Skip to main content
SpringerLink
Log in

Isogeometric Topology Optimization Based on Deep Learning

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

Topology optimization plays an important role in a wide range of engineering applications. In this paper, we propose a novel isogeometric topology optimization algorithm based on deep learning. Unlike the other neural network-based methods, the density distributions in the design domain are represented in the B-spline space. In addition, we use relatively novel technologies, U-Net and DenseNet, to form the neural network structure. The 2D and 3D numerical experiments show that the proposed method has an accuracy rate of over 97% for the final optimization results. After training, the new approach can save time greatly for the new topology optimization compared with traditional solid isotropic material with penalization method and IGA method. The approach can also overcome the checkerboard phenomenon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J.A., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Eng. 199(5–8), 229–263 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988)

    Article  MathSciNet  Google Scholar 

  3. Bendsøe, M.P., Sigmund, O.: Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69(9–10), 635–654 (1999)

    MATH  Google Scholar 

  4. Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Method and Applications. Springer, Berlin (2003)

    MATH  Google Scholar 

  5. Bengio, Y., Simard, P., Frasconi, P.: Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Netw. 5(2), 157–166 (1994)

    Article  Google Scholar 

  6. Bourdin, B., Chambolle, A.: Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9(9), 19–48 (2003)

    Article  MathSciNet  Google Scholar 

  7. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley (2009)

  8. Deng, H., To, A.C.: Topology optimization based on deep representation learning (DRL) for compliance and stress-constrained design. Comput. Mech. 66, 449–469 (2020)

    Article  MathSciNet  Google Scholar 

  9. Dunning, P.D., Kim, H.A.: A new hole insertion method for level set based structural topology optimization. Int. J. Numer. Methods Eng. 93(1), 118–134 (2013)

    Article  MathSciNet  Google Scholar 

  10. Fu, Y.F., Rolfe, B., Chiu, L.N.S., Wang, Y., Ghabraie, K.: Smooth topological design of 3D continuum structures using elemental volume fractions. Comput. Struct. 231, 106213 (2020)

    Article  Google Scholar 

  11. Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. The MIT Press (2015)

  12. Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J. Appl. Mech. 81(8), 081009 (2014)

    Article  Google Scholar 

  13. Hassani, B., Khanzadi, M., Tavakkoli, S.M.: An isogeometrical approach to structural topology optimization by optimality criteria. Struct. Multidiscip. Optim. 45(2), 223–233 (2012)

    Article  MathSciNet  Google Scholar 

  14. Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw. 2(5), 359–366 (1989)

    Article  Google Scholar 

  15. Hou, W., Gai, Y., Zhu, X., Wang, X., Zhao, C., Xu, L., Jiang, K., Hu, P.: Explicit isogeometric topology optimization using moving morphable components. Comput. Methods Appl. Mech. Eng. 326, 694–712 (2017)

    Article  MathSciNet  Google Scholar 

  16. Huang, G., Liu, Z., Maaten, L.V.D., Weinberger, K.Q.: Densely connected convolutional networks. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), vol. 1, pp. 2261–2269 (2017)

  17. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)

    Article  MathSciNet  Google Scholar 

  18. Kang, P., Youn, S.K.: Isogeometric topology optimization of shell structures using trimmed NURBS surfaces. Finite Elem. Anal. Des. 120, 18–40 (2016)

    Article  MathSciNet  Google Scholar 

  19. Krizhevsky, A., Sutskever, I., Hinton, G.: ImageNet classification with deep convolutional neural networks. Commun. ACM 60(6), 84–90 (2017)

    Article  Google Scholar 

  20. Lecun, Y., Bottou, L.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)

    Article  Google Scholar 

  21. Li, X., Wei, X., Zhang, Y.: Hybrid non-uniform recursive subdivision with improved convergence rates. Comput. Methods Appl. Mech. Eng. 352, 606–624 (2019)

    Article  MathSciNet  Google Scholar 

  22. Liu, K., Tovar, A.: An efficient 3D topology optimization code written in Matlab. Struct. Multidiscip. Optim. 50(6), 1175–1196 (2014)

    Article  MathSciNet  Google Scholar 

  23. Mcculloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5(4), 115–133 (1943)

    Article  MathSciNet  Google Scholar 

  24. Nguyen, T.H., Paulino, G.H., Song, J., Le, C.H.: A computational paradigm for multiresolution topology optimization (MTOP). Struct. Multidiscip. Optim. 41(4), 525–539 (2010)

    Article  MathSciNet  Google Scholar 

  25. Nie, Z., Jiang, H., Kara, L.B.: Stress field prediction in cantilevered structures using convolutional neural networks. J. Comput. Inf. Sci. Eng. 20(1), 011002 (2019)

    Article  Google Scholar 

  26. Qian, X.: Topology optimization in B-spline space. Comput. Methods Appl. Mech. Eng. 265, 15–35 (2013)

    Article  MathSciNet  Google Scholar 

  27. Rawat, S., Shen, M.H.H.: A novel topology design approach using an integrated Deep Learning network architecture. e-Print Archive. arXiv:1808.02334 (2018)

  28. Rawat, S., Shen, M.H.H.: A novel topology optimization approach using conditional deep learning. e-Print Archive. arXiv:1901.04859 (2019)

  29. Ronneberger, O., Fischer, P., Brox, T.: U-Net: convolutional networks for biomedical image segmentation. Med. Image Comput. Comput.-Assist. Interv. 2015, 234–241 (2015)

    Google Scholar 

  30. Saurabh, B., Harsh, G., Sanket, B., Sagar, P., Levent, K.: 3D topology optimization using convolutional neural networks. e-Print Archive. arXiv:1808.07440v1 (2018)

  31. Seo, Y.D., Kim, H.J., Youn, S.K.: Isogeometric topology optimization using trimmed spline surfaces. Comput. Methods Appl. Mech. Eng. 199(49–52), 3270–3296 (2010)

    Article  MathSciNet  Google Scholar 

  32. Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163(2), 489–528 (2000)

    Article  MathSciNet  Google Scholar 

  33. Sosnovik, I., Oseledets, I.: Neural networks for topology optimization. Russ. J. Numer. Anal. Math. Model. 34(4), 215–223 (2019)

    Article  MathSciNet  Google Scholar 

  34. Wang, M., Qian, X.: Efficient filtering in topology optimization via B-Splines. J. Mech. Des. 137(3), 031402 (2015)

    Article  Google Scholar 

  35. Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1–2), 227–246 (2003)

    Article  MathSciNet  Google Scholar 

  36. Wang, Y., Benson, D.J.: Isogeometric analysis for parameterized LSM-based structural topology optimization. Comput. Mech. 57(1), 19–35 (2016)

    Article  MathSciNet  Google Scholar 

  37. Xie, Y., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49(5), 885–896 (1993)

    Article  Google Scholar 

  38. Xie, Y., Yang, X., Steven, G.P., Querin, O.: The theory and application of evolutionary structural optimization method. Eng. Mech. 16(6), 70–81 (1999)

    Google Scholar 

  39. Yu, Y., Hur, T., Jung, J.: Deep learning for determining a near-optimal topological design without any iteration. Struct. Multidicip. Optim. 59, 787–799 (2019)

    Article  Google Scholar 

  40. Zhang, W., Li, D., Yuan, J., Song, J., Guo, X.: A new three-dimensional topology optimization method based on moving morphable components (MMCs). Comput. Mech. 59(4), 1–19 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Zheng, R., Kim, C.: An enhanced topology optimization approach based on the combined MMC and NURBS-curve boundaries. Int. J. Precis. Eng. Manuf. 21(2), 1529–1538 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the scholars for their valuable suggestions and useful comments contributed to the final version of this paper. The authors were supported by the National Key R &D Program of China (2020YFB1708900), NSF of China (No. 61872328), and the Youth Innovation Promotion Association CAS.

Author information

Authors and Affiliations

  1. School of Mathematical Science, University of Science and Technology of China, Hefei, People’s Republic of China

    Taining Zheng & Xin Li

Authors
  1. Taining Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, T., Li, X. Isogeometric Topology Optimization Based on Deep Learning. Commun. Math. Stat. 10, 543–564 (2022). https://doi.org/10.1007/s40304-021-00253-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-021-00253-8

Keywords

Mathematics Subject Classification

Search

Navigation