10` eme Congr` es Fran¸ cais d’Acoustique Lyon, 12-16 Avril 2010 Characterization of poroelastic materials with a bayesian approach Jean-Daniel Chazot, J´ erˆome Antoni, Erliang Zhang Universit´ e de Technologie de Compi` egne, CNRS UMR 6253 Roberval, 60205 Compi` egne Cedex, France, jean-daniel.chazot@utc.fr A characterization method of poroelastic intrinsic parameters is used and compared with other direct methods. This inverse characterization method enables to get all the parameters with a simple measure- ment in a standing wave tube. It is based on a bayesian approach that enables to get probabilistic data such as the maximum aposteriori value and confidence intervals on each parameter. In this approach, it is necessary to define prior information on the parameters depending on the studied material. This last point is very important to regularize the inverse problem of identification. In a first step, the direct problem formulation is presented. Then, the inverse characterization is developed and applied to an experimental case. 1 Introduction Sound insulation of various structures with porous mate- rials finds its interest in several industrial domains such as building acoustics, automotive, and aeronautics. It is therefore important, at design stage, to be able to characterize the vibroacoustic behavior of these poroe- lastic structures. In practice Biot’s model[1] is com- monly used. However eight parameters are necessary to describe the poroelastic material. Several kinds of char- acterization, direct or not, can be used to get acous- tical parameters [9, 5, 12] or elastic parameters of the solid phase [7, 14, 2, 8]. However these kinds of charac- terisation measurements require a specific experimental device for each parameter. Besides, uncertainties ob- tained on each measurement are sometimes very large and lead to some difficulties to get a good comparison of the model with standard experimental data obtained with a standing wave tube apparatus. Finally, for par- ticular materials like fibrous ones, some parameters like elastic coefficients are very difficult to identify even with direct methods. The non-linear dynamic behavior is in- deed well known for such materials where the apparent stiffeness depends on the strain level at which the sample is tested [13]. On the other hand, inverse acoustic mea- surements [6, 3, 4] as well as indirect methods [11, 10] can also be used to adjust these parameters. In this paper, a robust inverse method is presented. All the parameters, acoustic and elastic, are charac- terised with only one measurement in a standing wave tube. The baysian theory is used to improve the effi- ciency of the optimisation scheme to identify the param- eters. The resulting cost function is therefore a combi- nation of the likelihood function and the prior informa- tion. Estimated values of the parameters are obtained via a minimisation of the cost function performed with a global search and a refined local search. Descent opti- misation methods and MCMC methods have hence been tested with simulated data and have led to very good re- sults. One interesting feature of the MCMC method is to get the probability density functions of each param- eter as well as the joint probability density functions between parameters. Uncertainties can hence be given on estimated values of each parameter as well as depen- dencies between parameters. 2 Poroelastic model description The Biot’s model presented in this paper is used to cal- culate the reflexion and transmission coefficients of a porous material sample in a standing wave tube. Know- ing these coefficients it is possible to predict the pressure at any position inside the tube. 2.1 Biot’s model In Biot’s theory, poroelastic materials are defined as materials consisting of a fluid and a solid phase. Biot’s model takes into account three different interactions between the two phases. Elastic coupling is taken into account in Equ. 1:  σ s =2N ε s +(P - 2N ) tr(ε s ) I + Q tr(ε f ) I , σ f = R tr(ε f ) I + Q tr(ε s ) I , (1) where σ s (resp. σ f ) represents the solid (resp. fluid) stress tensor, and ε s (resp. ε f ) represents the solid (resp. fluid) strain tensor. I is the identity matrix, and P ,Q,R, and N are classical elastic coefficients used in Biot’s model and detailed in Equ. 2. P = 4 3 N + K b + (1 - φ) 2 φ K f , Q = R(1 - φ) φ , R = φK f . (2) N, K b and ρ s are respectively the shear modulus, the bulk modulus and the density of the solid frame. φ is the porisity, and K f is the fluid compressibility