COMSOL 6.3 - Porous Absorber

This is a model of acoustic absorption by a porous acoustic open cell foam. In porous materials the sound propagates in a network of small interconnected pores. Because the dimensions of the pores are small, losses occur due to thermal conduction and viscous friction. Acoustic foams are used to sound proof rooms and ducts as well as to treat reverberation problems in rooms (see Ref. 1).

The aim of the model is to characterize the absorption properties — more specifically, the specific surface impedance and the absorption coefficient — of a layer of melamine foam in terms of sound incidence angle and frequency. The melamine foam contains an air inclusion. An analytical solution exists in the case where the layer is uniform. The model uses a 2D geometry of such a system.

Model Definition

Figure 1 depicts the geometry of the modeled system, in which an incident sound field hits the porous melamine foam layer at angle θ. An air inclusion, circular domain of radius a, is present in the porous layer. The incident wave has wave vector k. In the figure, the dotted lined indicates the boundaries of the model domain. You only model a portion of width W and apply periodic Floquet boundary conditions on the left and right boundaries to extend the domain to infinity. The incident field is modeled by applying a background pressure field to the air domain. At the top, a perfectly matched layer (PML) domain is used to model an infinitely large air domain. The thickness of the porous melamine layer is Hp = 10 cm and the height of the modeled air region is H = 30 cm. The height of the PML domain is Hpml.

Model the melamine foam using the Pressure Acoustics interface’s Poroacoustics domain feature using the Johnson-Champoux-Allard model with a rigid frame. This is an equivalent fluid model for a rigid frame porous material, a so-called five parameter semi-empirical equivalent fluid model. See About the Poroacoustics Models in the Acoustics Module User’s Guide. The surrounding fluid is air, and the material parameters for the foam are as listed in Table 1 (following Ref. 2, material sample number 31).

Table 1: Melamine foam material parameters.
Symbol	Value	Description
εp	0.995	Porosity
Rf	10,500 Pa·s/m2	Flow resistivity
s	0.49	Viscous characteristic length parameter
Lth	470 μm	Thermal characteristic length
Lv	240 μm	Viscous characteristic length
τ	1.0059	Tortuosity factor

Figure 1: Geometry of the modeled system, the air inclusion has a radius a.

The incident background pressure field is given as

(1)

where θ is the incidence angle and k0 is the wave number in the free field (air domain). The pressure p solved for in the model is the total field and the scattered field pscat is given as pscat = p − pinc. Note that this expression for the scattered field is only valid in the air domain, as the incident field is not known a priori in the porous material.

Two parameters that characterize the absorption properties of the porous absorber are the specific surface impedance Z and the absorption coefficient α (see Ref. 1). The absorption coefficient, which represents the ratio of the absorbed and incident energy, is for a plane wave defined as

(2)

where R is the pressure reflection coefficient that gives the ratio of the scattered to the incident pressure. This expression is valid as long as there are no higher-order diffraction modes; higher-order modes start to occur at a given cutoff frequency. In general the absorption coefficient can be defined through its energetic definitions as

(3)

where Pscat is the total scattered power and Pinc is the total incident power. Both of these quantities can be computed from the intensity of the scattered and incident acoustic fields. The intensities of both fields exist as postprocessing variables when the option is used in the feature. Remember that in the nomenclature used in the Acoustics Module the incident field is called the background field, when setting up the model.

The surface specific impedance (normalized by the plane wave characteristic impedance) is defined as

(4)

where ρ is the density of air, c is the speed of sound, and un = u ⋅ n is the normal velocity at the surface of the melamine layer. When computing the expression it will be taken as the average at the surface of the porous layer. When averaging it is important to average the ratio p/un and not average p and un separately, the latter leads to incorrect results.

Both the absorption coefficient and the surface normal impedance are dependent on frequency and on the incidence angle.

Uniform Porous Layer Solution

In the case of a uniform porous layer (with no air inclusions) of thickness Hp backed by a sound hard wall, an analytical solution exists for the surface impedance, reflection coefficient, and absorption coefficient (see Ref. 1). The surface normal impedance (normalized by the characteristic plane wave impedance) is given by

(5)

where a subscript “c” represents complex-valued impedance and wave number variables from the Poroacoustic domain. From the normal impedance the absorption coefficient is deduced.

Results and Discussion

Figure 2 and Figure 3 plot the scattered and total pressure fields for an incidence angle of 45° and the frequency f = 10 kHz. Notice how the wave is absorbed in the porous layer.

Figure 2: Scattered pressure field for an incidence angle of 45o and frequency f = 10 kHz.

Figure 3: Total acoustic pressure for an incidence angle of 45o and frequency f = 10 kHz.

Figure 4 depicts the acoustic pressure (the real part) at the surface of the porous melamine layer for two angles of incidence. Figure 5 plots the specific acoustic impedance at the surface of the porous absorber (computed as the average across the surface). Figure 6 shows the absorption coefficient; comparing the energy based absorption coefficient (Equation 3) to the coefficient computed based on the reflection coefficient (Equation 2), and the analytical solution of a uniform porous layer. Finally, Figure 7 depicts the absorption coefficient for normal incidence in octave and 1/3 octave bands. The plot feature automatically creates tables, which can be found under > and easily exported as text or spreadsheet files.

Looking closely at Figure 6, it can be seen that the absorption coefficient based on the reflection coefficient, which assumes pure plane waves, start to differ slightly from the energy based values, at specific frequencies (1000 Hz and 1700 Hz, depending on the incidence angle). These are the cut on/off frequencies for higher order diffraction modes. In this particular setup, the difference is not large as the properties of the air inclusion are relatively close to the porous material. For other configurations, for example, having solid inclusions, the difference could be larger. The energy based absorption coefficient represents the actual absorbed energy metric.

The dependency of the surface specific impedance and/or absorption coefficient on incidence angle and frequency is important for modeling absorbers as impedance boundary conditions in, for example, a Ray Acoustics model. In larger model systems the present model could be used as a “submodel” to determine appropriate impedance boundary conditions. The real part of the impedance (the resistance) is associated with energy loss whereas the imaginary part (the reactance) is associated with phase changes of the field. The reciprocal value of the impedance is the admittance.

In this system, the absorption coefficient approaches 1 for increasing frequency. This corresponds to the frequency where the product between the porous absorber height Hp and kyπ−1 of the incident wave is equal to one. This is where half a wavelength fits into the absorbing layer.

Figure 4: Sound pressure level at the surface of the porous absorber.

Figure 5: Specific acoustic impedance at the surface of the porous absorber.

Figure 6: Absorption coefficient for the porous melamine absorber as function of frequency and incidence angle. Compared to the reflection coefficient based expression and the analytical solution of a uniform layer.

Figure 7: Absorption coefficient of the porous melamine absorber for normal incidence in octave bands, 1/3 octave bands, and continuous frequency.

Notes About the COMSOL Implementation

Periodic Floquet boundary condition

Apply a periodic Floquet boundary condition to model an infinite periodic structure. The periodicity is determined by the wave number of the background (incident) pressure field. The relation between the pressure at the left and right boundaries of the model domain is

(6)

where d = (W, 0) is a vector extending from the left to the right boundary and k is the wave vector defined in Equation 1. COMSOL Multiphysics automatically calculates the vector d when applying the Floquet periodicity.

Visualize periodic solution

To visualize the periodic solution, create an Array 2D dataset and enable under section. Enter the same as used in the periodic conditions.

Comparing to the Analytical Solution

In the results section the simulated absorption coefficient and surface impedance are compared to the analytical solution of a uniform porous layer. To make a verification of the model simply select the inclusion (the circular air domain) as a Poroacoustic domain and run the model again. You will find that the analytical and model results show perfect agreement.

References

1. T.J. Cox and P. D’Antonio, Acoustic Absorbers and Diffusers, Theory, Design and Applications, 2nd ed., Taylor and Francis, 2009.

2. N. Kino and T. Ueno, “Comparison between characteristic lengths and fiber equivalent diameter in glass fiber and melamine foam materials of similar flow resistivity”, J. App. Acoustics, vol. 69, pp. 325–331, 2008.

Acoustics_Module/Building_and_Room_Acoustics/porous_absorber

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Load the parameters for the model. The list of parameters include geometry definitions, definitions used in the mesh, and material parameters for the melamine foam.

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file porous_absorber_parameters.txt.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type H+Hpml.

Click to expand the section. Clear the checkbox.

Select the checkbox.

In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	Hpml

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type Hp.

Locate the section. In the text field, type -Hp.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a.

Locate the section. In the text field, type W/2.

In the text field, type -Hp/2.

In the toolbar, click

Click the

button in the toolbar.

The geometry should look like that in the figure below.

Definitions

Load the expressions defining the surface impedance and absorption coefficient, see Equation 2 and Equation 4, from a file.

Variables 1

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file porous_absorber_variables.txt.

Load the expressions defining the analytical expressions for a single porous layer with a sound hard backing, see Equation 5.

Variables 2

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file porous_absorber_analytical.txt.

Define two nonlocal integration couplings that act on points in the geometry. You will use them later to map (or probe) values from these points. One in the porous domain and one in the air domain.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Point 1 only.

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Point 3 only.

Define an average and integration coupling on the porous-air interface to help compute the average reflection coefficient as well as incident and reflected powers.

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 4 only.

Integration 3 (intop3)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 4 only.

Now proceed to set up the material properties. Add air as the default domain material and create a new material to define the melamine foam porosity.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Melamine Foam

In the window, under right-click and choose .

In the window for , type Melamine Foam in the text field.

Select Domain 1 only.

Click to expand the section. In the tree, select > .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Porosity	epsilon	epsilonP0	1	Basic

Locate the section. In the tree, select > > .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Flow resistivity	Rf_iso ; Rfii = Rf_iso, Rfij = 0	Rf0	Pa·s/m²	Poroacoustics model
Thermal characteristic length	Lth	Lth0	m	Poroacoustics model
Viscous characteristic length	Lv_iso ; Lvii = Lv_iso, Lvij = 0	Lv0	m	Poroacoustics model
Tortuosity factor	tau_iso ; tauii = tau_iso, tauij = 0	tau0	1	Poroacoustics model