TONR: an exploration for a novel way combining neural network with topology optimization. (English) Zbl 1507.74346
Summary: The rapid development of deep learning has opened a new door to the exploration of topology optimization methods. The combination of deep learning and topology optimization has become one of the hottest research fields at the moment. Different from most existing work, this paper conducts an in-depth study on the method of directly using neural networks (NN) to carry out topology optimization. Inspired by the idea from the field of “Inverting Representation of Image” and “Physics-Informed Neural Network”, a topology optimization via neural reparameterization framework (TONR) that can solve various topology optimization problems is formed. The core idea of TONR is Reparameterization, which means the update of the design variables (pseudo-density) in the conventional topology optimization method is transformed into the update of the NN’s parameters. The sensitivity analysis in the conventional topology optimization method is realized by automatic differentiation technology. With the update of NN’s parameters, the density field is optimized. Some strategies for dealing with design constraints, determining NN’s initial parameters, and accelerating training are proposed in the paper. In addition, the solution of the multi-constrained topology optimization problem is also embedded in the TONR framework. Numerical examples show that TONR can stably obtain optimized structures for different optimization problems, including the stress-constrained problem, structural natural frequency optimization problems, compliant mechanism design problems, heat conduction system design problems, and the optimization problem of hyperelastic structures. Compared with the existing methods that combine deep learning with topology optimization, TONR does not need to construct a dataset in advance and does not suffer from structural disconnection. The structures obtained by TONR can be comparable to the conventional methods.
MSC:
| 74P15 | Topological methods for optimization problems in solid mechanics |
Keywords:
topology optimization; neural network; reparameterization; automatic differentiation; nonlinearityReferences:
| [1] | Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 2, 197-224 (1988) · Zbl 0671.73065 |
| [2] | Bendsøe, M. P.; Sigmund, O., Material interpolation schemes in topology optimization, Arch. Appl. Mech. (Ingenieur Archiv), 69, 9-10, 635-654 (1999) · Zbl 0957.74037 |
| [3] | Sigmund, O., A 99 line topology optimization code written in matlab, Struct. Multidiscip. Optim., 21, 2, 120-127 (2001) |
| [4] | Xie, Y.; Steven, G., A simple evolutionary procedure for structural optimization, Comput. Struct., 49, 5, 885-896 (1993) |
| [5] | Querin, O.; Steven, G.; Xie, Y., Evolutionary structural optimisation (ESO) using a bidirectional algorithm, Eng. Comput., 15, 8, 1031-1048 (1998) · Zbl 0938.74056 |
| [6] | Wang, M. Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 1-2, 227-246 (2003) · Zbl 1083.74573 |
| [7] | Allaire, G.; Jouve, F.; Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 1, 363-393 (2004) · Zbl 1136.74368 |
| [8] | Guo, X.; Zhang, W.; Zhong, W., Doing topology optimization explicitly and geometrically—a new moving morphable components based framework, J. Appl. Mech., 81, 8, Article 081009 pp. (2014) |
| [9] | Zhang, W.; Yang, W.; Zhou, J.; Li, D.; Guo, X., Structural topology optimization through explicit boundary evolution, J. Appl. Mech., 84, 1, Article 011011 pp. (2016) |
| [10] | Du, B.; Yao, W.; Zhao, Y.; Chen, X., A moving morphable voids approach for topology optimization with closed b-splines, J. Mech. Des., 141, 8, Article 081401 pp. (2019) |
| [11] | Sigmund, O.; Maute, K., Topology optimization approaches: A comparative review, Struct. Multidiscip. Optim., 48, 6, 1031-1055 (2013) |
| [12] | Deaton, J. D.; Grandhi, R. V., A survey of structural and multidisciplinary continuum topology optimization: Post 2000, Struct. Multidiscip. Optim., 49, 1, 1-38 (2014) |
| [13] | van Dijk, N. P.; Maute, K.; Langelaar, M.; van Keulen, F., Level-set methods for structural topology optimization: A review, Struct. Multidiscip. Optim., 48, 3, 437-472 (2013) |
| [14] | Munk, D. J.; Vio, G. A.; Steven, G. P., Topology and shape optimization methods using evolutionary algorithms: A review, Struct. Multidiscip. Optim., 52, 3, 613-631 (2015) |
| [15] | Wein, F.; Dunning, P. D.; Norato, J. A., A review on feature-mapping methods for structural optimization, Struct. Multidiscip. Optim., 62, 4, 1597-1638 (2020) |
| [16] | Frankel, A.; Jones, R.; Alleman, C.; Templeton, J., Predicting the mechanical response of oligocrystals with deep learning, Comput. Mater. Sci., 169, Article 109099 pp. (2019) |
| [17] | Nguyen, T. N.; Lee, S.; Nguyen-Xuan, H.; Lee, J., A novel analysis-prediction approach for geometrically nonlinear problems using group method of data handling, Comput. Methods Appl. Mech. Engrg., 354, 506-526 (2019) · Zbl 1441.74116 |
| [18] | Li, X.; Liu, Z.; Cui, S.; Luo, C.; Li, C.; Zhuang, Z., Predicting the effective mechanical property of heterogeneous materials by image based modeling and deep learning, Comput. Methods Appl. Mech. Engrg., 347, 735-753 (2019) · Zbl 1440.74258 |
| [19] | Capuano, G.; Rimoli, J. J., Smart finite elements: A novel machine learning application, Comput. Methods Appl. Mech. Engrg., 345, 363-381 (2019) · Zbl 1440.65190 |
| [20] | Ghavamian, F.; Simone, A., Accelerating multiscale finite element simulations of history-dependent materials using a recurrent neural network, Comput. Methods Appl. Mech. Engrg., 357, Article 112594 pp. (2019) · Zbl 1442.65142 |
| [21] | Yao, H.; Gao, Y.; Liu, Y., FEA-Net: A physics-guided data-driven model for efficient mechanical response prediction, Comput. Methods Appl. Mech. Engrg., 363, Article 112892 pp. (2020) · Zbl 1436.74078 |
| [22] | Chen, G.; Li, T.; Chen, Q.; Ren, S.; Wang, C.; Li, S., Application of deep learning neural network to identify collision load conditions based on permanent plastic deformation of shell structures, Comput. Mech., 64, 2, 435-449 (2019) · Zbl 1468.74082 |
| [23] | Han, Z.; Rahul, T.; De, S., A deep learning-based hybrid approach for the solution of multiphysics problems in electrosurgery, Comput. Methods Appl. Mech. Engrg., 357, Article 112603 pp. (2019) · Zbl 1442.65472 |
| [24] | Bessa, M. A.; Glowacki, P.; Houlder, M., Bayesian machine learning in metamaterial design: fragile becomes supercompressible, Adv. Mater., 31, 48, Article 1904845 pp. (2019) |
| [25] | Chen, X.; Chen, X.; Zhou, W.; Zhang, J.; Yao, W., The heat source layout optimization using deep learning surrogate modeling, Struct. Multidiscip. Optim., 62, 6, 3127-3148 (2020) |
| [26] | Goodfellow, I.; Bengio, Y.; Courville, A., (Deep Learning. Deep Learning, Adaptive Computation and Machine Learning (2016), The MIT Press: The MIT Press Cambridge, Massachusetts) · Zbl 1373.68009 |
| [27] | Adeli, H.; Park, H. S., A neural dynamics model for structural optimization—theory, Comput. Struct., 57, 3, 383-390 (1995) · Zbl 0900.73502 |
| [28] | Papadrakakis, M.; Lagaros, N. D.; Tsompanakis, Y., Structural optimization using evolution strategies and neural networks, Comput. Methods Appl. Mech. Engrg., 156, 1-4, 309-333 (1998) · Zbl 0964.74045 |
| [29] | Lagaros, N. D.; Charmpis, D. C.; Papadrakakis, M., An adaptive neural network strategy for improving the computational performance of evolutionary structural optimization, Comput. Methods Appl. Mech. Engrg., 194, 30-33, 3374-3393 (2005) · Zbl 1101.74050 |
| [30] | Kodiyalam, S.; Gurumoorthy, R., Neural networks with modified backpropagation learning applied to structural optimization, AIAA J., 34, 2, 408-412 (1996) · Zbl 0894.73092 |
| [31] | Ulu, E.; Zhang, R.; Kara, L. B., A data-driven investigation and estimation of optimal topologies under variable loading configurations, Comput. Methods Biomech. Biomed. Eng. Imaging Visual., 4, 2, 61-72 (2016) |
| [32] | Sosnovik, I.; Oseledets, I., Neural networks for topology optimization, Russian J. Numer. Anal. Math. Modelling, 34, 4, 215-223 (2019) · Zbl 1420.68178 |
| [33] | Banga, S.; Gehani, H.; Bhilare, S.; Patel, S.; Kara, L., 3D topology optimization using convolutional neural networks (2018), arXiv:1808.07440 [physics, stat], arXiv:1808.07440 |
| [34] | Guo, T.; Lohan, D. J.; Cang, R.; Ren, M. Y.; Allison, J. T., An indirect design representation for topology optimization using variational autoencoder and style transfer, (2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2018), American Institute of Aeronautics and Astronautics: American Institute of Aeronautics and Astronautics Kissimmee, Florida) |
| [35] | 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) |
| [36] | Rawat, S.; Shen, M. H.H., A novel topology design approach using an integrated deep learning network architecture (2019), arxiv:1808.02334 [cs, stat], arXiv:1808.02334 |
| [37] | Rawat, S.; Shen, M.-H. H., A novel topology optimization approach using conditional deep learning (2019), arxiv:1901.04859 [cs, stat], arXiv:1901.04859 |
| [38] | Shen, M.-H. H.; Chen, L., A new CGAN technique for constrained topology design optimization (2019), arxiv:1901.07675 [cs, stat], arXiv:1901.07675 |
| [39] | 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) |
| [40] | Oh, S.; Jung, Y.; Kim, S.; Lee, I.; Kang, N., Deep generative design: integration of topology optimization and generative models, J. Mech. Des., 141, 11, Article 111405 pp. (2019) |
| [41] | Zhang, Y.; Peng, B.; Zhou, X.; Xiang, C.; Wang, D., A deep convolutional neural network for topology optimization with strong generalization ability (2020), arxiv:1901.07761 [cs, stat], arXiv:1901.07761 |
| [42] | Hoyer, S.; Sohl-Dickstein, J.; Greydanus, S., Neural reparameterization improves structural optimization (2019), arxiv:1909.04240 [cs, stat], arXiv:1909.04240 |
| [43] | Kallioras, N. A.; Kazakis, G.; Lagaros, N. D., Accelerated topology optimization by means of deep learning, Struct. Multidiscip. Optim., 62, 3, 1185-1212 (2020) |
| [44] | Kallioras, N. A.; Lagaros, N. D., DL-SCALE: A novel deep learning-based model order upscaling scheme for solving topology optimization problems, Neural Comput. Appl. (2020) |
| [45] | Hinton, G. E.; Osindero, S.; Teh, Y.-W., A fast learning algorithm for deep belief nets, Neural Comput., 18, 7, 1527-1554 (2006) · Zbl 1106.68094 |
| [46] | Deng, H.; To, A. C., Topology optimization based on deep representation learning (DRL) for compliance and stress-constrained design, Comput. Mech. (2020) · Zbl 1466.74033 |
| [47] | 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, Article 031715 pp. (2021) |
| [48] | Abueidda, D. W.; Koric, S.; Sobh, N. A., Topology optimization of 2D structures with nonlinearities using deep learning, Comput. Struct., 237, Article 106283 pp. (2020) |
| [49] | Qian, C.; Ye, W., Accelerating gradient-based topology optimization design with dual-model artificial neural networks, Struct. Multidiscip. Optim. (2020) |
| [50] | Chandrasekhar, A.; Suresh, K., TOuNN: topology optimization using neural networks, Struct. Multidiscip. Optim. (2020) |
| [51] | 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) |
| [52] | 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 |
| [53] | Lee, S.; Kim, H.; Lieu, Q. X.; Lee, J., CNN-based image recognition for topology optimization, Knowl.-Based Syst., 198, Article 105887 pp. (2020) |
| [54] | Nakamura, K.; Suzuki, Y., Deep learning-based topological optimization for representing a user-Specified Design Area (2020), arxiv:2004.05461 [cs], arXiv:2004.05461 |
| [55] | 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) |
| [56] | Wang, C.; Yao, S.; Wang, Z.; Hu, J., Deep super-resolution neural network for structural topology optimization, Eng. Optim., 1-14 (2020) |
| [57] | Zheng, S.; He, Z.; Liu, H., Generating three-dimensional structural topologies via a U-Net convolutional neural network, Thin-Walled Struct., 159, Article 107263 pp. (2021) |
| [58] | Keshavarzzadeh, V.; Alirezaei, M.; Tasdizen, T.; Kirby, R. M., Image-based multiresolution topology optimization using deep disjunctive normal shape model, Comput. Aided Des., 130, Article 102947 pp. (2021) |
| [59] | Deng, H.; To, A. C., A parametric level set method for topology optimization based on deep neural network (DNN) (2021), arxiv:2101.03286 [math], arXiv:2101.03286 |
| [60] | Xue, L.; Liu, J.; Wen, G.; Wang, H., Efficient, high-resolution topology optimization method based on convolutional neural networks, Front. Mech. Eng. (2021) |
| [61] | Ates, G. C.; Gorguluarslan, R. M., Two-stage convolutional encoder-decoder network to improve the performance and reliability of deep learning models for topology optimization, Struct. Multidiscip. Optim. (2021) |
| [62] | Behzadi, M.; Ilies, H. T., Real-time topology optimization in 3D via deep transfer learning (2021), arxiv:2102.07657 [cs], arXiv:2102.07657 |
| [63] | Liu, K.; Tovar, A.; Nutwell, E.; Detwiler, D., Towards nonlinear multimaterial topology optimization using unsupervised machine learning and metamodel-based optimization, (Volume 2B: 41st Design Automation Conference (2015), American Society of Mechanical Engineers: American Society of Mechanical Engineers Boston, Massachusetts, USA), Article V02BT03A004 pp. |
| [64] | Lei, X.; Liu, C.; Du, Z.; Zhang, W.; Guo, X., Machine learning-driven real-time topology optimization under moving morphable component-based framework, J. Appl. Mech., 86, 1, Article 011004 pp. (2019) |
| [65] | Zhou, Y.; Zhan, H.; Zhang, W.; Zhu, J.; Bai, J.; Wang, Q.; Gu, Y., A new data-driven topology optimization framework for structural optimization, Comput. Struct., 239, Article 106310 pp. (2020) |
| [66] | Jiang, X.; Wang, H.; Li, Y.; Mo, K., Machine learning based parameter tuning strategy for MMC based topology optimization, Adv. Eng. Softw., 149, Article 102841 pp. (2020) |
| [67] | Mahendran, A.; Vedaldi, A., Understanding deep image representations by inverting them, (2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2015), IEEE: IEEE Boston, MA, USA), 5188-5196 |
| [68] | Dosovitskiy, A.; Brox, T., Inverting visual representations with convolutional networks, (2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016), IEEE: IEEE Las Vegas, NV, USA), 4829-4837, arXiv:1506.02753 |
| [69] | Ulyanov, D.; Vedaldi, A.; Lempitsky, V., Deep image prior, Int. J. Comput. Vis., 128, 7, 1867-1888 (2020) |
| [70] | Raissi, M.; Karniadakis, G. E., Hidden physics models: machine learning of nonlinear partial differential equations, J. Comput. Phys., 357, 125-141 (2018) · Zbl 1381.68248 |
| [71] | Raissi, M.; Perdikaris, P.; Karniadakis, G., 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 (2019) · Zbl 1415.68175 |
| [72] | Raissi, M.; Yazdani, A.; Karniadakis, G. E., Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations, Science, 367, 6481, 1026-1030 (2020) · Zbl 1478.76057 |
| [73] | Lu, L.; Meng, X.; Mao, Z.; Karniadakis, G. E., DeepXDE: A deep learning library for solving differential equations, SIAM Rev., 63, 1, 208-228 (2021) · Zbl 1459.65002 |
| [74] | Kingma, D. P.; Ba, J., Adam: a method for stochastic optimization (2017), arxiv:1412.6980 [cs], arXiv:1412.6980 |
| [75] | Svanberg, K., The method of moving asymptotes— a new method for structural optimization, Internat. J. Numer. Methods Engrg., 24, 2, 359-373 (1987) · Zbl 0602.73091 |
| [76] | Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, B. S.; Sigmund, O., Efficient topology optimization in MATLAB using 88 lines of code, Struct. Multidiscip. Optim., 43, 1, 1-16 (2011) · Zbl 1274.74310 |
| [77] | Zienkiewicz, O. C.; Taylor, R. L.; Fox, D., The Finite Element Method for Solid and Structural Mechanics (2014), Elsevier/Butterworth-Heinemann: Elsevier/Butterworth-Heinemann Amsterdam ; Boston · Zbl 1307.74003 |
| [78] | Bendsøe, M. P.; Sigmund, O., (Topology Optimization: Theory, Methods, and Applications. Topology Optimization: Theory, Methods, and Applications, Engineering Online Library (2011), Springer: Springer Berlin Heidelberg) · Zbl 1059.74001 |
| [79] | Ledig, C.; Theis, L.; Huszar, 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 (2017), arxiv:1609.04802 [cs, stat], arXiv:1609.04802 |
| [80] | Zhuang, F.; Qi, Z.; Duan, K.; Xi, D.; Zhu, Y.; Zhu, H.; Xiong, H.; He, Q., A comprehensive survey on transfer learning, Proc. IEEE, 109, 1, 43-76 (2021) |
| [81] | Ronneberger, O.; Fischer, P.; Brox, T., U-net: convolutional networks for biomedical image segmentation, (Navab, N.; Hornegger, J.; Wells, W. M.; Frangi, A. F., Medical Image Computing and Computer-Assisted Intervention - MICCAI 2015, Vol. 9351 (2015), Springer International Publishing: Springer International Publishing Cham), 234-241 |
| [82] | Ioffe, S.; Szegedy, C., Batch normalization: accelerating deep network training by reducing internal covariate shift (2015), arxiv:1502.03167 [cs], arXiv:1502.03167 |
| [83] | Wang, F.; Lazarov, B. S.; Sigmund, O., On projection methods, convergence and robust formulations in topology optimization, Struct. Multidiscip. Optim., 43, 6, 767-784 (2011) · Zbl 1274.74409 |
| [84] | Xu, S.; Cai, Y.; Cheng, G., Volume preserving nonlinear density filter based on heaviside functions, Struct. Multidiscip. Optim., 41, 4, 495-505 (2010) · Zbl 1274.74419 |
| [85] | Huang, X.; Zuo, Z.; Xie, Y., Evolutionary topological optimization of vibrating continuum structures for natural frequencies, Comput. Struct., 88, 5-6, 357-364 (2010) |
| [86] | Zuo, Z. H.; Xie, Y. M.; Huang, X., Evolutionary topology optimization of structures with multiple displacement and frequency constraints, Adv. Struct. Eng., 15, 2, 359-372 (2012) |
| [87] | Fan, Z.; Xia, L.; Lai, W.; Xia, Q.; Shi, T., Evolutionary topology optimization of continuum structures with stress constraints, Struct. Multidiscip. Optim., 59, 2, 647-658 (2019) |
| [88] | Nocedal, J.; Wright, S. J., (Numerical Optimization. Numerical Optimization, Springer Series in Operations Research (1999), Springer: Springer New York) · Zbl 0930.65067 |
| [89] | Pathak, D.; Krahenbuhl, P.; Darrell, T., Constrained convolutional neural networks for weakly supervised segmentation, (2015 IEEE International Conference on Computer Vision (ICCV) (2015), IEEE: IEEE Santiago, Chile), 1796-1804 |
| [90] | Márquez-Neila, P.; Salzmann, M.; Fua, P., Imposing hard constraints on deep networks: promises and limitations (2017), arxiv:1706.02025 [cs], arXiv:1706.02025 |
| [91] | Kervadec, H.; Dolz, J.; Yuan, J.; Desrosiers, C.; Granger, E.; Ayed, I. B., Constrained deep networks: Lagrangian optimization via log-barrier extensions (2020), arxiv:1904.04205 [cs], arXiv:1904.04205 |
| [92] | Minkov, M.; Williamson, I. A.D.; Andreani, L. C.; Gerace, D.; Lou, B.; Song, A. Y.; Hughes, T. W.; Fan, S., Inverse design of photonic crystals through automatic differentiation, ACS Photonics, 7, 7, 1729-1741 (2020) |
| [93] | Dilgen, C. B.; Dilgen, S. B.; Fuhrman, D. R.; Sigmund, O.; Lazarov, B. S., Topology optimization of turbulent flows, Comput. Methods Appl. Mech. Engrg., 331, 363-393 (2018) · Zbl 1439.74265 |
| [94] | Nørgaard, S. A.; Sagebaum, M.; Gauger, N. R.; Lazarov, B. S., Applications of automatic differentiation in topology optimization, Struct. Multidiscip. Optim., 56, 5, 1135-1146 (2017) |
| [95] | Guest, J. K.; Prévost, J. H.; Belytschko, T., Achieving minimum length scale in topology optimization using nodal design variables and projection functions, Internat. J. Numer. Methods Engrg., 61, 2, 238-254 (2004) · Zbl 1079.74599 |
| [96] | Zhou, M.; Sigmund, O., On fully stressed design and P-norm measures in structural optimization, Struct. Multidiscip. Optim., 56, 3, 731-736 (2017) |
| [97] | Bruggi, M., On an alternative approach to stress constraints relaxation in topology optimization, Struct. Multidiscip. Optim., 36, 2, 125-141 (2008) · Zbl 1273.74397 |
| [98] | Sigmund, O., On the design of compliant mechanisms using topology optimization*, Mech. Struct. Mach., 25, 4, 493-524 (1997) |
| [99] | Park, J.; Nguyen, T. H.; Shah, J. J.; Sutradhar, A., Conceptual design of efficient heat conductors using multi-material topology optimization, Eng. Optim., 51, 5, 796-814 (2019) |
| [100] | Fowler, R. M.; Howell, L. L.; Magleby, S. P., Compliant space mechanisms: A new frontier for compliant mechanisms, Mech. Sci., 2, 2, 205-215 (2011) |
| [101] | Buhl, T.; Pedersen, C.; Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 19, 2, 93-104 (2000) |
| [102] | Bruns, T. E.; Tortorelli, D. A., An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms, Internat. J. Numer. Methods Engrg., 57, 10, 1413-1430 (2003) · Zbl 1062.74589 |
| [103] | Wallin, M.; Ivarsson, N.; Tortorelli, D., Stiffness optimization of non-linear elastic structures, Comput. Methods Appl. Mech. Engrg., 330, 292-307 (2018) · Zbl 1439.74053 |
| [104] | Huang, X. H.; Xie, Y., Bidirectional evolutionary topology optimization for structures with geometrical and material nonlinearities, AIAA J., 45, 1, 308-313 (2007) |
| [105] | Huang, X.; Xie, Y., Topology optimization of nonlinear structures under displacement loading, Eng. Struct., 30, 7, 2057-2068 (2008) |
| [106] | Zhang, Z.; Zhao, Y.; Du, B.; Chen, X.; Yao, W., Topology optimization of hyperelastic structures using a modified evolutionary topology optimization method, Struct. Multidiscip. Optim., 62, 6, 3071-3088 (2020) |
| [107] | Han, Y.; Xu, B.; Liu, Y., An efficient 137-line MATLAB code for geometrically nonlinear topology optimization using bi-directional evolutionary structural optimization method, Struct. Multidiscip. Optim. (2021) |
| [108] | Kwak, J.; Cho, S., Topological shape optimization of geometrically nonlinear structures using level set method, Comput. Struct., 83, 27, 2257-2268 (2005) |
| [109] | Chen, F.; Wang, Y.; Wang, M. Y.; Zhang, Y., Topology optimization of hyperelastic structures using a level set method, J. Comput. Phys., 351, 437-454 (2017) |
| [110] | Xue, R.; Liu, C.; Zhang, W.; Zhu, Y.; Tang, S.; Du, Z.; Guo, X., Explicit structural topology optimization under finite deformation via moving morphable void (MMV) approach, Comput. Methods Appl. Mech. Engrg., 344, 798-818 (2019) · Zbl 1440.74325 |
| [111] | Holzapfel, G. A., Nonlinear Solid Mechanics: A Continuum Approach for Engineering (2000), Wiley: Wiley Chichester ; New York · Zbl 0980.74001 |
| [112] | Klarbring, A.; Strömberg, N., Topology optimization of hyperelastic bodies including non-zero prescribed displacements, Struct. Multidiscip. Optim., 47, 1, 37-48 (2013) · Zbl 1274.74351 |
| [113] | de Borst, R.; Crisfield, M. A.; Remmers, J. J.C.; Verhoosel, C. V., Non-Linear Finite Element Analysis of Solids and Structures (2012), Wiley: Wiley Hoboken, NJ · Zbl 1300.74002 |
| [114] | Kim, N.-H., Introduction to Nonlinear Finite Element Analysis (2014), Springer: Springer New York |
| [115] | Lahuerta, R. D.; Simões, E. T.; Campello, E. M.B.; Pimenta, P. M.; Silva, E. C.N., Towards the stabilization of the low density elements in topology optimization with large deformation, Comput. Mech., 52, 4, 779-797 (2013) · Zbl 1311.74099 |
| [116] | Yoon, G. H.; Kim, Y. Y., Element connectivity parameterization for topology optimization of geometrically nonlinear structures, Int. J. Solids Struct., 42, 7, 1983-2009 (2005) · Zbl 1111.74035 |
| [117] | Wang, F.; Lazarov, B. S.; Sigmund, O.; Jensen, J. S., Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems, Comput. Methods Appl. Mech. Engrg., 276, 453-472 (2014) · Zbl 1423.74768 |
| [118] | Luo, Y.; Wang, M. Y.; Kang, Z., Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique, Comput. Methods Appl. Mech. Engrg., 286, 422-441 (2015) · Zbl 1423.74754 |
| [119] | Ortigosa, R.; Ruiz, D.; Gil, A. J.; Donoso, A.; Bellido, J. C., A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method, Comput. Methods Appl. Mech. Engrg., 364, 24 (2020) · Zbl 1442.74176 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
