Skip to main content
SpringerLink
Log in

Multi-stage deep neural network accelerated topology optimization

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

This paper introduces a novel deep learning approach for generating fine resolution structures that preserve all the information from the topology optimization (TO). The proposed approach utilizes neural networks (NNs) that map the desired engineering properties to seed for determining optimized structure. The main novelty of this framework is the utilization of parameters such as density and nodal deflections to predict optimized topologies for a wide range of design space. In this research, a three-stage NN framework is employed, wherein the first stage is a feedforward deep neural network. The second stage uses a convolutional neural network (CNN), and the final stage is the post-processing of results obtained from CNN. The first stage maps input design variables vector to output density distribution image. The image output of the first stage and design variables vector inputted to the first network serve as inputs to the second stage, which outputs optimized structure. Finally, post-processing of this optimized structure is done to ensure volume fraction constraint and optimal solution. These structures can be tessellated per the nodal displacement of an FEA solved larger structure. Because of this framework, the proposed approach makes TO computations faster and captures essential physics while using NNs to determine the final structure. The application of the proposed model to three-dimensional problems is also demonstrated.

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

  • Agnese J, Herrera J, Tao H, Zhu X (2020) A survey and taxonomy of adversarial neural networks for text-to-image synthesis. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, p e1345

  • Allaire G, Jouve F, Toader A.M (2004) Structural optimization using sensitivity analysis and a level-set method. Journal of computational physics 194(1) 363–393

    Article  MathSciNet  Google Scholar 

  • Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscipl Optim 43:1–16. https://doi.org/10.1007/s00158-010-0594-7

  • Ates G, Gorguluarslan R (2021) Two-stage convolutional encoder-decoder network to improve the performance and reliability of deep learning models for topology optimization. Struct Multidiscipl Optim. https://doi.org/10.1007/s00158-020-02788-w

  • Banga S, Gehani H, Bhilare S, Patel S, Kara L (2018) 3d topology optimization using convolutional neural networks. arXiv:1808.07440

  • Bendsøe M (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202. https://doi.org/10.1007/BF01650949

  • Bendsøe M (2013) Optimization of Structural Topology Shape and Material. Springer, Berlin Heidelberg

    MATH  Google Scholar 

  • Bendsøe M, Lund E, Olhoff N, Sigmund O (2005) Topology optimization–broadening the areas of application. Control Cybernet 34

  • Bendsøe M, Sigmund O (2003) Topology optimization—theory methods and applications. Springer Verlag, Germany

  • Bendsøe M.P., Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2) 197–224

    Article  MathSciNet  Google Scholar 

  • Bendsøe M.P., Sigmund O (1999) Material interpolation schemes in topology optimization. Archive of applied mechanics 69(9) 635–654

    MATH  Google Scholar 

  • Chapman CD, Saitou K, Jakiela MJ (1999) Genetic algorithms as an approach to configuration and topology design. J Mech Design 116(4):1005–1012. https://doi.org/10.1115/1.2919480

  • Creswell A, White T, Dumoulin V, Arulkumaran K, Sengupta B, Bharath A.A (2018) Generative adversarial networks: An overview. IEEE Signal Processing Magazine 35(1), 53–65

    Article  Google Scholar 

  • Dong C, Loy C.C., He K, Tang X (2015) Image super-resolution using deep convolutional networks. IEEE transactions on pattern analysis and machine intelligence 38(2) 295–307

    Article  Google Scholar 

  • Gibiansky L.V., Sigmund O (2000) Multiphase composites with extremal bulk modulus. Journal of the Mechanics and Physics of Solids 48(3), 461–498

    Article  MathSciNet  Google Scholar 

  • Goodfellow I, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Courville A, Bengio Y (2014) Generative adversarial nets. In: Advances in neural information processing systems, pp 2672–2680

  • Grihon S, Krog L, Marasco A (2008) Smart design of structures through topology optimisation

  • Guedes J, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Computer methods in applied mechanics and engineering 83(2) 143–198

    Article  MathSciNet  Google Scholar 

  • Hu X, Li F, Samaras D, Chen C (2019) Topology-preserving deep image segmentation

  • Ledig C, Theis L, Huszár F, Caballero J, Cunningham A, Acosta A, Aitken A, Tejani A, Totz J, Wang Z, et al (2017) Photo-realistic single image super-resolution using a generative adversarial network. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 4681–4690

  • Li Q, Steven G, Xie Y (1999) On equivalence between stress criterion and stiffness criterion in evolutionary structural optimization. Structural optimization 18(1) 67–73

    Article  Google Scholar 

  • Long J, Shelhamer E, Darrell T (2014) Fully convolutional networks for semantic segmentation. Arxiv 79

  • Mosinska A, Marquez-Neila P, Kozinski M, Fua P (2018) Beyond the pixel-wise loss for topology-aware delineation, pp 3136–3145 . https://doi.org/10.1109/CVPR.2018.00331

  • Petersson J (1999) A finite element analysis of optimal variable thickness sheets. SIAM journal on numerical analysis 36(6) 1759–1778

    Article  MathSciNet  Google Scholar 

  • Roche F, Hueber T, Limier S, Girin L (2018) Autoencoders for music sound modeling: a comparison of linear shallow deep recurrent and variational models. arXiv:1806.04096

  • Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. International Journal of Solids and Structures 31(17), 2313–2329

    Article  MathSciNet  Google Scholar 

  • Sigmund O (2000) New class of extremal composites. J Mech Phys Solids 48:397–428. https://doi.org/10.1016/S0022-5096(99)00034-4

  • Sigmund O (2001) A 99 line topology optimization code written in matlab. Structural and multidisciplinary optimization 21(2) 120–127

    Article  Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization 33(4–5), 401–424

    Article  Google Scholar 

  • Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids 45(6), 1037–1067

    Article  MathSciNet  Google Scholar 

  • Soille P (1999) Morphological Image Analysis Principles and Applications. Springer-Verlag

    Book  Google Scholar 

  • Sosnovik I, Oseledets I (2019) Neural networks for topology optimization. Russian Journal of Numerical Analysis and Mathematical Modelling 34(4), 215–223

    Article  MathSciNet  Google Scholar 

  • Ségonne F (2008) Active contours under topology control-genus preserving level sets. Int J Comput Vis 79:107–117. https://doi.org/10.1007/s11263-007-0102-8

  • Tanskanen P (2002) The evolutionary structural optimization method: theoretical aspects. Computer methods in applied mechanics and engineering 191(47–48) 5485–5498

    Article  Google Scholar 

  • Vatanabe SL, Lippi TN, de Lima CR, Paulino GH, Silva EC (2016) Topology optimization with manufacturing constraints: a unified projection-based approach. Advances in Engineering Software 100 97–112

    Article  Google Scholar 

  • Wang M.Y., Wang X, Guo D (2003) A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1–2) 227–246

    Article  MathSciNet  Google Scholar 

  • Wang SY, Tai K, Wang MY (2006) An enhanced genetic algorithm for structural topology optimization. Int J Numer Method Eng 65(1):18–44. https://doi.org/10.1002/nme.1435

  • Wang X, Yu K, Wu S, Gu J, Liu Y, Dong C, Qiao Y, Change Loy C (2018) Esrgan: enhanced super-resolution generative adversarial networks. In: Proceedings of the European conference on computer vision (ECCV)

  • Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in matlab. Structural and multidisciplinary optimization 52(6) 1229–1241

    Article  MathSciNet  Google Scholar 

  • Xie YM, Steven GP (1997) Evolutionary Structural Optimization. Springer-Verlag, London

    Book  Google Scholar 

  • Yan X, Yang J, Sohn K, Lee H (2016) Attribute2image: Conditional image generation from visual attributes. In: European conference on computer vision, pp 776–791. Springer, New York

  • Yildiz A, Öztürk N, Kaya N, Öztürk F (2003) Integrated optimal topology design and shape optimization using neural networks. Structural and Multidisciplinary Optimization 25(4), 251–260

    Article  Google Scholar 

  • Yu Y, Hur T, Jung J, Jang I.G (2019) Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799

    Article  Google Scholar 

  • Zhang H, Xu T, Li H, Zhang S, Wang X, Huang X, Metaxas DN (2017) Stackgan: text to photo-realistic image synthesis with stacked generative adversarial networks. In: Proceedings of the IEEE international conference on computer vision, pp 5907–5915

  • Zhang Y, Chen A, Peng B, Zhou X, Wang D (2019) A deep convolutional neural network for topology optimization with strong generalization ability. arXiv:1901.07761

Download references

Author information

Authors and Affiliations

  1. Department of Automotive Engineering, Clemson University, 4 Research Drive, Greenville, SC, 29607, USA

    Dustin Bielecki, Darshil Patel & Rahul Rai

  2. Department of Mechanical and Aerospace Engineering, University at Buffalo, 240 Bell Hall, Buffalo, NY, 14260, USA

    Gary F. Dargush

Authors
  1. Dustin Bielecki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Darshil Patel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rahul Rai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gary F. Dargush
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dustin Bielecki.

Ethics declarations

Funding Information

No funding was received to assist with the preparation of this manuscript.

Conflicts of Interests

The authors declare that they have no conflict of interest.

Replication of Results

MATLAB and Python codes were used to generate results in this paper. The datasets are available on the corresponding author’s github page.

Additional information

Responsible Editor: Palaniappan Ramu.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bielecki, D., Patel, D., Rai, R. et al. Multi-stage deep neural network accelerated topology optimization. Struct Multidisc Optim 64, 3473–3487 (2021). https://doi.org/10.1007/s00158-021-03028-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-021-03028-5

Keywords

Search

Navigation