Efficient, high-resolution topology optimization method based on convolutional neural networks

Abstract

Topology optimization is a pioneer design method that can provide various candidates with high mechanical properties. However, high resolution is desired for optimum structures, but it normally leads to a computationally intractable puzzle, especially for the solid isotropic material with penalization (SIMP) method. In this study, an efficient, high-resolution topology optimization method is developed based on the super-resolution convolutional neural network (SRCNN) technique in the framework of SIMP. SRCNN involves four processes, namely, refinement, path extraction and representation, nonlinear mapping, and image reconstruction. High computational efficiency is achieved with a pooling strategy that can balance the number of finite element analyses and the output mesh in the optimization process. A combined treatment method that uses 2D SRCNN is built as another speed-up strategy to reduce the high computational cost and memory requirements for 3D topology optimization problems. Typical examples show that the high-resolution topology optimization method using SRCNN demonstrates excellent applicability and high efficiency when used for 2D and 3D problems with arbitrary boundary conditions, any design domain shape, and varied load.

Keywords

topology optimization / convolutional neural network / high resolution / density-based

References

[1] Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224

[2] Bendsøe M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202

[3] Sigmund O A. 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127

[4] Rozvany G I N, Zhou M, Birker T. Generalized shape optimization without homogenization. Structural Optimization, 1992, 4(3–4): 250–252

[5] Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896

[6] Querin O M, Steven G P, Xie Y M. Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Engineering Computations, 1998, 15(8): 1031–1048

[7] Huang X, Xie Y M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in Analysis and Design, 2007, 43(14): 1039–1049

[8] Rozvany G I N. A critical review of established methods of structural topology optimization. Structural and Multidisciplinary Optimization, 2009, 37(3): 217–237

[9] Xia L, Zhang L, Xia Q, Stress-based topology optimization using bi-directional evolutionary structural optimization method. Computer Methods in Applied Mechanics and Engineering, 2018, 333: 356–370

[10] Wang M Y, Wang X, Guo D. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 227–246

[11] Wei P, Li Z, Li X, An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Structural and Multidisciplinary Optimization, 2018, 58(2): 831–849

[12] Xia Q, Shi T, Xia L. Stable hole nucleation in level set based topology optimization by using the material removal scheme of BESO. Computer Methods in Applied Mechanics and Engineering, 2019, 343: 438–452

[13] Xia Q, Shi T. Generalized hole nucleation through BESO for the level set based topology optimization of multi-material structures. Computer Methods in Applied Mechanics and Engineering, 2019, 355: 216–233

[14] Liu H, Zong H, Shi T, M-VCUT level set method for optimizing cellular structures. Computer Methods in Applied Mechanics and Engineering, 2020, 367: 113154

[15] Guo X, Zhang W, Zhong W. Doing topology optimization explicitly and geometrically—A new moving morphable components based framework. Journal of Applied Mechanics, 2014, 81(8): 081009

[16] Zhang W, Chen J, Zhu X, Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Computer Methods in Applied Mechanics and Engineering, 2017, 322: 590–614

[17] Lei X, Liu C, Du Z, Machine learning-driven real-time topology optimization under moving morphable component-based framework. Journal of Applied Mechanics, 2019, 86(1): 011004

[18] Cai S, Zhang W. An adaptive bubble method for structural shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112778

[19] Zhu J H, Zhang W H, Xia L. Topology optimization in aircraft and aerospace structures design. Archives of Computational Methods in Engineering, 2016, 23(4): 595–622

[20] Fu Y F, Rolfe B, Chiu L N S, Design and experimental validation of self-supporting topologies for additive manufacturing. Virtual and Physical Prototyping, 2019, 14(4): 382–394

[21] Meng L, Zhang W, Quan D, From topology optimization design to additive manufacturing: Today’s success and tomorrow’s roadmap. Archives of Computational Methods in Engineering, 2020, 27(3): 805–830

[22] Chin T W, Kennedy G J. Large-scale compliance-minimization and buckling topology optimization of the undeformed common research model wing. In: Proceedings of the 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. San Diego: AIAA, 2016

[23] Liu J, Ou H, He J, Topological design of a lightweight sandwich aircraft spoiler. Materials, 2019, 12(19): 3225

[24] Sutradhar A, Park J, Carrau D, Designing patient-specific 3D printed craniofacial implants using a novel topology optimization method. Medical & Biological Engineering & Computing, 2016, 54(7): 1123–1135

[25] Alexandersen J, Sigmund O, Aage N. Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection. International Journal of Heat and Mass Transfer, 2016, 100: 876–891

[26] Ye M, Gao L, Li H. A design framework for gradually stiffer mechanical metamaterial induced by negative Poisson’s ratio property. Materials & Design, 2020, 192: 108751

[27] Groen J P, Sigmund O. Homogenization-based topology optimization for high-resolution manufacturable microstructures. International Journal for Numerical Methods in Engineering, 2018, 113(8): 1148–1163

[28] Wu Z, Xia L, Wang S, Topology optimization of hierarchical lattice structures with substructuring. Computer Methods in Applied Mechanics and Engineering, 2019, 345: 602–617

[29] Zhu B, Skouras M, Chen D, Two-scale topology optimization with microstructures. ACM Transactions on Graphics, 2017, 36(4): 120b

[30] Wang Y, Xu H, Pasini D. Multiscale isogeometric topology optimization for lattice materials. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 568–585

[31] Li H, Luo Z, Gao L, Topology optimization for concurrent design of structures with multi-patch microstructures by level sets. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 536–561

[32] Li H, Luo Z, Xiao M, A new multiscale topology optimization method for multiphase composite structures of frequency response with level sets. Computer Methods in Applied Mechanics and Engineering, 2019, 356: 116–144

[33] Christiansen A N, Bærentzen J A, Nobel-Jørgensen M, Combined shape and topology optimization of 3D structures. Computers & Graphics, 2015, 46: 25–35

[34] Wang H, Liu J, Wen G. An efficient evolutionary structural optimization method with smooth edges based on the game of building blocks. Engineering Optimization, 2019, 51(12): 2089–2018

[35] Nguyen T H, Paulino G H, Song J, A computational paradigm for multiresolution topology optimization (MTOP). Structural and Multidisciplinary Optimization, 2010, 41(4): 525–539

[36] Nguyen-Xuan H. A polytree-based adaptive polygonal finite element method for topology optimization. International Journal for Numerical Methods in Engineering, 2017, 110(10): 972–1000

[37] Leader M K, Chin T W, Kennedy G J. High-resolution topology optimization with stress and natural frequency constraints. AIAA Journal, 2019, 57(8): 3562–3578

[38] Chin T W, Leader M K, Kennedy G J. A scalable framework for large-scale 3D multimaterial topology optimization with octree-based mesh adaptation. Advances in Engineering Software, 2019, 135: 102682

[39] Groen J P, Langelaar M, Sigmund O, Higher-order multi-resolution topology optimization using the finite cell method. International Journal for Numerical Methods in Engineering, 2017, 110(10): 903–920

[40] Gupta D K, van Keulen F, Langelaar M. Design and analysis adaptivity in multi-resolution topology optimization. 2018, arXiv:1811.09821v1

[41] Xiao M, Lu D, Breitkopf P, Multi-grid reduced-order topology optimization. Structural and Multidisciplinary Optimization, 2020, 61: 2319–2341

[42] Lieu Q X, Lee J. Multiresolution topology optimization using isogeometric analysis. International Journal for Numerical Methods in Engineering, 2017, 112(13): 2025–2047

[43] Xu M, Xia L, Wang S, An isogeometric approach to topology optimization of spatially graded hierarchical structures. Composite Structures, 2019, 225: 111171

[44] Wang Y, Liao Z, Ye M, An efficient isogeometric topology optimization using multilevel mesh, MGCG and local-update strategy. Advances in Engineering Software, 2020, 139: 102733

[45] Wang H, Liu J, Wen G. Achieving large-scale or high-resolution topology optimization based on modified BESO and XEFM. 2019, arXiv:1908.07157

[46] Kim Y Y, Yoon G H. Multi-resolution multi-scale topology optimization—A new paradigm. International Journal of Solids and Structures, 2000, 37(39): 5529–5559

[47] Stainko R. An adaptive multilevel approach to the minimal compliance problem in topology optimization. Communications in Numerical Methods in Engineering, 2006, 22(2): 109–118

[48] Liao Z, Zhang Y, Wang Y, A triple acceleration method for topology optimization. Structural and Multidisciplinary Optimization, 2019, 60(2): 727–744

[49] Suresh K. Generating 3D topologies with multiple constraints on the GPU. In: Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization. Orlando, 2013, 1–9

[50] Challis V J, Roberts A P, Grotowski J F. High resolution topology optimization using graphics processing units (GPUs). Structural and Multidisciplinary Optimization, 2014, 49(2): 315–325

[51] Aage N, Andreassen E, Lazarov B S, Giga-voxel computational morphogenesis for structural design. Nature, 2017, 550(7674): 84–86

[52] Long K, Gu C, Wang X, A novel minimum weight formulation of topology optimization implemented with reanalysis approach. International Journal for Numerical Methods in Engineering, 2019, 120(5): 567–579

[53] Wang Y, Liao Z, Shi S, Data-driven structural design optimization for petal-shaped auxetics using isogeometric analysis. Computer Modeling in Engineering & Sciences, 2020, 122(2): 433–458

[54] Zhou Y, Zhan H, Zhang W, A new data-driven topology optimization framework for structural optimization. Computers & Structures, 2020, 239: 106310

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

[56] Banga S, Gehani H, Bhilare S, 3D topology optimization using convolutional neural networks. 2018, arXiv:1808.07440v1

[57] Zhang Y, Chen A, Peng B, A deep convolutional neural network for topology optimization with strong generalization ability. 2019, arXiv:1901.07761v1

[58] Li B, Huang C, Li X, Non-iterative structural topology optimization using deep learning. Computer-Aided Design, 2019, 115: 172–180

[59] Dong C, Loy C C, He K, Image super-resolution using deep convolutional networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2016, 38(2): 295–307

[60] Bendsøe M P, Sigmund O. Topology Optimization: Theory, Methods, and Applications. Berlin: Springer, 2013, 37–40

[61] Andreassen E, Clausen A, Schevenels M, Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 2011, 43(1): 1–16

[62] Liu H, Wang Y, Zong H, Efficient structure topology optimization by using the multiscale finite element method. Structural and Multidisciplinary Optimization, 2018, 58(4): 1411–1430