A Review of Design Methods for Mechanical Metastructures

YAO Qian; YANG Zhao; WANG Xin; ZHAI Zhi; LI Zhen; GENG Xinyu; LI Bingyang; WANG Pengfei

doi:10.21656/1000-0887.450106

Volume 45 Issue 8

Aug. 2024

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2024 > 45(8): 974-1000

YAO Qian, YANG Zhao, WANG Xin, ZHAI Zhi, LI Zhen, GENG Xinyu, LI Bingyang, WANG Pengfei. A Review of Design Methods for Mechanical Metastructures[J]. Applied Mathematics and Mechanics, 2024, 45(8): 974-1000. doi: 10.21656/1000-0887.450106

Citation:

PDF( 15492 KB)

A Review of Design Methods for Mechanical Metastructures

doi: 10.21656/1000-0887.450106

YAO Qian^{1, 2
,},
YANG Zhao^{1, 2},
WANG Xin^{2
,
,},
ZHAI Zhi¹,
LI Zhen²,
GENG Xinyu²,
LI Bingyang²,
WANG Pengfei^{2
,
,}

1.
Space Intelligent Manufacturing Research Center, Xi'an Jiaotong University, Xi'an 710049, P.R.China
2.
Advanced Materials and Energy Center, China Academy of Aerospace Science and Innovation, Beijing 100088, P.R.China

Received Date: 2024-04-19
Rev Recd Date: 2024-06-19
Publish Date: 2024-08-01

Abstract

Abstract

Mechanical metastructures have superior mechanical properties coming from the topology of specially designed representative elements, and have attracted wide attention due to their mathematical basis, supernormal features and broad applications. To optimize the design procedure, grasp the future trends and promote the interdisciplinary innovation, the fundamental design concepts and research advancements of mechanical metastructures were reviewed. First, the design methods for mechanical metastructures were classified based on the forward and inverse design concepts. Second, within the forward design category, the principles, applicable fields, and optimization directions of periodic superstructures, surface defect superstructures, and mathematical model-based superstructures were discussed. For the reverse design category, the progresses and existing problems in the application of optimization algorithms and learning algorithms in the field of mechanical metastructures were analyzed, with future challenges and open issues concluded.
- mechanical metastructure,
- lattice structure,
- origami,
- TPMS,
- inverse design,
- machine learning

FullText(HTML)

References(116)

References

[1]	MONTEMAYOR L, CHERNOW V, GREER J R. Materials by design: using architecture in material design to reach new property spaces[J]. MRS Bulletin, 2015, 40(12): 1122-1129. doi: 10.1557/mrs.2015.263
[2]	GUPTA V, ADHIKARI S, BHATTACHARYA B. Exploring the dynamics of hourglass shaped lattice metastructures[J]. Scientific Reports, 2020, 10(1): 20943. doi: 10.1038/s41598-020-77226-4
[3]	JIAO P, MUELLER J, RANEY J R, et al. Mechanical metamaterials and beyond[J]. Nature Communications, 2023, 14(1): 6004. doi: 10.1038/s41467-023-41679-8
[4]	CRASTER R, GUENNEAU S, KADIC M, et al. Mechanical metamaterials[J]. Reports on Progress in Physics, 2023, 86(9): 094501. doi: 10.1088/1361-6633/ace069
[5]	于相龙, 周济. 力学超材料的构筑及其超常新功能[J]. 中国材料进展, 2019, 38(1): 14-21. YU Xianglong, ZHOU Ji. Mechanical metamaterials: architected materials and unexplored properties[J]. Materials China, 2019, 38(1): 14-21. (in Chinese)
[6]	BERTOLDI K, VITELLI V, CHRISTENSEN J, et al. Flexible mechanical metamaterials[J]. Nature Reviews Materials, 2017, 2(11): 1-11.
[7]	LI X, PENG W, WU W, et al. Auxetic mechanical metamaterials: from soft to stiff[J]. International Journal of Extreme Manufacturing, 2023, 5(4): 042003. doi: 10.1088/2631-7990/ace668
[8]	XU R, CHEN C, SUN J, et al. The design, manufacture and application of multistable mechanical metamaterials: a state-of-the-art review[J]. International Journal of Extreme Manufacturing, 2023, 5(4): 042013. doi: 10.1088/2631-7990/acf96a
[9]	YU X, ZHOU J, LIANG H, et al. Mechanical metamaterials associated with stiffness, rigidity and compressibility: a brief review[J]. Progress in Materials Science, 2018, 94: 114-173. doi: 10.1016/j.pmatsci.2017.12.003
[10]	AMBEKAR R S, KUSHWAHA B, SHARMA P, et al. Topologically engineered 3D printed architectures with superior mechanical strength[J]. Materials Today, 2021, 48: 72-94. doi: 10.1016/j.mattod.2021.03.014
[11]	CHEN Z, LIN Y T, SALEHI H, et al. Advanced fabrication of mechanical metamaterials based on micro/nanoscale technology[J]. Advanced Engineering Materials, 2023, 25(22): 2300750. doi: 10.1002/adem.202300750
[12]	SU R, CHEN J, ZHANG X, et al. 3D-printed micro/nano-scaled mechanical metamaterials: fundamentals, technologies, progress, applications, and challenges[J]. Small, 2023, 19(29): 2206391. doi: 10.1002/smll.202206391
[13]	JIAO P, MUELLER J, RANEY J R, et al. Mechanical metamaterials and beyond[J]. Nature Communications, 2023, 14(1): 6004. doi: 10.1038/s41467-023-41679-8
[14]	WANG H, LYU Y, BOSIAKOV S, et al. A review on the mechanical metamaterials and their applications in the field of biomedical engineering[J]. Frontiers in Materials, 2023, 10: 1273961. doi: 10.3389/fmats.2023.1273961
[15]	ZHOU X, REN L, SONG Z, et al. Advances in 3D/4D printing of mechanical metamaterials: from manufacturing to applications[J]. Composites (Part B): Engineering, 2023, 254: 110585. doi: 10.1016/j.compositesb.2023.110585
[16]	JIAO P, ALAVI A H. Artificial intelligence-enabled smart mechanical metamaterials: advent and future trends[J]. International Materials Reviews, 2021, 66(6): 365-393. doi: 10.1080/09506608.2020.1815394
[17]	BERGER J B, WADLEY H N G, MCMEEKING R M. Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness[J]. Nature, 2017, 543(7646): 533-537. doi: 10.1038/nature21075
[18]	LU C, HSIEH M, HUANG Z, et al. Architectural design and additive manufacturing of mechanical metamaterials: a review[J]. Engineering, 2022, 17: 44-63. doi: 10.1016/j.eng.2021.12.023
[19]	MEZA L R, DAS S, GREER J R. Strong, lightweight, and recoverable three-dimensional ceramic nanolattices[J]. Science, 2014, 345(6202): 1322-1326. doi: 10.1126/science.1255908
[20]	ZHENG X, LEE H, WEISGRABER T H, et al. Ultralight, ultrastiff mechanical metamaterials[J]. Science, 2014, 344(6190): 1373-1377. doi: 10.1126/science.1252291
[21]	KADER M A, HAZELL P J, BROWN A D, et al. Novel design of closed-cell foam structures for property enhancement[J]. Additive Manufacturing, 2020, 31: 100976. doi: 10.1016/j.addma.2019.100976
[22]	AL-KETAN O, REZGUI R, ROWSHAN R, et al. Microarchitected stretching-dominated mechanical metamaterials with minimal surface topologies[J]. Advanced Engineering Materials, 2018, 20(9): 1800029. doi: 10.1002/adem.201800029
[23]	WANG D, SHI S, MAO Y, et al. Biodegradable dual-network cellulosic composite bioplastic metafilm for plastic substitute[J]. Angewandte Chemie International Edition, 2023, 62(50): e202310995. doi: 10.1002/anie.202310995
[24]	FU M H, CHEN Y, HU L L. Bilinear elastic characteristic of enhanced auxetic honeycombs[J]. Composite Structures, 2017, 175: 101-110. doi: 10.1016/j.compstruct.2017.04.007
[25]	XU M, XU Z, ZHANG Z, et al. Mechanical properties and energy absorption capability of AuxHex structure under in-plane compression: theoretical and experimental studies[J]. International Journal of Mechanical Sciences, 2019, 159: 43-57. doi: 10.1016/j.ijmecsci.2019.05.044
[26]	LIU W, LI H, ZHANG J, et al. Theoretical analysis on the elasticity of a novel accordion cellular honeycomb core with in-plane curved beams[J]. Journal of Sandwich Structures & Materials, 2020, 22(3): 109963621876817.
[27]	CHEN Y, QIAN F, ZUO L, et al. Broadband and multiband vibration mitigation in lattice metamaterials with sinusoidally-shaped ligaments[J]. Extreme Mechanics Letters, 2017, 17: 24-32. doi: 10.1016/j.eml.2017.09.012
[28]	JEONG S, YOO H H. Shape optimization of bowtie-shaped auxetic structures using beam theory[J]. Composite Structures, 2019, 224: 111020. doi: 10.1016/j.compstruct.2019.111020
[29]	LI X, LU Z, YANG Z, et al. Yield surfaces of periodic honeycombs with tunable Poisson's ratio[J]. International Journal of Mechanical Sciences, 2018, 141: 290-302. doi: 10.1016/j.ijmecsci.2018.04.005
[30]	QI C, JIANG F, YANG S. Advanced honeycomb designs for improving mechanical properties: a review[J]. Composites (Part B): Engineering, 2021, 227: 109393. doi: 10.1016/j.compositesb.2021.109393
[31]	ZAISER M, ZAPPERI S. Disordered mechanical metamaterials[J]. Nature Reviews Physics, 2023, 5(11): 679-688. doi: 10.1038/s42254-023-00639-3
[32]	ASEFA S A, SHIM S, SEONG M, et al. Chiral metasurfaces: a review of the fundamentals and research advances[J]. Applied Sciences, 2023, 13(19): 10590. doi: 10.3390/app131910590
[33]	CHEN Y, MAI Y W, YE L. Perspectives for multiphase mechanical metamaterials[J]. Materials Science and Engineering R: Reports, 2023, 153: 100725. doi: 10.1016/j.mser.2023.100725
[34]	CHEN S, WANG B, ZHU S, et al. A novel composite negative stiffness structure for recoverable trapping energy[J]. Composites (Part A): Applied Science and Manufacturing, 2020, 129: 105697. doi: 10.1016/j.compositesa.2019.105697
[35]	GRIMA J N, CARUANA-GAUCI R, WOJCIECHOWSKI K, et al. Smart hexagonal truss systems exhibiting negative compressibility through constrained angle stretching[J]. Smart Materials and Structures, 2013, 22(8): 084015. doi: 10.1088/0964-1726/22/8/084015
[36]	MOHAMMADI K, MOVAHHEDY M R, SHISHKOSKY I, et al. Hybrid anisotropic pentamode mechanical metamaterial produced by additive manufacturing technique[J]. Applied Physics Letters, 2020, 117: 061901. doi: 10.1063/5.0014167
[37]	JIANG Y, LI Y. 3D printed auxetic mechanical metamaterial with chiral cells and re-entrant cores[J]. Scientific Reports, 2018, 8: 2397. doi: 10.1038/s41598-018-20795-2
[38]	LEE H, JANG Y, CHOE J K, et al. 3D-printed programmable tensegrity for soft robotics[J]. Science Robotics, 2020, 5(45): eaay9024. doi: 10.1126/scirobotics.aay9024
[39]	YANG H, D'AMBROSIO N, LIU P, et al. Shape memory mechanical metamaterials[J]. Materials Today, 2023, 66: 36-49. doi: 10.1016/j.mattod.2023.04.003
[40]	JI L, HU W, TAO R, et al. Compression behavior of the 4D printed reentrant honeycomb: experiment and finite element analysis[J]. Smart Materials and Structures, 2020, 29(11): 115016. doi: 10.1088/1361-665X/ababe4
[41]	SILVERBERG J L, EVANS A A, MCLEOD L, et al. Using origami design principles to fold reprogrammable mechanical metamaterials[J]. Science, 2014, 345(6197): 647-650. doi: 10.1126/science.1252876
[42]	SINHA P, MUKHOPADHYAY T. Programmable multi-physical mechanics of mechanical metamaterials[J]. Materials Science and Engineering R: Reports, 2023, 155: 100745. doi: 10.1016/j.mser.2023.100745
[43]	BERTAGNE C L, COGNATA T J, SHETH R B, et al. Testing and analysis of a morphing radiator concept for thermal control of crewed space vehicles[J]. Applied Thermal Engineering, 2017, 124: 986-1002. doi: 10.1016/j.applthermaleng.2017.06.062
[44]	LI X, GAO L, ZHOU W, et al. Novel 2D metamaterials with negative Poisson's ratio and negative thermal expansion[J]. Extreme Mechanics Letters, 2019, 30: 100498. doi: 10.1016/j.eml.2019.100498
[45]	MELONI M, CAI J, ZHANG Q, et al. Engineering origami: a comprehensive review of recent applications, design methods, and tools[J]. Advanced Science, 2021, 8(13): 2000636. doi: 10.1002/advs.202000636
[46]	夏进军, 李洁, 张雨萌, 等. 折纸结构及其特性的工程应用策略[J]. 材料导报, 2021, 35(11): 12. XIA Jinjun, LI Jie, ZHANG Yumeng, et al. Strategies for the engineering application of origami structures and their properties[J]. Material Reports, 2021, 35(11): 12. (in Chinese)
[47]	方虹斌, 吴海平, 刘作林, 等. 折纸结构和折纸超材料动力学研究进展[J]. 力学学报, 2022, 54(1): 1-38. FANG Hongbin, WU Haiping, LIU Zuolin, et al. Advances in the dynamics of origami structures and origami metamaterials[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(1): 1-38. (in Chinese)
[48]	MENG Z, CHEN W, MEI T, et al. Bistability-based foldable origami mechanical logic gates[J]. Extreme Mechanics Letters, 2021, 43: 101180. doi: 10.1016/j.eml.2021.101180
[49]	XIANG X, QIANG W, HOU B, et al. Quasi-static and dynamic mechanical properties of Miura-ori metamaterials[J]. Thin-Walled Structures, 2020, 157: 106993. doi: 10.1016/j.tws.2020.106993
[50]	LIU S, PENG G, JIN K. Design and characteristics of a novel QZS vibration isolation system with origami-inspired corrector[J]. Nonlinear Dynamics, 2021, 106: 255-277. doi: 10.1007/s11071-021-06821-5
[51]	FANG H, LI S, WANG K W. Self-locking degree-4 vertex origami structures[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2016, 472(2195): 20160682. doi: 10.1098/rspa.2016.0682
[52]	CHEN B G, LIU B, EVANS A A, et al. Topological mechanics of origami and kirigami[J]. Physical Review Letters, 2016, 116(13): 135501. doi: 10.1103/PhysRevLett.116.135501
[53]	LIU W, SONG Y, CHEN Y, et al. Reconfigurable thick-panel structures based on a stacked origami tube[J]. Journal of Mechanisms and Robotics, 2024, 16(12): 121005. doi: 10.1115/1.4064836
[54]	FILIPOV E T, TACHI T, PAULINO G H. Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials[J]. Proceedings of the National Academy of Sciences, 2015, 112(40): 12321-12326. doi: 10.1073/pnas.1509465112
[55]	YE H, LIU Q, CHENG J, et al. Multimaterial 3D printed self-locking thick-panel origami metamaterials[J]. Nature Communications, 2023, 14(1): 1607. doi: 10.1038/s41467-023-37343-w
[56]	WANG Y, YE H, HE J, et al. Electrothermally controlled origami fabricated by 4D printing of continuous fiber-reinforced composites[J]. Nature Communications, 2024, 15(1): 2322. doi: 10.1038/s41467-024-46591-3
[57]	YAN Z, ZHANG F, LIU F, et al. Mechanical assembly of complex, 3D mesostructures from releasable multilayers of advanced materials[J]. Science Advances, 2016, 2(9): e1601014. doi: 10.1126/sciadv.1601014
[58]	FU H, NAN K, BAI W, et al. Morphable 3D mesostructures and microelectronic devices by multistable buckling mechanics[J]. Nature Materials, 2018, 17(3): 268-276. doi: 10.1038/s41563-017-0011-3
[59]	TANG Y, YIN J. Design of cut unit geometry in hierarchical kirigami-based auxetic metamaterials for high stretchability and compressibility[J]. Extreme Mechanics Letters, 2017, 12: 77-85. doi: 10.1016/j.eml.2016.07.005
[60]	HWANG D G, BARTLETT M D. Tunable mechanical metamaterials through hybrid kirigami structures[J]. Scientific Reports, 2018, 8: 3378. doi: 10.1038/s41598-018-21479-7
[61]	SUSSMAN D M, CHO Y, CASTLE T, et al. Algorithmic lattice kirigami: a route to pluripotent materials[J]. Proceedings of the National Academy of Sciences, 2015, 112(24): 7449-7453. doi: 10.1073/pnas.1506048112
[62]	CHEN S, CHEN J, ZHANG X, et al. Kirigami/origami: unfolding the new regime of advanced 3D microfabrication/nanofabrication with "folding"[J]. Light: Science & Applications, 2020, 9(1): 75.
[63]	NAKAJIMA J, FAYAZBAKHSH K, TESHIMA Y. Experimental study on tensile properties of 3D printed flexible kirigami specimens[J]. Additive Manufacturing, 2020, 32: 101100. doi: 10.1016/j.addma.2020.101100
[64]	WANG L C, SONG W L, FANG D. Twistable origami and kirigami: from structure-guided smartness to mechanical energy storage[J]. ACS Applied Materials & Interfaces, 2019, 11(3): 3450-3458.
[65]	QIU N, WAN Y, SHEN Y, et al. Experimental and numerical studies on mechanical properties of TPMS structures[J]. International Journal of Mechanical Sciences, 2024, 261: 108657. doi: 10.1016/j.ijmecsci.2023.108657
[66]	BILLON K, ZAMPETAKIS I, SCARPA F, et al. Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[J]. Composite Structures, 2017, 160: 1042-1050. doi: 10.1016/j.compstruct.2016.10.121
[67]	MEZA L R, ZELHOFER A J, CLARKE N, et al. Resilient 3D hierarchical architected metamaterials[J]. Proceedings of the National Academy of Sciences, 2015, 112(37): 11502-11507. doi: 10.1073/pnas.1509120112
[68]	MAN X, LIU T, XIA B, et al. Space-coiling fractal metamaterial with multi-bandgaps on subwavelength scale[J]. Journal of Sound and Vibration, 2018, 423: 322-339. doi: 10.1016/j.jsv.2018.02.060
[69]	MICHALEWICZ Z, SCHOENAUER M. Evolutionary algorithms for constrained parameter optimization problems[J]. Evolutionary Computation, 1996, 4(1): 1-32. doi: 10.1162/evco.1996.4.1.1
[70]	MICHALEWICZ Z, JANIKOW C Z, KRAWCZYK J B. A modified genetic algorithm for optimal control problems[J]. Computers & Mathematics With Applications, 1992, 23(12): 83-94.
[71]	JANIKOW C Z, MICHALEWICZ Z. A specialized genetic algorithm for numerical optimization problems[C]//Proceedings of the 2nd International IEEE Conference on Tools for Artificial Intelligence. Herndon, VA, USA, 1990: 798-804.
[72]	SLOWIK A, KWASNICKA H. Evolutionary algorithms and their applications to engineering problems[J]. Neural Computing & Applications, 2020, 32(16): 12363-12379.
[73]	KATOCH S, CHAUHAN S S, KUMAR V. A review on genetic algorithm: past, present, and future[J]. Multimedia Tools and Applications, 2021, 80(5): 8091-8126. doi: 10.1007/s11042-020-10139-6
[74]	WANG L, LIU H T. Parameter optimization of bidirectional re-entrant auxetic honeycomb metamaterial based on genetic algorithm[J]. Composite Structures, 2021, 267: 113915. doi: 10.1016/j.compstruct.2021.113915
[75]	DOS R F, KARATHANASOPOULOS N. Inverse metamaterial design combining genetic algorithms with asymptotic homogenization schemes[J]. International Journal of Solids and Structures, 2022, 250: 111702. doi: 10.1016/j.ijsolstr.2022.111702
[76]	DONG J, HU C, HOLMES J, et al. Structural optimisation of cross-chiral metamaterial structures via genetic algorithm[J]. Composite Structures, 2022, 282: 115035. doi: 10.1016/j.compstruct.2021.115035
[77]	LIU S, ACAR P. Parameter space exploration of cellular mechanical metamaterials using genetic algorithms[J]. AIAA Journal, 2023, 61(8): 3633-3643. doi: 10.2514/1.J062864
[78]	贾宇翔, 王甲富, 陈维, 等. 基于智能算法的超材料快速优化设计方法研究进展[J]. 雷达学报, 2021, 10(2): 220-239. JIA Yuxiang, WANG Jiafu, CHEN Wei, et al. Research progress on rapid optimization design methods of metamaterials based on intelligent algorithms[J]. Journal of Radars, 2021, 10(2): 220-239. (in Chinese)
[79]	BENDSOE M P, LUND E, OLHOFF N, et al. Topology optimization-broadening the areas of application[J]. Control and Cybernetics, 2005, 34(1): 7-35.
[80]	CHENG K T, OLHOFF N. An investigation concerning optimal-design of solid elastic plates[J]. International Journal of Solids and Structures, 1981, 17(3): 305-323. doi: 10.1016/0020-7683(81)90065-2
[81]	BENDSOE M P, KIKUCHI N. Generaating optimal topologies in structural design using a homogenization method[J]. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197-224. doi: 10.1016/0045-7825(88)90086-2
[82]	陈小前, 赵勇, 霍森林, 等. 多尺度结构拓扑优化设计方法综述[J]. 航空学报, 2023, 44(15): 25-60. CHEN Xiaoqian, ZHAO Yong, HUO Senlin, et al. A review of topology optimization design methods for multi-scale structures[J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(15): 25-60. (in Chinese)
[83]	ZHOU M, ROZVANY G I N. The COC algorithm, part Ⅱ: topological, geometrical and generalized shape optimization[J]. Computer Methods in Applied Mechanics and Engineering, 1991, 89(1/3): 309-336.
[84]	STOLPE M, SVANBERG K. On the trajectories of penalization methods for topology optimization[J]. Structural and Multidisciplinary Optimization, 2001, 21(2): 128-139. doi: 10.1007/s001580050177
[85]	XIE Y M, STEVEN G P. A simple evolutionary procedure for sructure optimization[J]. Computers & Structures, 1993, 49(5): 885-896.
[86]	杨德庆, 刘正兴, 隋允康. 连续体结构拓扑优化设计的ICM方法[J]. 上海交通大学学报, 1999, 33(6): 734-736. doi: 10.3321/j.issn:1006-2467.1999.06.027 YANG Deqing, LIU Zhengxing, SUI Yunkang. An ICM method for topological optimization design of continuum structures[J]. Journal of Shanghai Jiao Tong University, 1999, 33(6): 734-736. (in Chinese)) doi: 10.3321/j.issn:1006-2467.1999.06.027
[87]	WANG M Y, WANG X M, GUO D M. A level set method for structural topology optimization[J]. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1/2): 227-246.
[88]	WANG M Y, ZHOU S W. Phase field: a variational method for structural topology optimization[J]. CMES-Computer Modeling in Engineering & Sciences, 2004, 6(6): 547-566.
[89]	GUO X, ZHANG W S, ZHONG W L. Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[J]. Journal of Applied Mechanics, 2014, 81(8): 081009. doi: 10.1115/1.4027609
[90]	ZHOU Y, ZHANG W H, ZHU J H, et al. Feature-driven topology optimization method with signed distance function[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 310: 1-32. doi: 10.1016/j.cma.2016.06.027
[91]	SONG J, WANG Y, ZHOU W, et al. Topology optimization-guided lattice composites and their mechanical characterizations[J]. Composites (Part B): Engineering, 2019, 160: 402-411. doi: 10.1016/j.compositesb.2018.12.027
[92]	DUAN S, XI L, WEN W, et al. Mechanical performance of topology-optimized 3D lattice materials manufactured via selective laser sintering[J]. Composite Structures, 2020, 238: 111985. doi: 10.1016/j.compstruct.2020.111985
[93]	BEN-YELUN I, SAUCEDO-MORA L, SANZ M A, et al. Topology optimization approach for functionally graded metamaterial components based on homogenization of mechanical variables[J]. Computers & Structures, 2023, 289: 107151.
[94]	GAO J, WANG L, XIAO M, et al. An isogeometric approach to topological optimization design of auxetic composites with tri-material micro-architectures[J]. Composite Structures, 2021, 271: 114163. doi: 10.1016/j.compstruct.2021.114163
[95]	SAURABH S, GUPTA A, CHOWDHURY R. Impact of parametric variation to achieve extreme mechanical metamaterials through topology optimization[J]. Composite Structures, 2023, 326: 117611. doi: 10.1016/j.compstruct.2023.117611
[96]	ZENG Q, DUAN S, ZHAO Z, et al. Inverse design of energy-absorbing metamaterials by topology optimization[J]. Advanced Science, 2022, 10(4): 2204977.
[97]	DU Z, BIAN T, REN X, et al. Inverse design of mechanical metamaterial achieving a prescribed constitutive curve[J]. Theoretical and Applied Mechanics Letters, 2024, 14(1): 100486. doi: 10.1016/j.taml.2023.100486
[98]	杨德庆, 秦浩星. 基于功能基元拓扑优化法的任意正泊松比超材料结构设计[J]. 上海交通大学学报, 2019, 53(7): 819-829. YANG Deqing, QIN Haoxing. Metamaterials design with arbitrary Poisson's ratio by functional element topology optimization[J]. Journal of Shanghai Jiao Tong University, 2019, 53(7): 819-829. (in Chinese)
[99]	孟军辉, 刘清洋, 金泽华. 基于复杂约束拓扑优化的负泊松比超材料设计[J]. 北京理工大学学报, 2023, 43(8): 852-862. MENG Junhui, LIU Qingyang, JIN Zehua. Negative Poisson's ratio metamaterial design based on topology optimization with complex constraints[J]. Transactions of Beijing Institute of Technology, 2023, 43(8): 852-862. (in Chinese)
[100]	KOLLMANN H T, ABUEIDDA D W, KORIC S, et al. Deep learning for topology optimization of 2D metamaterials[J]. Materials & Design, 2020, 196: 109098.
[101]	刘文俭, 熊家军, 范亚, 等. 机器学习在超材料智能设计中的研究现状[J]. 信阳师范学院学报, 2021, 34(3): 501-509. LIU Wenjian, XIONG Jiajun, FAN Ya, et al. Research status of intelligent design of metamaterials based on machine learning[J]. Journal of Xinyang Normal University, 2021, 34(3): 501-509. (in Chinese)
[102]	ALZUBI J, NAYYAR A, KUMAR A. Machine learning from theory to algorithms: an overview[J]. Journal of Physics: Conference Series, 2018, 1142: 012202.
[103]	CARLEO G, CIRAC I, CRANMER K, et al. Machine learning and the physical sciences[J]. Reviews of Modern Physics, 2019, 91(4): 045002. doi: 10.1103/RevModPhys.91.045002
[104]	CHALLAPALLI A, PATEL D, LI G. Inverse machine learning framework for optimizing lightweight metamaterials[J]. Materials & Design, 2021, 208: 109937.
[105]	TIAN J, TANG K, CHEN X, et al. Machine learning-based prediction and inverse design of 2D metamaterial structures with tunable deformation-dependent Poisson's ratio[J]. Nanoscale, 2022, 14(35): 12677-12691. doi: 10.1039/D2NR02509D
[106]	HA C S, YAO D, XU Z, et al. Rapid inverse design of metamaterials based on prescribed mechanical behavior through machine learning[J]. Nature Communications, 2023, 14(1): 5765. doi: 10.1038/s41467-023-40854-1
[107]	TAKAGI A, ICHIKAWA R, MIYAGAWA T, et al. Machine learning-based estimation method for the mechanical response of composite cellular structures[J]. Polymer Testing, 2023, 126: 108161. doi: 10.1016/j.polymertesting.2023.108161
[108]	罗仪豪, 张峻, 杜世银, 等. 基于深度学习的超材料设计及光纤光束控制研究进展[J]. 中国激光, 2023, 50(11): 104-122. LUO Yihao, ZHANG Jun, DU Shiyin, et al. Research progress in metamaterial design and fiber beam control based on deep learning[J]. Chinese Journal of Lasers, 2023, 50(11): 104-122. (in Chinese)
[109]	ZHU S, YU T, XU T, et al. Intelligent computing: the latest advances, challenges and future[J]. Intelligent Computing, 2022, 2: 0006. doi: 10.1007/s43674-021-00021-9
[110]	ZHENG X, CHEN T T, GUO X, et al. Controllable inverse design of auxetic metamaterials using deep learning[J]. Materials & Design, 2021, 211: 110178.
[111]	PAHLAVANI H, AMANI M, SALDIVAR M C, et al. Deep learning for the rare-event rational design of 3D printed multi-material mechanical metamaterials[J]. Communications Materials, 2022, 3: 46.
[112]	李想, 严子铭, 柳占立, 等. 基于仿真和数据驱动的先进结构材料设计[J]. 力学进展, 2021, 51(1): 82-105. LI Xiang, YAN Ziming, LIU Zhanli, et al. Advanced structural material design based on simulation and data-driven method[J]. Advances in Mechanics, 2021, 51(1): 82-105. (in Chinese)
[113]	QIAN C, CHEN H. A perspective on the next generation of invisibility cloaks-intelligent cloaks[J]. Applied Physics Letters, 2021, 118(18): 180501. doi: 10.1063/5.0049748
[114]	LIU Z C, ZHU D Y, RODRIGUES S P, et al. Generative model for the inverse design of metasurfaces[J]. Nano Letters, 2018, 18(10): 6570-6576. doi: 10.1021/acs.nanolett.8b03171
[115]	MELATI D, GRINBERG Y, DEZFOULI M K, et al. Mapping the global design space of nanophotonic components using machine learning pattern recognition[J]. Nature Communications, 2019, 10: 4775. doi: 10.1038/s41467-019-12698-1
[116]	CHEN J, HU S, ZHU S, et al. Metamaterials: from fundamental physics to intelligent design[J]. Interdisciplinary Materials, 2023, 2(1): 5-29. doi: 10.1002/idm2.12049