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Non-iterative structural topology optimization using deep learning
Computer-Aided Design 115 (2019) 172–180
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Computer-Aided Design journal homepage: www.elsevier.com/locate/cad
Non-iterative structural topology optimization using deep learning✩ , ✩✩ Baotong Li a , Congjia Huang a , Xin Li b , Shuai Zheng c , Jun Hong a ,
∗
a
Key Laboratory of Education Ministry for Modern Design & Rotor-Bearing System, Xi’an Jiaotong University, China School of Electrical Engineering and Computer Science, Louisiana State University, United States c school of software engineering, Xi’an Jiaotong University, China b
article
info
Article history: Received 25 April 2019 Accepted 26 May 2019 Keywords: Topology optimization Deep learning Generative adversarial network Hierarchical refinement Heat conduction
a b s t r a c t This paper presents a non-iterative topology optimizer for conductive heat transfer structures with the help of deep learning. An artificial neural network is trained to deal with the black-and-white pixel images and generate near-optimal structures. Our design is a two-stage hierarchical prediction– refinement pipeline consisting of two coupled neural networks: a generative adversarial network (GAN) for predicting a low resolution near-optimal structure and a super-resolution generative adversarial network (SRGAN) for predicting the refined structure in high resolution. Training datasets with given boundary conditions and the optimized pixel image structures are obtained after simulating a big amount of topology optimization procedures. For more effective training and inference, these datasets are generated with two different resolutions. Experiments demonstrated that our learning based optimizer can provide accurate estimation of the conductive heat transfer topology using negligible computational time. This effective incorporation of deep learning into topology optimization could enable promising applications in large-scale engineering structure design. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Effective design of integrated and small-scale electronic equipment presents a critical requirement on efficient heat conduction simulation. With the increasing cooling requirements, developing effective cooling technologies for the heat-generating volumes or surfaces within a limited space [1–5] has become an important task and attracted significant attentions. The cooling effectiveness is related to both high-quality cooling materials and the topological structure of the material distribution [6]. The layout optimization can be treated as a material distribution problem, and the topology optimization has become a powerful tool for structure design. The classical topology optimization algorithms mainly include the solid isotropic microstructures with penalization (SIMP) method, the Homogenization design method (HDM), the evolutionary structural optimization (ESO) method and level set (LST) ✩ This paper has been recommended for acceptance by Pierre Alliez, Yong-Jin Liu & Xin Li. ✩✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.cad. 2019.05.038. ∗ Corresponding author. E-mail address: [email protected] (J. Hong). https://doi.org/10.1016/j.cad.2019.05.038 0010-4485/© 2019 Elsevier Ltd. All rights reserved.
method, etc [7]. Although the mathematical principles used in the above methods are different, they have played an extremely important role in several fields as effective tools. The above methods are based on gradient algorithms and finite element calculation, they can accurately obtain the optimal material distribution in different tasks. In order to ensure near optimal structures, we have to increase the number of design iterations and variables, which results that large-scale topology optimization still comes at a high computational cost. Especially in three-dimensional problem, the computational cost can be beyond our limit. To overcome this limitation, tremendous efforts have been spent on reducing the computational cost in solving topology optimization. Borrval and Petersson [8] propose to solve the large-scale topology optimization in 3D using parallel computing combined with domain decomposition. Wu et al. [9] present a system equipped with a high-performance GPU solver to handle models comprising several millions of elements efficiently. Jang and Kwak [10] suggest a new topology optimization method called design space optimization to reduce the computing time significantly. Kim et al. [11] design an adaptive genetic algorithm to adjust the number of design variables adaptively during the topology optimization. Bruns and Tortorelli [12] remove low-density elements from a finite element model systematically to reduce the computing cost. Nguyen et al. [13] present a framework for improving multiresolution topology optimization via multiple distinct discretization. Groen and Sigmund [14] use
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a projection method to obtain high-resolution, manufacturable structures from efficient and coarse-scale homogenization-based topology optimization results, and this significantly reduces the cost of computation. Recently, deep learning has been successfully applied in various fields, such as self-driving cars, natural language processing, computer visions, healthcare, and financial sector. In many engineering problems, deep learning has also been applied. Snyder et al. [15] use machine learning to solve density functions. Brockherde et al. [16] use learning to study kinetic energy function, which is necessary to solve the Kohn–Sham equations. Mills et al. [17] train a deep neural network to predict an electron ground-state energy in four classes confining two-dimensional electrostatic potentials. The use of deep learning can be effective to circumvent the expensive computation in topology optimization. Inspired by the application of deep learning in computational physics and engineering, we propose a new topology optimization algorithm, or more specifically, prediction algorithm based on learned generative models, to more efficiently solve conductive heat transfer. The main idea is to replace the iterative optimization using an integrated pre-trained network. This network observes a large amount of topology optimization simulations, and it will predict the heat conduction structure when the boundary condition is given. To our knowledge, this is the first attempt to directly predict near-optimal structure under different boundary conditions. Our method can significantly reduce the computation cost of topology optimization.
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2.2. Topology optimization through deep learning Using deep learning for topology optimization has not been widely explored yet but has caused great interest in the past few years. Ulu et al. [30] explored the feasibility and performance of a data-driven approach to structural topology optimization problems. Lei and Guo et al. [31] intended to achieve real time structural topology optimization using machine learning (ML) techniques under Moving Morphable Component (MMC)-based framework. The MMC framework was used to generate training dataset. The supported vector regression (SVR) and the K-nearestneighbors (KNN) model are employed to establish the mapping between the design parameters and the external load. Sosnovik and Oseledets [32] employed the deep convolutional networks to map the intermediate structure obtained in each iteration to the final result. This approach can truly speed up the convergence process, but the deep convolutional networks are only used for classification and regression and cannot predict a new structure that does not exist in the datasets. The generative adversarial networks (GANs) have been explored for generating new structures in design domain [33,34]. Rawat et al. [35] presented a new topology optimization process to generate near-optimal structure using an integrated GANs and convolutional neural network architecture with 3024 samples of training datasets. However, their GANs cannot generate the structure according to boundary conditions. Our work can be considered as an extension of this design, where a novel topology optimization approach based on the integration of generative adversarial networks (GAN) and superresolution generative adversarial network (SRGAN) is developed to achieve computationally inexpensive topology optimization.
2. Related work 3. Problem formulation and basic idea 2.1. Classic topology optimization algorithms 3.1. Problem model Different methods have been developed to optimize the distribution and layout of a limited amount of high conductive material. In the past years, there are several notable alternative approaches for topology optimization. The solid isotropic microstructures with penalization (SIMP) method [18] can insure the constraint conditions and minimize the objective function simultaneously through changing the material densities of elements which are allowed to be either 0 or 1. Homogenization design method (HDM) [19] can gain the optimal design in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes. Evolutionary structural optimization (ESO) method [20] was proposed for shape and layout optimization of structure, where the low-value material can be progressively eliminated from the structure. Level set (LST) method [21] described the optimal topology through extracting the boundary of structure using the level set function. Additionally, there are other several method for topology optimization, such as Moving Morphable Components (MMC) Approach [22,23]. Topology optimization has been extensively applied to diverse optimization problems such as thermal engineering problems in a variety of industries [24–28]. Particularly, the SIMP method has been adopted broadly as a standard approach of topology optimization for heat conduction [29]. It can obtain the optimal material distribution in the design space under the prescribed boundary condition, and improve the system heat cooling performance significantly. However, it has an inherent limitation of its big computation cost due to a large number of design variables and iterations during the optimization process.
The volume-to-point problem in heat conduction has attracted considerable attention recently. To obtain a physically meaningful solution for the heat conduction, the SIMP solves a continuous optimization. The material distribution is described by a density variable, xj , which represents the material density from 0 to 1. The optimized structure is described by the pixel values defined on the design domain. The optimization is formulated as: minC =
N ∑
x
j=1
Ej (xj )tTj ktj ,
(1)
s.t. V (x)/V0 = f , KT = P , xj ∈ [0, 1], j = 1...N , where C is the thermal compliance, and the objective function to minimize. Ej (xj ) is the thermal conductivity for element j. tj is a temperature vector of element j. K and T are the global conductivity matrix (with components k) and the global temperature vectors, respectively. P is the global heat flow vector. V (x) and V0 are the material volume and design domain volume, respectively. f is the prescribed volume ratio. The computational cost of conventional topology optimization will increase dramatically when the resolution of design domain increases. As shown in Fig. 1(a), the iterative variable update, finite element solving, and sensitivity calculation often result in expensive and slow computation. In particular, the sensitivity calculation and finite element solution for the huge amount of design variables, require a large amount of computing resources. Hence, most existing continuous or combinatorial topology optimization algorithms cannot effectively handle such big data.
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Fig. 1. The comparison between the calculation process of the SIMP method and the proposed method. (a) is the optimization process of the SIMP method, (b) is the hierarchical prediction process of the proposed method.
3.2. Our basic idea As shown in Fig. 1(b), our idea is to develop a learning network to approximate the iterative solver, to predict the near-optimal structures through a non-iterative procedure which is significantly faster and requires much less online computing resource. To predict the near-optimal structure from existing examples, training a generative model is a natural and effective strategy. We can synthesize a large amount of training data by performing simulations. Finally, we aim to train a generative model to synthesize an optimal structure when a new boundary condition is given. Unlike the conventional generative models that directly build/train a generator function, a GAN simultaneously trains a generator G and a discriminator D that competes with each other. The generator G is responsible for generating new data that are undistinguishable from the sample data, while the discriminator D is a binary classifier to examine whether the data is ‘‘fake’’ (generated by G) or ‘‘real’’ (directly taken from sampling data). G and D improve each other iteratively, so that the final generator function G produces desirable data. To learn the distribution of the training data, Pg , it is necessary to predefine the latent random variables to be mapped to a data space G(z , θg ), where the G is a multi-layered perception generation network taking a latent variable z as input and θg are G’s network parameters. In addition, the discriminator D judges whether an image fed to it is made by the generator or from the training data. Simultaneously, D is trained to identify generated data, whereas G is trained to make its generated data consistent with the training data. The objective function for training the GAN is min max L(D, G) = Ex∼P(x) [logD(x)] + Ez ∼P(z) [log(1 − D(G(z)))] (2) G
D
where P(x) and P(z) are the probability density functions of the real data x and the latent random variables z, respectively. 4. Detailed approach Our design is illustrated in Fig. 2. First, a large volume of the heat conduction datasets with different boundary conditions are prepared through simulation using the conventional topology optimization method [36]. Each training data example is prepared (simulated) in two resolutions: one in low resolution and one in high resolution. First, we train a generative model using a GAN network to generate a low-resolution image under the given boundary constraint. Meanwhile, such a low-resolution structure is not fine enough for feasible cooling structure design. Therefore, next, we perform a refinement on this low-resolution structure
Fig. 2. Flow chart of the proposed method. The illustration of the proposed method. The preparation of training and testing datasets are shown firstly, then a two-phase prediction is implemented through step 1 and step 2, the performance evaluation is implemented finally.
image by doing a super-resolution enhancement. This step is also done by a GAN network [37] trained using the pairs of low-resolution structure and its corresponding high-resolution structure. 4.1. Datasets preparation for training and testing Encoding boundary/initial conditions. We need to encode the boundary or initial condition of the heat conduction into the latent variables. In the heat conduction problem, the boundary condition contains three main parameters: heat sink position, the heat source position, and the mass fraction. As illustrated in Fig. 3, the heat sink, heat source, and mass fraction are transformed into three matrices (i.e., three channels), and these channels are initialized to zero. At the phase of low-resolution (40 × 40) structure prediction, the input vectors are three 41 × 41 matrices, and the optimized/predicted structure is a 40 × 40 matrix. In the first and second channels, the value of 1 indicates the positions of heat sink and heat source. The values in the third channel is the mass fraction in the topology optimization. Generating training and testing data. We randomly generate a large amount of training and testing data with various boundary
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Fig. 3. The three main boundary conditions are encoded into three matrices (channels). (a) shows the encoding of heat source: the value of 1 denotes where the heat is generated, while the value 0 indicates that no heat is generated there. (b) shows the encoding of heat sinks: the value 1 indicates positions of heat sink. (c) shows the encoding of mass fraction: the value in the matrix indicates the material retention (mass fraction).
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Fig. 4. A standard normal distribution for latent variables. The boundary conditions include the heat sources, heat sinks, and mass fractions follow a multi-dimensional normal distribution. The training and testing datasets are prepared according to this fitted distribution.
conditions to ensure the generality of our method and avoid overfitting in the training. In particular, pseudo-random sampling is used to generate the heat sinks and heat sources. The N1 individual variables of the heat sink and N2 variables of heat source are sampled from the following distributions: N1 ∼ P(λ = 2),
(3)
N2 ∼ P(λ = N0 /3),
(4)
where P is the Poisson distribution, λ is the rate parameter, and N0 is the number of total pixel in the design domain. The mass fraction is sampled following a normal distribution: N3 ∼ N(µ, σ ), µ = 0.3, σ = 0.2,
(5)
where N is the normal distribution, µ and σ are the mean and standard deviation of the normal distribution. We set these parameters λ, µ, and σ following the experiments of [32]. When the boundary conditions are coded into latent variable fields, the latent variables of the final predicted results can be fitted into a standard normal distribution, as shown in Fig. 4. Then, the training and testing datasets are prepared according to these standard normal distributions. We prepare the datasets at two resolution levels. The design domain is discretized into the low 40 × 40 resolution and the high 160 × 160 resolution. The average time needed to generate a single optimized (simulated) structure using the conventional SIMP algorithm (a 88-line code) are 0.8 s and 18 s at the low and high resolutions, respectively. Finally, a total of 10,000 optimized structures are generated. The entire datasets are partitioned into training and testing datasets with a 99 : 1 ratio. 4.2. Predicting optimal structure at low resolution We train a GAN network to predict the optimized structure in low resolution. Figs. 5 and 6 illustrate the detailed GAN network design. The GAN consists of a generator G that generates an optimal structure image according to the input and a discriminator D that evaluates whether a given structure is a ‘‘real’’ optimized (groundtruth) structure or a ‘‘fake’’ one, i.e., a predicted structure
Fig. 5. The Architecture of G (Generator Network) in the first GAN for low-resolution optimal structure prediction.
generated by G. An effective G should generate structures that are similar to groundtruth and difficult for D to identify. In the generator, with the heat sink, heat source and mass fraction encoded into input vectors , the input vectors are fed into the generator network G through a series of convolution and deconvolution process to generate the predicted structure. As shown in Fig. 5, the convolution process includes one concatenation layer, six convolution layers, and three max-pooling layers. The deconvolution process includes nine convolutional layers and three upsampling layers. In addition, the rectified linear unit (ReLU) activation function was used in each convolution layer of the generator, and the output layer of the generator uses the Sigmoid function which can squeeze the result to the range of 0 to 1. Similarly, the architecture of the discriminator D includes seven convolution layers and two dense layers as shown in Fig. 6. Its output layer is a dense layer with a Sigmoid function. During the training process, the ADAM (adaptive moment estimation) optimization algorithm [38] is used to optimize the weights and biases of the networks to minimize the loss between the predicted structures and the simulated structures. The ADAM can dynamically adjust the learning rate of each parameter using the estimation of the first and second moment of the gradient. The loss of the generator G and the discriminator D are measured
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Fig. 8. The Architecture of GS (Generator Network) in the SRGAN for high-resolution optimal structure prediction.
Fig. 6. The Architecture of D (Discriminator Network) in the first GAN for low-resolution optimal structure prediction.
In particular, the training of DS and GS is formulated as DS maximizing its accuracy of classification (differentiating ‘‘real’’ simulated structure from a ‘‘fake’’ one), while GS generating as real as possible structures from given boundary conditions to fool DS . The objective function of the SRGAN is: min max V (DθD , GθG ) θG
θD
= EI HR ∼PD (I HR ) [logDθD (I HR )] + EI LR ∼PG (I LR ) [log(1 − DθD (I LR ))]
Fig. 7. The reduction of loss during the training of the first GAN that predicts low-resolution structures from given boundary conditions.
by the Mean Squared Error (MSE). Fig. 7 plots the reduction of MSE loss at the low resolution during the training. 4.3. Low-resolution to high-resolution structure refinement To refine the predicted low-resolution structure and get a high-resolution structure, we apply a super-resolution technology, because directly training a GAN to predict the high-resolution structure from given boundary conditions is difficult and often yields unstable results. Super-resolution is to use a low-resolution image to restore its most likely high-resolution image. It is recently an active research topic in the image processing and computer vision fields with broad applications in satellite imagery, surveillance, and medical imaging. One of the state-of-the-art and widely used super-resolution approach is the SRGAN [37], which is also a trained generative adversarial network (GAN). The generator GS of the SRGAN is trained to generate a highresolution structure image from a low-resolution structure image. The SRGAN’s discriminator DS again distinguishes whether a high-resolution structure is from a ‘‘real’’ simulation or a ‘‘fake’’ one generated by GS . Through the competition between the generator and discriminator during the training of SRGAN, with the help of DS , the GS can produce high-resolution structures similar to the simulated results.
(6)
where I LR and I HR are the low- and high-resolution structure images, DθD and GθG are the discriminator network DS and generator network GS , respectively. θG and θD are the weights and biases of GS and DS , respectively. For more detail about the design of SRGAN, please refer to [37]. A simplified SRGAN network is built in this paper to refine the low-resolution structure. Our network design is shown in Fig. 8. To achieve better refinement, seven residual networks, which are the cores of the generator network, are applied in GS . Every residual network includes two convolution layers with 3 × 3 convolution kernels. Two identity skip-connections are conducted to feed the features back into the convolutional layers for cycling to enhance the structure feature. Three convolutional layers and two upsampling layers are then alternately performed to expand the feature size and refine the resolution. Finally, a convolutional layer with 5 × 5 filter and the Sigmoid activation function is implemented to generate the high-resolution structure. As for DS , the real and fake 160 × 160 high-resolution structure images are fed to the DS network in two channels, as shown in Fig. 9. Then, ten convolution layers with different filters and strides are implemented alternatively. Finally, a fully connected layer and a dense layer with Sigmoid activation function are used for the true/false classification. To simplify the network model and reduce the calculation, the loss function is simplified to: LR LSR = lSR MSE + λE(log(1 − DθD (GθG (I )))),
(7)
and lSR MSE =
M N 1 ∑∑
MN
2
LR (RHR m,n − Gθ G (I )m,n ) ,
(8)
m=1 n=1
where I LR is the input low-resolution structure, λ is the loss weight empirically set 0.001, M , N are the horizontal and vertical pixel numbers, respectively, and RHR m,n is the pixel value at coordinate (m, n).
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is first set to the average gray-scale value. Then, we get a binarized hashing image through a simple thresholding: if the pixel value is smaller than Hthreshold , its hash value is set to 0, otherwise its hash value is set to 1. The similarity of two structures is then measured by their hashing images. Specifically, Hthreahold =
{ Hm,n =
M N 1 ∑∑
MN
1 0
Pm,n ,
(10)
m=1 n=1
if Pm,n ≥ Hthreahold , if Pm,n < Hthreahold
(11)
where Hthreshold is the threshold of the pixel value, Pm,n is the pixel value at (m, n), and M , N are the horizontal and vertical pixel numbers, respectively. Finally, the similarity between two structures a and b, is then measured by Fig. 9. The Architecture of DS (Discriminator Network) in the SRGAN for high-resolution optimal structure prediction.
Shash = 1 −
M N 1 ∑∑
MN
2
b a (Hm ,n − Hm,n ) ,
(12)
m=1 n=1
a b where Hm ,n and Hm,n are the hash values of structures a and b.
5. Experimental results
Fig. 10. The reduction of loss during the training of our SRGAN.
During the training process, the ADAM optimization algorithm [38] is used to optimize the parameters in the networks. Totally 9,900 pairs of low- and high-resolution data samples were used to train this networks. The plot of loss value in the training of SRGAN is shown in Fig. 10. 4.4. Evaluating performance of our networks We integrate the low-resolution image prediction GAN and the low-to-high resolution image refinement SRGAN and evaluate their performance. Each testing image sample and its boundary conditions are fed to this integrated network, and the output, a refined structure image, is compared with the optimized (simulated) structure image. To evaluate the similarity of predicted and optimized structures, we use two criteria: the MSE accuracy score and the perceptual hashing accuracy [39]. The MSE accuracy is defined as
SMSE = 1 −
M N 1 ∑∑
MN
(ϵm,n )2 ,
(9)
m=1 n=1
where M , N are the horizontal and vertical pixel numbers, and ϵm,n is the deviation between the predicted pixel value and the groundtruth value at pixel (m, n). The perceptual hashing algorithm generates a series of strings as the fingerprints for a structure. The similarity is measured by the similarity of the fingerprints. Specifically, a threshold Hthreshold
We compare our results with SIMP both qualitatively (by showing the images side-by-side) and quantitatively (using S in Eq. (12)), to evaluate our algorithm’s accuracy and efficiency. All experiments were performed on an Intel i7-8750H CPU with 8 GB RAM and NVidia GeForce GTX 1060 GPU. One hundred testing data of the classic volume-to-point heat conduction problem with randomly generated boundary conditions are used for evaluation. Fig. 11 illustrates an example where the heat sink is at the bottom center of the design field. The optimized structures at low and high resolutions are shown in (a) and (c), respectively. The predicted structures at both resolutions are shown in (b) and (d), correspondingly, and they are very similar to the simulated results. For quantitative measures of prediction accuracy, at the low resolution, the accuracy is 95.7% for the perceptual hash algorithm and 99.5% for the MSE. At the high resolution, the accuracy is 96.8% and 90.9% for the perceptual hash algorithm and the MSE, respectively. We also compare the prediction time needed for SIMP and our proposed method. The calculation time for generating the high-resolution structure using SIMP method is 22 s, while our two-phase network takes only 0.002 s. Note that we need a one-time offline training, which currently takes around 4 h. Fig. 12 shows more examples of predicted structures together with their corresponding simulated results. Sixteen testing data with random boundary conditions are used. We can see that the predicted structure and optimized structure are visually similar in both resolutions. The quantitative evaluation is summarized in Table 1 where the computational time and prediction accuracy are reported. The time to generate the high-resolution structure (160 × 160) using conventional method and proposed method are 23 s and 0.006 s, respectively. The time cost using our network is only 0.026% of the conventional SIMP method. As for the accuracy, the structure similarity is again evaluated using both MSE and perceptual hashing. The MSE and perceptual hashing error of the presented method is 1.3% 5.2%, respectively. Table 2 shows the perceptual hashing accuracy of the high-resolution predicted structures corresponding to these specific examples of different boundary conditions. We observe most predicted structures have prediction accuracy above 91%. One-phase prediction using GAN versus two-phase prediction using GAN + SRGAN. Fig. 13 shows a comparison of the structure prediction done by (1) a direct one-phase approach,
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B. Li, C. Huang, X. Li et al. / Computer-Aided Design 115 (2019) 172–180 Table 1 Comparison of computational efficiency and prediction accuracy between the conventional SIMP algorithm, the direct one-phase prediction using our trained GANs, and our proposed two-phase pipeline. The experiments run on 100 randomly sampled testing data. Average run-time (in seconds) Structural similarity
SIMP
Our method
One-phase prediction
18 s
0.006 s 98.7% (MSE)/94.8% (hash)
0.004 s 84.6% (MSE)/80.3%(hash)
Table 2 Accuracy of the predicted structures in high resolution. Sixteen randomly generated boundary conditions (BC) are used and their perceptual hashing accuracy scores are reported. BC
Accuracy
BC
Accuracy
1 2 3 4 5 6 7 8
91.8% 92.5% 90.6% 92.6% 91.7% 91.2% 90.8% 93.6%
9 10 11 12 13 14 15 16
91.1% 93.3% 91.5% 90.7% 91.3% 92.6% 93.4% 92.8%
Fig. 11. The comparison between the optimized and predicted structure. A intuitive similarity between the optimized structure and predicted structure under low and high resolution is shown.
Fig. 12. Comparison of optimized (groundtruth) structures and our predicted structures in both low and high resolutions. 16 random examples are shown.
where a GAN is trained directly to predict a high-resolution structure, and (2) our two-phase GAN + SRGAN strategy. We also use random samples from the testing dataset and eight examples are shown here. We can see that the proposed two-phase
Fig. 13. Comparing the structures generated using the direct one-phase prediction and structures generated using the proposed two-phase approach. The SIMP simulation (a) serves as the groundtruth, compared with the direct onephase prediction (b), the presented two-phase prediction–refinement strategy (c) produces structures with better accuracy and more detail. The accuracy is compared using MSE (d): the red and blue bars are MSE accuracy of the one-phase and two-phase strategies, respectively.
method consistently outperforms the direct one-phase prediction in accuracy (MSE). Comparison with other Traditional Topology Optimization Algorithms. Besides SIMP, many other iterative methods have been designed for efficient simulation on large-scale topology
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optimization problems. Notable recent approaches include the scalable GPU-enhanced solver [9], the design space optimization with design space adjustment and refinement [10], the reducible design variable method (RDVM) [11], and coarse-to-fine structure projection [14]. Unfortunately, because open-source programs are unavailable for these methods, we were unable to perform side-by-side quantitative comparisons with them on canonical datasets. Although the above methods can accelerate the topology optimization on large-scale data to a certain extent, they rely on iterative numerical optimization to search for the optimal solution, and the inevitable iterative finite element calculation still requires big amount of computing resources and is very time consuming. In contrast, our approach establishes a direct mapping from the initial boundary conditions to the final optimal solutions, avoiding the computational cost from the iterative simulations. Our pipeline produces comparable accuracy with these methods but clearly outperforms them in efficiency. 6. Conclusions We propose an efficient non-iterative topology optimization pipeline using deep learning for heat conduction structure design. Our pipeline contains two main steps: (1) near-optimal structure prediction in coarse resolution, and (2) structure refinement to obtain final structure in fine resolution. Both steps are done through trained Generative Adversarial Networks (GANs). To train the two GANs, training datasets consisting of low-resolution and high-resolution optimized structures are prepared through standard simulations of topology optimization. Boundary conditions, such as the heat sink, heat source, and the mass fraction are encoded using three matrices, and are fed to the integrated network. In coarse resolution, a classical GAN network is trained to predict the optimal structure given the boundary constraint. Then to obtain fine resolution optimal structure, a simplified SRGAN is trained to perform super-resolution and enhance the detail of the structure in the fine resolution. We evaluate the performance of this proposed method using testing data under various different boundary conditions. Experimental results demonstrated that this method can produce near-optimal heat conduction structure in a very efficient way. Compared with the conventional topology optimization methods, this approach has obvious advantages in the computation efficiency. Meanwhile, this pipeline is general and these learning modules can be adopted to predict near-optimal structural topology in other tasks. The framework is not limited to 2D. Our recent exploration of generalizing it to 3D topology optimization also shows very promising results. We will explore this in the near future. Limitations. The proposed method is a two-phase pipeline based on low-resolution prediction followed by a super-resolution refinement. The direct prediction runs stably in low resolutions (e.g. in our implementations/experiments, 40 × 40 pixels). The super-resolution refinement is currently restricted within 4 times, and too big upsampling factors could lead to unstable and inaccurate refinement. Therefore, if predictions on high-resolution design domains (e.g. 800 × 800) are needed, then our current two-phase pipeline may not provide enough prediction accuracy. A potential strategy is to develop a more effective integrated hierarchical or iterative prediction–refinement scheme. We will explore this design in the near future. Meanwhile, adopting other state-of-the-art GAN models and super-resolution schemes may also help improve our system’s performance. Acknowledgments The work reported in this paper is supported by the National Key R&D Program of China [2018YFB1700703], the National Natural Science Foundation of China [51822507] and [61728206].
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Computer-Aided Design journal homepage: www.elsevier.com/locate/cad
Non-iterative structural topology optimization using deep learning✩ , ✩✩ Baotong Li a , Congjia Huang a , Xin Li b , Shuai Zheng c , Jun Hong a ,
∗
a
Key Laboratory of Education Ministry for Modern Design & Rotor-Bearing System, Xi’an Jiaotong University, China School of Electrical Engineering and Computer Science, Louisiana State University, United States c school of software engineering, Xi’an Jiaotong University, China b
article
info
Article history: Received 25 April 2019 Accepted 26 May 2019 Keywords: Topology optimization Deep learning Generative adversarial network Hierarchical refinement Heat conduction
a b s t r a c t This paper presents a non-iterative topology optimizer for conductive heat transfer structures with the help of deep learning. An artificial neural network is trained to deal with the black-and-white pixel images and generate near-optimal structures. Our design is a two-stage hierarchical prediction– refinement pipeline consisting of two coupled neural networks: a generative adversarial network (GAN) for predicting a low resolution near-optimal structure and a super-resolution generative adversarial network (SRGAN) for predicting the refined structure in high resolution. Training datasets with given boundary conditions and the optimized pixel image structures are obtained after simulating a big amount of topology optimization procedures. For more effective training and inference, these datasets are generated with two different resolutions. Experiments demonstrated that our learning based optimizer can provide accurate estimation of the conductive heat transfer topology using negligible computational time. This effective incorporation of deep learning into topology optimization could enable promising applications in large-scale engineering structure design. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Effective design of integrated and small-scale electronic equipment presents a critical requirement on efficient heat conduction simulation. With the increasing cooling requirements, developing effective cooling technologies for the heat-generating volumes or surfaces within a limited space [1–5] has become an important task and attracted significant attentions. The cooling effectiveness is related to both high-quality cooling materials and the topological structure of the material distribution [6]. The layout optimization can be treated as a material distribution problem, and the topology optimization has become a powerful tool for structure design. The classical topology optimization algorithms mainly include the solid isotropic microstructures with penalization (SIMP) method, the Homogenization design method (HDM), the evolutionary structural optimization (ESO) method and level set (LST) ✩ This paper has been recommended for acceptance by Pierre Alliez, Yong-Jin Liu & Xin Li. ✩✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.cad. 2019.05.038. ∗ Corresponding author. E-mail address: [email protected] (J. Hong). https://doi.org/10.1016/j.cad.2019.05.038 0010-4485/© 2019 Elsevier Ltd. All rights reserved.
method, etc [7]. Although the mathematical principles used in the above methods are different, they have played an extremely important role in several fields as effective tools. The above methods are based on gradient algorithms and finite element calculation, they can accurately obtain the optimal material distribution in different tasks. In order to ensure near optimal structures, we have to increase the number of design iterations and variables, which results that large-scale topology optimization still comes at a high computational cost. Especially in three-dimensional problem, the computational cost can be beyond our limit. To overcome this limitation, tremendous efforts have been spent on reducing the computational cost in solving topology optimization. Borrval and Petersson [8] propose to solve the large-scale topology optimization in 3D using parallel computing combined with domain decomposition. Wu et al. [9] present a system equipped with a high-performance GPU solver to handle models comprising several millions of elements efficiently. Jang and Kwak [10] suggest a new topology optimization method called design space optimization to reduce the computing time significantly. Kim et al. [11] design an adaptive genetic algorithm to adjust the number of design variables adaptively during the topology optimization. Bruns and Tortorelli [12] remove low-density elements from a finite element model systematically to reduce the computing cost. Nguyen et al. [13] present a framework for improving multiresolution topology optimization via multiple distinct discretization. Groen and Sigmund [14] use
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a projection method to obtain high-resolution, manufacturable structures from efficient and coarse-scale homogenization-based topology optimization results, and this significantly reduces the cost of computation. Recently, deep learning has been successfully applied in various fields, such as self-driving cars, natural language processing, computer visions, healthcare, and financial sector. In many engineering problems, deep learning has also been applied. Snyder et al. [15] use machine learning to solve density functions. Brockherde et al. [16] use learning to study kinetic energy function, which is necessary to solve the Kohn–Sham equations. Mills et al. [17] train a deep neural network to predict an electron ground-state energy in four classes confining two-dimensional electrostatic potentials. The use of deep learning can be effective to circumvent the expensive computation in topology optimization. Inspired by the application of deep learning in computational physics and engineering, we propose a new topology optimization algorithm, or more specifically, prediction algorithm based on learned generative models, to more efficiently solve conductive heat transfer. The main idea is to replace the iterative optimization using an integrated pre-trained network. This network observes a large amount of topology optimization simulations, and it will predict the heat conduction structure when the boundary condition is given. To our knowledge, this is the first attempt to directly predict near-optimal structure under different boundary conditions. Our method can significantly reduce the computation cost of topology optimization.
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2.2. Topology optimization through deep learning Using deep learning for topology optimization has not been widely explored yet but has caused great interest in the past few years. Ulu et al. [30] explored the feasibility and performance of a data-driven approach to structural topology optimization problems. Lei and Guo et al. [31] intended to achieve real time structural topology optimization using machine learning (ML) techniques under Moving Morphable Component (MMC)-based framework. The MMC framework was used to generate training dataset. The supported vector regression (SVR) and the K-nearestneighbors (KNN) model are employed to establish the mapping between the design parameters and the external load. Sosnovik and Oseledets [32] employed the deep convolutional networks to map the intermediate structure obtained in each iteration to the final result. This approach can truly speed up the convergence process, but the deep convolutional networks are only used for classification and regression and cannot predict a new structure that does not exist in the datasets. The generative adversarial networks (GANs) have been explored for generating new structures in design domain [33,34]. Rawat et al. [35] presented a new topology optimization process to generate near-optimal structure using an integrated GANs and convolutional neural network architecture with 3024 samples of training datasets. However, their GANs cannot generate the structure according to boundary conditions. Our work can be considered as an extension of this design, where a novel topology optimization approach based on the integration of generative adversarial networks (GAN) and superresolution generative adversarial network (SRGAN) is developed to achieve computationally inexpensive topology optimization.
2. Related work 3. Problem formulation and basic idea 2.1. Classic topology optimization algorithms 3.1. Problem model Different methods have been developed to optimize the distribution and layout of a limited amount of high conductive material. In the past years, there are several notable alternative approaches for topology optimization. The solid isotropic microstructures with penalization (SIMP) method [18] can insure the constraint conditions and minimize the objective function simultaneously through changing the material densities of elements which are allowed to be either 0 or 1. Homogenization design method (HDM) [19] can gain the optimal design in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes. Evolutionary structural optimization (ESO) method [20] was proposed for shape and layout optimization of structure, where the low-value material can be progressively eliminated from the structure. Level set (LST) method [21] described the optimal topology through extracting the boundary of structure using the level set function. Additionally, there are other several method for topology optimization, such as Moving Morphable Components (MMC) Approach [22,23]. Topology optimization has been extensively applied to diverse optimization problems such as thermal engineering problems in a variety of industries [24–28]. Particularly, the SIMP method has been adopted broadly as a standard approach of topology optimization for heat conduction [29]. It can obtain the optimal material distribution in the design space under the prescribed boundary condition, and improve the system heat cooling performance significantly. However, it has an inherent limitation of its big computation cost due to a large number of design variables and iterations during the optimization process.
The volume-to-point problem in heat conduction has attracted considerable attention recently. To obtain a physically meaningful solution for the heat conduction, the SIMP solves a continuous optimization. The material distribution is described by a density variable, xj , which represents the material density from 0 to 1. The optimized structure is described by the pixel values defined on the design domain. The optimization is formulated as: minC =
N ∑
x
j=1
Ej (xj )tTj ktj ,
(1)
s.t. V (x)/V0 = f , KT = P , xj ∈ [0, 1], j = 1...N , where C is the thermal compliance, and the objective function to minimize. Ej (xj ) is the thermal conductivity for element j. tj is a temperature vector of element j. K and T are the global conductivity matrix (with components k) and the global temperature vectors, respectively. P is the global heat flow vector. V (x) and V0 are the material volume and design domain volume, respectively. f is the prescribed volume ratio. The computational cost of conventional topology optimization will increase dramatically when the resolution of design domain increases. As shown in Fig. 1(a), the iterative variable update, finite element solving, and sensitivity calculation often result in expensive and slow computation. In particular, the sensitivity calculation and finite element solution for the huge amount of design variables, require a large amount of computing resources. Hence, most existing continuous or combinatorial topology optimization algorithms cannot effectively handle such big data.
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Fig. 1. The comparison between the calculation process of the SIMP method and the proposed method. (a) is the optimization process of the SIMP method, (b) is the hierarchical prediction process of the proposed method.
3.2. Our basic idea As shown in Fig. 1(b), our idea is to develop a learning network to approximate the iterative solver, to predict the near-optimal structures through a non-iterative procedure which is significantly faster and requires much less online computing resource. To predict the near-optimal structure from existing examples, training a generative model is a natural and effective strategy. We can synthesize a large amount of training data by performing simulations. Finally, we aim to train a generative model to synthesize an optimal structure when a new boundary condition is given. Unlike the conventional generative models that directly build/train a generator function, a GAN simultaneously trains a generator G and a discriminator D that competes with each other. The generator G is responsible for generating new data that are undistinguishable from the sample data, while the discriminator D is a binary classifier to examine whether the data is ‘‘fake’’ (generated by G) or ‘‘real’’ (directly taken from sampling data). G and D improve each other iteratively, so that the final generator function G produces desirable data. To learn the distribution of the training data, Pg , it is necessary to predefine the latent random variables to be mapped to a data space G(z , θg ), where the G is a multi-layered perception generation network taking a latent variable z as input and θg are G’s network parameters. In addition, the discriminator D judges whether an image fed to it is made by the generator or from the training data. Simultaneously, D is trained to identify generated data, whereas G is trained to make its generated data consistent with the training data. The objective function for training the GAN is min max L(D, G) = Ex∼P(x) [logD(x)] + Ez ∼P(z) [log(1 − D(G(z)))] (2) G
D
where P(x) and P(z) are the probability density functions of the real data x and the latent random variables z, respectively. 4. Detailed approach Our design is illustrated in Fig. 2. First, a large volume of the heat conduction datasets with different boundary conditions are prepared through simulation using the conventional topology optimization method [36]. Each training data example is prepared (simulated) in two resolutions: one in low resolution and one in high resolution. First, we train a generative model using a GAN network to generate a low-resolution image under the given boundary constraint. Meanwhile, such a low-resolution structure is not fine enough for feasible cooling structure design. Therefore, next, we perform a refinement on this low-resolution structure
Fig. 2. Flow chart of the proposed method. The illustration of the proposed method. The preparation of training and testing datasets are shown firstly, then a two-phase prediction is implemented through step 1 and step 2, the performance evaluation is implemented finally.
image by doing a super-resolution enhancement. This step is also done by a GAN network [37] trained using the pairs of low-resolution structure and its corresponding high-resolution structure. 4.1. Datasets preparation for training and testing Encoding boundary/initial conditions. We need to encode the boundary or initial condition of the heat conduction into the latent variables. In the heat conduction problem, the boundary condition contains three main parameters: heat sink position, the heat source position, and the mass fraction. As illustrated in Fig. 3, the heat sink, heat source, and mass fraction are transformed into three matrices (i.e., three channels), and these channels are initialized to zero. At the phase of low-resolution (40 × 40) structure prediction, the input vectors are three 41 × 41 matrices, and the optimized/predicted structure is a 40 × 40 matrix. In the first and second channels, the value of 1 indicates the positions of heat sink and heat source. The values in the third channel is the mass fraction in the topology optimization. Generating training and testing data. We randomly generate a large amount of training and testing data with various boundary
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Fig. 3. The three main boundary conditions are encoded into three matrices (channels). (a) shows the encoding of heat source: the value of 1 denotes where the heat is generated, while the value 0 indicates that no heat is generated there. (b) shows the encoding of heat sinks: the value 1 indicates positions of heat sink. (c) shows the encoding of mass fraction: the value in the matrix indicates the material retention (mass fraction).
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Fig. 4. A standard normal distribution for latent variables. The boundary conditions include the heat sources, heat sinks, and mass fractions follow a multi-dimensional normal distribution. The training and testing datasets are prepared according to this fitted distribution.
conditions to ensure the generality of our method and avoid overfitting in the training. In particular, pseudo-random sampling is used to generate the heat sinks and heat sources. The N1 individual variables of the heat sink and N2 variables of heat source are sampled from the following distributions: N1 ∼ P(λ = 2),
(3)
N2 ∼ P(λ = N0 /3),
(4)
where P is the Poisson distribution, λ is the rate parameter, and N0 is the number of total pixel in the design domain. The mass fraction is sampled following a normal distribution: N3 ∼ N(µ, σ ), µ = 0.3, σ = 0.2,
(5)
where N is the normal distribution, µ and σ are the mean and standard deviation of the normal distribution. We set these parameters λ, µ, and σ following the experiments of [32]. When the boundary conditions are coded into latent variable fields, the latent variables of the final predicted results can be fitted into a standard normal distribution, as shown in Fig. 4. Then, the training and testing datasets are prepared according to these standard normal distributions. We prepare the datasets at two resolution levels. The design domain is discretized into the low 40 × 40 resolution and the high 160 × 160 resolution. The average time needed to generate a single optimized (simulated) structure using the conventional SIMP algorithm (a 88-line code) are 0.8 s and 18 s at the low and high resolutions, respectively. Finally, a total of 10,000 optimized structures are generated. The entire datasets are partitioned into training and testing datasets with a 99 : 1 ratio. 4.2. Predicting optimal structure at low resolution We train a GAN network to predict the optimized structure in low resolution. Figs. 5 and 6 illustrate the detailed GAN network design. The GAN consists of a generator G that generates an optimal structure image according to the input and a discriminator D that evaluates whether a given structure is a ‘‘real’’ optimized (groundtruth) structure or a ‘‘fake’’ one, i.e., a predicted structure
Fig. 5. The Architecture of G (Generator Network) in the first GAN for low-resolution optimal structure prediction.
generated by G. An effective G should generate structures that are similar to groundtruth and difficult for D to identify. In the generator, with the heat sink, heat source and mass fraction encoded into input vectors , the input vectors are fed into the generator network G through a series of convolution and deconvolution process to generate the predicted structure. As shown in Fig. 5, the convolution process includes one concatenation layer, six convolution layers, and three max-pooling layers. The deconvolution process includes nine convolutional layers and three upsampling layers. In addition, the rectified linear unit (ReLU) activation function was used in each convolution layer of the generator, and the output layer of the generator uses the Sigmoid function which can squeeze the result to the range of 0 to 1. Similarly, the architecture of the discriminator D includes seven convolution layers and two dense layers as shown in Fig. 6. Its output layer is a dense layer with a Sigmoid function. During the training process, the ADAM (adaptive moment estimation) optimization algorithm [38] is used to optimize the weights and biases of the networks to minimize the loss between the predicted structures and the simulated structures. The ADAM can dynamically adjust the learning rate of each parameter using the estimation of the first and second moment of the gradient. The loss of the generator G and the discriminator D are measured
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Fig. 8. The Architecture of GS (Generator Network) in the SRGAN for high-resolution optimal structure prediction.
Fig. 6. The Architecture of D (Discriminator Network) in the first GAN for low-resolution optimal structure prediction.
In particular, the training of DS and GS is formulated as DS maximizing its accuracy of classification (differentiating ‘‘real’’ simulated structure from a ‘‘fake’’ one), while GS generating as real as possible structures from given boundary conditions to fool DS . The objective function of the SRGAN is: min max V (DθD , GθG ) θG
θD
= EI HR ∼PD (I HR ) [logDθD (I HR )] + EI LR ∼PG (I LR ) [log(1 − DθD (I LR ))]
Fig. 7. The reduction of loss during the training of the first GAN that predicts low-resolution structures from given boundary conditions.
by the Mean Squared Error (MSE). Fig. 7 plots the reduction of MSE loss at the low resolution during the training. 4.3. Low-resolution to high-resolution structure refinement To refine the predicted low-resolution structure and get a high-resolution structure, we apply a super-resolution technology, because directly training a GAN to predict the high-resolution structure from given boundary conditions is difficult and often yields unstable results. Super-resolution is to use a low-resolution image to restore its most likely high-resolution image. It is recently an active research topic in the image processing and computer vision fields with broad applications in satellite imagery, surveillance, and medical imaging. One of the state-of-the-art and widely used super-resolution approach is the SRGAN [37], which is also a trained generative adversarial network (GAN). The generator GS of the SRGAN is trained to generate a highresolution structure image from a low-resolution structure image. The SRGAN’s discriminator DS again distinguishes whether a high-resolution structure is from a ‘‘real’’ simulation or a ‘‘fake’’ one generated by GS . Through the competition between the generator and discriminator during the training of SRGAN, with the help of DS , the GS can produce high-resolution structures similar to the simulated results.
(6)
where I LR and I HR are the low- and high-resolution structure images, DθD and GθG are the discriminator network DS and generator network GS , respectively. θG and θD are the weights and biases of GS and DS , respectively. For more detail about the design of SRGAN, please refer to [37]. A simplified SRGAN network is built in this paper to refine the low-resolution structure. Our network design is shown in Fig. 8. To achieve better refinement, seven residual networks, which are the cores of the generator network, are applied in GS . Every residual network includes two convolution layers with 3 × 3 convolution kernels. Two identity skip-connections are conducted to feed the features back into the convolutional layers for cycling to enhance the structure feature. Three convolutional layers and two upsampling layers are then alternately performed to expand the feature size and refine the resolution. Finally, a convolutional layer with 5 × 5 filter and the Sigmoid activation function is implemented to generate the high-resolution structure. As for DS , the real and fake 160 × 160 high-resolution structure images are fed to the DS network in two channels, as shown in Fig. 9. Then, ten convolution layers with different filters and strides are implemented alternatively. Finally, a fully connected layer and a dense layer with Sigmoid activation function are used for the true/false classification. To simplify the network model and reduce the calculation, the loss function is simplified to: LR LSR = lSR MSE + λE(log(1 − DθD (GθG (I )))),
(7)
and lSR MSE =
M N 1 ∑∑
MN
2
LR (RHR m,n − Gθ G (I )m,n ) ,
(8)
m=1 n=1
where I LR is the input low-resolution structure, λ is the loss weight empirically set 0.001, M , N are the horizontal and vertical pixel numbers, respectively, and RHR m,n is the pixel value at coordinate (m, n).
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is first set to the average gray-scale value. Then, we get a binarized hashing image through a simple thresholding: if the pixel value is smaller than Hthreshold , its hash value is set to 0, otherwise its hash value is set to 1. The similarity of two structures is then measured by their hashing images. Specifically, Hthreahold =
{ Hm,n =
M N 1 ∑∑
MN
1 0
Pm,n ,
(10)
m=1 n=1
if Pm,n ≥ Hthreahold , if Pm,n < Hthreahold
(11)
where Hthreshold is the threshold of the pixel value, Pm,n is the pixel value at (m, n), and M , N are the horizontal and vertical pixel numbers, respectively. Finally, the similarity between two structures a and b, is then measured by Fig. 9. The Architecture of DS (Discriminator Network) in the SRGAN for high-resolution optimal structure prediction.
Shash = 1 −
M N 1 ∑∑
MN
2
b a (Hm ,n − Hm,n ) ,
(12)
m=1 n=1
a b where Hm ,n and Hm,n are the hash values of structures a and b.
5. Experimental results
Fig. 10. The reduction of loss during the training of our SRGAN.
During the training process, the ADAM optimization algorithm [38] is used to optimize the parameters in the networks. Totally 9,900 pairs of low- and high-resolution data samples were used to train this networks. The plot of loss value in the training of SRGAN is shown in Fig. 10. 4.4. Evaluating performance of our networks We integrate the low-resolution image prediction GAN and the low-to-high resolution image refinement SRGAN and evaluate their performance. Each testing image sample and its boundary conditions are fed to this integrated network, and the output, a refined structure image, is compared with the optimized (simulated) structure image. To evaluate the similarity of predicted and optimized structures, we use two criteria: the MSE accuracy score and the perceptual hashing accuracy [39]. The MSE accuracy is defined as
SMSE = 1 −
M N 1 ∑∑
MN
(ϵm,n )2 ,
(9)
m=1 n=1
where M , N are the horizontal and vertical pixel numbers, and ϵm,n is the deviation between the predicted pixel value and the groundtruth value at pixel (m, n). The perceptual hashing algorithm generates a series of strings as the fingerprints for a structure. The similarity is measured by the similarity of the fingerprints. Specifically, a threshold Hthreshold
We compare our results with SIMP both qualitatively (by showing the images side-by-side) and quantitatively (using S in Eq. (12)), to evaluate our algorithm’s accuracy and efficiency. All experiments were performed on an Intel i7-8750H CPU with 8 GB RAM and NVidia GeForce GTX 1060 GPU. One hundred testing data of the classic volume-to-point heat conduction problem with randomly generated boundary conditions are used for evaluation. Fig. 11 illustrates an example where the heat sink is at the bottom center of the design field. The optimized structures at low and high resolutions are shown in (a) and (c), respectively. The predicted structures at both resolutions are shown in (b) and (d), correspondingly, and they are very similar to the simulated results. For quantitative measures of prediction accuracy, at the low resolution, the accuracy is 95.7% for the perceptual hash algorithm and 99.5% for the MSE. At the high resolution, the accuracy is 96.8% and 90.9% for the perceptual hash algorithm and the MSE, respectively. We also compare the prediction time needed for SIMP and our proposed method. The calculation time for generating the high-resolution structure using SIMP method is 22 s, while our two-phase network takes only 0.002 s. Note that we need a one-time offline training, which currently takes around 4 h. Fig. 12 shows more examples of predicted structures together with their corresponding simulated results. Sixteen testing data with random boundary conditions are used. We can see that the predicted structure and optimized structure are visually similar in both resolutions. The quantitative evaluation is summarized in Table 1 where the computational time and prediction accuracy are reported. The time to generate the high-resolution structure (160 × 160) using conventional method and proposed method are 23 s and 0.006 s, respectively. The time cost using our network is only 0.026% of the conventional SIMP method. As for the accuracy, the structure similarity is again evaluated using both MSE and perceptual hashing. The MSE and perceptual hashing error of the presented method is 1.3% 5.2%, respectively. Table 2 shows the perceptual hashing accuracy of the high-resolution predicted structures corresponding to these specific examples of different boundary conditions. We observe most predicted structures have prediction accuracy above 91%. One-phase prediction using GAN versus two-phase prediction using GAN + SRGAN. Fig. 13 shows a comparison of the structure prediction done by (1) a direct one-phase approach,
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B. Li, C. Huang, X. Li et al. / Computer-Aided Design 115 (2019) 172–180 Table 1 Comparison of computational efficiency and prediction accuracy between the conventional SIMP algorithm, the direct one-phase prediction using our trained GANs, and our proposed two-phase pipeline. The experiments run on 100 randomly sampled testing data. Average run-time (in seconds) Structural similarity
SIMP
Our method
One-phase prediction
18 s
0.006 s 98.7% (MSE)/94.8% (hash)
0.004 s 84.6% (MSE)/80.3%(hash)
Table 2 Accuracy of the predicted structures in high resolution. Sixteen randomly generated boundary conditions (BC) are used and their perceptual hashing accuracy scores are reported. BC
Accuracy
BC
Accuracy
1 2 3 4 5 6 7 8
91.8% 92.5% 90.6% 92.6% 91.7% 91.2% 90.8% 93.6%
9 10 11 12 13 14 15 16
91.1% 93.3% 91.5% 90.7% 91.3% 92.6% 93.4% 92.8%
Fig. 11. The comparison between the optimized and predicted structure. A intuitive similarity between the optimized structure and predicted structure under low and high resolution is shown.
Fig. 12. Comparison of optimized (groundtruth) structures and our predicted structures in both low and high resolutions. 16 random examples are shown.
where a GAN is trained directly to predict a high-resolution structure, and (2) our two-phase GAN + SRGAN strategy. We also use random samples from the testing dataset and eight examples are shown here. We can see that the proposed two-phase
Fig. 13. Comparing the structures generated using the direct one-phase prediction and structures generated using the proposed two-phase approach. The SIMP simulation (a) serves as the groundtruth, compared with the direct onephase prediction (b), the presented two-phase prediction–refinement strategy (c) produces structures with better accuracy and more detail. The accuracy is compared using MSE (d): the red and blue bars are MSE accuracy of the one-phase and two-phase strategies, respectively.
method consistently outperforms the direct one-phase prediction in accuracy (MSE). Comparison with other Traditional Topology Optimization Algorithms. Besides SIMP, many other iterative methods have been designed for efficient simulation on large-scale topology
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optimization problems. Notable recent approaches include the scalable GPU-enhanced solver [9], the design space optimization with design space adjustment and refinement [10], the reducible design variable method (RDVM) [11], and coarse-to-fine structure projection [14]. Unfortunately, because open-source programs are unavailable for these methods, we were unable to perform side-by-side quantitative comparisons with them on canonical datasets. Although the above methods can accelerate the topology optimization on large-scale data to a certain extent, they rely on iterative numerical optimization to search for the optimal solution, and the inevitable iterative finite element calculation still requires big amount of computing resources and is very time consuming. In contrast, our approach establishes a direct mapping from the initial boundary conditions to the final optimal solutions, avoiding the computational cost from the iterative simulations. Our pipeline produces comparable accuracy with these methods but clearly outperforms them in efficiency. 6. Conclusions We propose an efficient non-iterative topology optimization pipeline using deep learning for heat conduction structure design. Our pipeline contains two main steps: (1) near-optimal structure prediction in coarse resolution, and (2) structure refinement to obtain final structure in fine resolution. Both steps are done through trained Generative Adversarial Networks (GANs). To train the two GANs, training datasets consisting of low-resolution and high-resolution optimized structures are prepared through standard simulations of topology optimization. Boundary conditions, such as the heat sink, heat source, and the mass fraction are encoded using three matrices, and are fed to the integrated network. In coarse resolution, a classical GAN network is trained to predict the optimal structure given the boundary constraint. Then to obtain fine resolution optimal structure, a simplified SRGAN is trained to perform super-resolution and enhance the detail of the structure in the fine resolution. We evaluate the performance of this proposed method using testing data under various different boundary conditions. Experimental results demonstrated that this method can produce near-optimal heat conduction structure in a very efficient way. Compared with the conventional topology optimization methods, this approach has obvious advantages in the computation efficiency. Meanwhile, this pipeline is general and these learning modules can be adopted to predict near-optimal structural topology in other tasks. The framework is not limited to 2D. Our recent exploration of generalizing it to 3D topology optimization also shows very promising results. We will explore this in the near future. Limitations. The proposed method is a two-phase pipeline based on low-resolution prediction followed by a super-resolution refinement. The direct prediction runs stably in low resolutions (e.g. in our implementations/experiments, 40 × 40 pixels). The super-resolution refinement is currently restricted within 4 times, and too big upsampling factors could lead to unstable and inaccurate refinement. Therefore, if predictions on high-resolution design domains (e.g. 800 × 800) are needed, then our current two-phase pipeline may not provide enough prediction accuracy. A potential strategy is to develop a more effective integrated hierarchical or iterative prediction–refinement scheme. We will explore this design in the near future. Meanwhile, adopting other state-of-the-art GAN models and super-resolution schemes may also help improve our system’s performance. Acknowledgments The work reported in this paper is supported by the National Key R&D Program of China [2018YFB1700703], the National Natural Science Foundation of China [51822507] and [61728206].
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