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Intelligent optimization of stiffener unit cell via variational autoencoder-based feature extraction

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Abstract

Grid-stiffened structures are widely used in industrial equipment, where the layout of stiffener unit cells is critical in the structural performance. However, owing to the lack of design techniques, existing stiffener unit cell designs are often achieved only by comparative selection among several common cell configurations. In this study, a deep learning-based intelligent optimization framework for the stiffener unit cell of grid-stiffened panels was proposed. First, a database containing nearly 10,000 stiffener unit cells was generated by traversal while considering the manufacturability. Feature extraction was then performed on the generated database using a variational autoencoder and mapped to a 16-dimensional continuous latent design space according to geometric features. Subsequently, in this latent space, a Gaussian process model was established, and the maximum expected improvement criterion was utilized to drive the model update and optimization search, thus realizing data-driven optimization for the stiffener unit cell of grid-stiffened panels. In three typical numerical examples, compared with the best of the traditional stiffener unit cells, the obtained optimal designs were improved by 25.61%, 25.88%, and 10.66%, respectively, demonstrating the effectiveness of this method as an alternative stiffener unit cell design method. Furthermore, this study also indicates the significant potential of geometric feature extraction and further structural layout optimization via deep learning.

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References

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654

    MATH  Google Scholar 

  • Christensen PW, Klarbring A (2008) An Introduction to structural optimization. Springer, New York

    MATH  Google Scholar 

  • Doersch (2016) Tutorial on variational autoencoders. arXiv preprint arXiv:1606.05908

  • Dugré A, Vadean A (2016) Chausse. Challenges of using topology optimization for the design of pressurized stiffened panels. Struct Multidisc Optim 53(2):303–320

    Article  Google Scholar 

  • Gao J, Gao L, Luo Z, Li P (2019) Isogeometric topology optimization for continuum structures using density distribution function. Int J Numer Methods Eng 119(10):991–1017

    Article  Google Scholar 

  • Gao J, Wang L, Luo Z, Gao L (2021) IgaTop: an implementation of topology optimization for structures using IGA in MATLAB. Struct Multidisc Optim 64(3):1669–1700

    Article  Google Scholar 

  • Goodfellow I, Bengio Y, Courville A (2016) Deep learning. MIT Press, Cambridge

    MATH  Google Scholar 

  • Haftka RT, Gürdal Z (1992) Elements of structural optimization. Springer, New York

    Book  MATH  Google Scholar 

  • Han ZH, Zhang KS (2012) Surrogate-based optimization. In: Roeva O (ed) Real-world applications of genetic algorithms, InTech, pp 343–362

  • Hao P, Wang B, Li G (2012) Surrogate-based optimum design for stiffened shells with adaptive sampling. AIAA J 50(11):2389–2407

    Article  Google Scholar 

  • Hao P, Wang B, Li G, Meng Z, Tian K, Tang X (2014) Hybrid optimization of hierarchical stiffened shells based on smeared stiffener method and finite element method. Thin-Walled Struct 82:46–54

    Article  Google Scholar 

  • Hao P, Wang Y, Liu C, Wang B, Tian K, Li G, Wang Q, Jiang L (2018) Hierarchical nondeterministic optimization of curvilinearly stiffened panel with multicutouts. AIAA J 56(10):4180–4194

    Article  Google Scholar 

  • Hao P, Liu D, Zhang K, Yuan Y, Wang B, Li G, Zhang X (2021) Intelligent layout design of curvilinearly stiffened panels via deep learning-based method. Mater Design 197:109180

    Article  Google Scholar 

  • Hershey JR, Olsen PA (2007) Approximating the Kullback Leibler divergence between Gaussian mixture models. In: 2007 IEEE International Conference on Acoustics, Speech and Signal Processing-ICASSP'07. IEEE, 2007, vol 4, pp IV-317–IV-320

  • Huybrechts S, Meink TE (1997) Advanced grid stiffened structures for the next generation of launch vehicles. In: 1997 IEEE Aerospace Conference, vol 1, pp 263–270

  • Jackson PC (2019) Introduction to artificial intelligence. Courier Dover Publications, Mineola

    Google Scholar 

  • Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

    Article  MATH  Google Scholar 

  • Kingma DP, Welling M (2013) Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114

  • Lanzi L, Giavotto V (2006) Post-buckling optimization of composite stiffened panels: computations and experiments. Compos Struct 73(2):208–220

    Article  Google Scholar 

  • Larsen ABL, Sønderby SK, Larochelle H, Winther O (2015) Autoencoding beyond pixels using a learned similarity metric. arXiv:1512.09300

  • Li R, Wang S, Long Z, Gu D (2018) Undeepvo: monocular visual odometry through unsupervised deep learning. In: 2018 IEEE international conference on robotics and automation (ICRA), pp 7286–7291

  • Li S, Wei H, Yuan S, Zhu J, Zhang W (2021) Collaborative optimization design of process parameter and structural topology for laser additive manufacturing. Chin J Aeronaut

  • Liao Z, Wang Y, Gao L, Wang ZP (2022) Deep-learning-based isogeometric inverse design for tetra-chiral auxetics. Compos Struct 280:114808

    Article  Google Scholar 

  • Lim KH, Li X, Guan ZD (2013) Optimal design of advanced grid stiffened composite cylindrical shell. Appl Mech Mater 330:681–686

    Article  Google Scholar 

  • Liu D, Hao P, Zhang K, Tian K, Wang B, Li G, Xu W (2020) On the integrated design of curvilinearly grid-stiffened panel with non-uniform distribution and variable stiffener profile. Mater Design 190:108556

    Article  Google Scholar 

  • Noor AK, Venneri SL, Paul DB, Hopkins MA (2000) Structures technology for future aerospace systems. Comput Struct 74(5):507–519

    Article  Google Scholar 

  • Oune N, Bostanabad R (2021) Latent map Gaussian processes for mixed variable metamodeling. Comput Methods Appl Mech Eng 387:114128

    Article  MATH  Google Scholar 

  • Park C, Haftka RT, Kim NH (2017) Remarks on multi-fidelity surrogates. Struct Multidisc Optim 55(3):1029–1050

    Article  Google Scholar 

  • Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707

    Article  MATH  Google Scholar 

  • Seo J, Kapania RK (2022) Development of deep convolutional neural network for structural topology optimization. In: AIAA SCITECH 2022 Forum. 2351

  • Timoshenko SP, Gere JM (2009) Theory of elastic stability. Dover Publications, Mineola

    Google Scholar 

  • Walker M (2002) The effect of stiffeners on the optimal ply orientation and buckling load of rectangular laminated plates. Comput Struct 80(27–30):2229–2239

    Article  Google Scholar 

  • Wang B, Hao P, Li G, Tian K, Du K, Wang X, Zhang X, Tang X (2014) Two-stage size-layout optimization of axially compressed stiffened panels. Struct Multidisc Optim 50(2):313–327

    Article  Google Scholar 

  • Wang B, Tian K, Hao P, Cai YW, Li YW, Sun Y (2015) Hybrid analysis and optimization of hierarchical stiffened plates based on asymptotic homogenization method. Compos Struct 132(11):136–147

    Article  Google Scholar 

  • Wang B, Tian K, Hao P, Zheng Y, Ma Y, Wan J (2016) Numerical-based smeared stiffener method for global buckling analysis of grid-stiffened composite cylindrical shells. Compos Struct 152:807–815

    Article  Google Scholar 

  • Wang B, Tian K, Zhou C, Hao P, Zheng Y, Ma Y, Wang J (2017) Grid-pattern optimization framework of novel hierarchical stiffened shells allowing for imperfection sensitivity. Aerosp Sci Technol 62:114–121

    Article  Google Scholar 

  • Wang D, Yeo SY, Su Z, Wang ZP, Abdalla MM (2020a) Data-driven streamline stiffener path optimization (SSPO) for sparse stiffener layout design of non-uniform curved grid-stiffened composite (NCGC) structures. Comput Meth Appl Mech Eng 365:113001

    Article  MATH  Google Scholar 

  • Wang L, Chan YC, Ahmed F, Liu Z, Zhu P, Chen W (2020b) Deep generative modeling for mechanistic-based learning and design of metamaterial systems. Comput Methods Appl Mech Engrg 372:113377

    Article  MATH  Google Scholar 

  • Wang L, Beek A, Da D, Chan YC, Zhu P, Chen W (2022) Data-driven multiscale design of cellular composites with multiclass microstructures for natural frequency maximization. Compos Struct 280:114949

    Article  Google Scholar 

  • Wang X, Guo W (2016) Dynamic modeling and vibration characteristics analysis of submerged stiffened combined shells. Ocean Eng 127:226–235

    Article  Google Scholar 

  • Yang Z, Li X, Brinson LC, Choudhary AN, Chen W, Agrawal A (2018) Microstructural materials design via deep adversarial learning methodology. J Mech Des 140(11):111416

    Article  Google Scholar 

  • Zegard T, Paulino GH (2014) GRAND–Ground structure based topology optimization for arbitrary 2D domains using MATLAB. Struct Multidisc Optim 50(5):861–882

    Article  Google Scholar 

  • Zhang Y, Tao S, Chen W, Apley DW (2020) A latent variable approach to Gaussian process modeling with qualitative and quantitative factors. Technometrics 62(3):291–302

    Article  Google Scholar 

  • Zhao Y, Chen M, Yang F, Zhang L, Fang D (2017) Optimal design of hierarchical grid-stiffened cylindrical shell structures based on linear buckling and nonlinear collapse analyses. Thin-Walled Struct 119:315–323

    Article  Google Scholar 

  • Zhou H, Zhu J, Wang C, Zhang Y, Wang J, Zhang W (2022) Hierarchical structure optimization with parameterized lattice and multiscale finite element method. Struct Multidisc Optim 65(1):1–20

    Article  Google Scholar 

  • Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Method Eng 23(4):595–622

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Key Research and Development Program of China (2021YFF0306404), National Natural Science Foundation of China (U21A20429 and 11825202), and LiaoNing Revitalization Talents Program (XLYC1907142).

Author information

Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Key Laboratory of Digital Twin for Industrial Equipment, Liaoning Province, Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116023, China

    Dachuan Liu, Peng Hao, Tengfei Xu, Yingjie Zhu, Bo Wang & Gang Li

  2. China Academy of Launch Vehicle Technology, Beijing, 100076, China

    Xuanxiu Liu

Authors
  1. Dachuan Liu
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  2. Peng Hao
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  3. Tengfei Xu
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  4. Yingjie Zhu
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  5. Xuanxiu Liu
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  6. Bo Wang
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  7. Gang Li
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Corresponding author

Correspondence to Peng Hao.

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Replication of results

The details of the proposed methodology and of the specific values of the parameters considered have been provided in the paper. Hence, the results can be reproduced. The main functions of VAE-GAN for feature extraction can be downloaded from the website: https://github.com/xutengfei2000/VAE-GAN-for-feature-extraction/tree/main. However, because the engineering example involves the script-based finite element modeling code of the research group, it is not convenient to show them publicly.

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Appendix: Evolutionary history of optimal solution

Appendix: Evolutionary history of optimal solution

See Figs. 27, 28, 29, 30, 31, 32, and Table 7.

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Optimization process of optimization example 3

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Load vs. end-shortening curves of the optimal stiffener layout

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Comparison of the collapse load of different stiffener unit cells for optimization example 3

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Stiffener layouts of the best 5 stiffener unit cells obtained for optimization example 3

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Fig. 27
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Evolutionary history in optimization process 1 of optimization example 1

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Fig. 28
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Evolutionary history in optimization process 2 of optimization example 1

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Evolutionary history in optimization process 3 of optimization example 1

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Evolutionary history in optimization process 1 of optimization example 2

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Evolutionary history in optimization process 2 of optimization example 2

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Fig. 32
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Evolutionary history in optimization process 3 of optimization example 2

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Table 7 Specific number of optimization processes
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Liu, D., Hao, P., Xu, T. et al. Intelligent optimization of stiffener unit cell via variational autoencoder-based feature extraction. Struct Multidisc Optim 66, 8 (2023). https://doi.org/10.1007/s00158-022-03463-y

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