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Improving connectivity and accelerating multiscale topology optimization using deep neural network techniques

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Abstract

Rapid advancement in additive manufacturing capabilities and computational resources has catalyzed multiscale topology optimization (TO). Recently, TO in general and multiscale TO in particular have received increased attraction due to its systematic approach to design globally and locally optimized structures. The multiscale TO computational paradigm brings additional new challenges, including geometric frustration, non-smooth boundaries, and higher computational time that needs to be handled. In this paper, a novel deep learning-based computational pipeline is developed to overcome the challenges encountered in the multiscale TO framework. The outlined computational pipeline employs two neural networks (NNs) architecture: (1) SILONet and (2) ConnectivetyNet. SILONet predicts optimized microstructure for elements of macroscale optimized structure, and ConnectivetyNet improves connectivity for multiscale topology fitted with SILONet outputs. The main novelty of this paper is the utilization of two NNs architecture for alleviating challenges, such as geometric frustration, non-smooth edges, dangling structures at boundaries, and accelerating multiscale TO computations while capturing essential physics and ensuring optimized solutions. The effectiveness of the proposed approach is demonstrated by its application to several two-dimensional and three-dimensional test cases.

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Notes

  1. Geometric frustration results when neighboring macroscale elements are of different geometry, which leads to non-connectivity between neighboring macroscale elements.

  2. Morphological operators often take a binary image and a structuring element as input and combine them using a set operator (intersection, union, inclusion, complement).

References

  • Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393. https://doi.org/10.1016/j.jcp.2003.09.032

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method* 1. J Comput Phys 194:363–393

    Article  MathSciNet  Google Scholar 

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

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optimiz 1(4):193–202

    Article  Google Scholar 

  • Bendsøe M (2013) Optimization of structural topology, shape, and material. Springer, Berlin Heidelberg

    MATH  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

  • Bielecki D, Patel D, Rai R, Dargush GF (2021) Multi-stage deep neural network accelerated topology optimization. Struct Multidisc Optim 64(6):3473–3487

    Article  MathSciNet  Google Scholar 

  • Bielecki D, Patel D, Rai R, Dargush GF (2021) Multi-stage deep neural network accelerated topology optimization. Struct Multidisc Optim pp 1–15

  • Chapman CD, Saitou K, Jakiela MJ (1994) Genetic algorithms as an approach to configuration and topology design. J Mech Des 116(4):1005–1012. https://doi.org/10.1115/1.2919480

    Article  Google Scholar 

  • Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidisc Optim 35(2):107–115

    Article  Google Scholar 

  • Huang X, Zhou S, Xie Y, Li Q (2013) Topology optimization of microstructures of cellular materials and composites for macrostructures. Comput Mater Sci 67:397–407

    Article  Google Scholar 

  • Jenkins N, Maute K (2016) An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems. Struct Multidisc Optim. https://doi.org/10.1007/s00158-016-1467-5

    Article  MathSciNet  Google Scholar 

  • Kang Z, Wang Y (2011) Structural topology optimization based on non-local shepard interpolation of density field. Comput Methods Appl Mech Eng 200(49–52):3515–3525

    Article  MathSciNet  Google Scholar 

  • Kollmann HT, Abueidda DW, Koric S, Guleryuz E, Sobh NA (2020) Deep learning for topology optimization of 2d metamaterials. Materials Design 196:109098

    Article  Google Scholar 

  • Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet classification with deep convolutional neural networks. Adv Neural Inf Process Syst 25:1097–1105

    Google Scholar 

  • Krog L, Grihon S, Marasco A (2009) Smart design of structures through topology optimization. In: 8th World Congress on Structural and Multidisciplinary Optimization, Lisbon, Portugal, pp 1–5

  • Li B, Huang C, Li X, Zheng S, Hong J (2019) Non-iterative structural topology optimization using deep learning. Comput Aided Des 115:172–180

    Article  Google Scholar 

  • Li H, Luo Z, Gao L, Walker P (2018) Topology optimization for functionally graded cellular composites with metamaterials by level sets. Comput Methods Appl Mech Eng 328:340–364

    Article  MathSciNet  Google Scholar 

  • Liang C, Patel DR, Shelat MR (2018) Mathematical modeling of a long finned tube heating and cooling in a multizone furnace. J Thermal Sci Eng Appl 10(5)

  • Liu L, Yan J, Cheng G (2008) Optimum structure with homogeneous optimum truss-like material. Comput Struct 86(13–14):1417–1425

    Article  Google Scholar 

  • Luo Z, Zhang N, Wang Y, Gao W (2013) Topology optimization of structures using meshless density variable approximants. Int J Numer Meth Eng 93(4):443–464

    Article  MathSciNet  Google Scholar 

  • Niu B, Yan J, Cheng G (2009) Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidisc Optim 39(2):115–132

    Article  Google Scholar 

  • Rodrigues H, Guedes JM, Bendsoe M (2002) Hierarchical optimization of material and structure. Struct Multidisc Optim 24(1):1–10

    Article  Google Scholar 

  • Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidisc Optim 21:120–127. https://doi.org/10.1007/s001580050176

    Article  Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optimiz 16(1):68–75

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wang Y, Chen F, Wang MY (2017) Concurrent design with connectable graded microstructures. Comput Methods Appl Mech Eng 317:84–101

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wang Y, Zhang L, Daynes S, Zhang H, Feih S, Wang MY (2018) Design of graded lattice structure with optimized mesostructures for additive manufacturing. Mater Design 142:114–123

    Article  Google Scholar 

  • White DA, Arrighi WJ, Kudo J, Watts SE (2019) Multiscale topology optimization using neural network surrogate models. Comput Methods Appl Mech Eng 346:1118–1135

    Article  MathSciNet  Google Scholar 

  • Xia L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within fe2 nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542

    Article  Google Scholar 

  • Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896

    Article  Google Scholar 

  • Yoon G (2009) Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation. Int J Numer Meth Eng 82:591–616. https://doi.org/10.1002/nme.2777

    Article  MATH  Google Scholar 

  • Yu Y, Hur T, Jung J, Jang IG (2019) Deep learning for determining a near-optimal topological design without any iteration. Struct Multidisc Optim 59(3):787–799

    Article  Google Scholar 

  • Zhang W, Sun S (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Meth Eng 68(9):993–1011

    Article  Google Scholar 

  • Zhou M, Rozvany G (1991) The coc algorithm, part ii: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336

    Article  Google Scholar 

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Funding

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

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Authors and Affiliations

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

    Darshil Patel, Dustin Bielecki & Rahul Rai

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

    Gary Dargush

Authors
  1. Darshil Patel
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  2. Dustin Bielecki
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  3. Rahul Rai
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  4. Gary Dargush
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Correspondence to Darshil Patel.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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MATLAB and Python codes were used to generate results in this paper. The datasets can be made available for academic use from the corresponding author.

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Responsible Editor: Seonho Cho

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Appendix: Hyperparameters table

Appendix: Hyperparameters table

Table 4 U-Net architecture hyperparameters and network details for the 2D and 3D frameworks
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Patel, D., Bielecki, D., Rai, R. et al. Improving connectivity and accelerating multiscale topology optimization using deep neural network techniques. Struct Multidisc Optim 65, 126 (2022). https://doi.org/10.1007/s00158-022-03223-y

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