Designing phononic crystal with anticipated band gap through a deep learning based data-driven method

Available online at www.sciencedirect.com

ScienceDirect Comput. Methods Appl. Mech. Engrg. xxx (xxxx) xxx www.elsevier.com/locate/cma

Designing phononic crystal with anticipated band gap through a deep learning based data-driven method Xiang Li, Shaowu Ning, Zhanli Liu ∗, Ziming Yan, Chengcheng Luo, Zhuo Zhuang Applied Mechanics Lab., Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084 China Received 21 April 2019; received in revised form 20 September 2019; accepted 5 November 2019 Available online xxxx

Abstract Phononic crystal is a type of artificial heterogeneous material constituted by a periodic repetition of cells. This characteristic provides a possible solution to the accurate manipulation of acoustic and elastic waves. For this reason, phononic crystal is of application potentials in vibration and noise reduction, filtering, acoustic lens, acoustic imaging, and acoustic stealth, etc. It is thus of significance in the fields of information, communication, and medical applications. To design phononic crystal with anticipated manipulation characteristic has become a research hotspot in recent years. However, how to accurately manipulate acoustic and mechanical wave is still a major challenge for existing designing approaches. Assisted by image-based finite element analysis and deep learning, a data-driven approach is proposed in this study for designing phononic crystals. An autoencoder is trained to extract the topological features from sample images. Finite element analysis is employed to study the band gaps of samples. A multi-layer perceptron is trained to establish the inherent relation between band gaps and topological features. The trained models are ultimately employed to design phononic crystals with anticipated band gaps. Not limited to this material, the proposed method could be further extended to design various structured mechanical materials with specific functionalities. c 2019 Elsevier B.V. All rights reserved. ⃝ Keywords: Material design; Phononic crystal; Band gap; Deep learning; Finite element analysis

1. Introduction Micro-/nanostructured mechanical meta-materials (MNSM) has become a novel hotspot of materials science. An appealing feature of these materials is the possibility to engineer new or enhance material behaviors [1]. By purposeful structural design, these materials exhibit remarkable effective properties characteristics such as negative Poisson’s ratio, electromagnetic wave manipulation, negative dynamic modulus or density, and enhanced high strain rate behavior, etc [1]. Among various MNSM, phononic crystal is a type of artificial heterogeneous material consisted of periodic cells [2]. A cell is generally represented by a scattering inclusion embedded within a matrix [2–8], as shown in Fig. 1 [9]. Phononic crystal can be used to manipulate mechanical and acoustic waves from various perspectives [10,11], and it is of significance in the fields of information, communication, and medical applications [12]. ∗ Corresponding author.

E-mail address: [email protected] (Z. Liu). https://doi.org/10.1016/j.cma.2019.112737 c 2019 Elsevier B.V. All rights reserved. 0045-7825/⃝

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

2

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 1. The representations of phononic crystals and the dispersion curves [9].

Early researches are mainly focused on the scattering phononic crystal, which consists of inclusion and matrix. The concept of “resonant phononic crystal” was proposed in the year 2000 [13]. The scattering inclusion is normally covered by a layer of coating for this material. By comparison, the resonant phononic crystal is capable of manipulating acoustic waves with much lower frequencies and is thus of a broader application prospect. Based on it, phononic crystal materials of negative mass, negative density, and the so-named “double negative” have been further developed. These advancements extend acoustic wave manipulation from amplitude attenuation towards wave direction alteration, negative refraction, and acoustic focusing, etc [14,15]. Studies on dispersion curves have revealed that certain frequencies of acoustic waves are normally prohibited by a phononic crystal [16–20]. The prohibited frequencies are generally known as band gaps [5,6,8], as depicted in Fig. 1(b) [9]. They are found to be related to engineering parameters such as size, density, modulus, and inclusion geometry, etc [5]. The band gap can be tailored by purposeful selections of these parameters. Wang et al. [21] study to alter the band gap by introducing large deformation to phononic crystal. Huang et al. [22] investigate designing band gap by adopting models with internal resonators. Krodel et al. [23] study to broaden band gap of structures by using resonators with various eigenfrequencies. Nanthakumar et al. [24] investigate to inverse design phononic topological insulators based on quantum spin Hall effect. Nguyen et al. [25] propose a tunable quantum valley Hall phonoic topological insulator based on strain-driven mechanism. Besides these researches, topology optimization has been successfully employed to the design of phononic crystal [26,27]. Topology optimization is a computational approach to material distribution that creates structure layouts without preconceived shape information [28]. It has been applied to iteratively search for innovative material distribution within a prescribed design domain to achieve optimum structural performance [29,30]. The past thirty years have seen the substantial advancement of topology optimization approaches [29]. These approaches include density-based [30–34], level-set [35–37], phase-field [38–40], topology derivative [41,42], and evolutionary approach [43,44], etc [29]. Over the years, topology optimization has been utilized to inverse design porous, double phase, double negative phononic crystal materials and structures with various optimization objectives [26,45–50]. However, accurate manipulation of acoustic waves is still a challenge to current topology optimization methods. As pointed out by Bessa et al. [51], a major challenge to material design is the high-dimensionality of the engineering design space. The advancement of artificial neural networks and deep learning provides opportunities to tackle this challenge. The earliest study related to artificial neural network dates back to the 1940s [52]. From Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

3

Fig. 2. The workflow of designing phononic crystals with anticipated band gaps.

the 21st century, deep learning has been fundamentally developed and successfully applied to computer vision, natural language processing, biomedical engineering, material discovery, and strategy making [53–60], etc. Assisted by the back-propagation algorithm [61–63], deep learning models demonstrate outstanding fitting capabilities for situations where input and output mappings are extensively complex [55]. It has also attracted the attention of researchers in engineering mechanics in recent years. Machine learning based on the artificial neural network has been implemented in the study of the constitutive of solid materials [64–69], Kirchhoff plate analysis [70], fluid dynamics [71–76], complex heterogeneous materials [51,77–81], and solving differential equations [82–86], etc. Generative models have become a focused research area in artificial neural networks and deep learning. Common generative models include generative adversarial networks (GAN) [87], auto-encoder (AE) [62] and its variations models, such as variational auto-encoder (VAE) [88] and adversarial auto-encoder (AAE) [89]. In contrast to discriminative models, generative models aim to learn the distribution of real data to reproduce data with or without certain degrees of variations [57]. By training, meaningful latent representations can be extracted to identify abstract features of the training samples [62,88,89]. Generative models have been implemented to complex heterogeneous composite reconstructions [77,90–92], new drug discoveries [93–95], biological and genetic studies [96–98], and material designs [59], etc. Deep learning has been recently utilized in the inverse design of photonic structures and devices [99–101]. In this study, the authors propose a data-driven method to design phononic crystal with anticipated band gaps based on deep learning. Unlike conventional approaches, the proposed method does not require real-time mechanical calculation. Instead, deep learning models are trained to excavate the inherent mapping between topological structures and band gaps of phononic crystal RVEs (Representative volume element). The models are further utilized to output the corresponding topological structure given the anticipated band gap. The workflow of the study is depicted in Fig. 2. Analytical functions are employed to generate a huge number of phononic crystal RVE samples with opulent topologies. An auto-encoder is trained to extract abstract topological features from sample images. The band gaps are calculated by finite element analysis. A fully-connected multi-layer perceptron (MLP) is trained to excavate the mapping from band gaps to topological features. Combining the two networks, the inherent relation between band gaps and topological structures is established. The method can thus be utilized to output phononic crystal designs given anticipated band gaps, as shown in Fig. 2(b). Each portion of the diagram will be illustrated in the following sections. This paper is organized in the following scheme. The approach to generating phononic crystal RVE samples and the principal theory to dispersion characteristic calculation is introduced in Section 2. The fundamental theories Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

4

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

and architectural details of the adopted deep learning models are discussed in Section 3. In Section 4, the training and testing procedures are elaborated. The performance and accuracy of the proposed method are analyzed. Some characteristics of the proposed method are discussed in the final section. 2. Phononic crystal modeling and dispersion characteristics analysis As previously discussed, a data-driven method is proposed to generate phononic crystal RVE structures given anticipated band gaps. In this section, the procedure to generate RVE samples is illustrated. The theory to calculate dispersion characteristic is introduced. 2.1. Phononic crystal RVE modeling An RVE of a periodic phononic crystal material is shown in Fig. 3. The dimension of the crystal is L 0 by L 0 . It is constituted by a scattering inclusion (white color) within a matrix (black color). The outline geometry of the scattering inclusion is represented by the coordinates of (x1 , x2 ) governed by Eq. (1). { x1 = r (θ ) cos θ x2 = r (θ[) sin θ ] N (1) ∑ r (θ ) = r0 1 + ci cos (n i θ) i=1

The areas of the scattering inclusion and the phononic crystal RVE are ∫ 1 2π 2 Sc = r (θ ) dθ 2 0 and S0 = L 20

(2)

(3)

Thus, the filling ratio of the RVE is described by Sc φ= (4) S0 The distance from the maximum radius of the scattering to the boundary of the RVE needs to satisfy the following condition. tmin = L 0 − 2 max (x1 , x2 )

and

tmin > 0

(5)

In this study, phononic crystal RVE samples are generated by adopting the 4-order formulation of Eq. (1). { x1 = r (θ ) cos θ x2 = r (θ ) sin θ (6) r (θ ) = r0 [1 + c1 cos (n 1 θ ) + c2 cos (n 2 θ ) + c3 cos (n 3 θ ) + c4 cos (n 4 θ )] The values of parameters c1 to c4 range from −0.35 to 0.35, respectively. The values of n 1 to n 4 are selected among integers from 3 to 16. The filling ratio φ range from 0.1 to 0.35. Large quantities of distinguishable phononic crystal samples can be generated by appropriately assigning the parameters. Some of these samples are listed in Fig. 4. The value of n is related to the number of “extrusions”, while the value of c is related to the lengths of “extrusions”. Finite element analysis is then employed to calculate the dispersion characteristics of the generated samples, as will be illustrated in the following section. 2.2. The principal theory for analyzing dispersion characteristics In this study, a data-driven method is proposed to design the topological structures of phononic crystal RVE given anticipated band gaps. Numerous data that represent the inherent mapping between topological structures and band gaps are prerequisite to successfully training the deep learning models [57]. To generate training data, it is of importance to efficiently analyze the band gap given of a phononic crystal RVE sample. Various numerical Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

5

Fig. 3. A phononic crystal RVE and the outline coordinates of the scattering inclusion.

Fig. 4. Some phononic crystal RVE samples generated by using various parameter combinations.

approaches including plane wave expansion (PWE), finite difference time domain (FDTD), and finite element method (FEM) have been utilized to calculate band gaps [12]. FEM is employed in this study. The motion of a periodic structure is governed by the momentum equation in Eq. (7), DivS + ρ0 u¨ = 0

(7)

where S represents the first Piola–Kirchhoff stress. u represents the displacement, which is normally described by u (x, t) = u¯ (x) e−iωt

(8)

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

6

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 5. The representation of (a) the lattice vectors; (b) the simplified Brillouin zone.

In Eq. (8), u¯ denotes the magnitude of the displacement; ω is the angular frequency of the propagation wave. Eq. (8) is then substituted into Eq. (7) to obtain Eq. (9). DivS + ρ0 ω2 u¯ = 0

(9)

A function ϕ (x) can be written as the combination of a real part and an image part as ϕ (x) = ϕ (x)r e + iϕ (x)im

(10)

Thus, S and u¯ can be decomposed into a real and an image part and then substituted into Eq. (9) to obtain the following equations. { DivSr e + ρ0 ω2 u¯ r e = 0 (11) DivSim + ρ0 ω2 u¯ im = 0 To analyze the dispersion characteristics, the Bloch–Floquet periodic boundary condition is required. u¯ (x + R) = u¯ (x) ei k·R Similarly, Eq. (12) is also decomposed into a real and an image part as ( ) ( ) { re u¯ (x + R) = u¯ re (x) cos ( k0 · R) − u¯ im (x) sin (k0 · R ) u¯ im (x + R) = u¯ re (x) sin k0 · R + u¯ im (x) cos k0 · R

(12)

(13)

where k0 represents the wave number vector. Its value should be assigned within the Brillouin zone. The simplified Brillouin zone is governed by Eq. (14) ⎧ a2 × z ⎪ ⎪ ⎨ b1 = 2π ∥z∥2 z × a1 (14) ⎪ ⎪ ⎩ b2 = 2π ∥z∥2 z = a1 × a2 where a1 and a2 are the lattice vectors, as depicted in Fig. 5(a); b1 and b2 are the reciprocal lattice vectors. From Fig. 5(b), it can be observed that the simplified Brillouin zone is symmetric. Therefore, the exploration zone of wave number vector k0 is simplified to the irreducible Brillouin zone, as schemed within the yellow triangle in Fig. 5(b). In addition, k0 is only required to explore its value along the boundaries of the irreducible Brillouin zone. After the value of k0 is given, Eqs. (11) and (13) are employed to analyze the locations and widths of band gaps by finite element analysis. As previously discussed, the band gap distribution is related to the size, density, modulus, and the geometry of scattering inclusion. In this study, only the geometry of scattering inclusion is taken into consideration. The Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

7

Fig. 6. Dispersion curves and band gaps of several generated phononic crystal RVE samples.

frequency is normalized per Eq. (15) f nor mali zed =

f ·L f ·L =√ E vs ρ 2(1+ν)

(15)

where f is the natural frequency before normalization; L is the actual scale of a phononic crystal RVE; vs is shear wave velocity; E is Young’s modulus; ν is Poisson’s ratio; ρ is density. Fig. 6 depicts the calculated dispersion curves of several generated samples. Their band gaps can be further analyzed, as represented by the gray color rectangles in the figure. The essential problem to design phononic crystal RVE is to excavate the inherent mapping from the band gap distribution to the geometry of the scattering inclusion. The difficulty in exploring an analytical expression is due to the high-dimensionality of data representations [51]. Deep learning is thus adopted to establish the implicit mapping functions, as will be elaborated in the following sections. 3. Deep learning models In this section, the basic theories of the artificial neural network, convolutional neural network, and auto-encoder are introduced. The architectures of the deep learning models adopted in this study are elaborated. 3.1. Basis of artificial neural networks The structure of a traditional feedforward artificial neural network is shown in Fig. 7(a). It normally consists of an input layer, an output layer, and hidden layers. A circle in the diagram represents a neuron. A straight line that connects two neurons is weight. The activation of a neuron is calculated by {nl−1 } ∑( ) l−1 l l l σj = f w ji · σi + b j (16) i=1 Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

8

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 7. The structure of (a) a fully-connected artificial neural network; (b) a convolutional neural network.

In Eq. (16), w is weight; b is bias; l is layer number; i and j are neuron numbers; n is the number of neurons in a layer; f is an activation function. The neurons are fully-connected, which implies neurons interact with all neurons from an adjacent layer. The fully-connected structure often leads to a large weight matrix for complex network architecture. The idea of the convolutional neural network (CNN) is proposed to improve the efficiency of a machine learning system. The concept is originated from the “Neocognitron” proposed by Fukushima and Miyake [102]. The modern structure of CNN is established and refined by LeCun et al. [103]. Fig. 7(b) depicts the structure of CNN. In contrast to a fully-connected network, a unique feature of CNN is parameter sharing. It implies that weights and biases are shared by neurons. Another feature is sparse interactions, which means that a neuron only interacts with neurons of a small region from an adjacent layer. These features marginally reduce network parameters and thus boost up the training procedure [57]. The activation of a neuron in CNN is calculated by ( m m ) ∑∑ l σj = f wk,l ai+k, j+l + b (17) k=0 l=0

where m is the kernel size. The aim of an artificial network is to determine the network parameters w and b by minimizing the cost function. The cost function is adopted to identify the divergence between the network output and the data label. For example, a commonly adopted cost function is the mean square error function [104,105]. C=

n 1 ∑ ∥oi − l i ∥22 2n i=1

(18)

where n is the number of training samples. oi is the output value of the ith training sample, while l i is the label of the ith training sample. The notation ∥∥2 is the L2 norm. The parameters values are selected in a way so that the cost C should be minimized. The gradient descent algorithm is adopted to approximate the minimum of the cost function C. For illustration, C is assumed to be a function of all parameters θ . The variation of C can be approximated by the slight variation of θ as [106,107] ∆C ≈ ∇C · ∆θ

(19)

In Eq. (19), ∆θ is determined by Eq. (20) so that the cost function C is descended and approximated to a minimum. ∆θ = −η∇C

(20)

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

9

Fig. 8. An auto-encoder consists of an encoder and a decoder.

The value of the learning rate η should be discreetly selected so that the approximation in Eq. (19) holds. The value of the parameter tensor θ is iteratively updated by θ updated = θ − η∇C

(21)

3.2. Basis of the auto-encoder model The concept of auto-encoder has been within the scope of machine learning territory since the 1980s [108– 114]. An auto-encoder is a neural network that aims to replicate the input data as its output data [57]. It has been successfully employed to dimension reduction for feature learning, data compression, image denoising, and data reproducing [109,110,112,115,116]. An auto-encoder is normally comprised of an encoder and a decoder, as shown in Fig. 8. For illustration, the input data of the auto-encoder is identified by x ∈ Rdata ; the output is represented by x˜ ∈ Rdata . The encoder can be recognized as a function f that transforms the input data from the data space to the code h ∈ R f eatur e in the feature space. h = f (x)

(22)

The decoder serves as an inverse function g that transforms the code back to the data space. x˜ = g (h) = g [ f (x)]

(23)

In this study, the authors focus on the so-named “under-complete” auto-encoder, for which the dimension the feature space R f eatur e is lower than that of the data space Rdata [57]. The auto-encoder is normally trained via unsupervised learning by minimizing the cost function. The cost function is generally used to identify the distance between sample points in the data space Rdata , also known as the reconstruction error. A broadly adopted mean square error cost function is employed here for illustration. C=

n 1 ∑ ∥x i − g [ f (x i )]∥22 2n i=1

(24)

n in Eq. (24) represents the number of training samples. Via an unsupervised training, the under-complete encoder is forced to capture the essential features h of the training data. Hence, the role of the encoder f (x) is a feature extractor, while the role of the decoder g(h) is to reproduce the corresponding data sample from a feature vector h. Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

10

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 9. The architecture of the auto-encoder for extracting topological features from phononic crystal RVE samples.

3.3. Architectures of the deep learning models for phononic crystal RVE design In this study, an auto-encoder based on CNN structure is employed to extract topological features from phononic crystal RVE sample images. The architecture of the network is depicted in Fig. 9. The input x ∈ R128×128 is a 128 by 128 pixel one-channel image that represents the topology of an RVE sample. The input data is processed by the encoder Q via 5 convolutional layers to generate a feature vector f ∈ R1000×1 . The feature vector f is later transformed by 5 transposed convolutional layers of the decoder Q to generate the output image x˜ ∈ R128×128 . The function represented by the encoder Q converts an RVE sample x to its topological feature f . ( ) f = Q x; θ Q (25) where θ Q represents the parameters of encoder Q. The reverse function represented by the decoder Q reconstructs the image x˜ based on the feature f . [ ( ) ] x˜ = P( f ; θ P ) = P Q x; θ Q ; θ P (26) The auto-encoder is trained by unsupervised learning to minimize the following cost function that represents the reconstruction error. n n  [ ( ) ] 1 ∑ 1 ∑  P Q x i ; θ Q ; θ P − x i 2 ∥ x˜ i − x i ∥22 = C P;Q = (27) 2 2n i=1 2n i=1 A fully-connected multi-layer perceptron (MLP) is trained to establish the mapping from band gap distribution to topological feature. The architecture of the MLP is shown in Fig. 10. It is constructed by 6 fully-connected layers. The input of the network is the band gap distribution b. The output is the topological feature ˆf . Hence, the function represented by the MLP can be written as ˆf = F (b; θ F )

(28)

where θ F represents the parameters of the MLP. The cost function of the network F is the mean absolute error given by n n  1 ∑ 1∑   ∥Q (x i ) − F (bi ; θ F )∥1 CF = (29)  f i − ˆf i  = 1 n i=1 n i=1 where ∥∥1 represents the L1 norm. As previously mentioned, the frequency calculated by finite element analysis (FEA) is normalized by material properties and the model scale. It should be stressed that the data label for training the MLP (represented by F) is the output of the encoder Q (x i ). Only the value of Q (x i ) is used here to calculate the cost function C F and update θ F , as given in Eq. (30). Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

11

Fig. 10. The architecture of the fully-connected multi-layer perceptron.

Fig. 11. The distribution of pass and stop bands is discretized to generate a band gap vector.

The parameters of Q should not be altered here. θ F′ = θ F − η

∂C F ∂θ F

(30)

The continuous pass and stop bands distribution is discretized to generate the input of the MLP. The distribution is evenly divided into n segments, as depicted in Fig. 11(a). If the ith segment represents a pass band, the corresponding ith element of the band gap vector is filled by “1”. In contrast, if the ith segment represents a stop band, the corresponding ith element of the vector is assigned by “0”. The length n of the vector should be capable of reserving sufficient information of the band gap distribution. n is chosen to be 1000 in this study. Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

12

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 12. Random RVE samples for training the auto-encoder.

After the auto-encoder and the fully-connected MLP are trained, they can be utilized to generate phononic crystal RVE samples given anticipated band gaps per Eq. (31). The procedure is depicted in Fig. 11(b). x˜ = P( f ; θ P ) = P [F (b; θ F ) ; θ P ]

(31)

4. Results and discussions In this section, the procedures for training and testing the auto-encoder and MLP are elaborated. The errors of these deep learning models are analyzed. Then, the trained models are adopted to output the corresponding topological structures given anticipated band gaps. The performance of the method is discussed. 4.1. Training and testing the auto-encoder The auto-encoder is first trained to extract topological features from phononic crystal RVE sample images. Since finite element analysis is not involved in this process, a huge number of samples can be thus generated to train the auto-encoder. In this study, about 200,000 samples are created based on the 4-order analytical function given by Eq. (6). Besides, about 100,000 additional samples are generated using the random function in Eq. (32). Some of these random samples are depicted in Fig. 12. The random samples are adopted to ensure that abstract features can be successfully extracted from samples with opulent topological structures. The 300,000 samples are employed to train the auto-encoder. The reconstruction error versus training epoch is shown in Fig. 16(a). { x1 = r (θ ) cos θ x2 = r (θ ) sin θ (32) r (θ ) = r0 · rand(θ ) After the training procedure of the auto-encoder is finished, it is verified by reconstructing 20,000 testing samples. It should be mentioned that these 20,000 samples do not overlap with the 300,000 training samples. The approach to evaluating the error of the auto-encoder is depicted in Fig. 13(a). To calculate the testing accuracy, the reconstructed sample and the original samples are compared pixel by pixel. The number of incorrectly reconstructed pixels of a sample is identified by n err or . The number of all pixels of a sample is represented by n total . The testing error of a sample is calculated by n err or T esterr or = × 100% (33) n total Fig. 14 depicts the testing error distribution of the auto-encoder. The average testing error is 0.22%. The maximum and minimum errors are 0.50% and 0.03%, respectively. It demonstrates that the auto-encoder has successfully learned abstract features from sample topologies. It can be then adopted to train the fully-connected MLP. Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

13

Fig. 13. The procedures to evaluate errors.

4.2. Training, validating, and testing the MLP To train the MLP, about 29,000 training samples are employed. The band gap distributions of these samples are calculated based on finite element analysis. The band gap distributions are shown in Fig. 15. The x-axis is the normalized frequency; the y-axis is the sample number. White and black color represents the stop gap and the pass gap, respectively. The cost versus training epoch is shown in Fig. 16(b). The approach to evaluating the error of the MLP is depicted in Fig. 13(b). The band gaps of the testing samples are also calculated via finite element analysis. The band gap distributions are converted to vectors following the procedure shown in Fig. 11(b). The vectors are used as inputs of the MLP. The outputs of the MLP are then utilized as inputs of the decoder P. The outputs of P are the reconstructed phononic crystal RVE samples corresponding to the anticipated band gap vectors. In this study, the error between the output and input sample images, rather than the topological features, is used to evaluate the error of the MLP. The cross-validation (CV) error distribution of the fold with the maximum mean error is shown in Fig. 18(a) as an example. The mean CV error of this fold is 0.80%; the maximum and minimum CV error are 6.44% and 0.08%, respectively. The average CV error of all the folds is 0.72%. No over-fitting is observed in the cross-validation. Then, the MLP is further verified in the testing procedure. 595 samples are used in the testing procedure. The deep learning models have no pre-knowledge on the topological structures of the testing samples. The testing follows the same workflow of the CV procedure in Fig. 13(b). To evaluate the testing errors, the reconstructed samples are compared with the original testing samples. Some of the generated samples and the corresponding testing samples are listed in Fig. 17. The generated samples are mostly visually similar to the original testing samples. An interesting phenomenon is that some generated Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

14

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 14. The testing error distribution of the auto-encoder.

Fig. 15. Band gap distributions of the training samples.

samples appear to be mirror images of the original samples. It might be due to the information related to the “directions” of scattering inclusions is trimmed in the process of data abstraction. However, as an eigenvalue of the phononic crystal RVE sample, the band gap is not affected if the scattering inclusion’s direction is altered. Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

15

Fig. 16. The relation between the cost function and the training epochs of (a) the auto-encoder and (b) the MLP.

Fig. 17. Visual comparisons: the topological structures of some generated samples and the corresponding testing samples.

The error distribution of the testing samples is shown in Fig. 18(b). The mean testing error is 0.73%. The maximum and minimum testing errors are 6.23% and 0.08%, respectively. From the results, the proposed method appears to have the generalized capability to reconstruct samples based on anticipated band gaps. The deep learning models can be finally utilized to design phononic crystal RVEs. Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

16

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Fig. 18. The distributions of (a) the cross-validation errors and (b) the testing errors.

4.3. Design phononic crystal RVE based on the anticipated band gap The primary objective of this study is to design phononic crystal RVEs given anticipated band gaps. In this section, the trained deep learning models are finally utilized to output RVE structures given anticipated band gaps. Several anticipated band gap vectors are assigned. The corresponding samples are generated per the workflow in Fig. 11(b). The sample images are converted to finite element models in accordance with the workflow shown in Fig. 19. The details of this procedure are elaborated in the literature [81]. The dispersion characteristics are then calculated to obtain the actual band gap distributions. The results are compared with the anticipated band gaps, as depicted in Fig. 13(c). The topological structures of the designed phononic crystal RVEs and the corresponding dispersion curves are shown in Fig. 20(a) to (d). The anticipated band gaps are represented by red color, and the actual band gap of the designed RVEs are highlighted by blue color. The intersected regions are represented by magenta color. Based on these samples, the matches between the anticipated and the actual band gaps appear to be acceptable. 5. Conclusion A data-driven method for designing the topological structure of the two-phase phononic crystal is proposed in this study. Analytical functions are used to generate a huge amount of samples for training an auto-encoder. Abstract topological features are extracted by the auto-encoder. Finite element analysis is employed to calculate the band gaps of samples. The inherent relation between topological features and band gaps are established by training an MLP. The accuracies of the deep learning models are verified to be promising. The trained models are finally used to design phononic crystals with anticipated band gaps. Image processing, finite element analysis, and deep learning models are implemented in this study. The proposed method demonstrates acceptable performance in reverse-engineering novel phononic crystals with anticipated band Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

17

Fig. 19. The workflow to generate the finite element model based on a phononic crystal RVE sample image [81].

Fig. 20. The designed phononic crystal RVEs and the corresponding dispersion curves.

gaps. Compared to topological optimization, the proposed method does not require real-time computation for a specific optimization application. The deep learning models in this method manage to learn the implicit relationship between the input and output data. At a certain extent, the learned knowledge can accurately process data outside of the training set. Besides, continuously provided with training data, the deep learning models are able to self-evolve on the background. Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

18

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

Not limited to this material, the proposed method could be further applied to design various structured mechanical materials with specific functionalities. It is worth mentioning the limitations of the proposed method. In this research, a 9-parameter analytical function is employed to generate RVE samples. Hence, only a relatively small region of the design space is explored using this analytical function. These RVE samples are first used to verify the proposed method. For future work, the author will expand the design space using various modeling approaches and improve the performance of the proposed method. Acknowledgments This work is supported by the Science Challenge Project, China, No. TZ2018001, No. JCKY2016212A502, National Natural Science Foundation of China, under Grant No. 11722218, No. 11532008, the National Key Research and Development Program of China (No. 2017YFB0702003), Tsinghua University, China Initiative Scientific Research Program. References [1] J.H. Lee, J.P. Singer, E.L. Thomas, Micro-/nanostructured mechanical metamaterials, Adv. Mater. 24 (36) (2012) 4782–4810. [2] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (13) (1993) 2022. [3] R. Martínez-Sala, Sound attenuation by sculpture, Nature 378 (1995) 241. [4] J.H. Page, P. Sheng, H.P. Schriemer, I. Jones, X. Jing, D.A. Weitz, Group velocity in strongly scattering media, Science 271 (5249) (1996) 634–637. [5] Y. Pennec, B. Djafari-Rouhani, Fundamental Properties of Phononic Crystal, Phononic Crystals, Springer, 2016, pp. 23–50. [6] H.P. Schriemer, M.L. Cowan, J.H. Page, P. Sheng, Z. Liu, D.A. Weitz, Energy velocity of diffusing waves in strongly scattering media, Phys. Rev. Lett. 79 (17) (1997) 3166. [7] M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure, J. Sound Vib. 158 (1992) 377–382. [8] G.P. Srivastava, The Physics of Phonons, CRC press, 1990. [9] L.-Y. Zheng, Granular Monolayers: Wave Dynamics and Topological Properties, Université du Maine, 2017. [10] J. Vasseur, O.B. Matar, J. Robillard, A.-C. Hladky-Hennion, P.A. Deymier, Band structures tunability of bulk 2D phononic crystals made of magneto-elastic materials, AIP Adv. 1 (4) (2011) 041904. [11] Y. Xie, W. Wang, H. Chen, A. Konneker, B.-I. Popa, S.A Cummer, Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface, Nat. Commun. 5 (2014) 5553. [12] Y. Pennec, J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzy´nski, P.A. Deymier, Two-dimensional phononic crystals: Examples and applications, Surf. Sci. Rep. 65 (8) (2010) 229–291. [13] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials, Science 289 (5485) (2000) 1734–1736. [14] X. Hu, Y. Shen, X. Liu, R. Fu, J. Zi, Superlensing effect in liquid surface waves, Phys. Rev. E 69 (3) (2004) 030201. [15] F. Cervera, L. Sanchis, J. Sánchez-Pérez, R. Martinez-Sala, C. Rubio, F. Meseguer, C. López, D. Caballero, J. Sánchez-Dehesa, Refractive acoustic devices for airborne sound, Phys. Rev. Lett. 88 (2) (2001) 023902. [16] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (13) (1993) 2022. [17] M.S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites, Phys. Rev. B 49 (4) (1994) 2313. [18] M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems, Solid State Commun. 86 (3) (1993) 141–143. [19] J. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, M. Kushwaha, P. Halevi, Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite and some metallic systems, J. Phys.: Condens. Matter. 6 (42) (1994) 8759. [20] M. Sigalas, M.S. Kushwaha, E.N. Economou, M. Kafesaki, I.E. Psarobas, W. Steurer, Classical vibrational modes in phononic lattices: theory and experiment, Z. für Kristallographie-Crystalline Mater. 220 (9–10) (2005) 765–809. [21] P. Wang, J. Shim, K. Bertoldi, Effects of geometric and material nonlinearities on tunable band gaps and low-frequency directionality of phononic crystals, Phys. Rev. B 88 (1) (2013) 014304. [22] H. Huang, C. Sun, Continuum modeling of a composite material with internal resonators, Mech. Mater. 46 (2012) 1–10. [23] S. Krödel, N. Thomé, C. Daraio, Wide band-gap seismic metastructures, Extreme Mech. Lett. 4 (2015) 111–117. [24] S.S. Nanthakumar, X. Zhuang, H.S. Park, C. Nguyen, Y. Chen, T. Rabczuk, Inverse design of quantum spin hall-based phononic topological insulators, J. Mech. Phys. Solids 125 (2019) 550–571. [25] B.H. Nguyen, X. Zhuang, H.S. Park, T. Rabczuk, Tunable topological bandgaps and frequencies in a pre-stressed soft phononic crystal, J. Appl. Phys. 125 (9) (2019). [26] S. Halkjær, O. Sigmund, J.S. Jensen, Inverse design of phononic crystals by topology optimization, Z. Kristallogr.-Cryst. Mater. 220 (9–10) (2005) 895–905. [27] O. Sigmund, J.S. Jensen, Systematic design of phononic band–gap materials and structures by topology optimization, Phil. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 361 (1806) (2003) 1001–1019.

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

19

K. Liu, A. Tovar, An efficient 3D topology optimization code written in Matlab, Struct. Multidiscip. Optim. 50 (6) (2014) 1175–1196. O. Sigmund, K. Maute, Topology optimization approaches, Struct. Multidiscip. Optim. 48 (6) (2013) 1031–1055. M.P. Bendsøe, Optimal shape design as a material distribution problem, Struct. Optim. 1 (4) (1989) 193–202. G.I. Rozvany, M. Zhou, T. Birker, Generalized shape optimization without homogenization, Struct. Optim. 4 (3–4) (1992) 250–252. H. Mlejnek, Some aspects of the genesis of structures, Struct. Optim. 5 (1–2) (1992) 64–69. O. Sigmund, A 99 line topology optimization code written in Matlab, Struct. Multidiscip. Optim. 21 (2) (2001) 120–127. E. Andreassen, A. Clausen, M. Schevenels, B.S. Lazarov, O. Sigmund, Efficient topology optimization in MATLAB using 88 lines of code, Struct. Multidiscip. Optim. 43 (1) (2011) 1–16. G. Allaire, F. Jouve, A.-M. Toader, A level-set method for shape optimization, C. R. Math. 334 (12) (2002) 1125–1130. G. Allaire, F. Jouve, A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys. 194 (1) (2004) 363–393. M.Y. Wang, X. Wang, D. Guo, A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg. 192 (1–2) (2003) 227–246. B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization, ESAIM Control Optim. Calc. Var. 9 (2003) 19–48. A. Takezawa, S. Nishiwaki, M. Kitamura, Shape and topology optimization based on the phase field method and sensitivity analysis, J. Comput. Phys. 229 (7) (2010) 2697–2718. M. Burger, R. Stainko, Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim. 45 (4) (2006) 1447–1466. ˙ J. Sokołowski, A. Zochowski, Topological derivative in shape optimization, in: Encyclopedia of Optimization, Springer, 2001, pp. 2625–2626. J.A. Norato, M.P. Bendsøe, R.B. Haber, D.A. Tortorelli, A topological derivative method for topology optimization, Struct. Multidiscip. Optim. 33 (4–5) (2007) 375–386. Y.M. Xie, G.P. Steven, A simple evolutionary procedure for structural optimization, Comput. Struct. 49 (5) (1993) 885–896. C. Mattheck, S. Burkhardt, A new method of structural shape optimization based on biological growth, Int. J. Fatigue 12 (3) (1990) 185–190. H.-W. Dong, X.-X. Su, Y.-S. Wang, Multi-objective optimization of two-dimensional porous phononic crystals, J. Phys. D: Appl. Phys. 47 (15) (2014). H.-W. Dong, X.-X. Su, Y.-S. Wang, C. Zhang, Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm, Struct. Multidiscip. Optim. 50 (4) (2014) 593–604. H.-W. Dong, X.-X. Su, Y.-S. Wang, C. Zhang, Topology optimization of two-dimensional asymmetrical phononic crystals, Phys. Lett. A 378 (4) (2014) 434–441. Y.f. Li, X. Huang, F. Meng, S. Zhou, Evolutionary topological design for phononic band gap crystals, Struct. Multidiscip. Optim. 54 (3) (2016) 595–617. C.J. Rupp, A. Evgrafov, K. Maute, M.L. Dunn, Design of phononic materials/structures for surface wave devices using topology optimization, Struct. Multidiscip. Optim. 34 (2) (2006) 111–121. X. Zhang, J. He, A. Takezawa, Z. Kang, Robust topology optimization of phononic crystals with random field uncertainty, Internat. J. Numer. Methods Engrg. 115 (9) (2018) 1154–1173. M. Bessa, R. Bostanabad, Z. Liu, A. Hu, D.W. Apley, C. Brinson, W. Chen, W.K. Liu, A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality, Comput. Methods Appl. Mech. Engrg. 320 (2017) 633–667. W.S. McCulloch, W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys. 5 (4) (1943) 115–133. G.E. Hinton, S. Osindero, Y.-W. Teh, A fast learning algorithm for deep belief nets, Neural Comput. 18 (7) (2006) 1527–1554. G.E. Hinton, Learning multiple layers of representation, Trends Cogn. Sci. 11 (10) (2007) 428–434. Y. LeCun, Y. Bengio, G. Hinton, Deep learning, Nature 521 (7553) (2015) 436. J. Schmidhuber, Deep learning in neural networks: An overview, Neural Netw. 61 (2015) 85–117. I. Goodfellow, Y. Bengio, A. Courville, Y. Bengio, Deep Learning, MIT press Cambridge, 2016. P. Raccuglia, K.C. Elbert, P.D. Adler, C. Falk, M.B. Wenny, A. Mollo, M. Zeller, S.A. Friedler, J. Schrier, A.J. Norquist, Machine-learning-assisted materials discovery using failed experiments, Nature 533 (7601) (2016) 73. B. Sanchez-Lengeling, A. Aspuru-Guzik, Inverse molecular design using machine learning: Generative models for matter engineering, Science 361 (6400) (2018) 360–365. D. Silver, A. Huang, C.J. Maddison, A. Guez, L. Sifre, d.D.G. Van, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, Mastering the game of Go with deep neural networks and tree search, Nature 529 (7587) (2016) 484–489. Y. Le Cun, Learning Process in an Asymmetric Threshold Network, Disordered Systems and Biological Organization, Springer, 1986, pp. 233–240. D.E. Rumelhart, G.E. Hinton, R.J. Williams, Learning representations by back-propagating errors, Nature 323 (6088) (1986) 533. P. Werbos, Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences (Ph. D. dissertation), Harvard University, 1974. T. Furukawa, G. Yagawa, Implicit constitutive modelling for viscoplasticity using neural networks, Internat. J. Numer. Methods Engrg. 43 (2) (1998) 195–219. J. Ghaboussi, D.A. Pecknold, M. Zhang, R.M. Haj-Ali, Autoprogressive training of neural network constitutive models, Internat. J. Numer. Methods Engrg. 42 (1) (1998) 105–126. Y. Hashash, S. Jung, J. Ghaboussi, Numerical implementation of a neural network based material model in finite element analysis, Internat. J. Numer. Methods Engrg. 59 (7) (2004) 989–1005.

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

20

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

[67] G. Ji, F. Li, Q. Li, H. Li, Z. Li, A comparative study on Arrhenius-type constitutive model and artificial neural network model to predict high-temperature deformation behaviour in Aermet100 steel, Mater. Sci. Eng. A 528 (13–14) (2011) 4774–4782. [68] S. Jung, J. Ghaboussi, Neural network constitutive model for rate-dependent materials, Comput. Struct. 84 (15–16) (2006) 955–963. [69] Y. Sun, W. Zeng, Y. Zhao, Y. Qi, X. Ma, Y. Han, Development of constitutive relationship model of Ti600 alloy using artificial neural network, Comput. Mater. Sci. 48 (3) (2010) 686–691. [70] H. Guo, X. Zhuang, T. Rabczuk, A deep collocation method for the bending analysis of kirchhoff plate, Comput. Mater. Continua 58 (2) (2019) 433–456. [71] R. Beigzadeh, M. Rahimi, Prediction of heat transfer and flow characteristics in helically coiled tubes using artificial neural networks, Int. Commun. Heat Mass Transfer 39 (8) (2012) 1279–1285. [72] T. Butz, O. Von Stryk, Modelling and simulation of electro-and magnetorheological fluid dampers, ZAMM-J. Appl. Math. Mech. / Z. Angew. Math. Mech. Appl. Math. Mech. 82 (1) (2002) 3–20. [73] W.E. Faller, S.J. Schreck, Unsteady fluid mechanics applications of neural networks, J. Aircr. 34 (1) (1997) 48–55. [74] Y. Mi, M. Ishii, L. Tsoukalas, Flow regime identification methodology with neural networks and two-phase flow models, Nucl. Eng. Des. 204 (1–3) (2001) 87–100. [75] D. Wang, W. Liao, Modeling and control of magnetorheological fluid dampers using neural networks, Smart Mater. Struct. 14 (1) (2004) 111. [76] Z. Yuhong, H. Wenxin, Application of artificial neural network to predict the friction factor of open channel flow, Commun. Nonlinear Sci. Numer. Simul. 14 (5) (2009) 2373–2378. [77] R. Cang, H. Li, H. Yao, Y. Jiao, Y. Ren, Improving direct physical properties prediction of heterogeneous materials from imaging data via convolutional neural network and a morphology-aware generative model, Comput. Mater. Sci. 150 (2018) 212–221. [78] R. Koker, N. Altinkok, A. Demir, Neural network based prediction of mechanical properties of particulate reinforced metal matrix composites using various training algorithms, Mater. Des. 28 (2) (2007) 616–627. [79] R. Kondo, S. Yamakawa, Y. Masuoka, S. Tajima, R. Asahi, Microstructure recognition using convolutional neural networks for prediction of ionic conductivity in ceramics, Acta Mater. 141 (2017) 29–38. [80] Z. Liu, M. Bessa, W.K. Liu, Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials, Comput. Methods Appl. Mech. Engrg. 306 (2016) 319–341. [81] X. Li, Z. Liu, S. Cui, C. Luo, C. Li, Z. Zhuang, Engineering, Predicting the effective mechanical property of heterogeneous materials by image based modeling and deep learning, Comput. Methods Appl. Mech. Eng. 347 (2019) 735–753. [82] M. Dissanayake, N. Phan-Thien, Neural-network-based approximations for solving partial differential equations, Commun. Numer. Methods Eng. 10 (3) (1994) 195–201. [83] I.E. Lagaris, A. Likas, D.I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw. 9 (5) (1998) 987–1000. [84] P. Ramuhalli, L. Udpa, S.S. Udpa, Finite-element neural networks for solving differential equations, IEEE Trans. Neural Netw. 16 (6) (2005) 1381–1392. [85] J. Takeuchi, Y. Kosugi, Neural network representation of finite element method, Neural Netw. 7 (2) (1994) 389–395. [86] C. Anitescu, E. Atroshchenko, N. Alajlan, T. Rabczuk, Artificial neural network methods for the solution of second order boundary value problems, Comput. Mater. Continua 59 (1) (2019) 345–359. [87] I.J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, Y. Bengio, Generative adversarial networks, Adv. Neural Inf. Process. Syst. 3 (2014) 2672–2680. [88] D.P. Kingma, M. Welling, Auto-encoding variational bayes, 2013, arXiv preprint arXiv:1312.6114. [89] A. Makhzani, J. Shlens, N. Jaitly, I. Goodfellow, B. Frey, Adversarial autoencoders, 2015, arXiv preprint arXiv:1511.05644. [90] R. Cang, Y. Xu, S. Chen, Y. Liu, Y. Jiao, M.Y. Ren, Microstructure representation and reconstruction of heterogeneous materials via deep belief network for computational material design, J. Mech. Des. 139 (7) (2017) 071404. [91] X. Li, Y. Zhang, H. Zhao, C. Burkhart, L.C. Brinson, W. Chen, A transfer learning approach for microstructure reconstruction and structure–property predictions, 2018, arXiv preprint arXiv:1805.02784. [92] Z. Yang, X. Li, L.C. Brinson, A.N. Choudhary, W. Chen, A. Agrawal, Microstructural materials design via deep adversarial learning methodology, 2018, arXiv preprint arXiv:1805.02791. [93] H. Chen, O. Engkvist, Y. Wang, M. Olivecrona, T. Blaschke, The rise of deep learning in drug discovery, Drug Discov. Today 23 (6) (2018) 1241–1250. [94] A. Kadurin, A. Aliper, A. Kazennov, P. Mamoshina, Q. Vanhaelen, K. Khrabrov, A. Zhavoronkov, The cornucopia of meaningful leads: Applying deep adversarial autoencoders for new molecule development in oncology, Oncotarget 8 (7) (2017) 10883. [95] M.H. Segler, T. Kogej, C. Tyrchan, M.P. Waller, Generating focused molecule libraries for drug discovery with recurrent neural networks, ACS Cent. Sci. 4 (1) (2017) 120–131. [96] G. Eraslan, L.M. Simon, M. Mircea, N.S. Mueller, F.J. Theis, Single-cell RNA-seq denoising using a deep count autoencoder, Nat. Commun. 10 (1) (2019) 390. [97] C.H. Grønbech, M.F. Vording, P.N. Timshel, C.K. Sønderby, T.H. Pers, O. Winther, ScVAE: Variational auto-encoders for single-cell gene expression data, biorxiv, 2018, p. 318295. [98] A.J. Riesselman, J.B. Ingraham, D.S. Marks, Deep generative models of genetic variation capture mutation effects, 2017, arXiv preprint arXiv:.06527. [99] D. Liu, Y. Tan, Z. Yu, Training deep neural networks for the inverse design of nanophotonic structures, ACS Photonics 5 (4) (2017) 1365–1369. [100] K. Yao, R. Unni, Y. Zheng, Intelligent nanophotonics: merging photonics and artificial intelligence at the nanoscale, 2018, arXiv preprint arXiv:1810.11709.

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.

X. Li, S. Ning, Z. Liu et al. / Computer Methods in Applied Mechanics and Engineering xxx (xxxx) xxx

21

[101] M.H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, K. Parsons, Deep neural network inverse design of integrated nanophotonic devices, 2018, arXiv preprint arXiv:1809.03555. [102] K. Fukushima, S. Miyake, Neocognitron: A self-organizing neural network model for a mechanism of visual pattern recognition, in: Competition and Cooperation in Neural Nets, Springer, 1982, pp. 267–285. [103] Y. LeCun, Y. Bengio, Convolutional networks for images, speech, and time series, Handb. Brain Theory Neural Netw. 3361 (10) (1995) 1995. [104] R. Lippmann, An introduction to computing with neural nets, IEEE Assp Mag. 4 (2) (1987) 4–22. [105] B. Yegnanarayana, Artificial Neural Networks, PHI Learning Pvt. Ltd., 2009. [106] M.A. Nielsen, Neural Networks and Deep Learning, Determination Press, 2015. [107] M.H. Hassoun, Fundamentals of Artificial Neural Networks, MIT press, 1995. [108] D.H. Ballard, Modular learning in neural networks, in: AAAI, 1987, pp. 279–284. [109] P. Gallinari, Y. LeCun, S. Thiria, F. Fogelman-Soulie, Memoires associatives distribuees, Proc. Cogn. 87 (1987) 93. [110] Y. Le Cun, F.J.I. Fogelman-Soulié, Modèles Connexionnistes de L’Apprentissage, 2 (1), 1987, pp. 114–143. [111] D.E. Rumelhart, G.E. Hinton, R.J. Williams, Learning Internal Representations By Error Propagation, California Univ San Diego La Jolla Inst for Cognitive Science, 1985. [112] P. Vincent, H. Larochelle, Y. Bengio, P.-A. Manzagol, Extracting and composing robust features with denoising autoencoders, in: Proceedings of the 25th International Conference on Machine Learning, ACM, 2008, pp. 1096–1103. [113] Y. Bengio, Y. LeCun, Scaling learning algorithms towards AI, Large-scale Kernel Mach. 34 (5) (2007) 1–41. [114] Y. Bengio, Learning deep architectures for AI, Found. Trends Mach. Learn. 2 (1) (2009) 1–127. [115] L. Theis, W. Shi, A. Cunningham, F. Huszár, Lossy image compression with compressive autoencoders, arXiv preprint arXiv:.00395 (2017). [116] J. Ngiam, A. Khosla, M. Kim, J. Nam, H. Lee, A.Y. Ng, Multimodal deep learning, in: Proceedings of the 28th international conference on machine learning (ICML-11), 2011, pp. 689-696.

Please cite this article as: X. Li, S. Ning, Z. Liu et al., Designing phononic crystal with anticipated band gap through a deep learning based data-driven method, Computer Methods in Applied Mechanics and Engineering (2019) 112737, https://doi.org/10.1016/j.cma.2019.112737.