Staff

Abstract:
Although the elastic properties of porous materials depend mainly on the volume fraction of pores, the details of pore distribution within the material representative volume are also important and may be the subject of optimisation. To study their effect, experimental analyses were performed on samples made of a polymer material with a predefined distribution of spherical voids, but with various porosities due to different pore sizes. Three types of pore distribution with cubic symmetry were considered and the results of experimental analyses were confronted with mean-field estimates and numerical calculations. The mean-field ‘cluster’ model is used in which the mutual interactions between each of the two pores in the predefined volume are considered. As a result, the geometry of pore distribution is reflected in the anisotropic effective properties. The samples were produced using a 3D printing technique and tested in the regime of small strain to assess the elastic stiffness. The digital image correlation method was used to measure material response under compression. As a reference, the solid samples were also 3D printed and tested to evaluate the polymer matrix stiffness. The anisotropy of the elastic response of porous samples related to the arrangement of voids was assessed. Young’s moduli measured for the additively manufactured samples complied satisfactorily with modelling predictions for low and moderate pore sizes, while only qualitatively for larger porosities. Thus, the low-cost additive manufacturing techniques may be considered rather as preliminary tools to prototype porous materials and test mean-field approaches, while for the quantitative and detailed model validation, more accurate additive printing techniques should be considered. Research paves the way for using these computationally efficient models in optimising the microstructure of heterogeneous materials and composites.

Keywords:
Pore configuration, Anisotropy, Elasticity, Micro-mechanics, Additive manufacturing

Abstract:
An assembly of additively-manufactured modules to form two-dimensional networks of labyrinthine slits results in a sound absorber with extremely high tortuosity and thereby a relatively low frequency quarter wavelength resonance. Fully analytical modelling is developed for the generic design of such composite acoustic panels, allowing rapid exploration of various specific designs. In addition to labyrinthine channels in a non-porous solid skeleton, a case is also considered where the skeleton has microporosity such that its permeability is very much lower than that due to the labyrinthine channels alone. The analytical modelling is verified by numerical calculations, as well as sound absorption measurements performed on several 3D printed samples of modular composite panels. The experimental validation required overcoming the non-trivial difficulties related to additive manufacturing and testing samples of extreme tortuosity. However, due to the two-dimensionality and modularity of the proposed design, such absorbers can possibly be produced without 3D printing by assembling simple, identical modules produced separately. The experimental results fully confirmed the theoretical predictions that significant sound absorption, almost perfect at the peak, can be achieved at relatively low frequencies using very thin panels, especially those with double porosity.

Keywords:
Sound absorption,Extreme tortuosity,Double porosity,Acoustic composites,Additive manufacturing

Abstract:
The potential usefulness of relatively simple pore microstructures such as parallel, identical, inclined slits for creating broadband sound absorption has been argued through analytical models. In principle, such microstructures could be realised through budget additive manufacturing. However, validation of the analytical predictions through normal incidence impedance tube measurements on finite layers is made difficult by the finite size of the tube. The tube walls curtail the lengths of inclined slits and, as a result, prevent penetration of sound through the layer. As well as demonstrating and modelling this effect, this paper explores two manufacturing solutions. While analytical and numerical predictions correspond well to absorption spectra measured on slits normal to the surface, discrepancies between measured and predicted sound absorption are noticed for perforated and zigzag slit configurations. For perforated microgeometries this is found to be the case with both numerical and analytical modelling based on variable length dead-end pores. Discrepancies are to be expected since the dead-end pore model does not allow for narrow pores in which viscous effects are important. For zigzag slits it is found possible to modify the permeability used in the inclined slit analytical model empirically to obtain reasonable agreement with data.

Keywords:
slitted sound absorber, additive manufacturing, microstructure-based modelling

Abstract:
Accurate definition of boundary conditions is of crucial importance for room acoustic predictions because the wall impedance phase angle can affect the sound field in rooms and acoustic parameters applied to assess a room reverberation. In this paper, the issue was investigated theoretically using the convolution integral and a modal representation of the room impulse response for complex-valued boundary conditions. Theoretical considerations have been accompanied with numerical simulations carried out for a rectangular room. The case of zero phase angle, which is often assumed in room acoustic simulations, was taken as a reference, and differences in the sound pressure level and decay times were determined in relation to this case. Calculation results have shown that a slight deviation of the phase angle with respect to the phase equal to zero can cause a perceptual difference in the sound pressure level. This effect was found to be due to a change in modal frequencies as a result of an increase or decrease in the phase angle. Simulations have demonstrated that surface distributions of decay times are highly irregular, while a much greater range of the early decay time compared to the reverberation time range indicates that a decay curve is nonlinear. It was also found that a difference between the decay times predicted for the complex impedance and real impedance is especially clearly audible for the largest impedance phase angles because it corresponds approximately to 4 just noticeable differences for the reverberation metrics.

Keywords:
room acoustics, complex wall impedance, indoor sound field, room impulse response, reverberation parameters

Abstract:
At first glance, it seems that modern, inexpensive additive manufacturing (AM) technologies can be used to produce innovative, efficient acoustic materials with tailored pore morphology. However, on closer inspection, it becomes rather obvious that for now this is only possible for specific solutions, such as relatively thin, but narrow-band sound absorbers. This is mainly due to the relatively poor resolutions available in low-cost AM technologies and devices, which prevents the 3D-printing of pore networks with characteristic dimensions comparable to those found in conventional broadband sound-absorbing materials. Other drawbacks relate to a number of imperfections associated with AM technologies, including porosity or rather microporosity inherent in some of them. This paper shows how the limitations mentioned above can be alleviated by 3D-printing double-porosity structures, where the main pore network can be designed and optimised, while the properties of the intentionally microporous skeleton provide the desired permeability contrast, leading to additional broadband sound energy dissipation due to pressure diffusion. The beneficial effect of additively manufactured double porosity and the phenomena associated with it are rigorously demonstrated and validated in this work, both experimentally and through precise multi-scale modelling, on a comprehensive example that can serve as benchmark.

Keywords:
double porosity, additive manufacturing, sound absorption, pressure diffusion, multi-scale modelling

Abstract:
Absorbing properties of multi-layer acoustic absorbers were modeled using the impedance translation theorem and the Garai and Pompoli empirical model, which enables a determination of the characteristic impedance and propagation constant of fibrous sound-absorbing materials. The theoretical model was applied to the computational study of performance of single-layer acoustic absorber backed by a hard wall and the absorber consisting of one layer of absorbing material and an air gap between the rear of the material and a hard back wall. Simulation results have shown that a high thickness of absorbing material may cause wavy changes in the frequency relationship of the normal and random incidence absorption coefficients. It was also found that this effect is particularly noticeable for acoustic absorbers with a large thickness of air gap between the absorbing material and a hard back wall.

Keywords:
sound absorption, multi-layer absorber, surface impedance, fibrous materials, air gap

Abstract:
This paper investigates the mechanisms that can be used to enhance the absorption performance of poro-elastic materials in the viscous regime. It is shown that by adding small inclusions in a poro-elastic foam layer, a mass–spring effect can be introduced. If the poro-elastic material has relatively high viscous losses in the frequency range of interest, the mass–spring effect can enhance the sound absorption of the foam by introducing an additional mode in the frame and increasing its out-of-phase movement with respect to the fluid part. Moreover, different effects such as the trapped mode effect, the modified-mode effect, and the mass–spring effect are differentiated by decomposing the absorption coefficient in terms of the three energy dissipation mechanisms (viscous, thermal, and structural losses) in poro-elastic materials. The physical and geometrical parameters that can amplify or decrease the mass–spring effect are discussed. Additionally, the influence of the incidence angle on the mass–spring effect is evaluated and a discussion on tuning the inclusion to different target frequencies is given.

Keywords:
meta-poro-elastic material, Biot–Allard poroelastic model, mass–spring effect, viscous regime

Abstract:
Acoustic wave propagation in porous composites is investigated in this paper. The two-scale asymptotic homogenisation method is used to obtain the macroscopic description of sound propagation in such composites. The developed theory is both exemplified by introducing analytical models for the effective acoustical properties of porous composites with canonical inclusion patterns (i.e. a porous matrix with a periodic array of cylindrical or spherical inclusions) and validated by comparing the models predictions with the results of direct finite-element simulations and experimental testing, showing good agreement in all cases. It is concluded that the developed theory correctly captures the acoustic interaction between the constituents of the porous composite and elucidates the physical mechanisms underlying the dissipation of sound energy in such composites. These correspond to classical visco-thermal dissipation in the porous constituents, together with, for the case of composites made from constituents characterised by highly contrasted permeabilities, pressure diffusion which provides additional and tunable sound energy dissipation. In addition, this work determines the conditions for which a rigidly-backed porous composite layer can present improved sound absorption performance in comparison with that of layers made from their individual constituents. Hence, the presented results are expected to guide the rational design of porous composites with superior acoustic performance.

Keywords:
porous composites, wave propagation, acoustical properties, homogenisation, pressure diffusion

Abstract:
This paper investigates sound propagation in polydisperse heterogeneous porous composites. The two-scale asymptotic method of homogenization is used to obtain a macroscopic description of the propagation of sound in such composites. The upscaled equations demonstrate that the studied composites can be modeled as equivalent fluids with complex-valued frequency-dependent effective parameters (i.e., dynamic viscous permeability and compressibility) as well as unravel the sound energy dissipation mechanisms involved. The upscaled theory is both exemplified by introducing analytical and hybrid models for the acoustical properties of porous composites with different geometries and constituent materials (e.g., a porous matrix with much less permeable and/or impervious inclusions with simple or complex shapes) and validated through computational experiments successfully. It is concluded that the developed theory rigorously captures the physics of acoustic wave propagation in polydisperse heterogeneous porous composites and shows that the mechanisms that contribute to the dissipation of sound energy in the composite are classical visco-thermal dissipation together with multiple pressure diffusion phenomena in the heterogeneous inclusions. The results show that the combination of two or more permeable materials with highly contrasted permeabilities
can improve the acoustic absorption and transmission loss of the composite. This paper provides fundamental insights into the propagation of acoustic waves in complex composites that are expected to guide the rational design of novel acoustic materials.

Abstract:
Closed-cell metal foams are lightweight and durable materials resistant to high temperature and harsh conditions, but due to their fully closed porosity they are poor airborne sound absorbers. In this paper a classic method of drilling is used for a nearly closed-cell aluminium foam to open its porous interior to the penetration of acoustic waves propagating in air, thereby increasing the wave energy dissipation inside the pores of the perforated medium. The aim is to investigate whether it is possible to effectively approximate wave propagation and attenuation in industrial perforated heterogeneous materials with originally closed porosity of irregular shape by means of their simplified microstructural representation based on computer tomography scans. The applied multi-scale modelling of sound absorption in foam samples is confronted with impedance tube measurements. Moreover, the collected numerical and experimental data is compared with the corresponding results obtained for perforated solid samples to demonstrate a great benefit coming from the presence of an initially closed porous structure in the foam.

Keywords:
closed-cell metal foams, perforation, sound absorption, microstructure effects, dissipated powers

Abstract:
The purpose of this work is to check if additive manufacturing technologies are suitable for reproducing porous samples designed for sound absorption. The work is an inter-laboratory test, in which the production of samples and their acoustic measurements are carried out independently by different laboratories, sharing only the same geometry codes describing agreed periodic cellular designs. Different additive manufacturing technologies and equipment are used to make samples. Although most of the results obtained from measurements performed on samples with the same cellular design are very close, it is shown that some discrepancies are due to shape and surface imperfections, or microporosity, induced by the manufacturing process. The proposed periodic cellular designs can be easily reproduced and are suitable for further benchmarking of additive manufacturing techniques for rapid prototyping of acoustic materials and metamaterials.

Keywords:
porous materials, designed periodicity, additive manufacturing, sound absorption

Abstract:
This work presents benchmark examples related to the modelling of sound absorbing porous media with rigid frame based on the periodic geometry of their microstructures. To this end, rigorous mathematical derivations are recalled to provide all necessary equations, useful relations, and formulae for the so-called direct multi-scale computations, as well as for the hybrid multi-scale calculations based on the numerically determined transport parameters of porous materials. The results of such direct and hybrid multi-scale calculations are not only cross verified, but also confirmed by direct numerical simulations based on the linearised Navier-Stokes-Fourier equations. In addition, relevant theoretical and numerical issues are discussed, and some practical hints are given.

Keywords:
porous media, periodic microstructure, wave propagation, sound absorption

Abstract:
This work investigates a porous composite with modifiable micro-geometry so that its ability to absorb noise can be accommodated to different frequency ranges. The polymeric skeleton of the composite has a specific periodic structure with two types of pores (larger and smaller ones) and two types of channels (wide and narrow ones), and each of the large pores contains a small steel ball. Depending on the situation, the balls block different channels that connect the pores, and therefore alter the visco-inertial phenomena between the saturating air and solid skeleton which take place at the micro-scale level and are responsible for the dissipation of the energy of acoustic waves penetrating the porous composite. All this is studied numerically using advanced dual-scale modelling, and the results are verified by the corresponding experimental tests of 3D-printed samples. Particular attention is paid to the prototyping and additive manufacturing of such adaptive porous composites.

Keywords:
porous composite, adaptive sound absorber, microstructure-based modelling, additive manufacturing

Abstract:
This work discusses many practical and some theoretical aspects concerning modelling and design of plates with micro-slits, involving multi-scale calculations based on microstructure. To this end, useful mathematical reductions are demonstrated, and numerical computations are compared with possible analytical estimations. The numerical and analytical approaches are used to calculate the transport parameters for complex micro-perforated (micro-slotted) plates, which allow to determine the effective properties of the equivalent fluid, so that at the macro-scale level the plate can be treated as a specific layer of acoustic fluid. In that way, the sound absorption of micro-slotted plates backed with air cavities can be determined by solving a multi-layer system of Helmholtz equations. Two such examples are presented in the paper and validated experimentally. The first plate has narrow slits precisely cut out using a traditional technique, while the second plate - with an original micro-perforated pattern - is 3D-printed.

Keywords:
micro-slotted plates, micro-perforated plates, sound absorption, microstructure-based modelling, 3D-printing

Abstract:
The paper proposes and investigates computationally-efficient microstructure representations for sound absorbing fibrous media. Three-dimensional volume elements involving non-trivial periodic arrangements of straight fibres are examined as well as simple two-dimensional cells. It has been found that a simple 2D quasi-representative cell can provide similar predictions as a volume element which is in general much more geometrically accurate for typical fibrous materials. The multiscale modelling allowed to determine the effective speeds and damping of acoustic waves propagating in such media, which brings up a discussion on the correlation between the speed, penetration range and attenuation of sound waves. Original experiments on manufactured copper-wire samples are presented and the microstructure-based calculations of acoustic absorption are compared with the corresponding experimental results. In fact, the comparison suggested the microstructure modifications leading to representations with non-uniformly distributed fibres.

Keywords:
Sound absorption, Fibrous materials, Multiscale modelling, Microstructure representations

Abstract:
This paper proposes and discusses an approach for the design and quality inspection of the morphology dedicated for sound absorbing foams, using a relatively simple technique for a random generation of periodic microstructures representative for open-cell foams with spherical pores. The design is controlled by a few parameters, namely, the total open porosity and the average pore size, as well as the standard deviation of pore size. These design parameters are set up exactly and independently, however, the setting of the standard deviation of pore sizes requires some number of pores in the representative volume element (RVE); this number is a procedure parameter. Another pore structure parameter which may be indirectly affected is the average size of windows linking the pores, however, it is in fact weakly controlled by the maximal pore-penetration factor, and moreover, it depends on the porosity and pore size. The proposed methodology for testing microstructure-designs of sound absorbing porous media applies the multi-scale modeling where some important transport parameters—responsible for sound propagation in a porous medium—are calculated from microstructure using the generated RVE, in order to estimate the sound velocity and absorption of such a designed material.

Keywords:
Foams, Porous media, Viscosity, Acoustic absorption, Ceramics

Abstract:
This paper presents a methodology for the inverse characterization of sound absorbing rigid porous media, based on standard measurements of the surface acoustic impedance of a porous sample. The model parameters need to be normalized to have a robust identification procedure which fits the model-predicted impedance curves with the measured ones. Such a normalization provides a substitute set of dimensionless (normalized) parameters unambiguously related to the original model parameters. Moreover, two scaling frequencies are introduced, however, they are not additional parameters and for different, yet reasonable, assumptions of their values, the identification procedure should eventually lead to the same solution. The proposed identification technique uses measured and computed impedance curves for a porous sample not only in the standard configuration, that is, set to the rigid termination piston in an impedance tube, but also with air gaps of known thicknesses between the sample and the piston. Therefore, all necessary analytical formulas for sound propagation in double-layered media are provided. The methodology is illustrated by one numerical test and by two examples based on the experimental measurements of the acoustic impedance and absorption of porous ceramic samples of different thicknesses and a sample of polyurethane foam.

Keywords:
Acoustic modeling, Viscosity, Porous materials, Porous media, Acoustic impedance measurement

Abstract:
The paper presents the developed algorithm which implements the indirect variational boundary element method (IVBEM) for computation of the free-field acoustic radiation characteristics of vibrating rectangle-shaped plate structures with arbitrary boundary conditions. In order to significantly reduce the computational time and cost, the algorithm takes advantage of simple geometry of the considered problem and symmetries between the elements. The procedure of determining the distribution of acoustic pressure is illustrated on the example of thin, rectangular plate with a part of one edge clamped and all other edges free. The eigenfrequencies and the corresponding vibrational mode shapes of the plate are computed using the finite element method (FEM). The results of the numerical simulations are compared to the results of the experiments carried out in an anechoic chamber, proving good agreement between the predictions and the observations. The reliability of simulations and high computational efficiency make the developed algorithm a useful tool in analysis of the acoustic radiation characteristics of vibrating plate structures.

Keywords:
Acoustic radiation, Indirect variational Boundary Element Method, Plate structures

Abstract:
Acoustics of stiff porous media with open porosity can be very effectively modelled using the so-called Johnson-Champoux-Allard-Pride-Lafarge model for sound absorbing porous media with rigid frame. It is an advanced semi-phenomenological model with eight parameters, namely, the total porosity, the viscous permeability and its thermal analogue, the tortuosity, two characteristic lengths (one specific for viscous forces, the other for thermal effects), and finally, viscous and thermal tortuosities at the frequency limit of 0 Hz. Most of these parameters can be measured directly, however, to this end specific equipment is required different for various parameters. Moreover, some parameters are difficult to determine. This is one of several reasons for the so-called multiscale approach, where the parameters are computed from specific finite-element analyses based on some realistic geometric representations of the actual microstructure of porous material. Such approach is presented and validated for layers made up of loosely packed small identical rigid spheres. The sound absorption of such layers was measured experimentally in the impedance tube using the so-called two-microphone transfer function method. The layers are characterised by open porosity and semi-regular microstructure: the identical spheres are loosely packed by random pouring and mixing under the gravity force inside the impedance tubes of various size. Therefore, the regular sphere packings were used to generate Representative Volume Elements suitable for calculations at the micro-scale level. These packings involve only one, two, or four spheres so that the three-dimensional finite-element calculations specific for viscous, thermal, and tortuous effects are feasible. In the proposed geometric packings, the spheres were slightly shifted in order to achieve the correct value of total porosity which was precisely estimated for the layers tested experimentally. Finally, in this paper some results based on the self-consistent estimates are also provided.

Keywords:
Viscosity, Porous media, Acoustic absorption, Acoustic modeling, Tensor methods

Abstract:
Recently, a new class of ceramic foams with porosity levels up to 90% has been developed as a result of the association of the gelcasting process and aeration of the ceramic suspension. This paper presents and discusses original results advertising sound absorbing capabilities of such foams. The authors manufactured three types of alumina foams in order to investigate three porosity levels, namely: 72, 88, and 90%. The microstructure of foams was examined and typical dimensions and average sizes of cells (pores) and cell-linking windows were found for each porosity case. Then, the acoustic absorption coefficient was measured in a wide frequency range for several samples of various thickness cut out from the foams. The results were discussed and compared with the acoustic absorption of typical polyurethane foams proving that the alumina foams with high porosity of 88-90% have excellent sound absorbing properties competitive with the quality of sound absorbing PU foams of higher porosity.

Keywords:
Sound absorption, Porous materials, Alumina foams

Abstract:
The paper focuses on some issues regarding the utilization of small rectangle-shaped piezoelectric transducers as both sensors and actuators in active vibration and vibroacoustic control systems of beam, plate and panelled structures with arbitrary (non-homogeneous) boundary conditions. A new form of description of a simple proportional active control system with multiple independent feedback loops is proposed. The modal sensitivity functions of sensors and the modal selectivity functions of actuators are introduced to describe their ability for sensing and exciting specific structural modes of the structures. Basing on the assumed form of cost function and the derived equations of control system the influence of the modal characteristics of transducers on the stability of the system and on the performance of the active control is analyzed. The results of analytical solutions and numerical simulations are compared with the results of the experiments carried out on various beam and plate structures made up of aluminium or composite materials including the actual materials used in aviation, proving usefulness of the presented approach.

Keywords:
Vibroacoustics, Piezoelectric transducers, Modal sensitivity, Vibroacoustic control systems

Abstract:
The paper describes briefly some main aspects of the active feedback control system that has been developed and constructed for reduction of vibroacustic emission of vibrating plate structures with arbitrary boundary conditions. Relations between the forms and frequency of the vibrations induced by an external harmonic excitation and the distribution of the generated acoustic pressure field are investigated using the developed numerical model based on the Indirect Variational Boundary Element Method. The aim of the control is to minimize the sound pressure level in a given point of the ambient space. The system uses small, rectangle-shaped piezoelectric transducers as both sensors and actuators. The transducers are connected in a number of independent feedback loops, and the feedback gains are the control parameters which are optimized using the developed optimal control algorithm. The constructed active system has been tested for the stability and control performance during experimental research performed in an anechoic chamber. Results of experiments are presented in the paper, proving a high level of noise reduction and a good agreement with numerical predictions.

Keywords:
Vibrating plate structures, Active feedback control system, Vibrational modes

Abstract:
Active sandwich panels are an example of smart noise attenuators and a realization of hybrid active-passive approach for the problem of broadband noise reduction. The panels are composed of thin elastic faceplates linked by the core of a lightweight absorbent material of high porosity. Moreover, they are active, so piezoelectric actuators in the form of thin patches are fixed to their faceplates. Therefore, the passive absorbent properties of porous core, effective at high and medium frequencies, can be combined with the active vibroacoustic reduction necessary in a low frequency range. Important convergence issues for fully coupled finite-element modeling of such panels are investigated on a model of a disk-shaped panel under a uniform acoustic load by plane harmonic waves, with respect to the important parameter of the total reduction of acoustic transmission. Various physical phenomena are considered, namely, the wave propagation in a porous medium, the vibrations of elastic plate and the piezoelectric behavior of actuators, the acoustics-structure interaction and the wave propagation in a fluid. The modeling of porous core requires the usage of the advanced biphasic model of poroelasticity, because the vibrations of the skeleton of porous core cannot be neglected; they are in fact induced by the vibrations of the faceplates. Finally, optimal voltage amplitudes for the electric signals used in active reduction, with respect to the relative size of the piezoelectric actuator, are computed in some lower-to-medium frequency range.

Keywords:
Active sandwich panels, Multiphysics, Vibroacoustics, Poroelasticity, Piezoelectricity

Abstract:
The paper presents results of the theoretical and numerical investigation on acoustic radiation of vibrating plate structures submerged in water. The current state of the art on the considered issues is briefly reviewed. Then, the method for determining eigenmode shape functions and eigenfrequencies of plate vibrating in water that has been used in presented study is introduced. The constitutive equations for solid domain and the pressure acoustic equation for liquid domain are coupled via boundary conditions and solved numerically using the finite element method. Structural mode shapes and eigenfrequencies computed for plate submerged in water are compared to analogous results obtained for air and for the in vacuo case. It is ssumed, that the plate is rectangle shaped and that it is placed in an infinite rigid baffle. Three-dimensional near- and far-field acoustic radiation characteristics for the plate vibrating in water are introduced. Possibilities of implementation of the active control system for reduction of the hydroacoustic emission are briefly discussed.

Keywords:
Hydroacoustics, Plate structures, Hydrocoustic radiation

Abstract:
The paper presents numerical analysis – involving an advanced multiphysics modeling – of the concept of active porous composite sound absorbers. Such absorbers should be made up of a layer or layers of poroelastic material (porous foams) with embedded elastic inclusions having active (piezoelectric) elements. The purpose of such active composite material is to significantly absorb the energy of acoustic waves in a wide frequency range, particularly, at lower frequencies. At the same time the total thickness of composite should be very moderate. The active parts of composites are used to adapt the absorbing properties of porous layers to different noise conditions by affecting the so-called solid-borne wave – originating mainly from the vibrations of elastic skeleton of porous medium – to counteract the fluid-borne wave – resulting mainly from the vibrations of air in the pores; both waves are strongly coupled, especially, at lower frequencies. In fact, since the traction between the air and the solid frame of porous medium is the main absorption mechanism, the elastic skeleton is actively vibrated in order to adapt and improve the dissipative interaction of the skeleton and air in the pores. Passive and active performance of such absorbers is analyzed to test the feasibility of this approach.

Keywords:
Poroelasticity, Piezoelectricity, Active composites, Sound absorption, Finite-element modelling

Abstract:
This paper presents a fully coupled multiphysics modeling and experimental validation of the problem of active reduction of noise generated by a thin plate under forced vibration. The plate is excited in order to generate a significant low-frequency noise, which is then reduced by actuators in the form of piezoelectric patches glued to the plate with epoxy resin in locations singled out earlier during finite element (FE) analyses. To this end, a fully coupled FE system relevant for the problem is derived. The modeling is very accurate: The piezoelectric patches are modeled according to the electromechanical theory of piezoelectricity, the layers of epoxy resin are thoroughly considered, and the acoustic-structure interaction involves modeling of a surrounding sphere of air with the nonreflective boundary conditions applied in order to simulate the conditions found in anechoic chamber. The FE simulation is compared with many experimental results. The sound pressure levels computed in points at different distances from the plate agree excellently with the noise measured in these points. Similarly, the computed voltage amplitudes of controlling signal turn out to be very good estimations.

Keywords:
Low-frequency noise reduction, Multiphysics modeling, Active structural acoustic control

Abstract:
The paper discusses theoretical fundamentals necessary for accurate vibroacoustical modeling of structures or composites made up of poroelastic, elastic, and (active) piezoelectric materials, immersed in an acoustic medium (e.g. air). An accurate modeling of such hybrid active-passive vibroacoustic attenuators (absorbers or insulators) requires a multiphysics approach involving the finite element method to cope with complex geometries. Such fully-coupled, multiphysics model is given in this paper. To this end, first, the accurate PDE-based models of all the involved single-physics problems are recalled and, since a mutual interaction of these various problems is of the uttermost importance, the relevant couplings are thoroughly investigated and taken into account in the modeling. Eventually, the Galerkin finite element model is developed. This model should serve to develop designs of active composite vibroacoustic attenuators made up of porous foams with passive and active solid implants, or hybrid liners and panels made up of a core or layers of porous materials fixed to elastic faceplates with piezoelectric actuators, and coupled to air-gaps. A widespread design of such smart muffler s is still an open topic and should be addressed with accurate predictive tools based on the model proposed in the present paper. The model is accurate in the framework of kinematical and constitutive (material) linearity of behaviour. This is, however, the very case of the vibroacoustic application of elasto-poroelastic panels or composites, where the structural vibrations are induced by acoustic waves. The developed fully-coupled FE model is finally used to solve a generic two- dimensional example and some issues concerning finite element approximation and convergence are also discussed.

Keywords:
Poroelasticity, Piezoelectricity, Acoustics, Multiphysics, Weak formulation, Finite-element method

Abstract:
The article presents results of the acoustic measurements on open-cell porous media coated with a magnetorheological (MR) fluid. Sound absorption of polyurethane foams of different, single and dual porosity was tested in the impedance tube. The measurements were conducted in three stages using clean samples, the same samples moistened with MR fluid, and finally, exposing the MR fluid-coated samples to a constant magnetic field. The transfer function method was employed to determine the acoustic absorption coefficient. Two significant, controllable effects were observed in the curve illustrating the variation of the acoustic absorption coefficient with frequency, especially, for the foams of dual porosity. Namely, relative to the field-free conditions, or to the clean foams, the most substantial peak in the absorption curve could be shifted by applying a magnetic field. Moreover, a resulting significant increase in acoustic absorption yields, in a wide frequency range directly behind the peak.

Keywords:
Magnetorheological foams, Acoustic absorption, Adaptive sound insulator, Dual-porosity foams

Abstract:
This paper presents a complete finite-element description of a hybrid passive/active sound package concept for acoustic insulation. The sandwich created includes a poroelastic core and piezoelectric patches to ensure high panel performance over the medium/high and low frequencies, respectively. All layers are modelled thanks to a Comsol environment*. The piezoelectric/elastic and poroelastic/elastic coupling are fully considered. The study highlights the reliability of the model by comparing results with those obtained from the Ansys finite-element software and with analytical developments. The chosen shape functions and mesh convergence rate for each layer are discussed in terms of dynamic behaviour. Several layer configurations are then tested, with the aim of designing the panel and its hybrid functionality in an optimal manner. The differences in frequency responses are discussed from a physical perspective. Lastly, an initial experimental test shows the concept to be promising.

Keywords:
Poroelasticity, Piezoelectricity, Finite-element modelling, Acoustic insulation, Active-passive approach

Abstract:
Materials with open porosity are known to absorb sound very well. However, their efficiency in acoustic absorption and insulation is sometimes restricted to specific frequency ranges. It is possible to circumvent this drawback by designing a porous microstructure that can be modified on the fly and thereby enabling the change in its crucial geometrical parameters like tortuosity that influence the intensity of viscous energy dissipation phenomena taking place on a micro scale. A prototype of such a material consisting of relocatable small steel balls embedded in a periodic rigid skeleton is devised and additively manufactured in separate pieces in the stereolithography technology. The balls are inserted into proper places manually. The full sample is then assembled and its acoustic characteristics are determined computationally and experimentally using dual-scale, unit-cell analyses and impedance tube measurements, respectively. The resulting material is shown to possess two extreme spectra of normal incidence sound absorption coefficient and transmission loss that are dependent on the particular position of balls inside the microstructure. In consequence, acoustic waves from a much larger frequency range can be effectively absorbed or insulated by a relatively thin material layer compared to a similar design without movable balls.

Keywords:
sound absorption, sound transmission, modifiable porous microstructure, additive manufacturing

Abstract:
The low frequency peaks in the absorption spectra of layers of conventional porous materials correspond to quarter wavelength resonances and the peak frequencies are determined essentially by layer thickness. If the layer cannot be made thicker, the frequency of the peak can be lowered by increasing the tortuosity of the material. Modern additive manufacturing technologies enable exploration of pore network designs that have high tortuosity. This paper reports analytical models for pore structures consisting of geometrically complex labyrinthine networks of narrow slits resembling Greek meander patterns. These networks offer extremely high tortuosity in a non-porous solid skeleton. However, additional enhancement of the low frequency performance results from exploiting the dual porosity pressure diffusion effect by making the skeleton microporous with a significantly lower permeability than the tortuous network of slits. Analytical predictions are in good agreement with measurements made on two samples with the same tortuous slit pattern, but one has an impermeable skeleton 3D printed from a photopolymer resin and the other has a microporous skeleton 3D printed from a gypsum powder.

Keywords:
sound absorption, high tortuosity, dual porosity, 3D printed materials

Abstract:
This paper shows that specific additive manufacturing (AM) technology can be used to produce double-porosity acoustic materials where main pore networks are designed and a useful type of microporosity is obtained as a side effect of the 3D printing process. Here, the designed main pore network is in the form of annular pores set around the axis of the cylindrical absorber. In this way, the axial symmetry of the problem is ensured if only plane wave propagation under normal incidence is considered, which allows for modelling with purely analytical expressions. Moreover, the outermost annular pore is bounded by the wall of the impedance tube used to measure the sound absorption of the material, so that experimental tests can be easily performed. Two different AM technologies and raw materials were used to fabricate axisymmetric absorbers of the same design, in one case obtaining a material with double porosity, which was confirmed by the results of multi-scale calculations validated with acoustic measurements.

Abstract:
Low-frequency sound absorption by thin rigid porous hard-backed layers is enhanced if the geometrical tortuosity is increased. Increasing tortuosity increases the fluid flow path length through the porous layer thereby increasing the effective thickness. In turn, this reduces the effective sound speed within the layer and the frequency of the quarter wavelength layer resonance. One way of increasing tortuosity is through rectangular labyrinthine channel perforations. In addition to the tortuosity of the porous matrix, the bulk tortuosity value is influenced by the channel widths, lengths, and number of folds. A sample with an impervious skeleton and a sample in which the solid skeleton is perforated with oblique cylindrical holes evenly spaced in a rectangular pattern have been fabricated using conventional methods and an additive manufacturing technology, respectively. The sound absorption spectra resulting from these structures have been predicted analytically as well as numerically and compared with normal incidence impedance-tube measurements.

Abstract:
L’objectif de cette étude est de modéliser et de caractériser le comportement de matériaux réalisés en impression 3D lorsqu’ils sont placés en paroi d’un conduit. Le matériau considéré présente une structure périodique dont la cellule de base comporte une sphère reliée aux sphères des autres cellules par des canaux cylindriques. Le squelette rigide du matériau permet de le modéliser comme un Fluide Équivalent. Quand le matériau est placé en paroi de conduit, la modélisation par son impédance de surface n’est plus suffisante et la propagation dans le matériau doit être prise en compte. Trois modélisations du matériau sont étudiées. Les deux premières s’appuient sur une description macroscopique au moyen d’un Fluide Équivalent. Dans la première, il est décrit par ses fonctions caractéristiques dynamiques (densité et compressibilité), calculées au moyen d’un modèle numérique d’un tube de Kundt. Dans la seconde modélisation, les paramètres du modèle JCALP sont déduits par résolution des équations de Stokes, Laplace et Poisson pour une seule cellule du matériau. Le troisième modèle consiste à décrire le matériau dans sa globalité à l’échelle microscopique et à résoudre les équations de Navier-Stokes Linéarisées (NSL) dans le conduit et le matériau. Les résultats des trois modèles sont comparés en incidence normale et en paroi d’un conduit. Différentes techniques d’impression 3D ont été utilisées pour réaliser des échantillons, et montrent une variabilité importante des géométries effectivement réalisées et par suite des coefficients d’absorption mesurés en tube de Kundt. Les résultats d’expérimentations en paroi de conduit sont également comparés avec ceux de la modélisation.

Abstract:
The paper shows that acoustic materials with double porosity can be 3D printed with the appropriate design of the main pore network and the contrasted microporous skeleton. The microporous structure is obtained through the use of appropriate additive manufacturing (AM) technology, raw material, and process parameters. The essential properties of the microporous material obtained in this way are investigated experimentally. Two AM technologies are used to 3D print acoustic samples with the same periodic network of main pores: one provides a microporous skeleton leading to double porosity, while the other provides a single-porosity material. The sound absorption for each acoustic material is determined both experimentally using impedance tube measurements and numerically using a multiscale model. The model combines finite element calculations (on periodic representative elementary volumes) with scaling functions and analytical expressions resulting from homogenization. The obtained double-porosity material is shown to exhibit a strong permeability contrast resulting in a pressure diffusion effect, which fundamentally changes the nature of the sound absorption compared to its single-porosity counterpart with an impermeable skeleton. This work opens up interesting perspectives for the use of popular, low-cost AM technologies to produce efficient sound absorbing materials.

Abstract:
Sound absorption of polydisperse heterogeneous porous composites is investigated in this paper. The wave equation in polydisperse heterogeneous porous composites is upscaled by using the two-scale method of homogenisation, which allows the material to be modeled as an equivalent fluid with atypical effective parameters. This upscaled model is numerically validated and demonstrates that the dissipation of sound in polydisperse heterogeneous porous composites is due to visco-thermal dissipation in the composite constituents and multiple pressure diffusion in the polydisperse heterogeneous inclusions. Analytical and semi-analytical models are developed for the acoustical effective parameters of polydisperse heterogeneous porous composites with canonical geometry (e.g. porous matrix with cylindrical and spherical inclusions) and with complex geometries. Furthermore, by comparing the sound absorption coefficient of a hard-backed composite layer with that of layers made from the composite constituents alone, it is demonstrated that embedding polydisperse heterogeneous inclusions in a porous matrix can provide a practical way for significantly increasing low frequency sound absorption. The results of this work are expected to serve as a model for the rational design of novel acoustic materials with enhanced sound absorption properties.

Abstract:
Designs with uniformly distributed slits normal or inclined to the incident surface exhibit a great potential because of their simplicity and good acoustical performance. However, production of materials of this sort is challenging as the required fabrication precision is very high. This paper deals with additive manufacturing, modeling, and impedance tube testing of a few slitted geometries and their variations, including cases where the dividing walls between slits are perforated. They were designed to be producible with current 3D printing technology and provide reliable measurements using standardized equipment. The normal incidence sound absorption curves predicted analytically and numerically were verified experimentally. It is observed that such simple configurations may lead to absorption properties comparable to porous acoustic treatments with more complex microstructure. The good agreement between the predictions and measurements supports the validity of the multi-scale modeling employed.

Abstract:
A technique commonly used for improving room acoustics consists in increasing a total sound damping in a room. This objective can be achieved by using different configurations of a porous material for acoustical treatment of a room. In this work, that problem is analyzed theoretically by exploiting a modal representation of the impulse response (IR) function for steady-state sound field predictions. A formula for the IR function was obtained by solving a wave equation for an enclosure with complex-valued boundary conditions of walls. On the walls where the acoustic treatment is applied, these boundary conditions are related to the characteristic impedance, effective speed of sound and thickness of the porous material used for padding. Two different porous materials were considered in the analyses of the room with acoustic treatment, and to this end, the required effective properties were calculated for a rigid foam with a designed periodic microstructure, as well as for a poroelastic foam with specific visco-elastic properties of the skeleton.

Abstract:
This work presents a three-scale approach to modelling sound absorbing structures with non-uniform porosity, consisting of meso-patterns of localised micro-porosity. It can also be used for structures in which voids in a solid frame are filled with micro-fibres. The method involves double-scale, i.e. micro- and meso-scale, calculations of the effective properties of an equivalent homogenised medium, as well as macro-scale calculations of sound propagation and absorption in this medium, which at the macroscopic level can replace the entire absorbing structure of complex micro-geometry. The basic idea can be explained as follows: the mesoscale areas with localised micro-porosity are treated as homogenised meso-pores saturated with an equivalent visco-thermal fluid replacing the actual gas-saturated micro-porous medium, so that the macroscopic effective properties are finally calculated based on a simplified meso-scale geometry with homogenised mesopores.

Abstract:
This paper discusses the potential of meta-poro-elastic systems with small mass inclusions to create broadband sound absorption performance under the quarter-wavelength limit. A first feasibility study is done to evaluate whether embedding small mass inclusions in specific types of foam can lead to near-perfect absorption at tuned frequencies. This paper includes an optimization routine to find the material properties that maximize the losses due to the mass inclusion such that a near-perfect/perfect absorption coefficient can be achieved at specified frequencies. The near-perfect absorption is due to the mass-spring effect, which leads to an increase in the viscous loss. Therefore, it is efficient in the viscous regime. The well-known critical frequency, which depends on the porosity and flow resistivity of the material, is commonly used as a criteria to distinguish the viscous regime from the inertial regime. However, for the types of foam of interest to this work, the value of critical frequency is below the mass-spring resonance frequency. Hence, the inverse quality factor is used to provides a more accurate estimation on the frequency at which the transition from the viscous regime to the inertial regime.

Abstract:
Despite very good mechanical and physical properties such as lightness, rigidity and high thermal conductivity, closed-porosity metal foams alone are usually poor acoustic treatments. However, relatively low production cost weighs them in many applications in favour of their open-cell equivalents. In the present paper, this attractive and popular material is subject to consideration from the point of view of the improvement of its sound absorption characteristics. A classic method of perforation is proposed to open the porous interior of the medium to the penetration of acoustic waves and therefore enhance the dissipation of their energy. The interaction between the perforation diameter and closed-cell microstructure as well as its impact on the overall sound absorption of a similar foam were already studied in 2010 by Chevillotte, Perrot and Panneton, so these topics are not discussed much in this work. On the other hand, the objective here is to investigate if one can efficiently approximate the wave propagation phenomenon in real perforated heterogeneous materials with closed porosity of irregular shape by means of their simplified three-dimensional representation at the micro-level. The applied multi-scale modelling of sound absorption was confronted with measurements performed in an impedance tube. Moreover, as expected, numerical and experimental comparisons with relevant perforated solid samples show great benefit coming from the presence of a porous structure in the foam, although it was initially closed.

Abstract:
The paper presents numerical studies of wave propagation under normal and oblique incidence in sound-absorbing layers of poroelastic composites with active and passive inclusions embedded periodically along the composite layer surface. The purpose of active inclusions is to increase the mass-spring effect of passive inclusions attached to the viscoelastic skeleton of the poroelastic matrix of the composite in order to increase the dissipation of the energy of acoustic waves penetrating into such a layer of poroelastic composite. Finite element modelling is applied which includes the coupled models of Biot-Allard poroelasticity (for the poroelastic matrix), piezoelectricity and elastodynamics (for the active and passive inclusions, respectively), as well as the Helmholtz equation for the adjacent layer of air. The formulation based on the Floquet-Bloch theory is applied to allow for modelling of wave propagation at oblique incidence to the surface of the periodic composite layer. The actively exited piezoelectric inclusions may become additional (though secondary) sources for wave propagation. Therefore, a background pressure field in a wide adjacent air layer is used to simulate plane waves propagating from the specified direction, oblique or normal, onto the poroelastic layer surface, and a nonreflecting condition is applied on the external boundary of the air layer.

Abstract:
Recent investigations show that the normal incidence sound absorption in 3D-printed rigid porous materials is eminently underestimated by numerical calculations using standard models. In this paper a universal amendment to the existing mathematical description of thermal dispersion and fluid flow inside rigid foams is proposed which takes account of the impact of the additive manufacturing technology on the acoustic properties of produced samples. The porous material with a motionless skeleton is conceptually substituted by an equivalent fluid with effective properties evaluated from the Johnson-Champoux-Allard-Pride-Lafarge model. The required macroscopic transport parameters are computed from the microstructural solutions using the hybrid approach. A cross-functional examination of the quality (shape consistency, representative surface roughness, etc.) of two periodic specimens obtained from additive manufacturing processes is additionally performed in order to link it to the results of the multiscale acoustic modelling. Based on this study, some of the transport parameters are changed depending on certain quantities reflecting the actual quality of a fabricated material. The developed correction has a considerable influence on the predicted value of the sound absorption coefficient such that the original discrepancies between experimental and numerical curves are significantly diminished.

Keywords:
Rigid porous material, Additive manufacturing, Sound absorption

Abstract:
With a rapid development of modern Additive Manufacturing Technologies it seems inevitable that they will sooner or later serve for production of specific porous and meta-porous acoustic treatments. Moreover, these new technologies are already being used to manufacture original micro-geometric designs of sound absorbing media in order to test microstructure-based effects, models and hypothesis. In the view of these statements, this work reports differences in acoustic absorption measured for porous specimens which were produced from the same CAD-geometry model using several additive manufacturing technologies and 3D-printers. A specific periodic unit cell of open porosity was designed for the purpose. The samples were measured acoustically in the impedance tube and also subjected to a thorough microscopic survey in order to check their quality and look for the discrepancy reasons.

Keywords:
Sound absorption, Additive Manufacturing Technologies

Abstract:
Samples with periodic microstructures, designed for good sound absorption, have been manufactured by 3D printing. Typically, however, the acoustical properties of the resulting samples differ from those predicted. Two causes of the discrepancies are (1) inaccuracies related to the 3D-printing resolution and (2) imperfections resulting from micro-fibres, micro-pores, and pore surface roughness, created during manufacture. Discrepancies due to the first cause can be addressed, post hoc, by updating the idealised periodic geometric model used for creating the codes for fabrication on the basis of a survey using a scanning microscope, or through computerised micro-tomography scans. Reducing the discrepancies due to the second cause requires a relatively significant further modelling effort. Another cause for small discrepancies is when two layers of the same periodic porous material and thickness differ only by a relative shift of the internal geometry of the periodic Representative Volume Element (RVE). This causes the absorption peaks to be shifted in frequency. A modelling procedure is proposed to take this into account.

Keywords:
Sound absorption, Periodic porous media, Additive manufacturing

Abstract:
In this paper the possibility of enhancing the absorption coefficient of a poro-elastic material using small, elastic mass inclusions in frequencies lower than the quarter-wavelength resonance of the porous material is discussed. We show that absorption peaks can be achieved not only by what is known in literature as the trapped mode effect, but also by the resonance of small elastic inclusions at low frequencies, which can be interpreted as a mass-spring effect. In this work, the inclusion and the porous skeleton is considered elastic and fully coupled to each other, therefore accounting for all types of energy dissipation i.e. viscous, thermal, and structural losses and energy dissipated due to the relative motion of the fluid phase and the frame excited by the resonating inclusion. Additionally, the inclusions are also modeled as motionless and rigid to distinguish between the trapped mode and/or the modified frame mode effect and the mass-spring effect. Moreover, the distinction between these two effects are explained in more detail by comparing the dissipated energy by each mechanism (viscous, thermal and structural effect).

Keywords:
Meta-porous material, Biot-Allard poroelastic model, Mass-spring effect

Abstract:
When airborne acoustic waves penetrate porous media their carrier becomes the air in pores, but also the solid skeleton - provided that it is sufficiently soft. Then, there is a coupled propagation of fluid-borne and solid-borne waves in a poroelastic medium. The coupling of fluid and solid phases of such media can be responsible for significantly better or weaker sound absorption in medium and lower frequency ranges. It has been observed that adding some well-localised small mass inclusions inside a poroelastic layer may improve its acoustic absorption in some medium frequency range, however, at the same time the absorption is usually decreased at some slightly higher frequencies. This situation can be improved by applying additionally an active approach using small piezoelectric inclusions which actively influence the vibrations of the solid skeleton with added masses, so that the interaction between the solid-borne and fluid-borne waves is always directed for a better mutual energy dissipation of the both types of waves.

Abstract:
The purpose of this work is to present and investigate the concept of adaptive sound absorbers, that is, periodic porous media with modifiable micro-geometry, so that their ability of sound absorption or insulation can be changed in various frequency ranges. To demonstrate this concept, a simple periodic porous micro-geometry with small bearing balls inside pores is proposed. By a simple positioning of the periodic porous sample the gravity force is used for the small balls to close some of the windows linking the pores, changing in that way the flow path inside pores, which entails significant modifications of the relevant parameters of permeability and tortuosity. Also the viscous characteristic length is changed, while the porosity as well as the thermal characteristic length remain unchanged. Nevertheless, such significant changes of some crucial transport parameters strongly affect the overall acoustic wave propagation in the porous medium. All this is studied using an advanced dual-scale modelling as well as experimental testing of 3D-printed specimens.

Abstract:
Numerical simulation of absorption of sound in porous media is an important part of the design of the treatments for the environmental noise reduction. In the porous media, the mechanical energy carried by sound is dissipated by thermo-viscous interactions with the solid surface of the media frame, which usually has complicated geometry at the microscopic (sub-millimetre) scale. In order to be able to absorb the acoustic energy at the low frequencies of interest, a layer of porous material must be rather thick (at the order of centimetres). This is why direct numerical simulation (DNS) of the sound absorption in porous media is a rather computationally challenging task because small geometrical details must be properly resolved in a large computational domain. In order to avoid these difficulties, simplified semi-phenomenological models introducing so called effective fluid have been proposed. For example, the Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model is based on eight parameters which can be measured or calculated based on the media micro-structural geometry. Within this work, we compare the numerical results obtained by the 3D DNS with the prediction of the JCAPL model in case of several porous media represented by closely-packed spheres. The DNS calculations are performed using the linearised Navier-Stokes equations for layers of spheres of different thicknesses, the parameters for the JCAPL model are calculated subsequently using Laplace, Poisson, and Stokes flow analyses on a representative volume element of the media. Very good agreement between the results has been found.

Abstract:
Microstructure-based modelling of sound absorbing porous media has been recently successfully applied for various materials, however, still some questions concerning the reliability and accuracy of such predictions are open. These issues are investigated here for periodic foams with open porosity. First, a geometry of foam microstructure is generated using an algorithm which ensures periodic arrangements of pores in a cube. Then, the cube is appropriately scaled to various sizes and for each size case finite-element analyses are performed on the periodic fluid domain to calculate the so-called transport parameters. Finally, the effective speed of sound and density are determined for the so-called equivalent fluid, macroscopically suitable to describe wave propagation in such an open rigid foam filled with air. All this allows to estimate the sound absorption for periodic foam layers of various pore-sizes and thicknesses. This parametric study is confronted with some impedance-tube measurements carried out for a few foam samples produced using 3D-printing technology.

Keywords:
Sound absorbing foams, Microstructure, Micro-macro modelling, Acoustic testing, 3D printing

Abstract:
Realistic microstructure-based calculations have recently become an important tool for a performance prediction of sound absorbing porous media, seemingly suitable also for a design and optimization of novel acoustic materials. However, the accuracy of such calculations strongly depends on a correct choice of the representative microstructural geometry of porous media, and that choice is constrained by some requirements, like, the periodicity, a relative simplicity, and the size small enough to allow for the so-called separation of scales. This paper discusses some issues concerning this important matter of the representativeness of representative geometries (two-dimensional cells or three-dimensional volume elements) for sound absorbing porous and fibrous media with rigid frame. To this end, the accuracy of two- and three-dimensional cells for fibrous materials is compared, and the microstructure-based predictions of sound absorption are validated experimentally in case of a fibrous material made up of a copper wire. Similarly, the numerical predictions of sound absorption obtained from some regular Representative Volume Elements proposed for porous media made up of loosely-packed identical rigid spheres are confronted with the corresponding analytical estimations and experimental results. Finally, a method for controlled random generation of representative microstructural geometries for sound absorbing open foams with spherical pores is briefly presented.

Keywords:
Fibrous materials, Open-cell foams, Representative microstructure, Modelling of sound absorption

Abstract:
A methodology of inverse identification of parameters for the Johnson-Champoux-Allard-Lafarge model of porous sound absorbing materials (also with Pride and Lafarge enhancements) is advocated. The inverse identification is based on the measurements of surface acoustic impedance of porous samples. For a single sample of porous material set on a rigid backing wall such measurements provide two specific curves in the considered frequency range, namely, the real and imaginary parts of acoustic impedance. More data suitable for inverse identification can be gathered from additional measurements where the surface acoustic impedance is determined for the same sample yet with an air gap between the sample and the backing wall. As matter of fact, such measurements should be carried out for a few cases where the air gap varies in thickness. Eventually, a set of impedance curves is gained suitable for inverse simultaneous identification of model parameters. In the paper analytical solutions are given for both measurement configurations, namely, for a layer of porous material set on the rigid wall, and for the porous layer separated from the rigid wall by an air gap. These solutions are used by the identification procedure which minimises the difference between the experimental curves and the curves computed from the analytical solutions where the porous layer is modelled using some version of the mentioned poro-acoustic model. The minimisation is carried out with respect to the model parameters, however, not directly, since for this purpose the corresponding dimensionless parameters are introduced. Formulas for the dimensionless parameters are given with respect to the model parameters, and then conversely, for the model parameters with respect to the dimensionless ones. In the formulas two normalising frequencies are introduced which can be considered: one - as characteristic for viscous effects, and the other - as typical for thermal effects. It is claimed that they are not additional parameters, and can be set quite arbitrarily, however, reasonable values must be assumed to allow for very fast and robust identification with initial values for all dimensionless parameters set to 1. This feature is quite important in view of the fact that the choice of initial values for the actual model parameters is rather essential and can be often very problematic. The whole procedure is illustrated with a numerical example and by tests based on laboratory measurements of porous ceramic samples.

Keywords:
Sound absorbing porous media, Inverse identification, Acoustic impedance, Acoustic testing

Abstract:
Sound absorption in porous media with rigid structure and open porosity is most often modelled using the so-called fluid-equivalent approach, in which a porous medium is substituted by an effective dispersive fluid. There are many models of that kind. Perhaps the most frequently used and efficient one is the so-called Johnson-Champoux-Allard-Lafrage model, or its variations. This a rather advanced semi-phenomenological model with six to eight parameters; with enhancements by Pride and Lafarge, there are eight parameters, namely: the total open porosity, the tortuosity, the (viscous) permeability, the thermal permeability, the viscous and thermal characteristic lengths, and finally, the viscous and thermal tortuosities at low frequency limit (i.e., at 0 Hz). Although, most of these parameters can be measured, it is sometimes very problematic and requires various experimental facilities, which makes the idea of calculation of these parameters from the geometry of microstructure of porous medium very tempting – such multiscale modelling requires, however, some periodic yet sufficiently realistic representation of the actual porous geometry. In this paper such multiscale modelling is presented for the problem of sound absorption in layers composed of loosely-packed rigid spheres. Since the spheres are identical, the packing, although not dense, tends to be semi-regular. Therefore, some regular sphere packings are used to construct periodic Representative Volume Elements for such porous media – they are, however, modified a bit by shifting the spheres in order to fit exactly the actual measured porosity. Basing on such numerical representations of porous microstructure, all the necessary parameters are calculated from finite-element solutions of some relevant Boundary-Value Problems and the effective characteristics for equivalent fluid are determined. Then, the acoustic absorption coefficients are computed for a porous layer of specified thickness for some wide frequency range and the results are compared with the experimental curve obtained from the measurements of such layer carried out in the impedance tube using the so-called two-microphone transfer function method.

Keywords:
Granular media, Sound absorption, Multiscale modelling, Acoustic measurements

Abstract:
The micro-geometry of porous material is responsible for its sound absorption performance and should be now a design objective. Microstructural calculation of parameters and/or characteristic functions for acoustical models of porous materials with rigid frame requires the so-called Representative Volume Elements, that is, usually cubes which should contain several pores or fibres of typical sizes and distribution. The design of such RVEs, which correctly represent a typical micro-geometry of porous medium, is by no means an easy task since usually the RVE should be also periodic and 'isotropic' (identical with respect to the three mutually-perpendicular directions). The task is simpler in case of two-dimensional microscopic models of some fibrous materials, but such modelling is obviously rather approximative. Designs of periodic RVEs for porous foams and fibrous materials will be presented and used by FE analyses of microstructural problems defined by the application of the Multiscale Asymptotic Method to the problem of sound propagation and absorption in porous media with rigid skeleton. Moreover, a methodology of automatic generation of periodic RVEs with random arrangement of pores based on a simple bubble dynamics will be explained. Among other examples, designs of RVE cubes representative for a corrundum ceramic foam with porosity 90% will be shown and serve for microstructural calculation of some macroscopic parameters used in advanced acoustical modelling of porous media. The curves of acoustic impedance and absorption measured in the frequency range from 500Hz to 6.4kHz for two samples of the corrundum foam will be presented. These measurements will be used for inverse identification of relevant macroscopic parameters, namely: the tortuosity, the viscous and thermal permeabilities, and two characteristic lengths. The concurrence of some results obtained by the RVE-based micro-scale calculation and the measurement-based macro-scale identification will be shown.

Keywords:
Rigid porous media, Microstructure-based calculations, Inverse identification, Sound propagation

Abstract:
Samples of porous ceramics Al2O3, manufactured by a promising technology of gelcasting of cellural foams by using biopolymers as gel-formers, are examined in the impedance tube using the transfer function method. It is shown that the ceramics of total porosity around 90% forms an excellent sound absorbing material in the frequency range from 500 Hz to 6.4 kHz. Experimentally-determined curves of acoustic impedance and absorption are then used for inverse identification of relevant geometric parameters like: tortuosity, viscous and thermal permeability parameters and characteristic lengths. These parameters are required by some advanced models of sound propagation in rigid porous media, developed by Johnson, Koplik and Dashen, Champoux and Allard, with some variations introduced by Pride et al., and Lafarge et al. These models are utilized to produce curves of acoustic impedance and absorption that are used by the identification procedure which minimizes the objective function defined as a squared difference to the appropriate curves obtained experimentally. As a matter of fact, some experimental data are used for the determination of parameters while the other data-obtained for another sample of the same porous ceramics, yet having different thickness-serve for the validation purposes. Moreover, it is observed that the identified characteristic length for thermal effects corresponds very well to the average radius of pores, whereas the characteristic length for viscous forces is similar with the average size of

Keywords:
Sound-absorbing foams, Inverse identification, Micro-scale calculations, Porous ceramics

Abstract:
Vibroacoustic control of thin beam, plate and panelled structures with arbitrary boundary conditions is investigated. The study focuses on determining optimal locations of piezoelectric sensors and actuators on the surfaces of structures under vibroacoustic control. The work consists of three parts. In the first part, the undertaken assumptions and some governing equations are briefly introduced. Then, in the second part of the study, the piezo-transducers' locations which ensure optimal sensing/actuating capabilities for specific vibration modes are determined, basing on the derived analytical formulas and on some results of numerical simulations, as well as on the actuator/sensor equations given in the first part of the study. The relevant modes are selected by taking into account that the main purpose is to minimise the acoustic field generated by the vibrating structure. The third part of the work discusses some experimental investigations aimed for the verification of the results obtained theoretically. Some technical aspects of creating the composite structures for active control systems are briefly described in appendix.

Keywords:
Vibroacoustic panels, Vibroacoustic control, Piezoelectric transducers, Optimal placement

Abstract:
The paper discusses some issues concerning fully-coupled finite-element modelling of active-passive systems for vibroacoustic attenuation, involving porous, piezoelectric, and elastic materials, as well as 'acoustic' (inviscid) fluids. For porous materials, the advanced, bi-phasic model of poroelasticity is used, which allows to consider elastic vibrations of solid skeleton important at lower frequencies and for porous composites with active inclusions. A discrete finite-element model suitable for analysis of such multiphysics problems is briefly explained. The model is derived (using the Galerkin method) from the variational formulation of coupled problems of poroelasticity, piezoelectricity, elasticity, and acoustics. Finally, some relevant results obtained from a numerical analysis of a disk of active sandwich panel with poroelastic core, fitted into an acoustic waveguide, are presented.

Keywords:
Acoustics, Porous media, Smart materials, Vibrations, Coupled fields, Finite element methods, Numerical analysis, Elasticity

Abstract:
W artykule przedstawiono wyniki badań nad aktywną redukcją transmisji wibroakustycznej przez strukturę płytową. Zakres przedstawionych w pracy zagadnień obejmuje zarówno opis teoretyczny rozpatrywanych zjawisk i układów, jak i rezultaty symulacji numerycznych oraz badań doświadczalnych. Rozważanym obiektem jest płyta aluminiowa o grubości 2mm z przyklejonymi na jednej z jej powierzchni elementami piezoelektrycznymi. Część tych elementów pełni funkcję sensorów, pozostałe zaś stanowią aktywatory, za pomocą których realizowane jest sterowanie aktywne. Kontroler pracuje w układzie sprzężenia zwrotnego, na jego wejście podawany jest wzmocniony i odwrócony w fazie sygnał napięciowy z sensorów, będący jednocześnie sygnałem błędu. Algorytm sterowania realizowany jest w oparciu o klasyczny regulator proporcjonalno-całkująco-różniczkujący (PID), dla różnych konfiguracji połączeń poszczególnych członów. Fizyczna realizacja kontrolera wykonana została w formie układu analogowego bazującego na niskoszumnych wzmacniaczach operacyjnych.

Keywords:
Aktywna kontrola wibroakustyczna, Regulator PID, Aktywna redukcja drgań

Abstract:
A fully-coupled multiphysics modelling is applied for the problem of simultaneous active and passive reduction of noise generated by a thin panel under forced vibration providing many relevant results of various type (noise and vibration levels, necessary voltage for control signals, efficiency of the approach) which are validated experimentally. The panel is excited in order to generate a noise consisting of significant lower and higher frequency contributions. Then the low-frequency noisy modes are reduced by actuators in the form of piezoelectric patches glued with epoxy resin in locations chosen optimally thanks to the multiphysics analysis, whereas the emission of higher frequency noise is attenuated by well-chosen thin layers of porous materials. To this end, a fully-coupled finite element system relevant for the problem is derived. Such multiphysics approach is accurate: advanced models of porous media are used for the porous layers, the piezoelectric patches are modelled according to the fully-coupled electro-mechanical theory of piezoelectricity, the layers of epoxy resin are thoroughly considered, finally, the acoustic-structure interaction involves modelling of a surrounding sphere of air with the non-reflective boundary conditions applied in order to simulate the conditions found in anechoic chamber. The FE simulation is compared with many experimental results. The sound pressure levels computed in points at different distances from the panel agree excellently with the noise measured in these points. Similarly, the computed voltage amplitudes of controlling signal turn out to be very estimations.

Keywords:
Structural acoustics and vibration, Active noise reduction, Poroelasticity, Sandwich panels

Abstract:
W zagadnieniach identyfikacji i lokalizacji źródeł aktywności wibroakustycznej maszyn, istotnym problemem jest wizualizacja pól rozkładu wielkości akustycznych na wybranych powierzchniach oraz określenie udziału poszczególnych źródeł w bilansie energetycznym sygnału wibroakustycznego maszyny.
Stosowane w wibroakustyce metody formowania wiązki (beamforming) polegają na przestrzenno-czasowym przetwarzaniu sygnału rejestrowanego przez matrycę mikrofonową. Identyfikacja źródła odbywa się poprzez analizę zależności amplitudowo-fazowych sygnałów akustycznych padających na poszczególne przetworniki matrycy. Ponieważ z metodologicznego punktu widzenia interesujące jest określenie możliwości zastosowania metody kształtowania wiązki w przypadku złożonych urządzeń posiadających źródła o małej aktywności wibroakustycznej, obiektem wstępnych prac był zasilacz hydrauliczny typu Silentflo firmy MTS. Rezultaty otrzymane w wyniku przeprowadzonych badań w pełni potwierdzają zalety metody beamformingu w określeniu lokalizacji i identyfikacji źródeł aktywności wibroakustycznej maszyn.

Keywords:
Wibroakustyka, Lokalizacja źródeł akustycznych, Beamforming

Abstract:
The paper presents a design, accurate multiphysics modeling and analysis of active porous-composite sound absorbers. Such absorbers are made up of a layer of poroelastic material (a porous foam) with embedded elastic implants having active (piezoelectric) elements. The purpose of such active composite material is to significantly absorb the energy of acoustic waves in a wide frequency range, particularly, in low frequencies. At the same time the total thickness of composites should be very moderate. The active parts of composites are used to adapt the absorbing properties of porous layers to different noise conditions by affecting the so-called solid-borne wave (originating mainly from the vibrations of elastic skeleton of porous medium) to counteract the fluid-borne wave (resulting mainly from the vibrations of air in the pores); the both waves are strongly coupled, especially, in lower frequencies. Passive and active performance of the absorbers is analysed to test the feasibility of this approach.

Keywords:
Poroelasticity, Piezoelectricity, Weak formulation, Acoustic insulation, Active-passive approach

Abstract:
The paper presents the modelling and frequency analysis of poroelastic layers with heavy solid implants where an improvement of acoustic absorption at lower frequencies is observed. To model the porous material the Biot’s theory of poroelasticity is used while the solid implants are modelled in two ways: first, as small subdomains of elastic material (steel) situated inside the porouslayer, and for the second time, in a more virtual manner (mathematically equivalent to the presence of masses in the given points), as some adequate inertial terms added directly to the weak (variational) formulation of the problem. Since the solid implants are very small the both ways give similar results. Obviously, the second approach is much more efficient to carry out numerical tests where the influence of the distribution of masses for the acoustic absorption of layers can be analysed. It seems that the improvement by distributed masses (implants) may be greater than the one due to the mass effect alone.

Keywords:
Poroelasticity, Weak formulation, Acoustic absorption

Abstract:
The paper presents further development in modeling of active elasto-poroelastic sandwich panels. In fact, a new design of a demi-sandwich panel is proposed and analysed. A numerical model of panel is implemented in COMSOL Multiphysics environment using the most fundamental but very flexible Weak Form PDE Mode. Various physical problems are modeled using Finite Element Method: the wave propagation in acoustic and poroelastic medium, the vibrations of elastic plate, the piezoelectric behavior of actuator. All these problems interact. in the examined application of active panel. The presented results of FE analysis and some analytical solutions prove the necessity of modeling the panel's interaction with an acoustic medium. Again, confirmed is the fact that an active control is necessary for lower resonances while for the higher frequencies the passive reduction of vibroacoustic transmission performed by a well-designed poroelastic layer is sufficient.

Keywords:
Active sandiwch panels, Poroelasticity, Piezoelectricity, Vibroacoustics

Abstract:
The paper addresses the issue of an active sandwich panel made of elastic faceplates and a poroelastic core. The panel is supposed to be active thanks to piezoelectric patches glued to the one of elastic layers. This piezoelectric actuator is used to excite the panel vibrations in the low frequency range with the aim to reduce the transmitted wave. A complete description of the sandwich behaviour is obtained using a finite element model implemented in FEMLAB environment. The poroelastic material is modeled using a recent formulation (by Atalla et al.) valid for harmonic oscillations, but the classical Biot formulation is also implemented. Coupling occurring between poroelastic material and plates, and between elastic plate and piezoelectric patches is fully considered. The achieved numerical model allows prediction of transmission coefficient for plane waves under normal incidence. Hence, some numerical experiments can be offered for multiple assembly configurations whose ultimate goal is to determine the best assembly and the best control strategy for reducing the transmission over a wide frequency range.

Keywords:
Poroelacticity, Piezoelectricity, Active vibroacoustic panles

Abstract:
Absorbing properties of multi-layer acoustic absorbers were modeled using the impedance translation theorem and the Garai and Pompoli empirical model, which enables a determination of the characteristic impedance and propagation constant of fibrous sound-absorbing materials. The theoretical model was applied to the computational study of performance of single-layer acoustic absorber backed by a hard wall and the absorber consisting of one layer of absorbing material and an air gap between the rear of the material and a hard back wall. Simulation results have shown that a high thickness of absorbing material may cause wavy changes in the frequency relationship of the normal and random incidence absorption coefficients. It was also found that this effect is particularly noticeable for acoustic absorbers with a large thickness of air gap between the absorbing material and a hard back wall.

Abstract:
A composite additively manufactured material for controlled sound absorption is proposed. The operation of the material is based on its changeable microgeometry with steel balls that modify propagation of acoustic waves when subject to an external magnetic field. Both numerical predictions and experimental verification is provided.

Keywords:
sound absorption, porous material, multiscale modelling, coupled problem

Abstract:
An acoustical characterisation of additively manufactured rigid slitted structures is considered. A set of six JCAL microstructural parameters is deduced from dynamic density and bulk modulus obtained from normal incidence surface acoustic impedance experimental data. The results show that the characteristic lengths are the most difficult to characterise.

Abstract:
This paper presents the multiscale approach to the problem of acoustic waves propagating in a fluid (air) inside rigid fibrous of porous materials with open porosity. The approach essentially consists of the finite element analyses of three relevant problems defined on the representative fluid domain of a porous medium, the averaging and up-scaling techniques applied to calculate some necessary parameters from the porous microstructure which are used by a model for the effective properties of a homogenized fluid equivalent to the porous medium, and finally, the solution of a relevant Helmholtz problem on the macro-scale level in order to estimate, for example, the acoustic absorption of the porous medium. In the paper, this approach is illustrated by two examples: experimentally validated analyses of a fibrous material made up of a copper wire based on two Representative Volume Elements, and an analysis of a foam with spherical pores using a randomly generated periodic representative cell.

Keywords:
Multiscale modelling, Acoustic waves, Porous media, Fibrous materials

Abstract:
Many of fibrous materials are very good sound absorbers, because the acoustic waves, which propagate in air and penetrate a fibrous layer, interact with the fibres so that the wave energy is dissipated. The dissipation is related to some viscous and thermal effects occurring on the micro-scale level. On the macroscopic level, a fibrous medium can be treated as an effective inviscid fluid, provided that the fibres are stiff. Such a fluid-equivalent approach allows to use the Helmholtz equation for the macroscopic description of sound propagation and absorption. It is applied by the advanced Johnson-Allard models, which require from 5 to 8 parameters related to the micro-geometry of fibrous microstructure. These are the so-called transport parameters in porous media: the open porosity and tortuosity, the permeability and its thermal analogue, two characteristic lengths (for viscous forces and thermal effects), etc. Moreover, some parameters for air (which fills the medium) are also necessary.

Keywords:
Fibrous materials, Sound absorption, Multiphysics modelling

Abstract:
The paper presents an approach for a morphological design of foams with spherical pores. It involves an algorithm for random generation of foam microstructure (controlled by some parameters), which is used to compute the transport parameters, and then, the effective speed of sound. Eventually, the sound absorption of such designed foam can be estimated.

Keywords:
Foams with spherical pores, Microstructure design, Sound absorption, Micro-macro modelling

Abstract:
Sound absorption in porous materials with rigid frame and open porosity can be very effectively estimated by applying the Johnson-Allard model in order to substitute a porous medium with an equivalent effective fluid and then utilise the Helmholtz equation for time-harmonic acoustics. The model uses several parameters which characterize the micro-geometry of porous material from the macroscopic perspective; they are: the total porosity, the viscous permeability and its thermal analogue, the tortuosity, and two characteristic lengths - one specific for viscous forces, the other for thermal effects. These parameters can be measured experimentally, however, recent computational powers allow to calculate them from the microstructure of porous medium provided that a good representation of usually very complex micro-geometry can be found. Inverse identification of these parameters is also possible.
The microstructural approach is based on the Multiscale Asymptotic Method and leads to two uncoupled micro-scale Boundary-Value Problems (BVPs). The first one is a harmonic, viscous, incompressible flow, with no-slip boundary conditions on the skeleton walls, driven in the specified direction by the pressure gradient of unit amplitude, uniform in the whole fluid domain. The second one is a harmonic thermal flow with isothermal boundary conditions on the skeleton walls and the uniform source of unit amplitude in the whole fluid domain. A scaled Laplace problem should also be solved in order to calculate some of the parameters. All BVPs must be solved using the same periodic cell representative for the microstructure.
COMSOL Multiphysics offers two extremely useful features which makes this numerical environment very suitable for such microstructure-based modelling of periodic media representative for porous materials; they are: the periodic boundary conditions and a very convenient possibility of implementation of new mathematical models, or modification of the implemented ones, using symbolic expressions. To illustrate this the acoustic absorption for a layer of freely packed assemblies of small rigid spheres was measured and compared with the result calculated from the microsctructural analyses using COMSOL Multiphysics. The FE analyses were carried out using periodic representative volume elements (RVEs) of regular sphere packings, for example, the so-called body-centered cubic (BCC) adjusted to match the actual porosity of 42%. The discrepancies between the numerical and experimental results - although not very big - suggest that better microstructural representations are necessary. Such RVEs may be constructed, for example, by using techniques for random generation of periodic representative volume elements which have been recently advocated by Zielinski.

Keywords:
Multiphysics modelling, Rigid porous media, Sound absorption

Abstract:
Active vibroacoustic control of beam and plate structures with arbitrary boundary conditions is considered. The goal is to develop a method of minimizing sound radiation efficiency of such structures. Primary sound field arise as a result of vibrations, due to external disturbances. It is assumed that the control system is compact - it does not contain any additional, ambient microphones. Piezoelectric transducers, mounted on the surface of the controled object, are used as sensors and actuators. Accurate numerical model of the considered structure is needed to determine optimal parameters of the control system. Theoretical background and the results of numerical and experimental research are briefly introduced.

Keywords:
Active vibroacoustic control, Plate structures, Beams, Piezoelectric transducers

Abstract:
There are several widely-used acoustic models of porous media, starting from that simple, purely phenomenological, model proposed by Delany and Bazely, and finishing with semi-phenomenological propositions of Johnson et al., combined with the ones of Champoux and Allard, with some important variations proposed by Pride, Lafarge, and others. All these models use some average macroscopic parameters, namely: the total porosity and flow resistivity (or permeability) - for the Delany-Bazely model - which are supplemented by the average tortuosity of pores and their characteristic dimensions - in the case of more advanced semi-phenomenological models. These models allow to describe the acoustic wave propagation in porous media in a wide frequency range, provided that the skeleton is rigid. However, using some formulas derived for these models with the Biot’s theory of poroelasticity permits to describe correctly sound propagation in soft porous materials. Thus, the determination of the above-mentioned parameters is very important. For direct, experimental measurements specialistic equipment is required, different for various parameters. Therefore, an inverse identification based on curves of, for example, acoustic impedance or absorption (measured for samples of known thickness) can be used to estimate the model parameters. In this work, it will be shown that knowledge of microstructural geometry of porous medium is very helpful to validate correct estimation. Moreover, a periodic microscopic cell consisting of a few pores representing an average morphology of porous ceramics is proposed to serve for numerical analyses to estimate permeability parameters. The concurrence of such micro-scale derivation and inverse identification is discussed.

Keywords:
Porous media, Micro-scale calculations, Inverse indentification, Sound waves

Abstract:
The paper presents an accurate multiphysics modeling and analysis of active porous-composite sound absorbers composed of a layer of poroelastic material (a porous foam) with embedded elastic implants having active (piezoelectric) elements. The purpose of such active composite material is to significantly absorb the energy of acoustic waves in a wide frequency range. At the same time the total thickness of composites should be very small. The active parts of composites are used to adapt the absorbing properties of porous layers to different noise conditions by affecting the so-called solid-borne wave (originating mainly from the vibrations of elastic skeleton of porous medium) to counteract the fluid-borne wave (resulting mainly from the vibrations of air in the pores); the both waves are strongly coupled, especially, in lower frequencies. Passive and active performance of the absorbers is analysed to test the feasibility of this approach. Since the absorption should be actively improved by affecting the vibrations of the elastic skeleton of porous layers, it is apparent that the rigid-frame modelling cannot be used here. Instead, the advanced biphasic theory of poroelasticity must be used to model porous material of the active absorbers.

Keywords:
Poroelasticity, Active piezoelectric inclusions, Smart materials, Acoustic absorption