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Integrating material selection with design optimization via neural networks

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Abstract

The engineering design process often entails optimizing the underlying geometry while simultaneously selecting a suitable material. For a certain class of simple problems, the two are separable where, for example, one can first select an optimal material, and then optimize the geometry. However, in general, the two are not separable. Furthermore, the discrete nature of material selection is not compatible with gradient-based geometry optimization, making simultaneous optimization challenging. In this paper, we propose the use of variational autoencoders (VAE) for simultaneous optimization. First, a data-driven VAE is used to project the discrete material database onto a continuous and differentiable latent space. This is then coupled with a fully-connected neural network, embedded with a finite-element solver, to simultaneously optimize the material and geometry. The neural-network’s built-in gradient optimizer and back-propagation are exploited during optimization. The proposed framework is demonstrated using trusses, where an optimal material needs to be chosen from a database, while simultaneously optimizing the cross-sectional areas of the truss members. Several numerical examples illustrate the efficacy of the proposed framework. The Python code used in these experiments is available at github.com/UW-ERSL/MaTruss.

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References

  1. Eggert R (2005) Engineering design. Pearson/Prentice Hall, Hoboken

    Google Scholar 

  2. Rozvany GIN, Bendsoe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev 48(2):41–119

    Article  Google Scholar 

  3. Achtziger W (1996) Truss topology optimization including bar properties different for tension and compression. Struct Optim 12(1):63–74

    Article  Google Scholar 

  4. Rakshit S, Ananthasuresh GK (2008) Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization. Struct Multidiscip Optim 35(1):55–68

    Article  Google Scholar 

  5. Ashby MF, Cebon D (1993) Materials selection in mechanical design. Le Journal de Physique IV 3(C7):C7-1

    Google Scholar 

  6. Ashby MF, Johnson K (2013) Materials and design: the art and science of material selection in product design. Butterworth-Heinemann, Oxford

    Google Scholar 

  7. Ashby MF (2000) Multi-objective optimization in material design and selection. Acta Materialia 48(1):359–369

    Article  Google Scholar 

  8. Jahan A, Ismail MY, Sapuan SM, Mustapha F (2010) Material screening and choosing methods—a review. Mater Des 31(2):696–705

    Article  Google Scholar 

  9. Venkata Rao R (2006) A material selection model using graph theory and matrix approach. Mater Sci Eng A 431(1–2):248–255

    Google Scholar 

  10. Zhou C-C, Yin G-F, Hu X-B (2009) Multi-objective optimization of material selection for sustainable products: artificial neural networks and genetic algorithm approach. Mater Des 30(4):1209–1215

    Article  Google Scholar 

  11. Ananthasuresh GK, Ashby MF (2003) Concurrent design and material selection for trusses. Workshop: Optimal Design at Laboratoire de Mécanique des Solides, Ecole Polytechnique, Palaiseau, France. November 26-28, 2003

  12. Stolpe M, Svanberg K (2004) A stress-constrained truss-topology and material-selection problem that can be solved by linear programming. Struct Multidiscip Optim 27(1):126–129

    Article  MathSciNet  MATH  Google Scholar 

  13. Ching E, Carstensen JV (2021) Truss topology optimization of timber—steel structures for reduced embodied carbon design. Eng Struct 113540 (Vol: 252)

  14. Roy S, Crossley WA, Jain S (2021) A hybrid approach for solving constrained multi-objective mixed-discrete nonlinear programming engineering problems. IntechOpen, 2021 [Online]

  15. Arora JS, Huang MW, Hsieh CC (1994) Methods for optimization of nonlinear problems with discrete variables: a review. Struct Optim 8(2):69–85

    Article  Google Scholar 

  16. Martins JRRA, Ning A (2021) Engineering design optimization. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  17. Lee J, Leyffer S (2011) Mixed integer nonlinear programming. The IMA volumes in mathematics and its applications. Springer, New York

    Google Scholar 

  18. Köppe M (2012) On the complexity of nonlinear mixed-integer optimization. In: Mixed integer nonlinear programming. Springer, New York, pp 533–557

  19. Kingma DP, Welling M (2019) An introduction to variational autoencoders. arXiv:1906.02691

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Li X, Ning S, Liu Z, Yan Z, Luo C, Zhuang Z (2020) Designing phononic crystal with anticipated band gap through a deep learning based data-driven method. Comput Methods Appl Mech Eng 361:112737

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo T, Lohan DJ, Cang R, Ren MY, Allison JT (2018) An indirect design representation for topology optimization using variational autoencoder and style transfer. In: 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (p. 0804).

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

    MATH  Google Scholar 

  24. Schmidhuber J (2015) Deep learning in neural networks: an overview. Neural Netw 61:85–117

    Article  Google Scholar 

  25. Systèmes Dassault (2021) Solidworks. http://www.solidworks.com, Access date: 1 Oct 2021

  26. Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, Kopf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J, Chintala S (2019) Pytorch: an imperative style, high-performance deep learning library. In: Advances in neural information processing systems, vol 32. pp 8024–8035

  27. Kingma DP, Ba JL (2015) Adam: a method for stochastic optimization. In: 3rd International conference on learning representations, ICLR 2015—conference track proceedings, Dec 2015. arXiv:1412.6980

  28. Shi L, Li B, Hašan M, Sunkavalli K, Boubekeur T, Mech R, Matusik W (2020) Match: differentiable material graphs for procedural material capture. ACM Trans Graph 39(6):1–15

    Google Scholar 

  29. Hu Y, Anderson L, Li T-M, Sun Q, Carr N, Ragan-Kelley J, Durand F (2019) Difftaichi: differentiable programming for physical simulation. 2019 Oct 1. arXiv:1910.00935

  30. Suresh K (2021) Design optimization using MATLAB and SOLIDWORKS. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  31. Kervadec H, Dolz J, Yuan J, Desrosiers C, Granger E, Ayed IB (2019) Constrained deep networks: Lagrangian optimization via log-barrier extensions 2(3):4. 2019 Apr 8. arXiv:1904.04205

  32. Segerlind LJ (1984) Applied finite element analysis. John Wiley & Sons; 1991 Jan 16.

  33. Chandrasekhar A, Sridhara S, Suresh K (2021) Auto: a framework for automatic differentiation in topology optimization. Struct Multidiscip Optim 64(6):4355–4365

    Article  Google Scholar 

  34. Duchi J, Hazan E, Singer Y (2011) Adaptive subgradient methods for online learning and stochastic optimization. J Mach Learn Res 1:12(7)

  35. Ashby MF (2011) Chapter 1–introduction. In: Ashby MF (ed) Materials selection in mechanical design, 4th edn. Butterworth-Heinemann, Oxford, pp 1–13

    Google Scholar 

  36. Ward L, Agrawal A, Choudhary A, Wolverton C (2016) A general-purpose machine learning framework for predicting properties of inorganic materials. npj Comput Mater 2(1):1–7

    Article  Google Scholar 

  37. Ge X, Goodwin RT, Gregory JR, Kirchain RE, Maria J, Varshney LR (2019) Accelerated discovery of sustainable building materials. arXiv:1905.08222

  38. Design G (2018) CES Selector. Cambridge, UK: Material Universe. Zugriff unter. https://www.grantadesign.com

  39. Razavi A, Van den Oord A, Vinyals O (2019) Generating diverse high-fidelity images with vq-vae-2. Advances in neural information processing systems. 2019;32

  40. Vahdat A, Kautz J (2020) NVAE: a deep hierarchical variational autoencoder. Adv Neural Inf Process Syst 33:19667–19679

    Google Scholar 

  41. Leung FH-F, Lam H-K, Ling S-H, Tam PK-S (2003) Tuning of the structure and parameters of a neural network using an improved genetic algorithm. IEEE Trans Neural Netw 14(1):79–88

    Article  Google Scholar 

  42. Hou X, Shen L, Sun K, Qiu G (2017) Deep feature consistent variational autoencoder. In: 2017 IEEE winter conference on applications of computer vision (WACV). IEEE, (pp 1133–1141)

  43. Peng X, Tsang IW, Zhou JT, Zhu H (2018) k-meansnet: when k-means meets differentiable programming. arXiv:1808.07292

  44. Wang L, Dong X, Wang Y, Liu L, An W, Guo Y (2022) Learnable lookup table for neural network quantization. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 12423–12433

  45. Chandrasekhar A, Suresh K (2021) TOuNN: topology optimization using neural networks. Struct Multidiscip Optim 63(3):1135–1149

    Article  MathSciNet  Google Scholar 

  46. Giraldo-Londoño O, Mirabella L, Dalloro L, Paulino GH (2020) Multi-material thermomechanical topology optimization with applications to additive manufacturing: design of main composite part and its support structure. Comput Methods Appl Mech Eng 363:112812

    Article  MathSciNet  MATH  Google Scholar 

  47. Takenaka K (2012) Negative thermal expansion materials: technological key for control of thermal expansion. Sci Technol Adv Mater 13:013001

  48. Clausen A, Aage N, Sigmund O (2015) Topology optimization of coated structures and material interface problems. Comput Methods Appl Mech Eng 290:524–541

    Article  MathSciNet  MATH  Google Scholar 

  49. Chan Y-C, Da D, Wang L, Chen W (2021) Remixing functionally graded structures: data-driven topology optimization with multiclass shape blending. arXiv:2112.00648

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Acknowledgements

The authors would like to thank the support of National Science Foundation through Grant CMMI 1561899.

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Author notes

  1. Aaditya Chandrasekhar and Saketh Sridhara: contributed equally.

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI, USA

    Aaditya Chandrasekhar, Saketh Sridhara & Krishnan Suresh

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  1. Aaditya Chandrasekhar
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  2. Saketh Sridhara
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  3. Krishnan Suresh
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Correspondence to Krishnan Suresh.

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The Python code pertinent to this paper is available at https://github.com/UW-ERSL/MaTruss. The material data were sourced from SolidWorks.

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Chandrasekhar, A., Sridhara, S. & Suresh, K. Integrating material selection with design optimization via neural networks. Engineering with Computers 38, 4715–4730 (2022). https://doi.org/10.1007/s00366-022-01736-0

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