Metaheuristic optimisation of sound absorption performance of multilayered
porous materials
Vivek T. RAMAMOORTHY(1) , Ender ÖZCAN(1) , Andrew J. PARKES(1) , Luc JAOUEN(2) , François-Xavier BÉCOT(2)(1) University of Nottingham, UK, {vivek.ramamoorthy1, ender.ozcan, andrew.parkes}@nottingham.ac.uk(2) MATELYS, France, {luc.jaouen, fxb}@matelys.com

Abstract
The optimization of multilayered-sound-packaging is a challenging task which involves searching the best/optimal settings for a number of acoustic parameters. The search space size becomes too large to handle by brute
force, as the number of those parameters increases. In this study, we apply metaheuristics to configure multilayered porous materials for maximizing their overall sound absorption coefficient. The multi-layered system
is composed of a stack of facing screens (i.e. a woven or non-woven textile, a perforated plate) and/or thicker
porous material combinations, chosen from a database of common acoustic materials. Firstly, we consider a twolayered system with a fibrous porous material layer modeled using Delany-Bazley-Miki (DBM) model and afacing screen modeled by Johnson-Champoux-Allard (JCA) model, and optimize different parameters for maximizing the sound absorption. Secondly, for multilayered systems with more than two layers, a genetic algorithm
is presented to explore the thickness and acoustical parameters that maximize the sound absorption coefficientsfor a frequency range of interest.
Keywords: Metaheuristics, Multilayered porous materials, Genetic algorithms, Sound absorption.

1

INTRODUCTION

In recent years, composite structures have been widely used in aerospace, automobile and construction applications due to their higher strength to weight ratio. Using composite construction can significantly reduce theweight and costs in these applications potentially leading to massive reductions in fuel consumption and emissions in the coming decades [1]. In automobile and aerospace composite applications, to ensure passenger
comfort at similar levels as that of metallic structures, acoustic isolation design needs to be enhanced. Thisshould be so done that the weight saving benefits are retained. Multilayered porous sound packages are commonly used to ensure noise levels are in acceptable limits. However, optimally designing these packages is achallenging task due to the large number of parameters and constraints involved, and often sub-optimal designs
may be chosen for manufacture. Such problems can be tackled by using intelligent alternate metaheuristic optimisation techniques to provide near-optimal solutions in reasonable time [2]. Genetic algorithms (GA) make useof the principles of natural evolution such as, selection pressure, cross-over, and mutation to perform optimisation. After they were introduced by J. H. Holland [3] in the 1960s, they have become a widely used method for
optimisation across many fields [4]. A few studies are available on the performance of metaheuristics on porous
sound packaging optimisation. Tanneau et al. have used a GA for optimising transmission loss in aeronauticalfuselage application [5]. Lind-Nordgren et al. [6] presented an optimisation of multilayered systems with Biot’s
finite element model for porous materials [7, 8, 9] by using scaling laws to link micro and macro properties.
They show that the cost function landscapes have finite minima and also note that the optimisation greatly
depends on the cost function. They use the method of moving asymptotes (MMA) [10] to optimise specificproblem cases and have reported the optima. In this article, we give a note on the objective function landscape across different parameters of multilayered sound absorption problems modeled using a transfer matrix
method (TMM) and finite-size transfer matrix method (FTMM) solver for specific absorption problem cases,
and some applications of different metaheuristic techniques such as local search and GA on these problems.
Successful heuristics for solving sound absorption optimisation problems could also be exploited for solving

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Figure 1. Two layered sound absorption system with a fibrous porous material and a facing screen.
sound transmission loss problems.

2

TWO LAYERED POROUS AND SCREEN

Consider a two layered sound absorption system with a woven or non-woven textile or a perforated plate forthe first layer and a thick fibrous-porous material for the second layer backed by a rigid impervious wall asshown in the Figure 1. This type of system is commonly used in noise absorption packages. In order to efficiently optimize the acoustic performance of this two layered system, it is necessary to understand the objectivefunction landscape before a suitable heuristic can be chosen.
2.1 Modeling the fibrous-porous layer
To model the fibrous layer, a Delany-Bazley-Miki model [11] was considered. The model describes a given
porous material with the equations 1 and 2 for the characteristic impedance Z and wave number k.
"−0.632−0.632 #

3 f
3 f
(1)
− j8.43 10
Z = ρ0 c0 1 + 5.50 10
σf
σf
"

−0.618−0.618 #
2π f
3 f
3 f
k=
1 + 7.81 10− j11.41 10
c0
σf
σf

(2)

In the above equations, f is the frequency in Hz, σ f is the static airflow√ resistivity of the fibrous layer in
N·s·m−4 , ρ0 is the density of air, c0 is the speed of sound in air and j = −1. The properties of air computed
at T = 20◦ C were considered in all the following calculations. The surface impedance (ZM1 ) at the leading faceof the fibrous layer is then found for the oblique incidence scenario using equation 3.
ZM1 =

− jZk
k3tan(k3t f )(3)

In theqequation, t f is the thickness of the fibrous layer and k3 is the oblique wave number calculated from) 2
k3 = k2 − 2π f csin(θ, where θ is the angle of oblique incidence.
0
2.2 Modeling the screen
We consider the procedure for modeling facing screens from [12, 13] as briefed below. TMM with the complete
Johnson-Champoux-Allard model [14, 15, 16] was used to describe the fibrous porous layer. In this case, theequation 4 was used to compute the dynamic mass density of the screen (ρecs ) and the equation 5 was used toecs ).
compute its dynamic bulk modulus (K
s
"
#
α∞ ρ0
σs φs
4α∞2 ηρ0 ω
ρecs (ω) =1 + j 2 02 2
(4)
1− j
φs
ωρ0 α∞σs Λs φ3214

ecs (ω) =K

γP0 /φs
−1s02
Λ C p ρ0 ω 8η
γ − (γ − 1) 1 − j 02
1+ j sΛs C p ρ0 ω
16η


(5)

Here, γ is the adiabatic index for air, P0 is the atmospheric pressure, η is the dynamic viscosity of air, ω isthe angular frequency equal to 2π f , C p is the specific heat of air at constant pressure and φs is the porosity ofthe screen. We compute α∞ from the equation 6.
r


ε
φs
2
3 8r
α∞ = 1 + 2
where
ε = 1 − 1.13ξ − 0.09ξ + 0.27ξ
and
ξ =2
(6)
ts
3π
π
ε is the corrected length accounting forqflow distortions near the screen perforations and Λ0s = r. The valueof r is computed from the relation r = σ8η. Here, σs is the static air flow resistivity of the screen. Now,s φs
the equivalent characteristic
impedanceforthescreen (Zcs ) and equivalent wave number (kcs ) may be computedqq
ecs and kcs = ω ρ̃cs .
from the relations Zcs = ρ̃cs K
ecsK

For the two layered system, the surface impedance Zs can be computed from equation 7.
Zs = Zcs− jZM1 + Zcs tan(kcsts )ZM1 tan(kcsts ) − jZcs

(7)Here, ts is the thickness of the screen. To find the absorption, we use the formula for oblique incidence using
the expression 8.
ρ0 c0
Zs (ω) − sin(θ)(8)α(ω) = 1 −ρ0 c0
Zs (ω) + sin(θ)
2.3 Two parameter optimisation
We consider a test study where the objective function αrms is maximized for different values of σs and σ fwhile keeping the other parameters unchanged as stated in the equation 9.
s
N [α( f )]2
Σ
i
i=1

max αrms =(9)
σs ,σ fN
For this, the thickness of the fibrous-porous layer was set to 20 mm, the thickness of the facing screen to
0.5 mm, and the porosity of the screen to 0.04 and 0.90 (in two separate cases). The absorption coefficients
have been computed here for an oblique incidence of angle 34◦ (chosen arbitrarily for illustration purposes)
for frequencies from 20 Hz to 5000 Hz in linear steps of 20 Hz. The landscape of the objective function was
found to be unimodal or plateau-like in these two parameters as seen in the Figure 2. It is thus noted thata suitable local search heuristic [17] may be used for optimisation. The local search heuristics can be usedsolely or in conjunction with other metaheuristic methods as they exploit the neighbourhood of good solutions
to improvements during optimisation.
The best αrms obtained using hill climbing for fixed parameters, φs = 90%, t f = 20 mm, ts = 0.5 mm, at an
oblique incidence angle of 34◦ is 0.846 and it occurs at: σs = 19700 N·s·m−4 and σ f = 49000 N·s·m−4 .
2.4 Three parameter optimisation
The optimisation is extended to include the porosity of the screen φs as given in equation 10. When theobjective function (αrms ) landscapes are plotted versus σ f and σs similar to Figure 2 for various values of φs , it3215

Figure 2. Absorption landscape for two layered: porous material and screen, over resistivities σs and σ f for
φs = 0.04 (left) and φs = 0.90 (right) for fixed parameters t f = 20 mm, ts = 0.5 mm, at an oblique incidence
angle of 34◦ .

Figure 3. Three layered sound absorption system.
was observed that the landscapes grossly increased upto a certain φs , and then decrease, indicating the presenceof an optimum over this parameter. Another observation was that the presense of finite modes or plateau-likenature of the landscape was retained when tuning the parameter φs .
sN [α( f )]2
Σ
i
i=1

max αrms =(10)
σs ,σ f ,φs
N
For the three parameter problem, the optimum of αrms occured at a porosity of the screen value of φs = 52%,
at σs = 1 N·s·m−4 and σ f = 71100 N·s·m−4 (rounded to the nearest 100). The objective function value wasαrms = 0.862 at the optimum. It is noted that σs = 1 N·s·m−4 was the lower bound for that parameter in the
optimisation. This means that, for the studied configuration, not using a screen is better in terms of absorption,
an expected result, as the purpose of the screen is to protect the porous layer.

3

MULTI-LAYERED SOUND PACKAGE OPTIMISATIONIn this section, multilayered porous packages with three or more layers are considered. For the test studies, wewill ignore non-porous/impervious materials and consider only those which have open-porosities for the layers.
To ensure that non physical parameter combinations are avoided, a material database of 32 common porous
sound package materials is adopted. For assessing the performance of different metaheuristics in finding near
optimal sound packages, we make use of the AlphaCell solver, which computes the acoustic properties for
multilayered porous materials using TMM and FTMM, for evaluating the objective function.

3216

A genetic algorithm for three or more layered sound package:
A steady state GA for the multilayered absorption problem with diffuse field is described and their results are
discussed. This can be extended easily to sound transmission problems, or problems with other sound excitations(airborne or structure borne) or other phenomenological models for the system components. In a steady stateGA, replacement of the offspring into the population is done after every crossover operation.
Problem Representation: Although the acoustic parameters such as thicknesses can be real valued and continuous, sub-millimetre differences in thickness neither corresponds to a significant difference in acoustic properties
nor is it practical to ascertain in manufacturing the layer. Here, a discrete valued representation is proposed forthickness and other parameters, as this would reduce the search space from infinity to a finite number of configurations. As an illustration, a three layer system like shown in Figure 3 is considered with only the thickness
parameters optimised while the other parameters are fixed. The chromosome (representation of a multilayered
configuration) will be an array of 6 integers such as [10 6 15 20 20 20], where the first three integers represent
the material choice from the database of common materials and the last three integers represent allowed discrete thicknesses or simply thicknesses in millimeters. Each combination of these 6 integers will be a different
multilayered configuration. Each combination of these 6 integers will be a different multilayered configuration.
The following system gives how the chromosome integers represent the material choices:
1. The material which faces the incident wave will be chosen by the first integer. This can be any material
except for air and hence it can take values from 1 to Nm − 1, where Nm is the number of materials in the
database.
2. The second material can now be air but it should not be the same as the first material. Hence, the second
integer, selecting the second material, can also can take an integer value between 1 and Nm − 1, though
now from a slightly different set.
3. Each subsequent layer can similarly have any material which is not the same as the previous layer andthus, the corresponding integers can take integer values between 1 and Nm − 1. This is done until the lastlayer.
This representation inherently avoids using the same materials in adjacent layers. For thickness parameters, thecorresponding integers of the chromosome are simply used as thicknesses in millimeters.
Initialization: Initial population (group of multilayered configurations) is generated randomly using a pseudo
random number generator. The seed is chosen arbitrarily but then fixed, so that the GA experiments are repeatable.
Selection: In order to enforce a selection pressure for the evolution, a tournament selection was used witha tour size of 2. To implement tournament selection, two individuals (multilayered configurations) are randomly chosen from the initial population, and the individual with better fitness is chosen as the first parent forcrossover (p1 ). Then, two new individuals are randomly chosen again from the initial population and the one
with better fitness will be the other parent for crossover (p2 ). The tournament selection is repeated in successive
generations to choose parents for cross-over.
Crossover: A uniform crossover is applied to the two chosen parents from tournament selection. Every integer
from the parent chromosome is flipped with a probability of 0.5. The two new chromosomes so produced willbe the offspring which represent two new multilayered configurations which have characteristics from both theparent configurations.

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Mutation: Mutation is performed on the offspring of crossover and a mutation probability of 1/(chromosome
length) is used for each integer in the chromosome. The mutated offspring (o1 and o2 ) will then be used toevaluate the objective function which is αrms using the TMM solver.
Replacement and elitism: The two mutated offspring (o1 and o2 ) from a crossover operation and the twoparents (p1 and p2 ) which produced them are pooled, and the best two of these in terms of fitness (b1 and b2 )are chosen to replace the members in the population. Two elitist schemes, strong elitism and weak elitism, areimplemented and compared on the multilayered absorption problem under diffuse sound field. In strong elitism,
the two best individuals from the parent-offspring pool (b1 and b2 ) will replace the two weakest members of the
population (w1 and w2 ) in terms of their objective function value. Whereas, in weak elitism, the two selectedindividuals from the parent-offspring pool will replace the two parents (p1 and p2 ).
Results: The best fitnesses obtained from single trials of GA for strong and weak elitism were 0.887 and0.889 respectively. But, it would not be appropriate to conclude weak elitism is better than strong elitism fromthis single trial. In order to compare the performance of strong and weak elitism, the evolution of average ofbest fitnesses across generations is plotted for 30 GA trials in Figure 4. The initial population size was 10 andthe same initial random number seed was used for initializing strong and weak elitism trials essentially startingfrom the same initial population. We find that strong elitism in fact slightly converges faster on this problem
instance. For a total thickness constraint of 60 mm, the configuration with the highest sound absorption (diffuse
sound field) found from the trials, optimising only for the thicknesses and the material choice, is reported inTable 1. In the table, the layering order is from the material next to the rigid wall to the material facing theacoustic wave.
Table 1: The best configuration found for a total thickness constraint of 60 mm, with αrms = 0.919

4

Material

Model

Rockwool
Glasswool
FoamR1

JCAL
JCA
JCAPL

t
(mm)
19
23
18

φ
0.95
0.99
0.9

σ
(N·s·m−4 )120000
20000
7000

Λ
(µm)
12
26
129

Λ0
(µm)
20
135
440

α∞

k00
−10
(10
m2 )

α0

α00

1.08
1.11
1.12

10
83

1.22

1.13

CONCLUSIONS

A two layered porous and screen sound absorption system backed by a rigid impervious wall was characterizedusing DBM and JCA models, and the absorption performance landscape was presented. A hill climbing heuristic
was applied and the optimum parameters are reported for a specific problem case. A steady state genetic
algorithm based procedure was outlined for multilayer systems, using a TMM and FTMM solver to compute theacoustic indicators. An example of a three layered porous system was optimised. Also, a strong elitism seemed
to converge slightly faster than weak elitism over 30 trials. On studying the landscape further over different
thickness configuration for the same material configuration, the variation of the absorption performance wassmooth, continuous, with finite modes and plateaus. This study will be extended further and the effectiveness ofthe proposed approaches will be evaluated on more complicated problems to include more layers and materials,
to obtain optimal multilayered sound packaging.

ACKNOWLEDGEMENTS
This research was funded by the European Commission’s Horizon 2020 research and innovation programme
No2Noise under the Marie Skłodowska-Curie grant agreement no: 765472.

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Evolution of best fitnesses averaged over 30 trials

0.892

Strong elitism
Weak elitism

0.89

Fitness ( RMS )

0.888
0.886
0.884
0.882
0.88
0.878
0.876
0.874
0

5

10

15

Generations

Figure 4. Comparison of average of best across 30 trials between strong and weak elitism.

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