A Note on the Sound Absorption Characteristics of Microperforated Panels with Non-Circular Holes

Abstract

This study examines the characteristic parameters required for non-circular-hole microperforated panels (MPPs) to achieve sound absorption performance comparable to that of conventional circular-hole MPPs. Through numerical analysis, the flow resistivity and perforation ratio were found to be key parameters influencing the absorption characteristics of MPPs with square and equilateral triangular holes. The results indicate that for square-hole MPPs, matching either the flow resistivity alone or both the flow resistivity and perforation ratio to those of circular-hole MPPs leads to similar sound absorption characteristics. In contrast, for equilateral triangular-hole MPPs, both the above parameters must be matched to ensure comparable performance. Furthermore, this study explores MPPs incorporating a combination of circular and non-circular holes. It was confirmed that by appropriately matching the flow resistivity and perforation ratio, such mixed-hole MPPs can achieve sound absorption characteristics similar to those of MPPs composed solely of circular holes. These findings contribute to the broader design possibilities of MPPs, providing a foundation for optimising hole geometries in practical applications where manufacturing constraints or aesthetic considerations may necessitate non-circular hole patterns.

1. Introduction

The microperforated panel (MPP) is designed with an air cavity and a rigid backing, featuring a thin plate or film of arbitrary material with a thickness of less than 1 mm, perforated with numerous fine holes, typically with a diameter of approximately 1 mm or less at a perforation ratio of typically 1% or less. This configuration forms a Helmholtz-type resonator, where the combination of micro-perforations and the air cavity enables sound absorption. Generally, circular holes are arranged at uniform hole intervals [1,2].

The microperforated panel (MPP) was developed by Maa in the 1970s [3], and subsequently, he published predictive theories for its sound absorption characteristics and design theories [4,5,6]. Initially, the MPP was used in its classical application as a Helmholtz-type resonant sound absorber, as proposed by Maa. However, numerous studies have also been conducted on its applied forms, including its use as a space sound absorber [7,8,9,10,11,12].

Regarding variations in the MPP itself, recent studies have focused on heterogeneous MPPs. Lee et al. proposed a dot-art MPP, in which holes are arranged not in a uniformly spaced manner but to create illustrations or patterns [13]. In this case, the holes are positioned according to the design of the pattern and are therefore not uniformly spaced as in conventional MPPs. Consequently, they reported that conventional MPP prediction theories, such as Maa’s theory, cannot accurately predict the sound absorption characteristics of these structures.

On the other hand, studies on predictive theories for heterogeneous MPPs, including cases where holes of different diameters are mixed, have been conducted by Carbajo et al. [14] and Kusaka et al. [15]. Furthermore, Sakagami et al. proposed a design for dotted-art MPPs with improved predictability by distributing holes across the entire surface in addition to the pattern itself [16]. In addition to the above, other issues to be considered include predicting the sound absorption characteristics of MPPs that have become inhomogeneous due to variations in hole shape and dimensions caused by the effects of manufacturing accuracy. In particular, it has been pointed out that when MPPs and perforated plates are made by additive manufacturing, the effect of manufacturing accuracy on the acoustic properties is remarkable [17].

Apart from these studies, various efforts have been made to explore MPP designs that differ from conventional ones. For example, recent studies have explored novel MPP designs that differ even further from conventional MPPs. Miasa et al. [18] conducted an experimental study on MPPs with multiple hole sizes, while research on MPPs with non-circular holes has also emerged. Examples include studies on MPPs with square and equilateral triangular holes by Ning et al. [19] and investigations of petal-shaped perforations by Xu et al. [20].

Further investigation is needed for cases involving non-circular holes. For example, the findings of such studies could be utilised in situations where the holes are not perfectly circular due to manufacturing accuracy and exhibit irregularities.

From this perspective, this study first examines the sound absorption characteristics of MPPs with non-circular holes. Specifically, numerical analyses are conducted to determine and compare the characteristics of MPPs with square and equilateral triangular holes in addition to circular ones. In particular, the conditions under which MPPs with square and equilateral triangular holes exhibit the same sound absorption characteristics as those with circular holes are derived from numerical analysis results.

Furthermore, numerical analysis is performed on MPPs in which circular and non-circular holes coexist, and the extent to which their sound absorption characteristics differ from those of MPPs with only circular holes is evaluated using statistical indicators. The case where circular and non-circular holes coexist is expected to provide insights into the differences in sound absorption characteristics arising from variations in hole shapes due to manufacturing accuracy, as mentioned above. This study aims to obtain fundamental insights from a practical perspective. To focus on qualitative considerations, such as the effect of different hole shapes on the sound absorption coefficient, and to save computational cost, it targets cases featuring relatively large holes.

2. Sound Absorption Characteristics of Non-Circular MPPs

In this section, we present the fundamental sound absorption characteristics of micro-perforated panels (MPPs) with non-circular holes (specifically, square and equilateral triangular holes). Additionally, we investigate parameter-setting methods to approximate their absorption characteristics to those of conventional MPPs with circular holes. This study focuses on two types of non-circular holes: square and equilateral triangular.

As a specific approach, we set the hole dimensions and other parameters so that one or two characteristic parameters match between non-circular-hole MPPs and circular-hole MPPs. By comparing their sound absorption characteristics, we evaluate the degree of agreement. The characteristic parameters considered in this study include: (1) the hydraulic radius of a hole (defined as r_e = 2S/L, where S is the area of a hole and L is the hole perimeter), (2) the hole flow resistivity, which is the flow resistivity of a single hole [21], (3) the panel flow resistivity, which is the flow resistivity of the panel on the whole, and (4) the perforation ratio of the MPP. These characteristic parameters are matched by adjusting two parameters: the hole dimensions and the centre-to-centre distance of holes (hereinafter referred to as the hole interval).

In this study, we examine four cases, as summarised in

Table 1. Characteristic parameters to be matched to the circular-hole MPP and the controlled parameters of the non-circular-hole MPP.

	Characteristic Parameters to Match the Circular-Hole MPP	Parameters of the Non-Circular-Hole MPP to be Operated
1	Hydraulic radius	Hole dimension
2	Hole flow resistivity	Hole dimension
3	Hydraulic radius and panel flow resistivity	Hole dimension and interval
4	Hole flow resistivity and perforation ratio	Hole dimension and interval

below.

2.1. Model for Simulation (Single Hole Cases)

The analysis was conducted using the Pressure Acoustics Module of COMSOL Multiphysics^® Ver. 6.2 [22]. The constructed model is shown in Figure 1a–c. A single hole of an MPP was modelled and was placed within a waveguide with a square cross-section. The cross-sectional dimensions were set to 20 mm × 20 mm, corresponding to the hole interval, with a backing air layer thickness of 20 mm and an MPP thickness of 1 mm.

Additionally, a background sound field was defined, assuming a 200 mm-long incident sound field with a plane wave normally incident on the MPP. A Perfectly Matched Layer (PML) was applied to the boundary at the top end of the waveguide, i.e., the background sound field. The other boundaries were assumed to be rigid. The Thermoviscous Boundary Layer Impedance condition was assigned to the hole interior surfaces as well as to the front and back surfaces of the MPP. The Thermoviscous Boundary Layer Impedance condition adds the losses due to thermal and viscous dissipation in the acoustic boundary layers at a wall. The condition is applicable in cases where boundary layers are not overlapping. That is, it is not applicable in a very narrow waveguide (with dimensions comparable to the boundary layer thickness) or on very curved boundaries. In the present cases in which the dimension of the hole is larger than 1 mm, the thickness of the thermal boundary layer is around 0.1 mm at 500 Hz [22,23,24,25,26]. Therefore, the overlap is assumed to be not so severe. In order to confirm the accuracy of this condition, the solutions are compared with those derived by COMSOL’s thermoviscous acoustic interfaces. (See Appendix A for details).

The meshing was performed so as to ensure computational accuracy; a swept mesh was applied to the regions near the front and back surfaces of the MPP. Regarding the inside of the hole, finer meshing is applied as much as possible to consider the effect of the hole edges. A free tetrahedral mesh was used, and for circular-hole MPPs, the maximum element size was set to one-third of the hole radius, while for non-circular-hole MPPs, it was set to one-twelfth of the hole side length. For the front and back surfaces of the MPP, the mesh size was set to one-half of the hole dimension. In other regions, a free tetrahedral mesh was also used, with a maximum element size of one-fifth of the wavelength at the upper frequency limit of 4000 Hz (approximately 17.19 mm).

The maximum element size depends on the hole dimensions; thus, the total number of elements and degrees of freedom varied significantly depending on the hole size. In this analysis, the number of elements ranged from 28,729 to 1,013,384, and the degrees of freedom ranged from 63,763 to 1,909,417. The analysis was conducted in the frequency range of 100 Hz to 4000 Hz, with a resolution of 1/24 octave.

2.2. Evaluation Method

In order to evaluate the degree of agreement between the sound absorption characteristics of the circular-hole MPP and the non-circular-hole MPP, the following numerical indices are defined for the peak frequency, the peak sound absorption coefficient, and the difference in sound absorption coefficient over the entire frequency band of the two.

Relative deviation of the peak frequency of the sound absorption coefficient.

$$f_{difference}\left(\%\right)={\displaystyle \frac{f_{noncircular}-f_{circular}}{f_{circular}}}\times 100$$

Relative deviation of the maximum sound absorption coefficient

$${\alpha }_{difference}\left(\%\right)={\displaystyle \frac{{\alpha }_{noncurcular}-{\alpha }_{circular}}{{\alpha }_{circular}}}\times 100$$

RMS deviation of the sound absorption coefficient

$$RMS={\displaystyle \frac{1}{n}}\sum _{i=100}^{4000}\sqrt{{\left({\alpha }_{noncircular,i}-{\alpha }_{circular,i}\right)}^2}$$

where ${\alpha }_{circular}$ and $f_{circular}$ are the peak absorption coefficient and peak frequency of the circular MPP, and ${\alpha }_{noncircular}$ and $f_{noncircular}$ are those of the non-circular MPP. The subscript i associates with frequency, i Hz.

${\alpha }_{noncircular,i}$ denotes the sound absorption coefficient at i Hz for the irregular hole MPP, while ${\alpha }_{circular,i}$ signifies the sound absorption coefficient at i Hz for the circular-hole MPP. The number of sound absorption coefficient data analysed is denoted by n, and the sound absorption coefficient was calculated at every 1/24th of an octave from 100 Hz to 4000 Hz. Therefore, n = 129. The sign of relative deviations is indicated by a ± symbol; however, in the subsequent comparisons and discussions, the sign was disregarded in principle and the absolute values were evaluated as either large or small.

2.3. Degree of the Agreement of Sound Absorption Characteristics

In Figure 2a,b, the sound absorption characteristics of a circular-hole MPP and a non-circular-hole MPP are shown, with each characteristic parameter matched. The matched characteristic parameters are as follows: hydraulic radius (green), hole flow resistivity (red), hydraulic radius and panel flow resistivity (light blue), hole flow resistivity and perforation ratio (purple). For comparison, results for a 1 mm circular-hole MPP are shown (black). The findings reveal that the peak sound absorption coefficient and peak frequency exhibit an increase in response to the matching of the flow resistivity of the hole, the matching of the hydraulic radius, and the matching of the hydraulic radius and the overall flow resistivity, in that order.

The reference values for the difference in the deviation index to determine whether the same sound absorption characteristics as the circular-hole MPP can be obtained when the respective characteristics are matched are 10% for the relative deviation f_difference of the peak frequency of the sound absorption coefficient and the relative deviation α_difference of the peak sound absorption coefficient, and 0.1 for the RMS deviation of the sound absorption coefficient. When these reference values are exceeded, a clear visual deviation can be observed between the sound absorption characteristics of the circular-hole MPP and the non-circular-hole MPP whose characteristics are matched to those of the circular-hole MPP. Therefore, it was judged possible to obtain sound absorption characteristics equivalent to those of the circular-hole MPP by matching the characteristics if the deviation indices for all three did not exceed these reference values when certain characteristics were matched. The deviation indices for the non-circular-hole MPPs with each characteristic matched to the 1 mm diameter circular-hole MPP are shown in

Table 2. Index of deviation of square-hole MPPs with each characteristic property matched to a 1 mm diameter circular-hole MPP.

Characteristic Parameter to Be Fixed	fdifference (%)	αdifference (%)	RMS Deviation
Hydraulic radius	+10.42	+3.806	0.050672
Hole flow resistivity	+4.167	−0.043	0.024892
Hydraulic radius and panel flow resistivity	+16.67	+5.581	0.083242
Hole flow resistivity and perforation ratio	−1.667	−2.581	0.004727

and

Table 3. Index of deviation of equilateral triangular hole MPPs with each characteristic property matched to a 1 mm diameter circular-hole MPP.

Characteristic Parameter to Be Fixed	fdifference (%)	αdifference (%)	RMS Deviation
Hydraulic radius	+22.92	+6.559	0.104885
Hole flow resistivity	+13.33	+2.516	0.067118
Hydraulic radius and panel flow resistivity	+35.42	+7.484	0.149535
Hole flow resistivity and perforation ratio	−1.667	−3.968	0.007807

. From these tables it can be seen that the characteristics for which a good match of the sound absorption characteristics can be obtained by matching the circular-hole MPPs are the flow resistivity of the hole; the flow resistivity of the hole and the perforation ratio for the square-hole MPPs; and the flow resistivity of the hole and the perforation ratio for the equilateral triangular hole MPPs.

3. MPPs with a Mixture of Non-Circular and Circular Holes

In this section, we investigate whether a microperforated panel (MPP) with a mixture of circular and non-circular holes can achieve acoustic absorption characteristics equivalent to those of an MPP composed solely of circular holes. This investigation is based on the results from the previous section, which demonstrated that by appropriately setting the characteristic parameters of non-circular-hole MPPs, their sound absorption properties can be made close to those of circular-hole MPPs.

Based on the findings from the previous section, we examine the following cases:

• For MPPs with a mixture of circular and square holes, we consider two cases: 1 and 2.
• For MPPs with a mixture of circular and equilateral triangular holes, we consider case 1.

Thus, the cases analysed are as follows:

• When the non-circular holes are designed to match both the flow resistivity and perforation ratio of the circular-hole MPP.
• When the non-circular holes are designed to match only the flow resistivity of the circular-hole MPP.

As in the previous section, the analysis is conducted using the Pressure Acoustics Module of COMSOL Multiphysics^® Ver. 6.2.

3.1. Model for Simulation

As shown in Figure 3, a microperforated panel (MPP) with two circular holes and two non-circular holes arranged at equal intervals is placed inside a square cross-section waveguide. The cross-section length is set to twice the hole interval, and other conditions are the same as in the previous section. The mesh division is performed using the meshing function of COMSOL Multiphysics^®.

The mesh settings were the same as those used for the model in Section 2.1.

This setting is necessary because, when matching the characteristic parameters, the radius of the circular holes in an MPP with mixed circular and non-circular holes is, in many cases, smaller than the side length of the non-circular holes. Therefore, setting the maximum element size based on the smaller dimension is essential for maintaining computational accuracy.

For other regions, a free tetrahedral mesh is used, with the maximum element size set to one-fifth of the wavelength at the upper analysis limit of 4000 Hz, which is approximately 17.19 mm. As in Section 2, since the maximum element size depends on the hole dimensions, the total number of elements and degrees of freedom vary significantly depending on the hole size. In this analysis, these values ranged from 143,157 to 1,115,580 elements and 309,670 to 2,458,872 degrees of freedom, respectively. The simulations were conducted over a frequency range of 100 Hz to 4000 Hz, with calculations performed at 1/24-octave intervals.

3.2. Results and Discussion

An example of the absorption characteristics of an MPP with circular and non-circular holes (hereafter mixed-hole MPP) is shown in Figure 4. In this example, a mixed-hole MPP with 1 mm diameter circular holes and equilateral triangular holes (side 1.561 mm, hole interval 21.65 mm) is compared with an MPP with 1 mm circular holes only, and an MPP with equilateral triangular holes (side 1.561 mm, hole interval 23.18 mm) only. As is observed, by mixing non-circular holes, the absorption characteristics slightly deviate from those of an MPP with circular holes only. In this case, the characteristics of the mixed-hole MPP are lower than those of the MPP with circular holes only and higher than those of the MPP with triangular holes only. Later, we discuss the deviation due to the effect of the combination with non-circular holes quantitatively by the indices presented before.

Table 4. Index of deviation of absorption characteristics of MPPs with a mixture of square and circular holes with hole flow resistivity and perforation ratio matched to circular-hole MPPs.

Subject for Calculating Deviation	fdifference (%)	αdifference (%)	RMS Deviation
Circular-hole MPP and mixed-hole MPP	−0.627	+2.903	0.00477
Circular-hole MPP and square-hole MPP	−3.639	+5.737	0.00701

presents an index of deviation in the sound absorption characteristics of a microperforated panel (MPP) with a combination of circular and square holes, where the square holes have the same flow resistivity and perforation ratio as the circular holes.

Table 5. Index of deviation of sound absorption characteristics of MPPs with a mixture of equilateral triangular and circular holes with hole flow resistivity and perforation ratio matched to circular-hole MPPs.

Subject for Calculating Deviation	fdifference (%)	αdifference (%)	RMS Deviation
Circular-hole MPP and mixed-hole MPP	−1.413	+1.205	0.00631
Circular-hole MPP and equilateral triangular hole MPP	−4.860	+6.121	0.01110

provides an index of deviation in the sound absorption characteristics of an MPP with a combination of circular and equilateral triangular holes, where the triangular holes have the same flow resistivity and perforation ratio as the circular holes.

Table 6. Index of deviation of sound absorption characteristics of MPPs with a mixture of square and circular holes with hole flow resistivity matched to the circular-hole MPPs.

Subject for Calculating Deviation	fdifference (%)	αdifference (%)	RMS Deviation
Circular-hole MPP and mixed-hole MPP	−2.368	−1.835	0.00879
Circular-hole MPP and square-hole MPP	+1.715	+1.342	0.01690

presents an index of deviation in the sound absorption characteristics of an MPP with a combination of circular and square holes, where the square holes have the same flow resistivity as the circular holes.

From Table 4, Table 5 and Table 6, it is evident that the deviation in the sound absorption characteristics of MPPs with a combination of circular and non-circular holes is small compared to that of circular-hole MPPs.

Furthermore, from Table 4 and Table 5, it can be observed that the sound absorption coefficient of MPPs with a combination of circular and non-circular holes, where the non-circular holes have the same flow resistivity and perforation ratio as the circular holes, is approximately the intermediate value between the corresponding circular-hole MPPs and non-circular-hole MPPs across the entire frequency range.

In other words, if the sound absorption characteristics of the corresponding circular-hole MPP and non-circular-hole MPP are known, the sound absorption characteristics of the MPP with a combination of circular and non-circular holes can be estimated as an intermediate value between them. Therefore, it has been found that a more detailed prediction of the sound absorption characteristics is possible.

4. Concluding Remarks

In this study, the characteristic parameters that should be matched to achieve sound absorption performance comparable to that of circular-hole MPPs for non-circular-hole MPPs were identified through numerical analyses by FEM. In the analyses, as characteristic parameters to match, the following parameters and combinations of parameters were chosen: hydraulic radius of the hole, hole flow resistivity, hydraulic radius and panel flow resistivity, and hole flow resistivity and perforation ratio. In order to control them, the hole dimensions and hole intervals were adjusted. The sound absorption characteristics of a non-circular-hole MPP with the aforementioned characteristic parameters matched to those of a circular-hole MPP were analysed numerically using FEM and compared with those of a circular-hole MPP. The degree of agreement between the two was evaluated by the relative error of the peak frequency, the relative error of the peak value, and the RMS error of the sound absorption coefficient at all frequencies.

As a result, it was found that for square-hole MPPs, matching the hole flow resistivity, or both the hole flow resistivity and perforation ratio, to that of circular-hole MPPs leads to a good match in terms of sound absorption characteristics. Similarly, for equilateral triangular-hole MPPs, matching both the hole flow resistivity and perforation ratio to those of circular-hole MPPs ensures a good match in terms of sound absorption characteristics.

Furthermore, it was confirmed that by applying this parameter matching, MPPs with a combination of circular and non-circular holes can achieve sound absorption characteristics similar to those of MPPs composed solely of circular holes. The sound absorption characteristics of the mixed-hole MPP are approximately intermediate between those of the MPP with circular holes only and those of the MPP with non-circular holes.

These findings not only provide a basis for the design of non-circular-hole MPPs and the investigation of their properties but also provide practical understanding of the effects on sound absorption properties of, for example, non-circular hole shapes in the manufacturing process, especially errors caused by problems of manufacturing accuracy in the production process, such as those seen in additive manufacturing.