HAL Id: hal-03235444
https://hal.archives-ouvertes.fr/hal-03235444
Submitted on 12 Jun 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Investigation on inverse estimation of 9 poroelastic
material parameters based on absorption coefficient
Felix Egner, Elke Deckers, Wim Desmet
To cite this version:
Felix Egner, Elke Deckers, Wim Desmet. Investigation on inverse estimation of 9 poroelastic material
parameters based on absorption coefficient. Forum Acusticum, Dec 2020, Lyon, France. pp.3119-3124,
�10.48465/fa.2020.0361�. �hal-03235444�
INVESTIGATIONS ON INVERSE ESTIMATION OF 9 POROELASTIC
MATERIAL PARAMETERS BASED ON ABSORPTION COEFFICIENT
Felix Simeon Egner
Elke Deckers
Wim Desmet
Department of Mechanical Engineering, KU Leuven, Belgium
DMMS Core lab, Flanders Make, Belgium
felix.egner@kuleuven.be, elke.deckers@kuleuven.be, wim.desmet@kuleuven.be
ABSTRACT
The characterization of poroelastic materials typically re-quires various dedicated tests to estimate all necessary ma-terial parameters. Therefore, a characterization is linked with a lot of effort and usually it is not possible to perform in a standard acoustic laboratory. However, for designing vibro-acoustic systems that include poroelastic materials these parameters are essential. Once a material is char-acterized, the material’s impact in sound insulation and soundscape design can be simulated. Typical application fields are sound insulation in various means of transport and industrial machinery as well as building acoustics.
In this study we investigate an inverse method based on the transfer matrix formulation to estimate 9 material pa-rameters (acoustic and elastic) at once. As input a measure-ment of the material’s absorption coefficient is required. In order to overcome problems with low parameter sen-sitivity and multi-modality we split the inverse problem into sequential sub-problems based on models of differ-ent complexity and with increasing numbers of parame-ters. Furthermore, we use the resonance frequency of the solid phase, which, if excited, may be determined from the absorption coefficient measurement, to constrain the prob-lem.
The proposed inverse estimation procedure is described and an application to the characterization of three porous materials is shown.
1. INTRODUCTION
Poroelastic materials are characterized in order to obtain parameters for material models that describe sound propa-gation through the material. In the past, multiple material models have been proposed. They vary in their underly-ing assumptions and model complexity (number of param-eters). An overview of the most common models is given in table 1.
The parameters we consider here are:
acoustic static air flow resistivity σ, open porosity φ, high frequency limit of tortuosityα∞, viscous
character-istic length Λ and thermal charactercharacter-istic length Λ elastic Young’s modulus E, Poisson coefficient ν, solid
densityρ1, loss factorη1
model assumption parameter Delany-Bazely-Miki (DBM)
[1]
rigid frame σ Johnson-Champoux-Allard
(JCA(-PL), additions by Pride and Lafarge) [2]
rigid/limb frame
σ, φ, α∞,
Λ, Λ
Biot (poroelastic) [3, 4] elastic frame
JCA +
E, ν, ρ1,
η1
Table 1: Material models for sound propagation in porous media.
A main distinction of the models is whether the solid part of the porous material (i.e. the frame) is assumed to be rigid or elastic. If the frame is rigid the porous material can be seen as equivalent fluid, no elastic parameters are involved. In that case one compressional wave is able to propagate. In a material with elastic frame coupling be-tween the solid and fluid phase exists. Two compressional and one shear wave are able to propagate. Apart from reso-nance behaviour, the equivalent fluid description is a good assumption for frequencies higher than the decoupling fre-quency [2]
fd= φ
2σ
2πρ1.
(1) Material characterization methods can be divided into three categories [5]: direct-, indirect and inverse methods. A direct measurement refers to the measurement of a quan-tity with an apparatus made for the this specific purpose (e.g. a measurement of static airflow resistivity according to ISO 9053 [6]). Indirect measurements take analytical models into account to derive the desired quantity from a direct measurement of another quantity (e. g. the determi-nation of 6 acoustic parameters based on the JCAL model from a measurement of equivalent density and equivalent bulk modulus [7]). Yet another approach is taken by in-verse methods where a cost function is minimized in order to obtain the desired parameters.
In contrast to direct and indirect measurements where the measurement effort increases directly with the number of parameters, inverse measurements allow a determina-tion of all parameters at once. They can be designed such that the required measurement data originates from robust, standardized and inexpensive measurements. Furthermore,
the influence of user choices is limited. Typical shortcom-ings of inverse measurements are a low parameter sensi-tivity (i.e. a large parameter variation change the model output only slightly), multi-modality of the cost function and high computational effort.
A first inverse approach to characterize porous materi-als was taken to avoid complex, delicate and destructive direct measurements [8]. Here, three parameters (tortuos-ity, viscous and thermal characteristic length) were esti-mated based on the JCA model from a measurement of the complex valued surface impedance and known static air-flow resistivity and porosity. In [9] all 6 parameters of the JCA-L model were estimated using deterministic as well as statistical approaches. The latter resulting not only in a parameter estimate but also in an estimation of the un-certainty of the resulting parameters. A similar approach was taken in [5] for the characterization of poroelastic ma-terials. The authors report accurate results for the acous-tic parameters but variations due to a low sensitivity for elastic parameters. Acoustic and elastic parameters were estimated numerically in [10] based on near field pressure measurements in the impedance tube. This method works under the condition that the shear wave is excited. There-fore, the sample displacement has to be constraint around the circumference. A full characterization of the poroelas-tic parameters was done in [11] by means of a finite ele-ment model. The authors suggest to constrain the Young’s modulus by setting boundaries on the λ4-resonance:
fr= 1 4h (1− ν)E (1 + ν)(1 − 2ν)ρ1, (2)
which can be obtained from the absorption coefficient (h is
the directly measured sample thickness). Additionally, the authors suggest to provide the solid density for improved convergence behaviour of the estimation.
In our investigation we focus on robustness of the in-verse estimation. To this extend we propose a step-wise procedure. The reasoning is equivalent to [12], where different frequency regions are exploited subse-quentially. First, the global behaviour is estimated by a low-complexity model and subsequently the parameter fit is improved by taking advantage of complexer models. This method has shown to avoid finding local minima.
The remainder of the paper is organized as follows. In section 2 our stepwise inverse approach is defined: model, cost function and constraints are explained. Furthermore, a numerical benchmark study is shown. In section 3 the approach is validated experimentally and in section 4 con-clusions are drawn.
2. INVERSE PROBLEM
In this section, the inverse problem we propose for estimat-ing the poroelastic material parameters is defined.
2.1 Model
Our model evaluates the surface impedanceZsof a rigidly
backed material sample inside the impedance tube using
the transfer matrix method as described in [13]:
Zs= TMM(f, model, θ, ξ). (3)
The evaluation depends on frequencyf , the material model
(DBM, JCA, poroelastic), the material parameters θ and
environmental conditions ξ (properties of the
surround-ing air: temperature, ambient pressure and subsequently density ρ0 and speed of sound c0). From the surface
impedance and the characteristic impedance of airZ0 = ρ0c0the absorption coefficientα is computed
α = 1 −Zs− Z0 Zs+ Z0
2. (4) As the transfer matrix constitutes a one-dimensional model, laterally, ideal sliding edge boundary conditions are assumed.
2.2 Cost function
We define the inverse problem as minimization of the cost functionF with regards to the material parameters:
min θ F (f, θ) = S ⎛ ⎝ 1 Nf f (|α(f, θ)| − |˜α(f)|)2 ⎞ ⎠ (5) Thereby, ˜α is the measured absorption coefficient and α
the parameter dependent model output. The sum is eval-uated over the frequency,Nf is the number of frequency points and S(x) is a scaling function. In order to give
more weight to differences close to the minimum the scal-ing function is chosen to:
S(x) = log10 1− 1 1 + x (6)
Thus, values of the cost function are in the range [−∞, 0]. MATLAB’s gradient based solver employing the interior point algorithm (fmincon) is used to solve the inverse prob-lem. For a better convergence, all parameters are scaled to the interval [−1, 1].
2.3 Parameter boundaries and constraints
To constrain the model parameters, boundaries are set ac-cording to values suggested in the literature [8,9] (table 2). Thereby, the shape factorsc and c are introduced to re-place viscous and thermal characteristic length [8]:
c = 1 Λ 8α∞η σφ , c = 1 Λ 8α∞η σφ . (7)
With the geometric constraintc > cand the dynamic vis-cosity of airη. If present, the λ4-resonance frequency is used to constrain the elastic parameters. Additionally, with the measured massm of the sample porosity and solid
den-sity are constrained by
ρ1= 1 1− φ 4m d2πh− ρ0φ , (8)
σ φ α∞ c c N s m−4 - - - -lb 103 0.1 1 0.1 0.1 ub 105 1 4 3.3 3.3 E ν ρ1 η1 Pa - kg m−3 -lb 103 0 0 0 ub 106 0.5 103 1
Table 2: Parameter boundaries.
final values
θfinal
random initial conditions
θinitial= [σ, φ, α∞, c, c, E, ν, ρ1, η1]
θ = [σ]
DBM
JCA θ = [σ, φ, α∞, c, c]
JCA final values as initial conditions reduce boundaries
include resonance
poroelastic θ = [σ, φ, α∞, c, c, E, ν, ρ1, η1]
DBM final values as initial conditions reduce boundaries
Figure 1: Optimization procedure.
2.4 Stepwise procedure
Three material models are used. A first estimation of the static airflow resistivity is done by means of the DBM model. This estimate is subsequently used as initial con-dition for the JCA model in order to estimate all acoustic parameters. Finally, the acoustic parameters are used as intitial condition for the poroelastic model to estimate the full set of parameters. A schematic sketch of the procedure is shown in fig. 1. To increase the confidence of the previ-ously estimated parameters, the parameter boundaries are decreased as
lb = max(lb∗, θ∗− 0.5(ub∗− lb∗)) (9)
ub = min(ub∗, θ∗+ 0.5(ub∗− lb∗)) (10)
when stepping to the next model.lb and ub being lower and
upper boundary respectively and ∗ indicates values from the previous step. Additionally, if present, theλ4-resonance is excluded in the frequency range for DBM and JCA es-timation because these models cannot account for elastic effects. 200 1000 4500 Frequency (Hz) 0 0.2 0.4 0.6 0.8 1 Absorption coefficient (1) Numerical measurement Considered frequency points
Figure 2: Numerical measurement.
c c E 1 1
10-2 100 102 104
Relative parameter error (%)
Stepwise Direct
Mean Median
Figure 3: Relative parameter error for 100 initial condi-tions.
2.5 Numerical investigation
In a numerical study we compare our stepwise approach to a direct optimization with the poroelastic model. A forward evaluation of the poroelastic model was used as measurement (fig. 2). To account for measurement noise we added normal distributed noise with standard devia-tion of 0.005 and zero mean. In the example
measure-ment, the elastic resonance is found within the consid-ered frequency range of 200 Hz to 4500 Hz. 100 logarith-mically distributed frequency points were considered over this range and additionally 30 around the elastic resonance. As previously shown for acoustic parameters, it is impor-tant to cover all three characteristic frequency regions of the absorption coefficient [8]. That is because the sensitiv-ity of the parameters is frequency dependent.
In fig. 3 the relative parameter error is shown for 100 randomly chosen initial conditions. Regarding the acoustic parameters the stepwise approach results in lower median errors (we look at the median in order to avoid influence by local minima, which are present in both optimization approaches). The same yields for solid density and loss factor. For Youngs modulus and Poisson coefficient both approaches result in similar errors. Generally, the parame-ters Youngs modulus and Poisson coefficient show estima-tion errors of more than 20 %. We attribute this to the low parameter sensitivity, as reported in [11].
Even though, the estimation with our stepwise approach takes on average 14 s longer than a direct estimation with the poroelastic model (on average 90 s vs 76 s), local min-ima are found less often (16 out of 100 runs for stepwise vs. 73 out of 100 for direct). Figure 4 visualizes the
ef-Figure 4: Evolution of the (acoustic) parameter values throughout the optimization.
Parameter Material B Material G Material Y
σ (kN s m−4) 18.8 (±2.2) 13.6 (±3.8) 67.8 (±10.6) φ (-) 0.96 (±0.01) 0.95 (±0.00) 0.94 (±0.01) α∞(-) 1.13 (±0.19) 1.88 (±0.21) 2.4 (±0.14) Λ (μm) 65 (±9) 92 (±28) 41 (±3) Λ(μm) 145 (±33) 303 (±97) 275 (±93) E (kPa) 7.2 (±2.6) 183 (±30) 112 (±77) ν (-) 0.07 (-) 0.43 (±0.04) 0.39 (±0.05) ρ1(kg m−3) 985 (±5) 509 (±2) 704 (±2) η1(-) 0.16 (±0.06) 0.11 (±0.01) 0.65 (±0.04) h (mm) 14.0 (±1.4) 30.3 (±0.5) 30.0 (±0.0) fr(Hz) 49 267 147
Table 3: Reference material parameters and standard de-viation [14].
fect of the stepwise approach. The scaled parameter value at each iteration step of the optimization is shown (as the elastic parameters are only estimated with one model they are removed in the figure for a better overview). The value of the static airflow resistivity at the final step of the DBM model already approximates the final estimate (at the end of the poroelastic phase) well. Similarly, the other acous-tic parameters are nearly determined at the end of the JCA phase and only slight changes happen afterwards.
The error on the final values originates from the noise that was added to the reference absorption coefficient.
3. EXPERIMENTAL INVESTIGATION The approach presented is validated on three differ-ent porous materials that were characterized by the re-search lab Matelys and three other laboratories special-ized on porous material characterization. The materials are a polyester felt of 14 mm thickness (Material B), a polyurethane foam of 30.3 mm thickness (Material G) and
a visco-elastic foam of 30 mm thickness (material Y) [14]. An overview of the parameter values is given in table 3.
The absorption coefficient of five samples of each mate-rial has been measured. Figure 5 shows the average and an envelope consisting of the single measurements for each material. Generally the variance between samples of a material is small. There were no extra measures taken to prevent resonance effects of the tube (like the peak at 550 Hz), but these frequency regions were excluded in the
Figure 5: Measured absorption coefficient.
Parameter Material B Material G Material Y
σ (%) 18.06 1.62 35.76 φ (%) 6.32 4.49 1.68 α∞(%) 11.10 20.30 34.65 Λ (%) 18.46 7.61 11.66 Λ(%) 16.06 10.08 19.27 E (%) 6.22 65.55 309.44 ν (%) 64.31 39.70 35.77 ρ1(%) 99.47 54.80 42.02 η1(%) 99.52 99.84 60.72
Table 4: Relative error on the estimated parameters, values that are neither within the measurement standard deviation nor below 10 % are highlighted.
inverse estimation. The parameters are estimated based on the average absorption coefficient for each material respec-tively. In all three cases, no λ4-resonance was observed in the frequency range considered and thus no constraint on the resonance was set. We confirmed the absence of the
λ
4-resonance by increasing the sample stiffness artificially
(with needles). The expectation was, that any possible res-onance would shift to higher frequencies, however, no dif-ference in the measurement was observed.
For the inverse problem, 500 frequency points have been taken into account and the computational time was around 10 min for each material. The higher number of frequency points was selected for an increased averaging of the measurement noise in the fit. Table 4 shows the rel-ative error for each parameter estimate. The errors of the elastic parameters are high since there were no elastic ef-fects present in the measurement. Thus, there is not much influence of the elastic parameters on the absorption coef-ficient. In contrast, the acoustic parameters are estimated within a limited error range.
Figure 6 shows the absorption coefficient as predicted from the reference material parameters (table 3) as well as the envelope which accounts for the uncertainty on the pa-rameters. The envelope was computed with the sampling technique suggested in [15]. Additionally, our measure-ment and estimation are shown in the same figure. For material B the mean value of the prediction, measurement and estimation coincide well, apart from noise in the mea-surement and resonance effects due to our impedance tube. For material G a similar situation is found in the low fre-quency region. Above 1000 Hz the predicted mean value
(a) Material B.
(b) Material G.
(c) Material Y.
Figure 6: Overview on reference, measured and estimated absorption coefficients.
and our measurement deviate. Therefore, also the estima-tion, which follows the measurement, deviates from the predicted mean value. This explains the large error on the estimated tortuosity value, as simulations showed that tor-tuosity influences the absorption coefficient mostly in the high frequency region. For material Y our measurement deviates from the predicted envelope in certain frequency ranges. Therefore, the estimation error is generally higher than for the other materials.
The deviation of the measurement evokes the question of measurement accuracy for absorption coefficient mea-surements. Inter-laboratory studies have shown deviations of absorption coefficient measurements of the same mate-rial between different laboratories with up to 19 % relative error [16]. For an inverse estimation such a deviation leads to notably different parameter estimates. If the values we obtain from acoustic measurements of poroelastic materi-als vary to this large extent, perhaps the necessary ques-tion for material parameter estimaques-tion is not: how can we increase the accuracy of the estimation? But rather: is the estimation accurate enough for the given application and how to deal with uncertainty in the estimation?
Besides the experimental validation on three different
porous materials described here, the proposed approach was applied in the characterization of the fibrous material
Ecophon Industry Modus S [17].
4. CONCLUSION
We presented an investigation on the inverse estimation of 9 poroelastic material parameters. Thereby, we took advantage of a stepwise approach employing DBM, JCA and poroelastic models. It was shown numerically that the stepwise procedure reduces the likelihood for finding local minima. Applied to the characterization of three porous materials, the estimation procedure resulted in feasible es-timates for the acoustic parameters, given the measurement was close to the reference. It was concluded that the source for errors in parameter estimates is mainly the uncertainty in absorption coefficient measurements and not the opti-mization procedure itself.
5. ACKNOWLEDGEMENTS AND FUNDING The authors would like to express their gratitude to the re-search lab Matelys for providing material samples and ref-erence material parameters.
The research of Felix Egner is funded by an Early Stage Researcher grant within the European Project ACOU-TECT, Marie Curie Initial Training Network (GA 721536). Furthermore, the Research Fund KU Leuven is acknowl-edged for its support.
6. REFERENCES
[1] Yasushi Miki, “Acoustical properties of porous mate-rials. Modifications of Delany-Bazley models.,” JASJ, vol. 11, no. 1, pp. 19–24, 1990.
[2] J.-F. Allard and N. Atalla, Propagation of Sound in
Porous Media: Modelling Sound Absorbing Materials.
Hoboken, N.J: Wiley, 2nd ed ed., 2009.
[3] M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range,” JASA, vol. 28, pp. 168–178, Mar. 1956. [4] M. A. Biot, “Theory of Propagation of Elastic Waves
in a Fluid-Saturated Porous Solid. II. Higher Frequency Range,” JASA, vol. 28, pp. 179–191, Mar. 1956. [5] J.-D. Chazot, E. Zhang, and J. Antoni, “Acoustical and
mechanical characterization of poroelastic materials using a Bayesian approach,” JASA, vol. 131, pp. 4584– 4595, June 2012.
[6] DIN, “DIN EN ISO 9053 - Bestimmung des statischen Str¨omungswiderstandes,” 2017.
[7] L. Jaouen, “6-parameter acoustical characterization of porous media using a classical impedance tube,” in
[8] Y. Atalla and R. Panneton, “Inverse Acoustical Characterization Of Open Cell Porous Media Using Impedance Tube Measurements,” CAA-ACA, vol. 33, 2005.
[9] M. Niskanen, J.-P. Groby, A. Duclos, O. Dazel, J. C. Le Roux, N. Poulain, T. Huttunen, and T. L¨ahivaara, “Deterministic and statistical characterization of rigid frame porous materials from impedance tube measure-ments,” JASA, vol. 142, pp. 2407–2418, Oct. 2017. [10] J. Vanhuyse, E. Deckers, S. Jonckheere, B. Pluymers,
and W. Desmet, “Global optimisation methods for poroelastic material characterisation using a clamped sample in a Kundt tube setup,” MSSP, vol. 68-69, pp. 462–478, Feb. 2016.
[11] K. Verdi`ere, R. Panneton, N. Atalla, and S. Elk-oun, “Inverse Poroelastic Characterization of Open-Cell Porous Materials Using an Impedance Tube,” vol. 2017-01-1878, SAE Technical Paper, June 2017. [12] P. G¨oransson, J. Cuenca, and T. L¨ahivaara, “Parameter
estimation in modelling frequency response of coupled systems using a stepwise approach,” MSSP, vol. 126, pp. 161–175, July 2019.
[13] E. Deckers, S. Jonckheere, D. Vandepitte, and W. Desmet, “Modelling Techniques for Vibro-Acoustic Dynamics of Poroelastic Materials,” Arch.
Comput. Methods Eng, vol. 22, pp. 183–236, Apr.
2015.
[14] F. Chevillotte, F. Ronzio, C. Bertolini, M. T. Hoang, L. Dejaeger, L. lei, A. Duval, and P. Mordillat, “Inter-Laboratory Characterization of Biot Parameters of Poro-Elastic Materials for Automotive Applications,” in International Styrian Noise, Vibration & Harshness
Congress, vol. 11, (Graz).
[15] J. Rodenas, F. Chevillotte, and F. X. B´ecot, “On the use of uncertainties from the characterization to the com-putation of dispersion envelope of poro-elastic media,” in Symposium on the Acoustics of Poro-Elastic
Materi-als, vol. 5, p. 52, Dec. 2017.
[16] F. Pompoli, P. Bonfiglio, K. V. Horoshenkov, A. Khan, L. Jaouen, F.-X. B´ecot, F. Sgard, F. Asdrubali, F. D’Alessandro, J. H¨ubelt, N. Atalla, C. K. Am´edin, W. Lauriks, and L. Boeckx, “How reproducible is the acoustical characterization of porous media?,” JASA, vol. 141, pp. 945–955, Feb. 2017.
[17] N. B. Roozen, E. A. Piana, C. Micallef, F. S. Eg-ner, E. Deckers, and C. Glorieux, “ON THE EXPERI-MENTAL CHARACTERIZATION OF THE ACOUS-TIC PROPERTIES OF ECOPHON POROUS MATE-RIAL,” in Forum Acusticum 2020, (Lyon/online), p. 8, Dec. 2020.
