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Comparative study on manifold learning approaches for parametric topology optimization problem via unsupervised deep learning

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Abstract

The study designs iterative and non-iterative network frameworks that can predict solutions for topology optimization. The modified solid isotropic material with the penalization method is applied for topology optimization and is used to generate the input and output of the network. The optimal solution was predicted with respect to the variation in the cross-sectional thickness of the considered structure. The network frameworks were constructed using various neural network architectures, that is, convolutional autoencoders, convolutional neural networks, and artificial neural networks. Herein, the structure is defined as a subdomain to improve training efficiency and parameterization. The density distribution and strain energy of the structure were expressed as reduced data using a convolutional autoencoder. Two numerical examples were considered to evaluate the accuracy and efficiency of the proposed network frameworks. The test data were constructed to evaluate the performance of the proposed network framework. It was confirmed that the non-iterative network framework shows a more benign performance than the iterative one.

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Data availability statements

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by the Korea Electric Power Corporation research grant for the basic research and development project initiated in 2021 (R21XO01-6) and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2022M1A3B8076744).

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Authors and Affiliations

  1. Department of Aerospace Engineering, Jeonbuk National University, 567 Baekje-Daero, Deokjin-Gu, Jeonju, 54896, Republic of Korea

    Seongwoo Cheon & Haeseong Cho

  2. Future Air Mobility Research Center, Jeonbuk National University, 567 Baekje-Daero, Deokjin-Gu, Jeonju, 54896, Republic of Korea

    Haeseong Cho

  3. Smart Grid Research Center, Jeonbuk National University, 567 Baekje-Daero, Deokjin-Gu, Jeonju, 54896, Republic of Korea

    Haeseong Cho

Authors
  1. Seongwoo Cheon
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  2. Haeseong Cho
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Contributions

SC: Methodology, data curation, writing and editing. HC: Conceptualization, supervision, writing—review and editing.

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Correspondence to Haeseong Cho.

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Appendix A: Summary of the neural networks in the proposed frameworks

Appendix A: Summary of the neural networks in the proposed frameworks

This section briefly describes the architecture of each network framework and the hyperparameters. Moreover, the relevant convergence history of the loss function corresponding to the numerical examples in Sect. 5 is also provided.

1.1 A.1: Iterative network framework of MBB beam

See Tables 8, 9, 10 and Fig. 34.

Table 8 MBB beam example: CAE 1, 2
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Table 9 MBB beam example: DNN
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Table 10 MBB beam example: CNN 1, 2
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Fig. 34
figure 34

Convergence history of the loss function of iterative network framework corresponding to MBB beam example

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1.2 A.2: Non-iterative network framework of MB

1.2.1 B beam

See Tables 11, 12, 13 and Fig. 35.

Table 11 MBB beam example: convolutional autoencoder
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Table 12 MBB beam example: convolutional autoencoder
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Table 13 MBB beam example: DNN
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Fig. 35
figure 35

Convergence history of the loss function of non-iterative network framework corresponding to MBB beam example

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1.3 A.3: Iterative network framework of L-shaped beam

See Tables 14, 15, 16, 17 and Fig. 36.

Table 14 L-shaped beam example: CAE 1, 2
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Table 15 L-shaped beam example: DNN
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Table 16 L-shaped beam example: CNN 1
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Table 17 L-shaped beam example: CNN 2
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Fig. 36
figure 36

Convergence history of the loss function of iterative network framework corresponding to L-shaped beam example

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1.4 A.4: Non-iterative network framework of L-shaped beam

See Tables 18, 19, and Fig. 37.

Table 18 L-shaped beam example: Convolutional autoencoder
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Table 19 L-shaped beam example: DNN
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Fig. 37
figure 37

Convergence history of the loss function of non-iterative network framework corresponding to L-shaped beam example

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Cheon, S., Cho, H. Comparative study on manifold learning approaches for parametric topology optimization problem via unsupervised deep learning. Optim Eng 25, 325–371 (2024). https://doi.org/10.1007/s11081-023-09806-y

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