De-homogenization using convolutional neural networks. (English) Zbl 1507.74263
Summary: This paper presents a deep learning-based de-homogenization method for structural compliance minimization. By using a convolutional neural network to parameterize the mapping from a set of lamination parameters on a coarse mesh to a one-scale design on a fine mesh, we avoid solving the least square problems associated with traditional de-homogenization approaches and save time correspondingly. To train the neural network, a two-step custom loss function has been developed which ensures a periodic output field that follows the local lamination orientations. A key feature of the proposed method is that the training is carried out without any use of or reference to the underlying structural optimization problem, which renders the proposed method robust and insensitive wrt domain size, boundary conditions, and loading. A post-processing procedure utilizing a distance transform on the output field skeleton is used to project the desired lamination widths onto the output field while ensuring a predefined minimum length-scale and volume fraction. To demonstrate that the deep learning approach has excellent generalization properties, numerical examples are shown for several different load and boundary conditions. For an appropriate choice of parameters, the de-homogenized designs perform within 7–25% of the homogenization-based solution at a fraction of the computational cost. With several options for further improvements, the scheme may provide the basis for future interactive high-resolution topology optimization.
MSC:
| 74P05 | Compliance or weight optimization in solid mechanics |
| 74P15 | Topological methods for optimization problems in solid mechanics |
References:
| [1] | Bendsøe, M. P.; Sigmund, O., Topology Optimization: Theory, Methods and Applications (2004), Springer · Zbl 1059.74001 |
| [2] | LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning, Nature, 521, 7553, 436-444 (2015) |
| [3] | S. Hoyer, J. Sohl-Dickstein, S. Greydanus, Neural reparameterization improves structural optimization, Pre-print https://arxiv.org/abs/1909.04240. |
| [4] | Wang, D.; Xiang, C.; Pan, Y.; Chen, A.; Zhou, X.; Zhang, Y., A deep convolutional neural network for topology optimization with perceptible generalization ability, Eng. Optim., 1-16 (2021) |
| [5] | Cang, R.; Yao, H.; Ren, Y., One-shot generation of near-optimal topology through theory-driven machine learning, Comput. Aided Des., 109, 12-21 (2019) |
| [6] | Yu, Y.; Hur, T.; Jung, J.; Jang, I. G., Deep learning for determining a near-optimal topological design without any iteration, Struct. Multidiscip. Optim., 59, 3, 787-799 (2019) |
| [7] | Chandrasekhar, A.; Suresh, K., Tounn: Topology optimization using neural networks, Struct. Multidiscip. Optim., 63, 3, 1135-1149 (2021) |
| [8] | Nie, Z.; Lin, T.; Jiang, H.; Kara, L. B., Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain, J. Mech. Des., 143, 3 (2021) |
| [9] | Kallioras, N. A.; Kazakis, G.; Lagaros, N. D., Accelerated topology optimization by means of deep learning, Struct. Multidiscip. Optim., 62, 3, 1185-1212 (2020) |
| [10] | Lin, Q.; Hong, J.; Liu, Z.; Li, B.; Wang, J., Investigation into the topology optimization for conductive heat transfer based on deep learning approach, Int. Commun. Heat Mass Transfer, 97, 103-109 (2018) |
| [11] | Li, B.; Huang, C.; Li, X.; Zheng, S.; Hong, J., Non-iterative structural topology optimization using deep learning, Comput. Aided Des., 115, 172-180 (2019) |
| [12] | Li, B.; Tang, W.; Ding, S.; Hong, J., A generative design method for structural topology optimization via transformable triangular mesh algorithm, Struct. Multidiscip. Optim., 62, 3, 1159-1183 (2020) |
| [13] | Abueidda, D. W.; Almasri, M.; Ammourah, R.; Ravaioli, U.; Jasiuk, I. M.; Sobh, N. A., Prediction and optimization of mechanical properties of composites using convolutional neural networks, Compos. Struct., 227, Article 111264 pp. (2019) |
| [14] | Tan, R. K.; Zhang, N. L.; Ye, W., A deep learning-based method for the design of microstructural materials, Struct. Multidiscip. Optim., 61, 4, 1417-1438 (2020) |
| [15] | Wang, L.; Chan, Y.-C.; Ahmed, F.; Liu, Z.; Zhu, P.; Chen, W., Deep generative modeling for mechanistic-based learning and design of metamaterial systems, Comput. Methods Appl. Mech. Engrg., 372, Article 113377 pp. (2020) · Zbl 1506.74216 |
| [16] | Kollmann, H. T.; Abueidda, D. W.; Koric, S.; Guleryuz, E.; Sobh, N. A., Deep learning for topology optimization of 2d metamaterials, Mater. Des., 196, Article 109098 pp. (2020) |
| [17] | Chen, C. T.; Gu, G. X., Generative deep neural networks for inverse materials design using backpropagation and active learning, Adv. Sci., 7, 5, Article 1902607 pp. (2020) |
| [18] | Oh, S.; Jung, Y.; Kim, S.; Lee, I.; Kang, N., Deep generative design: Integration of topology optimization and generative models, Trans. ASME, J. Mech. Des., 141, 11, Article e4044229 pp. (2019) |
| [19] | Ulyanov, D.; Vedaldi, A.; Lempitsky, V., Deep image prior, Int. J. Comput. Vis., 128, 1867-1888 (2020) |
| [20] | Ledig, C.; Theis, L.; Huszár, F.; Caballero, J.; Cunningham, A.; Acosta, A.; Aitken, A.; Tejani, A.; Totz, J.; Wang, Z.; Shi, W., Photo-realistic single image super-resolution using a generative adversarial network, (Proceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition. Proceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017 2017- (2017)), 105-114 |
| [21] | Chi, H.; Zhang, Y.; Tang, T. L.E.; Mirabella, L.; Dalloro, L.; Song, L.; Paulino, G. H., Universal machine learning for topology optimization, Comput. Methods Appl. Mech. Engrg., 375, Article 112739 pp. (2021) · Zbl 1506.74267 |
| [22] | Pantz, O.; Trabelsi, K., A post-treatment of the homogenization method for shape optimization, SIAM J. Control Optim., 47, 3, 1380-1398 (2008) · Zbl 1161.49042 |
| [23] | Groen, J. P.; Sigmund, O., Homogenization-based topology optimization for high-resolution manufacturable micro-structures, Internat. J. Numer. Methods Engrg., 113, 8, 1148-1163 (2018) · Zbl 07874747 |
| [24] | Allaire, G.; Geoffroy-Donders, P.; Pantz, O., Topology optimization of modulated and oriented periodic microstructures by the homogenization method, Comput. Math. Appl., 78, 7, 2197-2229 (2019) · Zbl 1443.74246 |
| [25] | Groen, J. P.; Stutz, F. C.; Aage, N.; Bærentzen, J. A.; Sigmund, O., De-homogenization of optimal multi-scale 3d topologies, Comput. Methods Appl. Mech. Engrg., 364, Article 112979 pp. (2020) · Zbl 1442.74148 |
| [26] | M.P. Bendsøe, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng. · Zbl 0671.73065 |
| [27] | Shelhamer, E.; Long, J.; Darrell, T., Fully convolutional networks for semantic segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 39, 4, Article 7478072 pp. (2017), 640-651 |
| [28] | Nobel-Jørgensen, M.; Aage, N.; Christiansen, A.; Igarashi, T.; Bærentzen, J.; Sigmund, O., 3d interactive topology optimization on hand-held devices, Struct. Multidiscip. Optim., 51, 6, 1385-1391 (2015) |
| [29] | Bendsøe, M. P., Optimal shape design as a material distribution problem, J. Struct. Optim., 1, 193-202 (1989) |
| [30] | Bendsøe, M. P.; Sigmund, O., Material interpolation schemes in topology optimization, Arch. Appl. Mech., 69, 9-10, 635-654 (1999) · Zbl 0957.74037 |
| [31] | Baandrup, M.; Sigmund, O.; Polk, H.; Aage, N., Closing the gap towards super-long suspension bridges using computational morphogenesis, Nature Commun., 11, 2735 (2020) |
| [32] | Giele, R.; Groen, J.; Aage, N.; Andreasen, C.; Sigmund, O., On approaches for avoiding low stiffness regions in variable thickness sheet and homogenization-based topology optimization, Struct. Multidiscip. Optim., 64, 39-52 (2021) |
| [33] | Wang, F.; Lazarov, B. S.; Sigmund, O., On projection methods, convergence and robust formulations in topology optimization, Struct. Multidiscip. Optim., 43, 767-784 (2011) · Zbl 1274.74409 |
| [34] | Norvig, P.; Russell, S. J., Artificial Intelligence: A Modern Approach (2010), Pearson |
| [35] | Ghahramani, Z., Unsupervised learning, (Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), Vol. 3176 (2004)), 72-112 · Zbl 1120.68434 |
| [36] | He, K.; Zhang, X.; Ren, S.; Sun, J., Deep residual learning for image recognition, (Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition 2016- (2016)), Article 7780459 pp. |
| [37] | Mordvintsev, A.; Olah, C.; Tyka, M., Inceptionism: Going deeper into neural networks (2015), https://ai.googleblog.com/2015/06/inceptionism-going-deeper-into-neural.html |
| [38] | Groen, J.; Wu, J.; Sigmund, O., Homogenization-based stiffness optimization and projection of 2d coated structures with orthotropic infill, Comput. Methods Appl. Mech. Engrg., 349, 722-742 (2019), (also arXiv:1808.04740 August 2018) · Zbl 1441.74149 |
| [39] | Harris, F. J., On the use of windows for harmonic analysis with the discrete fourier transform, Proc. IEEE, 66, 1, 51-83 (1978) |
| [40] | Träff, E.; Sigmund, O.; Groen, J., Simple single-scale microstructures based on rank-3 optimal laminates, Struct. Multidiscip. Optim., 59, 4, 1021-1031 (2019) |
| [41] | Zhang, T. Y.; Suen, C. Y., A fast parallel algorithm for thinning digital patterns, Commun. ACM, 27, 3, 236-239 (1984) |
| [42] | Stutz, F. C.; Groen, J. P.; Sigmund, O.; Bærentzen, J. A., Singularity aware de-homogenization for high-resolution topology optimized structures, Struct. Multidiscip. Optim., 62, 5, 2279-2295 (2020) |
| [43] | Wagner, M. G., Real-time thinning algorithms for 2d and 3d images using gpu processors, J. Real-Time Image Process., 17, 5, 1255-1266 (2020) |
| [44] | F.D.A. Zampirolli, L. Filipe, A fast cuda-based implementation for the euclidean distance transform, in: Proceedings - 2017 International Conference on High Performance Computing and Simulation, HPCS 2017, http://dx.doi.org/10.1109/HPCS.2017.123. |
| [45] | Sigmund, O.; Aage, N.; Andreassen, E., On the (non-)optimality of michell structures, Struct. Multidiscip. Optim., 54, 2, 361-373 (2016) |
| [46] | Aage, N.; Andreassen, E.; Lazarov, B. S., Topology optimization using petsc: An easy-to-use, fully parallel, open source topology optimization framework, Struct. Multidiscip. Optim., 51, 3, 565-572 (2015) |
| [47] | Amir, O.; Aage, N.; Lazarov, B. S., On multigrid-cg for efficient topology optimization, Struct. Multidiscip. Optim., 49, 5, 815-829 (2014) |
| [48] | Duysinx, P.; Bendsøe, M. P., Topology optimization of continuum structures with local stress coinstraints, Internat. J. Numer. Methods Engrg., 43, 8, 1453-1478 (1998) · Zbl 0924.73158 |
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