Materials & Design 207 (2021) 109855
Contents lists available at ScienceDirect
Materials & Design
journal homepage: www.elsevier.com/locate/matdes
Accelerated topological design of metaporous materials of broadband
sound absorption performance by generative adversarial networks
Hongjia Zhang a,⇑, Yang Wang a, Honggang Zhao a, Keyu Lu b, Dianlong Yu a, Jihong Wen a,⇑
a
Vibration and Acoustics Research Group, Laboratory of Science and Technology on Integrated Logistics Support, College of Intelligence Science and Technology, National University
of Defense Technology, Changsha 410073, China
b
Institute of Unmanned Systems, College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
GANs newly used for topological
design of metaporous materials for
sound absorption.
Huge acceleration (~0.04s/design) for
design process enabling
instantaneous designing.
Successful designs of broadband
sound absorption checked by
simulation and experiment.
Creative configurations and rich local
features generated in GANs-designed
patterns.
AI-guided designing/optimizing as
new possibility for AI-materials
interdiscipline.
a r t i c l e
i n f o
Article history:
Received 10 March 2021
Revised 23 May 2021
Accepted 26 May 2021
Available online 28 May 2021
Keywords:
GANs
Topological design
Metaporous sound absorption materials
AI-materials interdiscipline
a b s t r a c t
The topological design and optimization of metaporous materials is one of the key challenges in the field
of sound absorption. Limited by the expensive computational cost, it is particularly disadvantaged when
instantaneous multiple designs are required. In recent years, an increasing number of research fields are
harnessing machine learning approaches thanks to their experience-free manner and outstanding efficiency. Generative Adversarial Networks (GANs), as a type of machine learning algorithms, enjoy the special benefit of powerful generative capability, making them brilliantly suitable for designing purposes.
Additionally, it can fully explore the data distribution space with enormous computational power and
create brand new designs. In this work, GANs are newly employed for the topological design of metaporous materials for sound absorption. Trained with numerically prepared data, they successfully propose designs with high-standard broadband absorption performance, verified by simulation and
experiment. The designing process is dramatically accelerated by hundreds of times using GANs (100
designs in 4.372 s). This allows GANs to easily provide more structures and configurations, and achieve
instantaneous multiple solutions, giving designers more choices to satisfy various constraints such as
mass or porosity. In addition, GANs are demonstrated remarkably capable of generating creative configurations and rich local features. This work proposes a new designing principle, illustrates the value of
machine learning in guiding the designing and optimizing process in the mechanical world, and opens
new possibilities for the future of AI-materials interdisciplinary research.
Ó 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
⇑ Corresponding authors.
E-mail addresses: h_zhang00@163.com (H. Zhang), wenjihong@vip.sina.com (J. Wen).
https://doi.org/10.1016/j.matdes.2021.109855
0264-1275/Ó 2021 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
H. Zhang, Y. Wang, H. Zhao et al.
Materials & Design 207 (2021) 109855
dimensional architectures that approach the Hashin-Shtrikman
upper bounds [29]. Conditional GANs were used to design
nanophotonic antennae with desired optical properties [30].
In this paper, GANs are employed to carry out topological
design of metaporous materials for sound absorption in a significantly accelerated and efficient manner. Trained with numerically
prepared data, GANs are demonstrated capable of generating
designs with high-standard broadband absorption performance,
verified by Finite Element Modeling (FEM) simulation and experiment. Meantime, brand new configurations and local features are
seen in the designs by GANs, illustrating its ability of ‘creativity’
and potential to further improve the existing materials for
enhanced performance.
1. Introduction
Materials for high-efficient airborne-sound absorption have
been extensively pursued for decades. Classical porous materials,
e.g., foam, glass fiber and mineral wool, are unsuitable for low frequency sound absorption as bulky and heavy materials are
required in this case [1]. To overcome the shortcoming of the classic porous materials, using metaporous materials, i.e., a porous
matrix embedded with inner scatterers, provides a promising
alternative. The last twenty years has seen a growing number of
works devoted to the designing of metaporous materials [2–10].
However, limited by intuition, the reported designs feature mostly
regular shapes, leaving the potential of the metaporous materials
largely unexploited. As an inverse design strategy, topology optimization was capable of producing more complex designs that
achieve far more enhanced absorption at multiple target frequencies owing to the possession of different resonance mechanisms
in the same design [11]. Nevertheless, the existing topology optimization methods suffer inherent drawbacks in terms of computational time and cost due to extensive variables and numerous
iterations needed. Particularly, it becomes even more expensive if
a number of designs are required. Moreover, instantaneous designing is hardly possible. To this end, the emerging machine learning
algorithms are considered to possess great potential to substantially boost the efficiency of designing and exploring new
topologies.
The last decade has witnessed the booming of machine learning
technology. It has become not only popular in computer science
related subjects but also other fields such as mechanics, biology
and material science [12–22]. It enjoys the advantage of being an
experience-free approach as well as allowing instantaneous multiple designs simultaneously with extremely high efficiency [23]. In
addition, it has been favorable for its capability of discovering complex intrinsic pattern of large datasets with the prevailing big data
era [13].To the best of our knowledge, machine learning methods
have so far not been employed for the design of sound absorption
materials. However, we are seeing a rising trend of machinelearning-based designing of two-dimension (2D) architectures
[24]. He et al. achieved inverse design of topological metaplates
for flexural waves through finding the non-linear mapping
between the key parameters and the topology via neural networks
[23]. A framework integrating topology optimization and generative models was proposed and validated on the case of 2D wheel
design using Boundary Equilibrium Generative Adversarial Networks (GANs) [25]. A convolutional neural network (CNN)-based
encoder and decoder network together with conditional GANs
was employed as a two-stage refinement to realize near-optimal
topological design [26].
Most of the existing works on topological design or optimization adopt basic convolutional neural networks. Those algorithms
are usually effective at indirectly designing the structure by deciding its individual parameters (such as the length of an edge or the
radius of a circle), but can hardly make a sophisticated design
based on complete patterns. As a contrast, our work can directly
generate topological patterns using a generative network - GANs.
GANs feature a generator and a discriminator that learn simultaneously in competition with each other [27,28]. The generator can
generate brand new data (e.g. the topological structure of metaporous materials in this paper) based on the captured potential
distribution of the training data. Since the generator is extraordinarily powerful computationally, it can thoroughly explore the
data space and produce creative designs that potentially exceed
the limitation of the existing ones. Previous works have successfully employed GANs to obtain over 400 complex two-
2. Dataset
The training set prepared with FEM simulation consists of 1832
images of a a = 64 64 pixels. Each image represents a pattern
for metaporous material. Typical patterns are shown in Fig. 1
where the solid black domain denotes the scatterer and the white
area is filled with porous media. The patterns are stored as binary
images composed of 0 and 1, corresponding to porous media and
scatterer, respectively. The shapes of the scatterer in the training
set are composed of three types of basic primitives, i.e., slotted
cube, slotted elliptic cylinder and thin partitions. For the slotted
cube, the lower (upper) bounds of the outer side lengths and slot
width are 0.4a (0.8a) and 0.08a (0.48a), respectively. The minimum
(maximum) lengths of outer semi-axis and slot are 0.2a (0.4a) and
0.08a (0.48a) for the slotted elliptic cylinder. As for the thin partition, the lower and the upper bounds of the length are respectively
0.2a and 0.8a. The thickness for all three primitives is no larger
than 0.0625a. The generation procedure for all the patterns starts
from a porous media (64 64 null matrix). The scatterer is subsequently ‘embedded’ into the porous media at a random location,
imitating the mutation process in the genetic algorithm [31], i.e.,
some pixels of porous media (0) are changed to that of scatterer
(1). To increase the diversity of the patterns, a perturbation operator is applied to the pattern where the pixels inside the scatterer
are reversed with 20% probability. Finally, in order to improve
the geometric topology and eliminate checkerboard pattern, the
abuttal filter is employed [32], i.e., some isolated porous elements
are changed to scatterer elements and vice versa.
Fig. 1. Typical patterns in the training set.
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Fig. 2. Schematic diagram of the full-wave simulation model
solid line at the right end is the hard acoustic boundary. The porous
media in the metaporous material is melamine foam as in our previous work [9]. The acoustic parameters of the melamine foam are
described using Johnson-Champoux-Allard (JCA) model [33] in
Poroacoustics Domain, as listed in Table 1, where U is the porosity,
r the flow resistivity, a1 the tortuosity, K the viscous length and K0
the thermal characteristic length. The sample frequencies in the
simulation are from 300 Hz to 3000 Hz at the interval of 50 Hz.
Since GANs generate new patterns based on what they are feed
with, we prepared 1832 patterns whose average absorption coefficients are larger than 0.9 to form the training set, aiming to obtain
patterns with at least equally high absorption performance.
The patterns in the training data contain up to three primitives.
Fig. 1(a)-(d) plots those with only one primitive. The employment
of perturbation operator during the generation process leads to the
non-smooth peripheries of the slotted scatterer as well as the discrete dotted black pixels inside. It is also the reason for the existence of the closed cylinder shown in Fig. 1(c). The patterns
consisting two or three primitives are shown in Fig. 1(e)-(l). For
instance, Fig. 1(e) and Fig. 1(g) show the patterns with two primitives of the same type. The examples for multiple primitives with
different types are given in Fig. 1(k) and (l). Particularly, we shall
pay attention to the correlation between primitives in the same
pattern. They can be simply apart as in Fig. 1(g), half intersected
in Fig. 1(f) or internally tangent in Fig. 1(h).
To obtain the absorption coefficients of each pattern, full-wave
simulation is performed via Pressure Acoustic and Frequency
domain module with the commercial finite element software
COMSOLÒ. As shown in Fig. 2, a plane wave is incident to the pattern with the amplitude of 1 Pa from the background pressure
field. Perfectly matched layer (PML) is applied at the left boundary
of the background pressure field to simulate anechoic termination
for outgoing wave (i.e., reflective wave). The horizontal black
dashed lines denote the periodic boundary while the green black
3. Methods
3.1. Generative adversarial networks
GANs feature a pair of networks, a generator G and a discriminator D, competing with each other. As a two-player zero-sum
game, G aims to generator ‘fake’ data that are as ‘real’ as possible
to ‘fool’ G, while G learns to discriminate fake data generated by
G from real ones as accurately as possible. This is essentially a minmax process where G and D can both respectively sharpen their
generative and discriminative skills [34]. Importantly, G does not
directly access real images and it learns only through interacting
with D [28]. Consequently, G learns to generate usable data (patterns in our case) for the purpose of the specific model setup. Eventually, a Nash equilibrium is reached, meaning each player cannot
obtain a better result without changing the other players’ strategy,
and two players stay equally good [35].
Table 1
Acoustical parameters of the melamine foam for FEM simulation.
U
a1
K
K0
r
0.95
1.42
180 lm
360 lm
8900N s m4
Fig. 3. Schematic illustration of the design procedures of metaporous materials with GANs.
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Materials & Design 207 (2021) 109855
accelerated tensor computation and performance at the same time.
With packages from Pytorch, procedures in deep learning such as
network building and gradient descent can be programed with a
couple of code lines, making it far more efficient.
In general, the architectures of G and D are described as follows.
The input of G is a greyscale image with 64 64 size. G contains
ten convolutional layers. Each convolutional layer is followed by
a batch normalization layer and a ReLU layer. The output of G is
still a grey image with the same size as the input one. D takes
the grey image generated by G as the input. D consists of five convolutional layers and a softmax layer. Each convolutional layer is
also followed by a ReLU layer. The output of D is a binary classification label which indicates if the generated image is legitimate or
not.
3.2. Design procedures
Fig. 3 demonstrates the procedure to design and verify the
metaporous materials. Three main steps are necessary for a complete and reliable design: 1) training data preparation; 2) generating designs using trained GANs; 3) verify those designs with FEM
simulation and experiments.
The left part of Fig. 3 shows the design procedures of metaporous materials with GANs in this work. The model is built with
Pytorch. Pytorch is a commonly used open-source deep learning
framework, basically a library that provides building blocks for
designing, training and validating deep neural networks via a
high-level programing interface [36,37]. Being a python library,
Pytorch enjoys the advantages of possessing both GPU-
Fig. 4. Designs proposed by GANs and their absorption performance simulated by FEM: (a) I: designs 1–4; (b) II: designs 5–8; (c) Absorption coefficients of Design 1–4; (d)
Absorption coefficients of Design 5–8.
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Specifically, with a random noise (64 64 size) as input, G outputs a ‘fake’ pattern (64 64 size) which is subsequently input
into D together with the ‘real’ patterns – the training data. D judges
whether the generated pattern is legitimate by comparing it with
training data. With the advancement of the minmax process, G
finally learns to generate designs that are comparable with the
training data. Delightfully, patterns generated this way possess
new features since G is computationally extremely powerful to
explore the full distribution of the data space (i.e., full
possibilities).
Note that the criterion of D’s judgement on whether a pattern is
legitimate or not is actually learned via training, instead of adopting a handcrafted criterion. The criterion is essentially represented
by the weights in the convolutional kernels that are constantly
applied to the input images to extract features at each convolutional layer in the neural networks. Starting with a random D with
a cross-entropy loss that calculates the difference between the output pattern of the G with the input pattern, D’s aim is to maximize
the loss while G’s aim is to minimize it, which forms a minmax
‘competition’. D’s capability of judging whether the generated pattern is legitimate is basically equal to its capability to maximize the
loss, which relies on the choice of weights. The same logic goes to
G’s capability of generating legitimate patterns. During training,
the ‘competition’ advances, where D and G continuously adjusts
their own weights based on the loss until a balance is reached
(i.e. the Nash equilibrium). Eventually, a capable D is obtained with
a set of learned weights to represent the criterion that is good at
‘judging’.
4. Results
4.1. GANs-designed patterns and numerical verification
GANs bring extreme acceleration of the designing process,
enhancing the efficiency by hundreds of times compared with
Table 2
Average absorption coefficient for designs.
Design 1
Design2
Design3
Design4
Design5
Design6
Design7
Design8
0.925
0.929
0.930
0.926
0.925
0.931
0.927
0.927
Fig. 5. Pressure distribution inside Design 3 at the characteristic frequencies of: (a) f1 = 950 Hz; (b) f2 = 1300 Hz; (c) f3 = 1960 Hz; (d) f4 = 2600 Hz.
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Materials & Design 207 (2021) 109855
It is illustrated that GANs are skilled at generating and alternating local features. Being greatly capable at manipulating input
information, it can comprehensively assemble different shapes/
primitives in creative configurations. For instance, the model has
learned from Fig. 1(h) that one primitive can be inside another
one. Meanwhile, it has captured several different primitives out
of the training data including slotted cylinders and thin rigid partitions. Design 5 in Fig. 4(b) is a result of putting thin partitions
inside a slotted cylinder as well as making some modification
and variation on the cylinder’s periphery. Hence, we reckon that
GANs are capable of capturing different shapes in the training data,
treating them as separate features and generating brand new combinations and create rich variation on the features themselves.
Particularly noteworthy is the position of the opening on the
cavity. The training data only contain patterns with opening positioned towards the incident end of the sound wave, i.e., the top of
the pattern. However, GANs have created new opening positions
towards the backing (the bottom of the pattern) as shown in
Design 3 and 4 in Fig. 4(a). This operation is of great importance
for sound absorption and is often highlighted in the existing nonmachine-learning-based designing methods [39].
the existing approaches such as density-based approach and
genetic algorithm [38]. The generation of 100 patterns takes
approximately only 4.372 s (with commonly used Nvidia
RTX3090) once the model is well trained. More details regarding
the time needed for data preparation and model training are
described in Appendix A.
More than a hundred designs of metaporous materials are proposed by GANs out of which eight typical ones are presented in
Fig. 4. The height and the width of a pattern are both 64 mm
(and 64 pixels). So, one pixel inside the pattern has the size of
1 mm 1 mm. Overall, all the designs can be classified into two
categories: I - multiple slotted cavities with different orientations
and creative variation on the periphery (Design 1–4 in Fig. 4(a));
II - a single slotted cavity with large opening and rich local features
inside (Design 5–8 in Fig. 4(b)). Note that ‘annulus’ here is a general term describing the circle-like or rectangle-like outlines in
the pattern. Additionally, to differentiate GAN-generated patterns
from the training data, patterns below are highlighted in blue
(scatterer) and pink (foam).
The FEM-calculated absorption coefficient curves of the eight
designs are illustrated in Fig. 4(c) and (d). Overall, all the eight
designs show broadband absorption capability. The average
absorption coefficients (between 300 Hz and 3000 Hz with the
interval of 50 Hz) are listed in Table 2. It is shown that the average
absorption coefficients of all designs are larger than 0.925, which is
a rather high standard.
4.2. Broadband absorption mechanism
To explore the mechanism behind the broadband absorption
performance, we choose Design 3 and Design 8 as representatives
Fig. 6. Pressure distribution inside Design 8 at the characteristic frequency of (a) f1 = 930 Hz; (b) f2 = 1280 Hz; (c) f3 = 1820 Hz; (d) f4 = 2400 Hz.
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Fig. 7. Sample preparation process.
Design 3 shows a different coupled mode at the frequency of f4.
Notable acoustic scattering is observed between the two slotted
cylinders near the incident end and partial acoustic energy also
enters the left slotted cylinder. It could be concluded that the collaboration of resonance, acoustic trapping and scattering effect
contributes to the overall broadband absorption of Design 3.
Fig. 6(a)-(d) further illustrates the pressure distributions of
Design 8 at four representative frequencies. At low frequency f1,
of the two categories of designs for deeper examination. Fig. 5(a)Fig. 6(d) plot the pressure distribution of Design 3 at four characteristic frequencies. It is noted that the two slotted cylinders are
sole ‘active’ at the frequencies of f1 and f3, i.e., maximum pressure
is located at the two slotted cylinders and resonance state exists
near the two frequencies. At the frequency of f2, the pressure is
trapped between the two slotted cylinders and the backing, while
both two slotted cylinders are simultaneously ‘active’. However,
Fig. 8. Experimental setup: (a) schematic illustration; (b) photo.
Fig. 9. Experimental results compared with FEM simulation for (a) Design 3 and (b) Design 8.
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Materials & Design 207 (2021) 109855
approach and alternating primitives (as well as adding local features) for the assembly to form a new design. The other is that
GANs have created new opening positions for the design, which
is of crucial significance in practice.
Overall, this work has for the first time utilized GANs to design
metaporous materials for sound absorption. The accelerated
designing process and the satisfying results demonstrate that AImaterials interdisciplinary approach is of excellent potential and
importance in guiding the designing and optimizing process in
the mechanical world.
the main acoustic energy enters the full big cavity while partially
trapped between the cavity and the backing. At frequencies f2
and f4, it is interesting to note that two different local resonance
modes are triggered by the thin partitions inside the cavity. The
two local resonance modes together with the scattering outside
the cavity also contribute to the sound absorption of Design 8 at
frequency of f3. Different form Design 3, Design 8 possesses more
complex local modes inside the cavity, thanks to the rich local features of thin partition generated by GANs.
4.3. Experimental verification
Declaration of Competing Interest
Fig. 7 illustrates the process of sample preparation. Overall, the
metaporous material is composed of three components, i.e., a cover
layer, an inner scatterer and melamine foam. The inner scatterer
and the cover layer are 3D printed with polylactic acid (PLA) with
the parameters of density q = 1250 kg/m3, Young’s modulus
E = 1.2GPa and Poisson’s ratio v = 0.35. The thickness of the cover
layer is 1 mm. Compared with the size of the cubic sample 60 mm,
the influence of the cover layer is neglectable. To facilitate the
embedding of the scatterer, a pattern with the shape of the scatterer is etched into the melamine foam. The metaporous material
is an assembly of all the components. Note that the sound wave
is incident into the top face along the -y direction shown in Fig. 7.
The experiment is carried out in a home-made impedance tube
using the two-microphone method [40]. As shown in Fig. 8(a), the
impedance tube features a square cross-section with the side
length of d = 60 mm. The sample is attached to a rigid backing.
The distances between the sample and the two microphones are
L1 = 200 mm and L2 = 150 mm, respectively.
Design 3 and Design 8 are chosen for experimental verification.
Fig. 9(a) and (b) compare results obtained by the experiment and
FEM simulation. Overall, a good agreement is reached. However,
some discrepancies still exist, especially for Design 8. We consider
that the mismatch is a result of two reasons: (1) the uncertainty of
fabrication for both 3D-printed frame and the etched melamine
foam; (2) the inaccuracies occurred during the assembling
process.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (Grant No. 52001325, 51775549, 11991032 and
11991030).
Appendix A. Computation time
The 1832 patterns with >0.9 average absorption coefficients are
selected from a larger database with 7636 patterns in total via
examining all their average absorption coefficients. That means
we use COMSOL (full-wave simulation) to calculate the average
absorption coefficient of all the 7636 patterns. Each calculation
takes about 3 s, making it 22,908 s (that’s roughly 382 min or
6.36 h) in total. Note that the 7636 patterns are generated following the generation procedure described in ‘section 2 Dataset’,
which is rather fast. The generation of 7636 patterns takes around
234.7 s (roughly 3.9 min) in total. So, the preparation of the training dataset of 1832 patterns takes approximately 386 min
(~6.43 h), including 2 parts: the generation of 7636 patterns
(3.9 min) and the COMSOL FEM calculation of 7636 patterns
(382 min).
Model training time is related to the GPU computation power.
With NVIDIA RTX3090, the whole training takes about 8 h. The
training process can easily speed up via training in parallel on multiple GPUs.
In total, the pre-designing stage (including training set preparation and model training) need <15 h.
On the other hand, based on our experience of the existing
topology optimization methods, one generation averagely takes
about 3 s and it needs around 200 generations for the algorithm
to converge to give out a design, i.e. 600 s for one pattern. Therefore, to design 100 well-performed patterns with genetic algorithms takes about 1000 min (~16.7 h), which is already longer
than the GANs-based approach. The above comparison is only for
designing 100 patterns. When more patterns are needed, the
advantage of GANs-based approach becomes a lot more obvious
as it only needs multiple seconds (for generation) while the existing methods need hours.
Therefore, the GANs-based approach has dramatically accelerated the designing procedure especially when instantaneous multiple solutions are needed.
5. Conclusions
In this work, GANs are used to design metaporous materials for
sound absorption. The training set is prepared with FEM simulation, containing 1832 binary images of 64 64 pixels. With
Pytorch framework, trained GANs can generate 100 patterns in
roughly 4.372 s, bringing tremendous acceleration of the designing
process, boosting the efficiency by hundreds of times compared
with the existing approaches. Eight typical designs proposed by
GANs are chosen to be verified by FEM simulation, out of which
two are printed out with a 3D printer for experimental evaluation.
The advantage in the computation time enables this approach to
provide more structures and configurations, and achieve instantaneous multiple solutions with high absorption performance, compared to a unique solution as the case with the existing methods.
The results illustrate that GANs are capable of generating metaporous material designs with satisfying broadband absorption performance. The FEM simulation of the eight designs demonstrated
that their average absorption coefficients are all larger than
0.925. The experiments again confirm the quality of the two 3Dprinted designs. Additionally, the numerical and experimental
results show satisfying agreement.
As to the generative capability, GANs are demonstrated skilled
at capturing input information and generating brand new configurations and rich local features comprehensively. Two typical creations of GANs’ are observed. One is to learn a new assembling
Data availability
The raw/processed data required to reproduce these findings
cannot be shared at this time as the data also forms part of an
ongoing study.
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