Noureddine Atalla

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...

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.

A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings

This paper discusses the prediction of the low frequency diffuse field transmission loss through double-wall sound barriers with elastic porous linings. The studied sound barriers are made up from a porous-elastic decoupling material sandwiched between an elastic skin and a septum. The prediction approach is based on a finite element model for the different layers of the sound barrier coupled to a variational boundary element method to account for fluid loading. The diffuse field is modeled as a combination of uncorrelated freely propagating plane waves with equal amplitude, no two of which are traveling in the same direction. The corresponding vibroacoustic indicators are calculated efficiently using a Gauss integration scheme. Also, a power balance is presented to explain the dissipation mechanisms in the different layers. Typical results showing the effects on the transmission loss of several parameters such as the septum mass, the decoupling porous layer properties and the multi-layer mounting conditions are presented.This paper discusses the prediction of the low frequency diffuse field transmission loss through double-wall sound barriers with elastic porous linings. The studied sound barriers are made up from a porous-elastic decoupling material sandwiched between an elastic skin and a septum. The prediction approach is based on a finite element model for the different layers of the sound barrier coupled to a variational boundary element method to account for fluid loading. The diffuse field is modeled as a combination of uncorrelated freely propagating plane waves with equal amplitude, no two of which are traveling in the same direction. The corresponding vibroacoustic indicators are calculated efficiently using a Gauss integration scheme. Also, a power balance is presented to explain the dissipation mechanisms in the different layers. Typical results showing the effects on the transmission loss of several parameters such as the septum mass, the decoupling porous layer properties and the multi-layer mounting conditi...

Review of numerical solutions for low-frequency structural-acoustic problems

Abstract Numerical methods for predicting the behavior of structural acoustic systems are becoming increasingly practical. In this paper, the theoretical basis of the most popular state-of-the-art numerical structural-acoustic analysis techniques, the finite element methods and boundary element methods, are described in general terms. Typical approximations are described to identify both the best utilization and the limitations of the methods. Models for cases of heavy and light fluid loading are discussed. The methods are compared to each other and to analytical methods. A comprehensive literature review is included for identification of more complete references describing the methods.

Effect of the microstructure closed pore content on the acoustic behavior of polyurethane foams

The present paper proposes to investigate the links between the microstructure of polyurethane foams and their sound absorbing efficiency, and more specifically the effect of membranes closing the cells. This study is based on the complete characterization of 15 polyurethane foam with various cell sizes and reticulation rates (i.e., open pore content): (i) characterization of the microstructure properties (cell size Cs, strut thickness t, reticulation rate Rw…) from SEM pictures, (ii) characterization of nonacoustic parameters (porosity Φ, airflow resistivity σ, tortuosity α∞…) from direct and indirect methods. Existing analytical links between microstructure properties and nonacoustic parameters are first applied to fully reticulated materials. Then, they are improved empirically to account for the presence of the closed pore content. The proposed expressions associated to the Johnson-Champoux-Allard porous model allow a good estimation of the sound absorbing behavior of all polyurethane foams, fully ret...

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.

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...