Adaptation of the equivalent-fluid model to the additively manufactured
acoustic porous materials
Kamil C. OPIELA(1) , Tomasz G. ZIELIŃSKI(1)(1) Institute of Fundamental Technological Research of the Polish Academy of Sciences, Poland, tzielins@ippt.pan.pl

ABSTRACT
Recent investigations show that the normal incidence sound absorption in 3D-printed rigid porous materials iseminently 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 propertiesof 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 inorder 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 coefficientsuch that the original discrepancies between experimental and numerical curves are significantly diminished.
Keywords: rigid porous material, additive manufacturing, sound absorption

1.

INTRODUCTION

It is popularly known that porous materials are effective acoustic insulators. They possess superior damping
properties due to a specific internal structure consisting of a solid skeleton and voids (pores) filled with fluid.
By modifying the spatial distribution and geometry of voids, among others, one can directly influence theoverall acoustic performance of a porous layer. Thus, knowing the correspondence between the design and therelevant characteristics of such media, it is possible to develop a particular configuration of a required behavior. Scientists have excogitated a multiscale theory to effectively predict the acoustic wave propagation andsound absorption in rigid porous materials using microstructure-based solutions for various cases, namely: foropen-cell aluminium foams [13, 14], perforated closed-cell metallic foams [2], ceramic foams with sphericalpores [17], polymeric foams [4, 5, 10, 11], double-porosity foams [3], syntactic hybrid foams (i.e., open-cell
polyurethane foams with embedded hollow microbeads) [12], 3D-printed foams [18], granular [6, 7, 16] andfibrous media [8, 15, 19]. Nevertheless, many important issues still need to be investigated, especially vis-à-vis
the usage of this approach to estimate the thermal and viscous effects connected with the wave propagation inside materials produced by popular, nowadays widely spread and continuously evolving additive manufacturing
techniques. Since they are actually one of the few strategies that allow forming porous samples with a definite
microstructure albeit not flawless, adopting appropriate model enhancements that will accomodate computations
to reality should eventually lead to substantial benefits from practical point of view.
A sequence of calculations and measurements is performed to figure out the possibilities in estimating
sound absorption of 3D-printed porous geometries with high precision. Two periodic samples are designed,
fabricated, and investigated in terms of their quality and acoustic properties. The experimental and numericalnormal incidence absorption curves are compared, and a study of the corresponding transport parameters ismade for each geometry. The discussion is additionally based on the computed effective density and bulk
modulus plots.

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2.

DESIGN AND MANUFACTURING OF PERIODIC POROUS MATERIALS

Prototyping of customised and complex three-dimensional geometries has recently become feasible with an additive manufacturing technique popularly known as 3D printing [9]. The strategy relies on the principle thatthe layers of material are successively formed on top of each other and fused together, eventually leading to
the final object. The relevant production capabilities of such a method, however, depend to a large extent on
its particular variant used. Manufacturing of sound-absorbing porous samples entails the necessity of using ahigh precision approach due to modest pore characteristic sizes. Most prevalent and relatively cheap machines
operate based on the fused deposition modelling (FDM) printing technology [9]. In it a continuous filament
made of a thermoplastic polymer is heated to the melting temperature at the nozzle, and extruded either onto
the platform or on top of an already existing element. As a rather primitive method it often results in inconsistent dimensions and degenerate shapes with significant deviations (at least from the microscopic point of view)
as well as other tough issues. Moreover, the achievable detail size usually do not exceed a few millimetres due
to large layer thicknesses available.
The FDM printing technology was used to fabricate periodic porous samples in this research. Two fairly
different but simple morphologies were proposed. The first one consists of a 10×10×12 array of representative
cubic cells (see Figure 1a) of side length equal 3 mm that contain only one central pore and six mutuallyperpendicular vertical as well as horizontal cylindrical channels (in Figures 1a and 1b the cell is shifted in thevertical direction by half of its characteristic size). In the following analysis they are therefore referred to as
the One-Pore-Cells (OPC) with detailed dimensions listed in Table 1. In fact, the values shown there are notconsistent with the designed, original values (for both geometries), but were decreased a bit (0.12 mm fromeach radius) in order to reflect the material shrinkage in 3D-printed objects.
(a)

(b)

(c)

(d)

Figure 1. The designed Representative Unit Cells: (a) the OPC geometry; (b) the domain of fluid in the OPC
geometry; (c) the NPC geometry; and (d) the domain of fluid in the NPC geometry.
Table 1. Diameters of pores and channels in the periodic cubic cells of size 3 mm (OPC) and 5 mm (NPC).
Geometry
OPC
NPC

large pore
2.46
4.36

Diameter [mm]small pore wide channel
–
0.96
1.76
1.36

narrow channel–
0.76

The second devised representative element is more complicated. It is called the Nine-Pore-Cell (NPC)
because it is comprised of nine spherical pores (or their parts) in total (one central, and eight one-eights ofsmall corner ones) connected by cylindrical oblique narrow channels (see Figure 1c). It forms a 6 × 6 × 12
array of 5 mm cubic cells that are additionally linked by wide channels in three directions. All the dimensions
describing the geometry are given in Table 1. Both arrays of OPC and NPC cells were virtually cut such thatto obtain a 36 and 60 mm-long cylinders, respectively, of diameter 29 mm that fit well to the impedance tube
for acoustic material testing.
The manufactured porous specimens were examined in twofold way: the microgeometry of the produced
sample was verified by the aid of a microscope, and the true porosity measurements were conducted on threedifferent devices. As it was expected, the surface of the samples is quite rough and a little deformed (see, forexample, the channel edges or ‘wrinkles’ in Figure 2), whereas the open porosity has greatly increased during
the FDM process compared to its theoretical, numerically calculated value. It was evaluated as a consequenceof force measurements and by weighting the samples on electronic scales of different quality after estimating

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(a)

(b)

Figure 2. Microscopic verification of the quality of the fabricated samples: (a) top view (zoom) of the OPC
sample; (b) top view (zoom) of the NPC sample.
the material mass density from six regularly-shaped objects of various sizes (the value agreed with its analogue
given in product specifications), and the results are consistent, as shown in Table 2.
Table 2. Open porosity in computations and measurements: calculated numerically and measured by means of
the gravity force as well as low (A) and high quality scales (B).
Porosity, φ [–]Geometry Calculated Measured—force Measured—mass A Measured—mass B
OPC
0.33980.44880.44800.4493NPC
0.43450.5394
0.54880.5513

3.

MODELLING, EXPERIMENTS, AND INVERSE CHARACTERISATION

There exist three basic techniques based on the Finite Element Method for a numerical prediction of soundpropagation and absorption in rigid porous media:
1. the Direct Simulation founded on solving a harmonic compressible viscous flow of fluid through theporous region (the Navier-Stokes equations);
2. the Direct Multiscale method in which viscous and thermal harmonic analyses are made for each frequency, f , on the representative volume of fluid in a periodic cell; and3. the Hybrid Multiscale approach, where the region of fluid corresponding to a given unit periodic cell is
fed to several geometrical and static finite element analyses of microstructural problems, the output of
which then constitutes the basis for further considerations on macroscopic acoustic characteristics of thisclass of materials—the so called transport parameters.
In the latter case, one calculates the porosity, φ , and thermal characteristic length, Λ0 , directly from the porous
morphology (e.g., the one shown in Figures 1b and 1d, as it was done in this work), from the character of an
incompressible viscous flow (the Stokes problem) the static viscous permeability, k0 , and static thermal tortuosity, α0 , are deduced, the solution to the electric conduction problem (the Laplace equation) gives necessary
information about the inertial tortuosity, α∞ , and viscous characteristic length, Λ, whereas static thermal permeability, k00 , and static thermal tortuosity, α00 , are found from the Poisson problem (thermal dispersion). The
damping behaviour of the material backed by a rigid wall is then efficiently assessed by the properties of anequivalent fluid, that is the complex-valued dynamic effective mass density, ρeff ( f ), and the dynamic effective
bulk modulus, Keff ( f ), functions, according to the following formulae:A( f ) = 1 − |R( f )|2 ,andZ( f ) = −ipR( f ) =Z( f ) − Z0
Z( f ) + Z0 2π f `pρ ( f ) effpρeff ( f )Keff ( f ) cot,
Keff ( f )

(1)

(2)

where A( f ) is the acoustic absorption coefficient, R( f ) is the reflection coefficient, Z0 = ρ0 c0 is the characteristic impedance of the pore-fluid, ρ0 and c0 are the density and speed of sound in the pore-fluid, and, finally,1218

` is the thickness of a porous medium. Knowing thatρeff ( f ) =ρ0 α( f )φandKeff ( f ) =K0φβ( f )

β(f) = γ −

with

γ −1
,
α 0( f )(3)

the surface acoustic impedance, Z( f ), can be equivalently expressed as:s 2π f ` p

iZ0 α( f )Z( f ) = −cotα( f )β ( f ) .
φ
β(f)
c0

(4)

Here K0 = γ p0 is the bulk modulus of the fluid in pores, γ is the heat capacity ratio of fluid in pores, p0
denotes the ambient mean pressure, and α( f ) as well as α 0 ( f ) stand for the dynamic (visco-inertial) tortuosity and the dynamic thermal tortuosity functions, respectively. Various models were developed and are
utilised to approximate the dynamic tortuosity functions. Among them one finds the most advanced semiphenomenological model by Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) that requires all eight transportparameters and some crucial physical properties of the fluid in pores like the dynamic viscosity, µ, the Prandtl
number, NPr , and the density, ρ0 . It looks as follows:"s#
2α 2 kφ2iπ f ρ0 2α∞ k0 2µ
2
+b −b+1 ,b= 2 ∞ 0,(5)α( f ) = α∞ +2iπ f ρ0 k0µΛφΛ φ (α0 − α∞ )
"s
#


2k00
φ
2iπ f ρ0 NPr 2k00 2
µ
0
0
0
2
0
−
b
+
1
,
b
=
.
(6)
+
b
α (f) = 1+2iπ f ρ0 NPr k00µ
Λ0 φ
Λ0 2 φ (α00 − 1)
In this work, this particular model was incorporated for the purposes of the evaluation of the effective dynamic
functions ρeff ( f ) and Keff ( f ) needed to compute the coefficient, A( f ).
The overall sound absorption properties of the analysed specimens were determined from experiments carried out in the 29 mm impedance tube following the procedure described in the ISO 10534-2 Standard [1]published in 1998. Furthermore, the acoustic transport parameters were found from the respective 3D problemsof the Hybrid Multiscale method. Figures 3 and 4 present the convergence of these parameters with respect to
the Finite Element meshes generated on the representative fluid regions (see Figures 1b and 1d once again) for
the One-Pore-Cell and Nine-Pore-Cell geometries, respectively. The final, reference computations shown lateron in this section were done on the finest grids, where the values of transport parameters are nearly converged.
(a)

φ
Λ0
1

0.5

1.5

2

Number of vertices

0.86672

1.4474

k00
α00

34.35
1

0.5

2

1.5

1.4472
1.447
6

·10−9

(c)

0.506

(d)
3.05455.35

k0 [m2 ]α∞Λ
1.5

1.4476

34.4

Number of degrees of freedom ·10

0.508

1

1.4478

34.3

0.51

0.5

(b)

34.45

2.5
·105
·10−3

2.1893
2.18922.1891
2.189
2.18892.1888

·10−9

3.054

5.34

3.0535

5.33

k0
α0

5.32

0.504
6

Number of degrees of freedom ·10

1

2

3

4

5

6

3.053

α0 [–]

0.339

0.867

k00 [m2 ]0.33920.8673

Λ [m]

φ [–]

0.8676

0.3394

Λ0 [m]

0.86790.3396α∞ [–]

34.5

α00 [–]·10−3

0.33983.0525
6

Number of degrees of freedom ·10

Figure 3. Mesh convergence with respect to the computed transport parameter values of the OPC domain:(a) the geometrical analysis, (b) the Poisson problem, (c) the Laplace problem, and (d) the Stokes problem.

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0.5

1

1.5

2

Number of vertices

2.5

0.6
0.599
0.598
0.597
0.596
0.595

α∞
Λ
0.5

1

1.5

2

1.608521

1.60851.608482

1.5

Number of degrees of freedom·10

2.048

2.047

121.9
0.5

2.04852.04751.60854

k00
α00

122

6

Number of degrees of freedom ·10

6

·10−9

(c)

(b)

1.60856

122.1

3
·105
·10−3

2.049

k00 [m2 ]1.1983

(d)
3.393

11.92

k0 [m2 ]0.4336

1.1986

Λ0 [m]

φ
Λ0

Λ [m]

φ [–]

1.19890.4338α∞ [–]

122.2

1.19920.4342

1.60858

α00 [–]

1.19950.4344
0.434

·10−9

(a)

3.391

11.9

3.389

11.88

k0
α0

11.86
11.84
1

2

3

4

3.387

α0 [–]

·10−3

3.385
3.383

7
6
Number of degrees of freedom ·10
56Figure 4. Mesh convergence with respect to the computed transport parameter values of the NPC domain:(a) the geometrical analysis, (b) the Poisson problem, (c) the Laplace problem, and (d) the Stokes problem.
It has been already shown that the wave attenuation effects in fabricated rigid-porous layers are eminently
underestimated by numerical calculations using standard models (see, e.g., [18]). The obtained results do confirm this observation. Table 3 collects the determined values of transport parameters for the OPC and NPC
samples leading to the curves shown in Figures 5–7. A large discrepancies between computed and experimental plots of ρeff ( f )/ρ0 , Keff ( f )/K0 and A( f ) are clearly visible. In order to get to know right values, aninverse characterisation procedure was undertaken. It was based on the least squares problem arising from theminimisation of the difference between the measurements of sound absorption and their analytically constructed
counterparts. Indeed, the achieved agreement between the respective curves is very good (see Figures 6 and
7). The credibility of the gathered results (see Table 3) is gained by realising the fact, that the values of theidentified porosities in both cases (OPC and NPC) are very close to the measured ones (cf. Table 2). Moreover,its high level is sustained by performing the following consideration. Surface acoustic impedance is a periodic
complex function with a period of π. Therefore, there exists a similarity number, n, for which it takes exactly
the same value for a given argument f (cf. (4)):
cot 2π f ` p 2π f ` p 2π f ` pα( f )β ( f ) = cotα( f )β ( f ) + nπ = cot|n̂| α( f )β ( f ) , n ∈ Z .
c0
c0
c0

(7)

Hence, the similarity number n̂ that scales both α( f ) and β ( f ) such that Z( f ) remains unchanged can beevaluated as:nc
p 0n̂ = 1 +.
(8)
2 f ` α( f )β ( f )We denote the family of density and bulk modulus effective functions yielding identical values for different
Table 3. Computed and identified transport parameters. For each geometry the relative change of the computed
values during the identification procedure is also presented.
Transport parameters
Geometry, caseφ [–]k0 [10−9 m2 ] k00 [10−9 m2 ] α∞ [–] α0 [–] α00 [–] Λ [mm] Λ0 [mm]
OPC, computed 0.3398
5.359
34.51
2.189
3.052
1.447
0.50350.8681
OPC, identified 0.4546
2.824
21.74
2.155
2.707
1.410
0.3245
1.294
OPC, relative 1.3379
0.52690.6301
0.9844 0.8869 0.9742
0.64451.4905
NPC, computed 0.4345
11.94
122.3
2.047
3.383
1.609
0.59401.200
NPC, identified 0.5567
17.42
140.7
1.923
3.301
1.697
0.36050.8386
NPC, relative 1.2811
1.45891.15050.9395 0.9758 1.0551
0.60680.6991

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5

0

−5

1000

2000

3000

4000

5000

Normalised effective density [–]

(b)
4
2
0
−2
−4
1000

6000

2000

3000

4000

5000

6000

Frequency, f [Hz]Frequency, f [Hz]Applies to subplots (a) and (c)Computed transport parameters:
Re (ρeff ( f )/ρ0 )Im (ρeff ( f )/ρ0 )Identified transport parameters:
Re (ρ̂eff ( f , 0)/ρ0 )Im (ρ̂eff ( f , 0)/ρ0 )Re (ρ̂eff ( f , 1)/ρ0 )Im (ρ̂eff ( f , 1)/ρ0 )Re (ρ̂eff ( f , −1)/ρ0 )Im (ρ̂eff ( f , −1)/ρ0 )Identified with computed thermal parameters:
Re (ρeff ( f )/ρ0 )Im (ρeff ( f )/ρ0 )Applies to subplots (b) and (d)Computed transport parameters:
Re (Keff ( f )/K0 )Im (Keff ( f )/K0 )Identified transport parameters:
Re (K̂eff ( f , 0)/K0 )Im (K̂eff ( f , 0)/K0 )Re (K̂eff ( f , 1)/K0 )Im (K̂eff ( f , 1)/K0 )Im (K̂eff ( f , −1)/K0 )Re (K̂eff ( f , −1)/K0 )Identified with computed thermal parameters:
Re (Keff ( f )/K0 )Im (Keff ( f )/K0 )

10

(c)

5

0

−5

Normalised effective bulk modulus [–]

(a)

1000

2000

3000

4000

5000

6000

Normalised effective bulk modulus [–]Normalised effective density [–]

10

(d)
4
2
0
−2
−4
1000

Frequency, f [Hz]

2000

3000

4000

5000

6000

Frequency, f [Hz]Figure 5. Normalised effective mass density (a,c) and bulk modulus (b,d) of a porous layer made of either the
OPC (a,b) or NPC (c,d) geometries.
factors n asρ̂eff ( f , n) =

ρ0 n̂α( f )φandK̂eff ( f , n) =K0.
n̂φ β ( f )(9)Plotting the functions ρ̂eff ( f , n)/ρ0 as well as K̂eff ( f , n)/K0 for n ∈ {−1, 0, 1} corresponding to the identified
transport parameters and comparing them with the calculated ρeff ( f )/ρ0 and Keff ( f )/K0 by means of the Hybrid
Multiscale method reveals the efficacy of the adopted inverse characterisation algorithm—the results for n = 0
are closer to the reference, ‘ideal’ curves (ρeff ( f )/ρ0 and Keff ( f )/K0 with computed transport parameters) thanthose pertaining to n = 1 and n = −1.
After obtaining the identified transport parameters, a simple sensitivity analysis was performed by modifying only particular computed values in accordance with the identification procedure and plotting the resulting
absorption curves. It was found out that the change of the three thermal transport parameters, namely thestatic thermal permeability, k00 , static thermal tortuosity, α00 , and thermal characteristic length, Λ0 , does not havemuch impact on the related sound absorption, albeit the relative decrease or increase of k00 and Λ0 during theidentification procedure was considerable for both geometries (see the relative values in Table 3). The plots
drawn using the full set of eight identified parameters are virtually the same as those curves proceeded fromfive identified and three computed parameters, being the thermal ones (Figures 6 and 7).

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Acoustic absorption coefficient, A( f ) [–]

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

Experimental

Computed

Identified

Identified with computed thermal parameters

0
400

800

1200

1600

2000

2400

2800

3200

3600

4000

4400

4800

5200

5600

6000

6400

Frequency, f [Hz]Figure 6. Normal incidence sound absorption of a rigidly backed 35.72 mm-thick porous layer made of the
OPC geometry.

Acoustic absorption coefficient, A( f ) [–]

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

Experimental
Computed

0.1

Identified
Identified with computed thermal parameters

0
400

800

1200

1600

2000

2400

2800

3200

3600

4000

4400

4800

5200

5600

6000

6400

Frequency, f [Hz]Figure 7. Normal incidence sound absorption of a rigidly backed 59.7 mm-thick layer made of the NPC cells.

4.

CONCLUSIONS

It is rather understandable that the fabricated objects can mirror the designed ideal geometry of representative
cells to just a certain degree, depending on the resolution and precision of a given technology. It is believed,that the inconsistency between experimental and simulation results is due to various reasons, including surface
defects, material shrinkage, and bounded accuracy. At the current stage of research it has been found thatporous materials 3D-printed in the FDM technology have much higher porosity and are characterised by different visco-inertial transport parameters than those calculated from the microstructural analyses. The thermaltransport parameters, on the other hand, that is the static thermal permeability, k00 , static thermal tortuosity,
α00 , and thermal characteristic length, Λ0 , seem to remain unchanged during the manufacturing process. Thisknowledge may open new possibilities in the (still ongoing) development of the universal amendment to theexisting mathematical models describing thermal dispersion and fluid flow inside 3D-printed rigid foams.

ACKNOWLEDGEMENTS
The financial support of Project No. 2015/19/B/ST8/03979: “Relations between the micro-geometry and soundpropagation and absorption in porous and poroelastic media”, financed by the Polish National Science Centre(NCN), is gratefully acknowledged.

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