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Topology optimization based on deep representation learning (DRL) for compliance and stress-constrained design

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Abstract

This paper proposed a new topology optimization method based on geometry deep learning. The density distribution in design domain is described by deep neural networks. Compared to traditional density-based method, using geometry deep learning method to describe the density distribution function can guarantee the smoothness of the boundary and effectively overcome the checkerboard phenomenon. The design variables can be reduced phenomenally based on deep learning representation method. The numerical results for three different kernels including the Gaussian, Tansig, and Tribas are compared. The structural complexity can be directly controlled through the architectures of the neural networks, and minimum length is also controllable for the Gaussian kernel. Several 2-D and 3-D numerical examples are demonstrated in detail to demonstrate the effectiveness of proposed method from minimum compliance to stress-constrained problems.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. 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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. Sigmund O, Bondsgc M (2003) Topology optimization. State-of-the-art future perspectives. Technical University of Denmark, Copenhagen

    Google Scholar 

  5. Lazarov BS, Wang F, Sigmund O (2016) Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86(1–2):189–218

    Google Scholar 

  6. Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413

    MathSciNet  MATH  Google Scholar 

  7. Zhang W et al (2018) Topology optimization with multiple materials via moving morphable component (MMC) method. Int J Numer Methods Eng 113(11):1653–1675

    MathSciNet  Google Scholar 

  8. Zhang W et al (2017) Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Comput Methods Appl Mech Eng 322:590–614

    MathSciNet  MATH  Google Scholar 

  9. Norato J, Bell B, Tortorelli D (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comput Methods Appl Mech Eng 293:306–327

    MathSciNet  MATH  Google Scholar 

  10. Zhang S, Gain AL, Norato JA (2017) Stress-based topology optimization with discrete geometric components. Comput Methods Appl Mech Eng 325:1–21

    MathSciNet  MATH  Google Scholar 

  11. Watts S, Tortorelli DA (2017) A geometric projection method for designing three-dimensional open lattices with inverse homogenization. Int J Numer Methods Eng 112(11):1564–1588

    MathSciNet  Google Scholar 

  12. White DA, Stowell ML, Tortorelli DA (2018) Toplogical optimization of structures using Fourier representations. Struct Multidiscipl Optim 58(3):1205–1220

    MathSciNet  Google Scholar 

  13. 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

    MathSciNet  Google Scholar 

  14. Gulian M, Raissi M, Perdikaris P, Karniadakis G (2019) Machine learning of space-fractional differential equations. SIAM J Sci Comput 41(4):A2485–A2509

    MathSciNet  MATH  Google Scholar 

  15. 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

    MathSciNet  MATH  Google Scholar 

  16. Raissi M, Karniadakis GE (2018) Hidden physics models: machine learning of nonlinear partial differential equations. J Comput Phys 357:125–141

    MathSciNet  MATH  Google Scholar 

  17. Raissi M, Wang Z, Triantafyllou MS, Karniadakis GE (2019) Deep learning of vortex-induced vibrations. J Fluid Mech 861:119–137

    MathSciNet  MATH  Google Scholar 

  18. Raissi M, Perdikaris P, Karniadakis GE (2017) Machine learning of linear differential equations using Gaussian processes. J Comput Phys 348:683–693

    MathSciNet  MATH  Google Scholar 

  19. Alber M et al. (2019) Multiscale modeling meets machine learning: what can we learn? arXiv:.11958

  20. Mao Z, Jagtap AD, Karniadakis GE (2020) Physics-informed neural networks for highspeed flows. Comput Methods Appl Mech Eng 360:112789

  21. Zhang D, Lu L, Guo L, Karniadakis GE (2019) Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. J Comput Phys 397:108850

    MathSciNet  Google Scholar 

  22. Meng X, Li Z, Zhang D, Karniadakis GE (2019) PPINN: parareal physics-informed neural network for time-dependent PDEs. arXiv:.10145

  23. Yang L et al. (2019) Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs. arXiv:.13444

  24. Lei X, Liu C, Du Z, Zhang W, Guo X (2019) Machine learning-driven real-time topology optimization under moving morphable component-based framework. J Appl Mech 86(1):011004

    Google Scholar 

  25. Oh S, Jung Y, Kim S, Lee I, Kang N (2019) Deep generative design: integration of topology optimization and generative models. J Mech Des 141(11):111405

  26. 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

    MathSciNet  MATH  Google Scholar 

  27. Bengio Y, Courville A, Vincent P (2013) Representation learning: a review and new perspectives. IEEE Trans Pattern Anal 35(8):1798–1828

    Google Scholar 

  28. LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521(7553):436

    Google Scholar 

  29. Qi CR, Su H, Mo K, Guibas LJ (2017) Pointnet: deep learning on point sets for 3D classification and segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 652–660

  30. Litany O, Bronstein A, Bronstein M, Makadia A (2018) Deformable shape completion with graph convolutional autoencoders. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 1886–1895

  31. Park JJ, Florence P, Straub J, Newcombe R, Lovegrove S (2019) Deepsdf: learning continuous signed distance functions for shape representation. arXiv:.05103

  32. Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3D surface construction algorithm. ACM Siggraph Comput Graph 21(4):163–169

    Google Scholar 

  33. Zhou P, Du J, Lü Z (2018) A generalized DCT compression based density method for topology optimization of 2D and 3D continua. Comput Methods Appl Mech Eng 334:1–21

    MathSciNet  MATH  Google Scholar 

  34. Luo Y, Bao J (2019) A material-field series-expansion method for topology optimization of continuum structures. Comput Struct 225:106122

    Google Scholar 

  35. White DA, Choi Y, Kudo J (2019) A dual mesh method with adaptivity for stress-constrained topology optimization. Struct Multidiscipl Optim 61:749–762

    MathSciNet  Google Scholar 

  36. Luo Y, Xing J, Kang Z (2020) Topology optimization using material-field series expansion and Kriging-based algorithm: an effective non-gradient method. Comput Methods Appl Mech Eng 364:112966

    MathSciNet  MATH  Google Scholar 

  37. Benkő P, Martin RR, Várady T (2001) Algorithms for reverse engineering boundary representation models. Comput Aided Des 33(11):839–851

    Google Scholar 

  38. Hart JC (1998) Morse theory for implicit surface modeling. In: Hart JC (ed) Mathematical visualization. Springer, Berlin, pp 257–268

  39. Gomes A, Voiculescu I, Jorge J, Wyvill B, Galbraith C (2009) Implicit curves and surfaces: mathematics, data structures and algorithms. Springer, Berlin

    MATH  Google Scholar 

  40. Ucicr T (1992) Feature-based image metamorphosis. Comput Graph 26:2

    Google Scholar 

  41. Li Q, Hong Q, Qi Q, Ma X, Han X, Tian J (2018) Towards additive manufacturing oriented geometric modeling using implicit functions. Vis Comput Ind Biomed Art 1(1):1–16

    Google Scholar 

  42. Yoo DJ (2011) Porous scaffold design using the distance field and triply periodic minimal surface models. Biomaterials 32(31):7741–7754

    Google Scholar 

  43. Turk G, Levoy M (1994) Zippered polygon meshes from range images. In: Proceedings of the 21st annual conference on computer graphics and interactive techniques, pp 311–318

  44. Tripathi Y, Shukla M, Bhatt AD (2019) Implicit-function-based design and additive manufacturing of triply periodic minimal surfaces scaffolds for bone tissue engineering. J Mater Eng Perform 28(12):7445–7451

    Google Scholar 

  45. Goodfellow I et al. (2014) Generative adversarial nets. In: Advances in neural information processing systems, pp 2672–2680

  46. Dai A, Ruizhongtai Qi C, Nießner M (2017) Shape completion using 3D-encoder-predictor cnns and shape synthesis. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 5868–5877

  47. Tan SM, Michael L (1995) Reducing data dimensionality through optimizing neural network inputs. AIChE J 41(6):1471–1480

    Google Scholar 

  48. Schoen AH (1970) Infinite periodic minimal surfaces without self-intersections. National Aeronautics and Space Administration

  49. Eldan R, Shamir O (2016) The power of depth for feedforward neural networks. In: Conference on learning theory, pp 907–940

  50. Lin HW, Tegmark M, Rolnick D (2017) Why does deep and cheap learning work so well? J Stat Phys 168(6):1223–1247

    MathSciNet  MATH  Google Scholar 

  51. Liang S, Srikant R (2016) Why deep neural networks for function approximation? arXiv preprint arXiv:1610.04161

  52. Telgarsky M (2015) Representation benefits of deep feedforward networks. arXiv:.08101

  53. Mhaskar HN, Poggio T (2016) Deep vs. shallow networks: an approximation theory perspective. Anal Appl 14(06):829–848

    MathSciNet  MATH  Google Scholar 

  54. Matsugu M, Mori K, Mitari Y, Kaneda YJNN (2003) Subject independent facial expression recognition with robust face detection using a convolutional neural network. Neural Netw 16(5–6):555–559

    Google Scholar 

  55. Mhaskar H, Liao Q, Poggio T (2016) Learning functions: when is deep better than shallow. arXiv:.00988

  56. Park JJ, Florence P, Straub J, Newcombe R, Lovegrove S (2019) DeepSDF: learning continuous signed distance functions for shape representation. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 165–174

  57. Vogl TP, Mangis J, Rigler A, Zink W, Alkon D (1988) Accelerating the convergence of the back-propagation method. Biol Cybern 59(4–5):257–263

    Google Scholar 

  58. Pujol J (2007) The solution of nonlinear inverse problems and the Levenberg–Marquardt method. Geophysics 72(4):W1–W16

    MathSciNet  Google Scholar 

  59. Biegler LT, Conn AR, Coleman TF, Santosa FN (1997) Large-scale optimization with applications: optimal design and control. Springer, Berlin

    MATH  Google Scholar 

  60. Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscipl Optim 43(1):1–16

    MATH  Google Scholar 

  61. Lee E, James KA, Martins JR (2012) Stress-constrained topology optimization with design-dependent loading. Struct Multidiscipl Optim 46(5):647–661

    MathSciNet  MATH  Google Scholar 

  62. Picelli R, Townsend S, Brampton C, Norato J, Kim H (2018) Stress-based shape and topology optimization with the level set method. Comput Methods Appl Mech Eng 329:1–23

    MathSciNet  MATH  Google Scholar 

  63. Kiyono C, Vatanabe S, Silva E, Reddy J (2016) A new multi-p-norm formulation approach for stress-based topology optimization design. Compos Struct 156:10–19

    Google Scholar 

  64. Lian H, Christiansen AN, Tortorelli DA, Sigmund O, Aage N (2017) Combined shape and topology optimization for minimization of maximal von Mises stress. Struct Multidiscipl Optim 55(5):1541–1557

    MathSciNet  Google Scholar 

  65. Zhou M, Sigmund O (2017) On fully stressed design and p-norm measures in structural optimization. Struct Multidiscipl Optim 56(3):731–736

    MathSciNet  Google Scholar 

  66. Cai S, Zhang W (2015) Stress constrained topology optimization with free-form design domains. Comput Methods Appl Mech Eng 289:267–290

    MathSciNet  MATH  Google Scholar 

  67. Xia L, Zhang L, Xia Q, Shi T (2018) Stress-based topology optimization using bi-directional evolutionary structural optimization method. Comput Methods Appl Mech Eng 333:356–370

    MathSciNet  MATH  Google Scholar 

  68. Wang MY, Li L (2013) Shape equilibrium constraint: a strategy for stress-constrained structural topology optimization. Struct Multidiscipl Optim 47(3):335–352

    MathSciNet  MATH  Google Scholar 

  69. Le C, Norato J, Bruns T, Ha C, Tortorelli D, Optimization M (2010) Stress-based topology optimization for continua. Struct Multidiscipl Optim 41(4):605–620

    Google Scholar 

  70. Bradley AM (2013) PDE-constrained optimization and the adjoint method. Technical Report. Stanford University. https://cs.stanford.edu/~ambrad/adjoint_tutorial.pdf

  71. Baydin AG, Pearlmutter BA, Radul AA, Siskind JM (2018) Automatic differentiation in machine learning: a survey. J Mach Learn Res 18(153):5595–5637

    MathSciNet  MATH  Google Scholar 

  72. Bartholomew-Biggs M, Brown S, Christianson B, Dixon L, Mathematics A (2000) Automatic differentiation of algorithms. J Comput 124(1–2):171–190

    MathSciNet  MATH  Google Scholar 

  73. Andersson JA, Gillis J, Horn G, Rawlings JB, Diehl M (2019) CasADi: a software framework for nonlinear optimization and optimal control. J Math Program Comput 11(1):1–36

    MathSciNet  MATH  Google Scholar 

  74. Holmberg E, Torstenfelt B, Klarbring A, Optimization M (2013) Stress constrained topology optimization. Struct Multidiscipl Optim 48(1):33–47

    MathSciNet  MATH  Google Scholar 

  75. Rahmatalla S, Swan C (2004) A Q4/Q4 continuum structural topology optimization implementation. Struct Multidiscipl Optim 27(1–2):130–135

    Google Scholar 

  76. Girosi F, Jones M, Poggio T (1995) Regularization theory and neural networks architectures. Neural Comput 7(2):219–269

    Google Scholar 

  77. Baudat G, Anouar F (2001) Kernel-based methods and function approximation. In: IJCNN’01. International joint conference on neural networks. Proceedings (Cat. No. 01CH37222), vol. 2. IEEE, pp 1244–1249

  78. Elleuch K, Chaari A (2011) Modeling and identification of hammerstein system by using triangular basis functions. Int J Electr Comput Eng 1:1

    Google Scholar 

  79. Wand MP, Jones MC (1994) Kernel smoothing. Chapman and Hall, London

    MATH  Google Scholar 

  80. Yakubovich S, Zayed AI (1997) Handbook of function and generalizedfunction transformations. Academic Press, New York

    Google Scholar 

  81. Carstensen JV, Guest JK (2018) Projection-based two-phase minimum and maximum length scale control in topology optimization. Struct Multidiscipl Optim 58(5):1845–1860

    MathSciNet  Google Scholar 

  82. Carstensen JV, Guest JK (2014) New projection methods for two-phase minimum and maximum length scale control in topology optimization. In: 15th AIAA/ISSMO multidisciplinary analysis and optimization conference, p 2297

  83. Guest JK, Smith Genut LC (2010) Reducing dimensionality in topology optimization using adaptive design variable fields. Int J Numer Methods Eng 81(8):1019–1045

    MATH  Google Scholar 

  84. Guest JK (2009) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199(1–4):123–135

    MathSciNet  MATH  Google Scholar 

  85. Guest JK (2009) Imposing maximum length scale in topology optimization. Struct Multidiscipl Optim 37(5):463–473

    MathSciNet  MATH  Google Scholar 

  86. Guest J, Prevost J (2006) A penalty function for enforcing maximum length scale criterion in topology optimization. In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, p 6938

  87. Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254

    MathSciNet  MATH  Google Scholar 

  88. Lazarov BS, Wang F (2017) Maximum length scale in density based topology optimization. Comput Methods Appl Mech Eng 318:826–844

    MathSciNet  MATH  Google Scholar 

  89. Zhou M, Lazarov BS, Wang F, Sigmund O (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Eng 293:266–282

    MathSciNet  MATH  Google Scholar 

  90. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscipl Optim 43(6):767–784

    MATH  Google Scholar 

  91. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscipl Optim 33(4–5):401–424

    Google Scholar 

  92. Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sin 25(2):227–239

    MATH  Google Scholar 

  93. Lazarov BS, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765–781

    MathSciNet  MATH  Google Scholar 

  94. Svanberg K (2007) MMA and GCMMA-two methods for nonlinear optimization. Optim Syst Theory 1:1–15

    Google Scholar 

  95. Rozvany GI (2009) A critical review of established methods of structural topology optimization. Struct Multidiscipl Optim 37(3):217–237

    MathSciNet  MATH  Google Scholar 

  96. Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet classification with deep convolutional neural networks. In: Advances in neural information processing systems, pp 1097–1105

  97. Freund Y, Mason L (1999) The alternating decision tree learning algorithm. In: icml, vol 99, pp 124–133

  98. Ho TK (1995) Random decision forests. In: Proceedings of 3rd international conference on document analysis and recognition, vol 1. IEEE, pp 278–282

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Acknowledgements

The authors would like to acknowledge the support from National Science Foundation (CMMI-1634261).

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  1. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Hao Deng & Albert C. To

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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). https://doi.org/10.1007/s00466-020-01859-5

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