads to cumbersome calculations for large finite element models and spectral analyses. To overcome this difficulty, an efficient algorithm is proposed. It is based on low-frequency approximations of the frequency-dependent dissipation mechanisms in poroelastic media. This efficient algorithm allows the poroelastic materials to be modeled with classical FEM codes. Also, the acoustic–poroelastic and the poroelastic–poroelastic coupling conditions are presented. The proposed model is compared to existing literature for both two-dimensional and three-dimensional problems. Excellent compari...

Acoustical determination of the parameters governing thermal dissipation in porous media

In this paper, the question of the acoustical determination of macroscopic thermal parameters used to describe heat exchanges in rigid open-cell porous media subjected to acoustical excitations is addressed. The proposed method is based on the measurement of the dynamic bulk modulus of the material, and analytical inverse solutions derived from different semiphenomenological models governing the thermal dissipation of acoustic waves in the material. Three models are considered: (1) Champoux-Allard model [J. Appl. Phys. 20, 1975-1979 (1991)] requiring knowledge of the porosity and thermal characteristic length, (2) Lafarge et al. model [J. Acoust. Soc. Am. 102, 1995-2006 (1997)] using the same parameters and the thermal permeability, and (3) Wilson model [J. Acoust. Soc. Am. 94, 1136-1145 (1993)] that requires two adjusted parameters. Except for the porosity that is obtained from direct measurement, all the other thermal parameters are derived from the analytical inversion of the models. The method is applied to three porous materials-a foam, a glass wool, and a rock wool-with very different thermal properties. It is shown that the method can be used to assess the validity of the descriptive models for a given material.

Enhanced weak integral formulation for the mixed (u_,p_) poroelastic equations

Recently Atalla et al. [J. Acoust. Soc. Am. 104, 1444–1452 (1998)] and Debergue et al. [J. Acoust. Soc. Am. 106, 2383–2390 (1999)] presented a weak integral formulation and the general boundary conditions for a mixed pressure-displacement version of the Biot’s poroelasticity equations. Finite element discretization was applied to the formulation to solve 3D vibro-acoustic problems involving elastic, acoustic, and poroelastic domains. In this letter, an enhancement of the weak integral formulation is proposed to facilitate its finite element implementation. It is shown that this formulation simplifies the assembly process of the poroelastic medium, the imposition of its boundary conditions, and its coupling with elastic and acoustic media.

Boundary conditions for the weak formulation of the mixed (u,p) poroelasticity problem

This paper presents the boundary conditions that apply to the weak integral formulation of the Biot mixed (u_,p) poroelasticity equations. These boundary conditions are derived from the classical boundary conditions of the Biot displacement (u_,U_) poroelasticity equations. They are applied to the surface integrals of the associated weak form to account for exterior excitations, supports, and couplings with exterior elastic, acoustic, poroelastic media, and a septum. It will be shown that the derived boundary conditions for the (u_,p) formulation lead to simpler finite element equations compared to those obtained from the (u_,U_) formulation. Finally, two numerical examples are presented to validate the poroelastic-septum coupling condition, and to highlight the limitations of the free edge condition on a poroelastic medium.

Evaluation of the acoustic and non-acoustic properties of sound absorbing materials using a three-microphone impedance tube

This paper presents a straightforward application of an indirect method based on a three-microphone impedance tube setup to determine the non-acoustic properties of a sound absorbing porous material. First, a three-microphone impedance tube technique is used to measure some acoustic properties of the material (i.e., sound absorption coefficient, sound transmission loss, effective density and effective bulk modulus) regarded here as an equivalent fluid. Second, an indirect characterization allows one to extract its non-acoustic properties (i.e., static airflow resistivity, tortuosity, viscous and thermal characteristic lengths) from the measured effective properties and the material open porosity. The procedure is applied to four different sound absorbing materials and results of the characterization are compared with existing direct and inverse methods. Predictions of the acoustic behavior using an equivalent fluid model and the found non-acoustic properties are in good agreement with impedance tube measurements.

Comments on the limp frame equivalent fluid model for porous media

In this letter, the low and high frequency limits of the effective density characterizing a limp frame porous medium are investigated. These theoretical limits are compared to the ones found for a classical rigid frame porous medium, and to experimental measurements. While the high frequency asymptotic behaviors of both limp and rigid effective densities are usually only slightly different, their low frequency behaviors are significantly different. Compared to experimental measurements performed on a limp frame fibrous layer, only the limp frame effective density yields good correlations over the whole frequency range.

Polynomial relations for quasi-static mechanical characterization of isotropic poroelastic materials

This paper proposes a quasi-static method for the characterization of the elastic properties of poroelastic materials. The method is based on the development of polynomial relations among compression stiffness, Young’s modulus, Poisson’s ratio, and shape factor derived from high order axisymmetrical finite element simulations on a disk-shaped poroelastic sample under static compression. The shape factor is defined as half the radius to thickness ratio of the sample. The polynomial relations account for the fact that the disk sample “wants” to bulge sideways when compressed between two rigid plates on which it is bonded. A compression test setup is used to measure the compression stiffness of two disk samples of different large shape factors. The measured stiffnesses together with the polynomial relations lead to a system of two equations and two unknowns. The solution of the system yields the Young’s modulus and Poisson’s ratio of the poroelastic material. Employing the proposed quasi-static method, Young...

Periodic unit cell reconstruction of porous media: Application to open-cell aluminum foams

In this article, the issue of reconstructing an idealized periodic unit cell (PUC) to represent a porous medium is examined by means of microcomputed tomography (μCT). Using μCT, three-dimensional images of open-cell foam are collected and used to characterize the representative parameters of its cellular morphology. These parameters are used in order to reconstruct the porous medium by means of an idealized PUC: a tetrakaidecahedron with ligaments of triangular cross sections, whose characteristic dimensions have been measured on the μCT images. The proposed reconstruction of the idealized PUC is applied to four aluminum foams. The averaged macroscopic properties of the foams (open porosity and thermal characteristic length) are deduced from their respective PUC model and compared to experimental measurements and literature data. Good correlations are obtained. For each of the foams, this provides a parameterized idealized periodic unit cell on which the partial differential equations governing sound dis...