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Accelerated topology optimization by means of deep learning

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Abstract

This study is focused on enhancing the computational efficiency of the solid isotropic material with penalization (SIMP) approach implemented for solving topology optimization problems. Solving such problems might become extremely time-consuming; in this direction, machine learning (ML) and specifically deep neural computing are integrated in order to accelerate the optimization procedure. The capability of ML-based computational models to extract multiple levels of representation of non-linear input data has been implemented successfully in various problems ranging from time series prediction to pattern recognition. The later one triggered the development of the methodology proposed in the current study that is based on deep belief networks (DBNs). More specifically, a DBN is calibrated on transforming the input data to a new higher-level representation. Input data contains the density fluctuation pattern of the finite element discretization provided by the initial steps of SIMP approach, and output data corresponds to the resulted density values distribution over the domain as obtained by SIMP. The representation capabilities and the computational advantages offered by the proposed DBN-based methodology coupled with the SIMP approach are investigated in several benchmark topology optimization test examples where it is observed more than one order of magnitude reduction on the iterations that were originally required by SIMP, while the advantages become more pronounced in case of large-scale problems.

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References

  • Aage N, Andreassen E, Lazarov BS (2015) Topology optimization using PETSc: an easy-to-use, fully parallel, open source topology optimization framework. Struct Multidiscip Optim 51(3):565–572

    MathSciNet  Google Scholar 

  • Adeli H, Park HS (1995) A neural dynamics model for structural optimization-theory. Comput Struct 57(3):383–390

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  • Bendsoe MP, Sigmund O (2013) Topology optimization: theory, methods and applications. Springer.

  • Bengio Y (2009) Learning deep architectures for AI, foundations and trends® in. Mach Learn 2(1):1–127

    MATH  Google Scholar 

  • Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158

    MathSciNet  MATH  Google Scholar 

  • Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459

    MATH  Google Scholar 

  • Christensen PW, Klarbring A (2009) An introduction to structural optimization. Springer

  • Collobert R, Weston J (2008) A unified architecture for natural language processing: deep neural networks with multitask learning. In: In proceedings of the 25 international conference on machine learning (ICML'08). Helsinki, Finland, pp 160–167

    Google Scholar 

  • Duarte LS, Celes W, Pereira A, Menezes IFM, Paulino GH (2015) PolyTop++: an efficient alternative for serial and parallel topology optimization on CPUs & GPUs. Struct Multidiscip Optim 52(5):845–859

    MathSciNet  Google Scholar 

  • Fischer A, Igel C (2012) An introduction to restricted Boltzmann machines. In: Progress in pattern recognition, image analysis, computer vision and applications. Springer, Buenos Aires, pp 14–36

    Google Scholar 

  • Freund Y, Haussler D, (1992) Unsupervised learning of distributions on binary vectors using two layer networks. In Proceedings of the 4th International Conference on Neural Information Processing Systems. 912–919

  • Gallagher RH, Zienklewicz OC (1973) Optimum structural design: theory and applications, New York, John Wiley & Sons

  • Gholizadeh S, Salajegheh E (2009) Optimal design of structures subjected to time history loading by swarm intelligence and an advanced metamodel. Comput Methods Appl Mech Eng 198(37–40):2936–2949

    MATH  Google Scholar 

  • Greenspan H, Van Ginneken B, Summers RM (2016) Guest editorial deep learning in medical imaging: overview and future promise of an exciting new technique. IEEE Trans Med Imaging 35(5):1153–1159

    Google Scholar 

  • Hajela P, Lee E, Lin CY, (1993) Genetic algorithms in structural topology optimization. In: M.P. Bendsøe, C.A.M. Soares (eds) Topology Design of Structures. NATO ASI Series (Series E: Applied Sciences), Springer, Dordrecht 227, pp. 117–134

  • Haug EJ, Arora JS, (1974) Optimal mechanical design techniques based on optimal control methods. In Proceedings of the 1st ASME Design Technology Transfer Conference, New York, 65–74

  • Hinton GE (2002a) Training products of experts by minimizing contrastive divergence. Neural Comput 14(8):1711–1800

    MATH  Google Scholar 

  • Hinton GE (2002b) Training products of experts by minimizing contrastive divergence. Neural Comput 14(8):1711–1800

    MATH  Google Scholar 

  • Hinton GE (2007) Boltzmann machine. Scholarpedia. 2(5):1668

    Google Scholar 

  • Hinton GE (2009) Deep belief networks. Scholarpedia. 4(5):5947

    Google Scholar 

  • Hinton GE (2012) A practical guide to training restricted Boltzmann machines. Lect Notes Comput Sci 7700:599–619

    Google Scholar 

  • Hinton GE, Salakhutdinov RR (2006) Reducing the dimensionality of data with neural networks. Science. 313:504–507

    MathSciNet  MATH  Google Scholar 

  • Hinton GE, Osindero S, Teh Y (2006) A fast learning algorithm for deep belief nets. Neural Comput 18(7):1527–1554

    MathSciNet  MATH  Google Scholar 

  • Kazakis G, Kanellopoulos I, Sotiropoulos S, Lagaros ND (2017) Topology optimization aided structural design: interpolation, computational aspects and 3D printing. Heliyon. 3(10):e00431

    Google Scholar 

  • Krizhevsky A, Sutskever I, Hinton G (2012) ImageNet classification with deep convolutional neural networks. In: In proceeding of the 25th international conference on neural information processing systems (NIPS'12). Lake Tahoe, Nevada, pp 1097–1105

    Google Scholar 

  • Labanda SR, Stolpe M (2015) Automatic penalty continuation in structural topology optimization. Struct Multidiscip Optim 52:1205–1221

    MathSciNet  Google Scholar 

  • Lagaros ND (2018) The environmental and economic impact of structural optimization. Struct Multidiscip Optim 58(4):1751–1768

    Google Scholar 

  • Li L, Khandelwal K (2015) Volume preserving projection filters and continuation methods in topology optimization. Eng Struct 85:144–161

    Google Scholar 

  • Mahdavi A, Balaji R, Frecker M, Mockensturm EM (2006) Topology optimization of 2D continua for minimum compliance using parallel computing. Struct Multidiscip Optim 32(2):121–132

    Google Scholar 

  • Martinez-Frutos J, Herrero-Perez D (2016) Large-scale robust topology optimization using multi-GPU systems. Comput Methods Appl Mech Eng 311:393–414

    MathSciNet  MATH  Google Scholar 

  • Mlejnek HP (1992) Some aspects of the genesis of structures. Struct Multidiscip Optim 5(1–2):64–69

    Google Scholar 

  • Moller O, Ricardo OF, Laura MQ, Rubinstein M (2009) Structural optimization for performance-based design in earthquake engineering: applications of neural networks. Struct Saf 31:490–499

    Google Scholar 

  • Moses F (1974) Mathematical programming methods for structural optimization, applied mechanics division. ASME. 5:35–47

    Google Scholar 

  • Papadrakakis M, Lagaros ND (2002) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491–3507

    MATH  Google Scholar 

  • Papadrakakis M, Lagaros ND, Tsompanakis Y (1998) Structural optimization using evolution strategies and neural networks. Comput Methods Appl Mech Eng 156(1–4):309–333

    MATH  Google Scholar 

  • Papadrakakis M, Lagaros ND, Tsompanakis Y, Plevris V (2001) Large scale structural optimization: computational methods and optimization algorithms. Arch Comput Methods Eng (State of the art reviews) 8(3):239–301

    MathSciNet  MATH  Google Scholar 

  • Querin OM, Steven GP, Xie YM (1998) Evolutionary structural optimization using a bi-directional algorithm. Eng Comput 15(8):1031–1048

    MATH  Google Scholar 

  • Rumelhart DE, McClelland J (1986) Parallel distributed processing: explorations in the microstructure of cognition. MIT Press, Cambridge (Britian)

    Google Scholar 

  • Sakata S, Ashida F, Zako M (2003) Structural optimization using Kriging approximation. Comput Methods Appl Mech Eng 192(7–8):923–939

    MATH  Google Scholar 

  • Sheu CY, Prager W (1968) Recent development in optimal structural design. Appl Mech Rev 21(10):985–992

    Google Scholar 

  • Sigmund O, (1994) Design of material structures using topology optimization, PhD thesis. DCAMM S-report S69 Department of solid mechanics. Technical University of Denmark

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech strct Mach 25(4):493–524

    Google Scholar 

  • Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21(2):120–127

    Google Scholar 

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

    Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches a comparative review. Struct Multidiscip Optim 48:1031–1055

    MathSciNet  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 Multidiscip Optim 16(1):68–75

    Google Scholar 

  • Smolensky P (1986) Information processing in dynamical systems: foundations of harmony theory. In: Parallel distributed processing: explorations in the microstructure of cognition. MIT Press Cambridge, Cambridge, pp 194–281

    Google Scholar 

  • Sosnovik I, Oseledets I (2017) Neural networks for topology optimization. arXiv preprint arXiv:1709.09578

    MATH  Google Scholar 

  • Spunt L (1971) Optimum structural design New Jersey USA. Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    MathSciNet  MATH  Google Scholar 

  • Talischi C, Paulino GH, Pereira A, Menezes IFM (2012a) PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45:329–357

    MathSciNet  MATH  Google Scholar 

  • Talischi C, Paulino GH, Pereira A, Menezes IFM (2012b) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. J Struct Multidiscip Optim 45(3):309–328

    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

    MathSciNet  MATH  Google Scholar 

  • Wu J, Dick CH, Westermann R (2016) A system for high-resolution topology optimization. IEEE Trans Vis Comput Graph 22(3):1195–1208

    Google Scholar 

  • Xie Y, Steven G (1992) Shape and layout optimization via an evolutionary procedure. In Proceedings of the International Conference Computational Engineering Science, Hong Kong

    Google Scholar 

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

    Google Scholar 

  • Xingjun G, Lijuan L, Haitao M (2017) An adaptive continuation method for topology optimization of continuum structures considering buckling constrains. Int J Appl Mech 9(7):1750092 –1-24

    Google Scholar 

  • Yoo J, Lee I, (2017) Efficient density based topology optimization using dual-layer element and variable grouping method for large 3D applications. World Congress Struct Multidiscip Optim

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

    Google Scholar 

Download references

Funding

This research has been supported by the OptArch project: “Optimization Driven Architectural Design of Structures” (no, 689983) belonging to the Marie Skłodowska-Curie Actions (MSCA) Research and Innovation Staff Exchange (RISE) H2020-MSCA-RISE-2015.

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Authors and Affiliations

  1. Institute of Structural Analysis & Antiseismic Research, Department of Structural Engineering, School of Civil Engineering, National Technical University of Athens, 9, Heroon Polytechniou Str., Zografou Campus, GR-15780, Athens, Greece

    Nikos Ath. Kallioras, Georgios Kazakis & Nikos D. Lagaros

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  1. Nikos Ath. Kallioras
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  2. Georgios Kazakis
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  3. Nikos D. Lagaros
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Correspondence to Nikos D. Lagaros.

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Replication of results

As far as the topology optimization part, in order to replicate the results presented above, the TOP88 2D and 3D variants together with PolyTop++ for the case of the unstructured mesh need to be used (that are freely available). While for the case of the deep learning part of the work, the reader should contact the corresponding author (nlagaros@central.ntua.gr) for providing the trained DBM network.

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Responsible Editor: Felipe A. C. Viana

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Kallioras, N.A., Kazakis, G. & Lagaros, N.D. Accelerated topology optimization by means of deep learning. Struct Multidisc Optim 62, 1185–1212 (2020). https://doi.org/10.1007/s00158-020-02545-z

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