Design and validation of microperforated panel absorbers using Occam’s razor and causal inference

Ning Xiang 1 , Michael Hoeft, and Cameron J. Fackler Graduate Program in Architectural Acoustics, Rensselaer Polytechnic Institute 110 8th Street, Troy, New York 12180, USA

ABSTRACT Microperforated panel absorbers in forms of micro-pores / -slits (MPSP) can achieve high absorption. However, noise control practice usually requires broadband high absorption. Single- or double-layer MPSP absorbers often cannot meet practical requirements. Multilayers become a natural option. Multiple layers of MPSP absorbers inherently complicates the design process upon a given design scheme. This work applies a Bayesian framework using a potentially multilayered prediction model. The Bayesian design involves two-levels of probabilistic inference to design a parsimonious number of layers using the model-selection solution, a quantitative implementation of Occam’s razor, while the parameter estimation is used to estimate MPSP parameters given the selected number of multilayers. This probabilistic design process rapidly hones in on the MPSP parameters of each individual layer so that the overall composite meets the design goal. When experimentally validating the designed prediction upon the design scheme, manufacturing inaccuracies may lead to deviations. In analyzing reasons of the deviations, this work further applies causal inference to analyze the cause-e ff ect relationship. To this end a causal model has also been established. Using both parametric prediction model for the absorption performance and the causal model for causal inference of the deviation causes, the Bayesian multilayer design can be satisfactorily validated.

1. INTRODUCTION

Micro-perforated and micro-slit (MPSP) panel sound absorbers are made of sub-millimeter pores / slits penetrating through panel surfaces. They are widely applied in a range of acoustic research and practice [1, 2]. Single-layer configurations of MPSP often yield high absorption over a very limited frequency range [3]. By extension, multiple individual MPSP layers may be combined into a multi-layer configuration to obtain broadband, high sound absorption. The MPSP absorbers can also be manufactured using transparent materials. For the transparent sound absorbers, microslits panels (MSP) will overperform over the micrpore panels (MPP) to retain desirable transparancy / translucency. When designing these transparent multilayer MPSP absorbers, the design process needs to consider both sound absorption requirements and optical transparency. This work evaluates an approach to design and constructs this type of multilayer MPSP sound absorbers. This work focuses on design process, which is an inversion process from the design requirements to

1 xiangn@rpi.edu

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

the selection of a suitable number of multi-layered MPSP absorbers, followed by the estimation of MPSP parameters of each individual layers. This design applies Bayesian inference to quantitatively implement Occam’s razor to penalise overlayered configurations. When experimentally validating the Bayesian design, it may be challenging to determine what parameters of which layers contribute to undesirable deviations from the design scheme leading to an absorber that does not meet the initial design requirements. For the first time, this work applies causal inference to tackle this challenge.

2. MICRO-PERFORATED / SLIT PANELS IN MULTILAYERS

We first introduce prediction models for single-layer microperforated panel (MPP) absorbers based on Maa’s work [3,4], followed by configuring multiple layers in a predictable manner. According to Daa-You Maa, air is swung back-and-forth in these sub-millimeter tubes, as an impinging sound wave moves air around the panel surface. The air mass inside the microtubes encounters viscous frictions at perforated microtube walls / boundaries, exhibiting acoustic resistance. Atalla and Sgard [5] published a prediction model for the acoustic behaviors of an MPP, having a similar prediction ability to Maa’s model [3]. They treated the MPP panel by a modified Johnson-Allard rigid frame porous model. The model considers the viscous boundary-layer e ff ect of the micro-pores into account. Their model predicts the panel impedance using three parameters as well, the perforation rate σ , the diameter of the micro-pores d , and panel thickness t ,

Z MPP = t + 2 ϵ e

" (1 + j) 4 R s

d + j ωρ # , (1)

σ

ϵ e = 0 . 425 d (1 − 1 . 14 √ σ ) , and R s = p

η ωρ/ 2 (2)

are the correction length ϵ e and the surface resistance R s with ω being the angular frequency, j = √

− 1. The air surrounding the panel and filling the micro-pores has density ρ and dynamic viscosity η . In 2001, Maa [4] derived micro-slit panels (MSP), treating each microslit as an infinite number of ‘micro-tubes’ arranged along a line. For normal incidence sound waves, this prediction model for acoustic impedance of the MSP panel is given by

√

! 1 / 2

 + j ω t



1 + k 2 0 12

Z MS P = 12 η t σ ρ c d 2

" 1 + (5 2 + 2 k 2 0 ) − 1 / 2 + F ( e ) d

# , (3)

2 k 0 d 12 t

σ c

2 t

where c is the sound speed in air, and ρ the density of air, ω is the angular frequency, t is the panel thickness, σ is the porosity (or perforation rate), and d is the diameter / width of the slits. The elliptical integral F ( e ), and the perforate constant k 0 as given below in Eqs. (4,5) are also needed to predict the acoustic impedance of an MSP

! 2 e 2 + 1 · 3

! 2 e 4 + 1 · 3 · 5

! 2 e 6 + . . .  , (4)

 1 + 1

F ( e ) = π

2 · 4

2 · 4 · 6

2

2

ω ρ/η, and e = p

k 0 = p

1 − ( a / ld ) 2 , (5)

with a = d / 2 being the slit ’radius’, and l being half the length of the slit. The resulting surface impedance, Z s consists of the acoustic impedance of the panel and the acoustic impedance of the air cavity between the MSP and a hard termination [6], namely

Z s = Z MPSP + Z cav , with Z cav = − j  ω D

 . (6)

c

Z MPSP is the panel impedance of MPSP (either MPP or MSP), D represents the cavity depth behind the panel.

Figure 1: Multilayered setting of micro-perforated / slit panels (MPSP). Each layer is made of a MPSP panel which is backed by an air cavity depth d n . Z MPSP , n is the panel impedance of the n -th layer MPSP. The acoustic impedance of the cavity behind the n -th panel is denoted as Z cav . The first (right most) layer is terminated by a rigid backing.

Figure 1 sketches multiple MPSP layers combined into a multilayered setting for potentially broadening the absorbing frequency range. The very left panel will exhibit a normal surface impedance Z s , N of the multilayer configuration

N Y

 =

 P s , N V s , N



T MPSP , n T cav , n
 T rigid , (7)

Z s , N = P s , N V s , N , with

n = 1

where each cavity and each MPSP panel are written in a matrix form [6] as

 cos( β d n ) j ρ c sin( β d n )

 , (8)

T cav , n =

j sin( β d n ) /ρ c cos( β d n )

 and T rigid =

 1

 , (9)

 1 Z MPSP , n 0 1

T MPSP , n =

0

where d n is n th air cavity depth, β = ω/ c, and ρ c is the characteristic resistance of the air. The normal incidence absorption coe ffi cient of the multilayer MPSP absorber can then be written as

α = 1 −| R | 2 , with R = Z s , N − ρ c

Z s , N + ρ c . (10)

3. MODEL-BASED DESIGN USING BAYESIAN INFERENCE

This paper discusses a design procedure of the sound MPSP absorber using Bayesian probability theory. The design proceeds from a design scheme (denoted as D ) described in this section and the prediction models (denoted as M ) described previously.

3.1. Design Scheme Practical applications often require sound absorbers to absorb sound energy specified by certain values of absorption coe ffi cients in a certain broad frequency range. Figure 2 illustrates one possible

WW. _ 2? noma _ Z - 2s a o s _ 5 N Ni 2 oY N|7_ = | io 8 S a a NI “ = 7-——— — INI NI Z psp. Z wpsp.N-1 @AEM JUSpIOU|

Figure 2: Design scheme required in a practical application. A standard deviation curve is also plotted for assignment of the likelihood function within the model-based Bayesian framework.

scheme of the absorption behavior required from a practical application. In the following it is termed as design scheme representing shaded areas within the upper and the lower design bounds. To be more precise, this specific application requires an absorption coe ffi cient higher than (including) 0.8 within a frequency range between 500 Hz and 2.5 kHz. Outside this frequency range, no specifics are required. Di ff erent design schemes may be specified by di ff erent applications.

3.2. Bayesian Design In the current task of the sound absorber design, once the application sets a design scheme D , there are tentatively a number of possible models M = { M 1 , · · · , M N } . Each model predicts an n - layer of MPSP panel absorber. Among the model set M , tentative N models might be able to meet the practical design requirement. Therefore, a high-level question needs to be answered first: given the design scheme D , what is a parsimonious number of layers to meet the design scheme? Once this question can be answered, the acoustic designer will also need to specify the MPSP parameters for the number of layers selected. This task represents two levels of inferential processes. Model selection of the number of layers and MPSP parameter estimation. This design task posts considerable challenges that a tentatively arbitrary number of multiple ( N ) layers with each layer requiring four MPSP parameters. That is to say, n × 4 parameters are included in model M n ( θ ) required to accomplish the specific design with n = 1 , 2 , . . . , N , namely a considerable high dimensionality. Parameter vector θ includes all n × 4 MPSP parameters needed to specify an n -layer absorber. Bayes’ theorem unifies the two levels of the inferential design within one framework as

p ( θ | D , M , I ) × Z = L ( θ ) × Π ( θ ) , (11)

where M represents one specific model to be selected among model set M . For simplicity, the following descriptions ignore subscript n , but always bear in mind that the model M predicts n -layers of the MPSP panels. Z is termed marginal likelihood or Bayesian evidence , being the key quantity to the two levels of inference with the unified Bayesian inferential framework; selection of the number of MPSP layers, while p ( θ | D , M , I ) represents the posterior probability, critical for MPSP parameter estimation. L ( θ ) represents the so-called likelihood function which needs to be assigned based on

Absorption coefficient [1] Design scheme --- Std. deviation lower bound 1 15 2 25 3 Frequency (kHz) 35 4 45 Standard deviation

available information in terms of the principle of maximum entropy (MaxEnt) [7] as

K Y

2 π exp − E 2 k 2 δ 2 k

! , (12)

1

L ( θ ) =

δ k √

k = 1

where k = 1 , . . . , K specify individual data points over discrete frequency range as defined by the design scheme D and E k is an error function expressed as



B l ( f k ) − α ( f k ) for α ( f k ) < B l ( f k ) ,

E k =

α ( f k ) − B u ( f k ) for α ( f k ) > B u ( f k ) ,

(13)

0 otherwise ,

where B l ( f k ), B u ( f k ) are the lower and upper bounds as illustrated by the design scheme D in Fig. 2. The multilayered model expressed in Eq.(10), via Eqs.(7-9) provides a predicted absorption coe ffi cient α as a function of frequency. Note that Eq.(12) is essentially a Gaussian likelihood, this is assigned upon the only available information, namely, the prediction model describes the absorption coe ffi cient well, so that a square of the error function as defined in Eq.(13) is bounded to an unknown, finite value. δ k represents the standard deviation of the Gaussian likelihood. Figure 2 also illustrates this quantity as a dash-line curve. We assign this due to the design accuracies required in applications. We denote Π ( θ ) as prior probabilities of all the MPSP parameters. We also apply the MaxEnt to assign them to be uniform with some prior information on individual parameters, namely

– Π ( d ) = uniform (0.2, 0.5) mm (micropore / slit diameter) – Π ( σ ) = uniform (0.01, 10) % (perforation rate) – Π ( D n ) = uniform (0.5, 5) cm (cavity depth)

Table 1: Parameters of a three-layered microslit panel absorber resulted from the two-levels of Bayesian design. All panels are 2.56 mm thick, being available for manufacturing the panels.

Parameters Layer 1 Layer 2 Layer 3

Slit diameter (mm) 0.213 0.215 0.356

Perforation rate (%) 1.96 4.11 7.41

Air cavity depth (cm) 1.55 1.78 1.90

We proceed the Bayesian design using a Makov Chain Monte Carlo sampling method [8] to evaluate Bayesian evidence Z for each individual models from M = { M 2 , · · · , M N } with N = 5. For the given design scheme as illustrated in Fig. 2, a three-layer model ( M 3 ) is selected among the four di ff erent models M . The selection is based on estimations of Bayesian evidences using the sampling method. This model selection of the number of layers represents a quantitative implementation of Occam razor, that is to penalize overlayered models. The selected (three-layer) model represents the most parsimonious one. This parsimonious number of layers benefits manufacture cost and reduces the overall thickness of the sound absorber with respect to overlayered designs. During Bayesian evidence estimations, the posteriors p ( θ | D , M , I ) on the left-hand side of Eq. (11) have also been sampled, so that the MPSP parameters for three-layers are readily estimated using p ( θ | D , M , I ). One set of estimated parameters using the MSP model as given in Eqs. (3-5) is listed in Table 1. One of the design results in form of absorption coe ffi cient of the three-layer MSP is illustrated in Fig. 3. In the figure, absorption coe ffi cients of each single-layers of MSP are also shown when each

Figure 3: Absorption coe ffi cient of a three-layer microslit panel absorber. The designed three-layer behavior is compared with the design scheme along with three individual single-layer microslit panels when backed rigidly [8].

of the single MSP panel were backed by a rigid termination. The overall absorption coe ffi cient of the three-layered absorber can only be obtained when layering them as sketched in Fig.1. One can see that simply averaging three individual single-layers would not yield designed results.

0.8 0.6 Absorption coefficient — 3 layers _ Design scheme —-— Layer 1 cf - — Layer 3 re 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (kHz)

Figure 4: Experimental validation using an impedance tube with manufactured panel samples. Left: Microslit panels (MSP) with the air cavity keepers. Right: Impedance tube end when the three layer MSPs are attached for the experimental validations. [8].

4. CAUSAL INFERENCE IN EXPERIMENTAL VALIDATIONS

The design parameters as listed in Table 1 are used to manufacture the MSP panel samples. Figure 4 (Left) shows a photograph for one set of MSP panels and air cavity keepers, while Figure 4 (Right) shows how the sample panels are mounted on the tube end. The tube diameter is 5.2 cm being suitable for normal incident absorption measurements up to 5 kHz. Two-microphone transfer function method is used for the validation e ff ort. A laser-cutter is used to create the MSP samples. The laser-cutter is often subject to small manufacturing inaccuracies when cutting the microslits, the inaccuracies may lead to unacceptable deviations. We have conducted causation analysis [9] in an iterative manner during the validation e ff ort [10, 11]. Figure 5 illustrates the causal models established for the MPSP validations. Each time when deviations between the absorption coe ffi cient of single-layered panel absorbers in

Figure 5: Causal models for the causal inference in the experimental validation. (a) The microperforated / slipt panel (MPSP) in single layered configuration. (b) Multi MPSP configuration with N layers.

comparison with the single-layer model prediction is observed in Fig. 3, the single-layered absorption coe ffi cients from experimental measurement are used to estimate the resulting MSP parameters for the manufactured panels to infer how much they deviate from the original design as listed in Table 1. This causal inference is once again carried out using model-based Bayesian parameter estimation (for one single layer at a time in this work). Possible corrections for guiding the laser-cutter are then applied for another iteration of manufacturing the samples.

Absorption Single layer rigid backing - Measured“ -- Initial model Frequency Absorption Frequency

Figure 6: Absorption coe ffi cients during iterative experimental validations. (a) The first iteration with unacceptable derivations. (b) The third iteration with acceptable derivations.

Figure 6 illustrates the absorption coe ffi cient of a three-layer MSP absorber from the validation measurements obtained after three such iterations. At initial experimental measurements, Figure 6 (a) reveals an unacceptable absorption performance away from the initial design scheme. Through the causation analysis using a model-based Bayesian estimation, based on the causal model as shown in Fig. 5, the cause of such deviations is analyzed simply using single-layer rigidly backed measurements. After the Bayesian estimations we find out causes of such deviations, that helps instruct the laser cutter to correct the previous manufacture. With such a cause-e ff ect analysis,

- - Initial Model — Measured - - Initial Model — Measured JUSIDIe09 UOI}diosqy Frequency (kHz) Frequency (kHz)

the created samples during the third iteration lead to the resulting absorption coe ffi cient as shown in Fig. 6 (b), demonstrating a desired absorption performance largely meeting the design scheme, although there are still derivations slightly outside the design scheme. Further iterations of corrections could be applied for guiding the laser-cutter for even closer results if desirable.

5. CONCLUDING REMARKS

We have successfully applied two levels of Bayesian inference to the design of multilayer micro- perforated / slit panels (MPSP) for sound absorbers to obtain both high sound absorption and wide bandwidth simultaneously. Based on Maa’s model of the micro-slit panel (MSP) absorber, we demonstrate that the two-levels of Bayesian inference help us design the MSP parameters e ffi ciently with multiple layers, e ff ectively implements Occam’s razor in favoring a parsimonious number of layers. This work shows, a three-layer MSP absorber can largely meet the specific design scheme. Experimental validations are carried out using the two-microphone transfer function method in an impedance tube. A laser-cutter is used to create the three-layer MSP samples. During the experimental validation, if deviations are observed, the cause of these deviations is then analysed by establishing causal models. Based on the causal models the cause of deviations is further estimated by a model-based Bayesian estimator. A few iterations of the corrected laser-cuttings lead to the three-layer samples to an acceptable performance.

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