多尺度结构拓扑优化设计方法综述

doi:10.7527/S1000-6893.2023.28863

综述

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多尺度结构拓扑优化设计方法综述

陈小前¹, 赵勇²(), 霍森林², 张泽雨², 都柄晓²

^1.军事科学院，北京　100195
^2.国防科技大学空天科学学院，长沙　410073

收稿日期:2023-04-13 修回日期:2023-04-28 接受日期:2023-05-24 出版日期:2023-08-15 发布日期:2023-06-06
通讯作者: 赵勇 E-mail:zhaoyong@nudt.edu.cn
基金资助:
国家杰出青年科学基金(11725211);空天前沿培育项目(KY0505072113);湖南省研究生科研创新项目(CX20220021)

A review of topology optimization design methods for multi-scale structures

Xiaoqian CHEN¹, Yong ZHAO²(), Senlin HUO², Zeyu ZHANG², Bingxiao DU²

^1.Chinese Academy of Military Science，Beijing 　100195，China
^2.College of Aerospace Science and Engineering，National University of Defense Technology，Changsha 　410073，China

Received:2023-04-13 Revised:2023-04-28 Accepted:2023-05-24 Online:2023-08-15 Published:2023-06-06
Contact: Yong ZHAO E-mail:zhaoyong@nudt.edu.cn
Supported by:
National Science Fund for Distinguished Young Scholars(11725211);Aerospace Frontier Inspiration Project(KY0505072113);Postgraduate Scientific Research Innovation Project of Hunan Province(CX20220021)

摘要/Abstract

摘要：

多尺度结构是下一代结构轻量化的核心方向之一，在航空航天领域有巨大的发展潜力和应用价值。面向多尺度结构的拓扑优化技术得到了广泛研究，取得了长足的进步和诸多成果，但将其应用于实际工程仍有不少挑战和难题。首先介绍了单尺度结构拓扑优化的主流方法和特点、均匀化方法的基本思想和特殊地位、多尺度结构拓扑优化方法和单尺度结构拓扑优化方法的关系。然后以结构分布特征作为分类标准，系统地梳理了周期、功能梯度和异构3种分布形式的多尺度结构的拓扑优化设计方法。接着针对多尺度结构拓扑优化中的结构连通性、多尺度方法适用性和机器学习技术的应用等重要问题的相关研究进展进行了综述和讨论。最后对领域亟待突破的问题及重要研究方向进行了总结和展望。

关键词: 轻量化结构, 多尺度, 结构优化, 均匀化方法, 机器学习

Abstract:

Multi-scale structure is one of the important directions of next-generation structural lightweighting and has great potential and strategic value for application in the aerospace field. Topology optimization techniques for multi-scale structures have also been extensively researched， leading to great progress and achievements. However， many challenges and difficulties remain before realizing actual engineering practice. This paper firstly introduces the mainstream methods and characteristics of single-scale structural topology optimization， the basic idea and special position of homogenization methods， as well as the relationship between topology optimization methods for multi-scale structures and topology optimization methods for single-scale structures. Then， using the structural distribution characteristics as the classification criteria， this paper systematically sorted out the topology optimization design methods for three distribution forms of multi-scale structures： periodic， functional gradient and heterogeneous. For the important issues， including structural connectivity， applicability of multi-scale methods， and application of machine learning technology in multi-scale structural topology optimization， relevant research progresses are reviewed and discussed. Finally， pressing challenges and important research directions in the field are summarized and prospected.

Key words: lightweight structures, multi-scale, structural optimization, homogenization methods, machine learning

中图分类号:

陈小前, 赵勇, 霍森林, 张泽雨, 都柄晓. 多尺度结构拓扑优化设计方法综述[J]. 航空学报, 2023, 44(15): 528863-528863.

Xiaoqian CHEN, Yong ZHAO, Senlin HUO, Zeyu ZHANG, Bingxiao DU. A review of topology optimization design methods for multi-scale structures[J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(15): 528863-528863.

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参考文献 275

1	AIZENBERG J， WEAVER J C， THANAWALA M S， et al. Skeleton of Euplectella sp.： Structural hierarchy from the nanoscale to the macroscale［J］. Science， 2005， 309（5732）： 275-278.
2	LAKES R. Materials with structural hierarchy［J］. Nature， 1993， 361（6412）： 511-515.
3	GIBSON L J， ASHBY M F. Cellular solids： structure and properties［M］. 2nd ed. Cambridge： Cambridge University Press， 1997： 2-3.
4	PLOCHER J， PANESAR A. Review on design and structural optimisation in additive manufacturing： Towards next-generation lightweight structures［J］. Materials & Design， 2019， 183： 108164.
5	BENDSØE M， SIGMUND O. Topology optimization： Theory， methods， and applications［M］. Berlin： Springer Science & Business Media， 2004： 1-2.
6	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.
7	BENDSØE M P， KIKUCHI N. Generating optimal topologies in structural design using a homogenization method［J］. Computer Methods in Applied Mechanics and Engineering， 1988， 71（2）： 197-224.
8	ORME M E， GSCHWEITL M， FERRARI M， et al. Additive manufacturing of lightweight， optimized， metallic components suitable for space flight［J］. Journal of Spacecraft and Rockets， 2017， 54（5）： 1050-1059.
9	AAGE N， ANDREASSEN E， LAZAROV B S， et al. Giga-voxel computational morphogenesis for structural design［J］. Nature， 2017， 550（7674）： 84-86.
10	MENG L， ZHANG W H， QUAN D L， et al. From topology optimization design to additive manufacturing： Today’s success and tomorrow’s roadmap［J］. Archives of Computational Methods in Engineering， 2019， 27（3）： 805-830.
11	TOMLIN M， MEYER J. Topology optimization of an additive layer manufactured （ALM） aerospace part［C］∥ Proceeding of the 7th Altair CAE Technology Conference. 2011： 1-9.
12	XU H， ZHAO Y， YAO W， et al. Multi-objective structural optimization and design of microsatellite supporting legs［C］∥ World Congress of Structural and Multidisciplinary Optimisation. Cham： Springer， 2018： 1273-1282.
13	SENHORA F V， SANDERS E D， PAULINO G H. Optimally-tailored spinodal architected materials for multiscale design and manufacturing［J］. Advanced Materials， 2022， 34（26）： 2109304.
14	WANG C， ZHU J H， WU M Q， et al. Multi-scale design and optimization for solid-lattice hybrid structures and their application to aerospace vehicle components［J］. Chinese Journal of Aeronautics， 2021， 34（5）： 386-398.
15	JIA Z A， LIU F， JIANG X H， et al. Engineering lattice metamaterials for extreme property， programmability， and multifunctionality［J］. Journal of Applied Physics， 2020， 127（15）： 150901.
16	YUAN X J， CHEN M J， YAO Y， et al. Recent progress in the design and fabrication of multifunctional structures based on metamaterials［J］. Current Opinion in Solid State and Materials Science， 2021， 25（1）： 100883.
17	SALEH B， JIANG J H， FATHI R， et al. 30 years of functionally graded materials： An overview of manufacturing methods， Applications and Future Challenges［J］. Composites Part B： Engineering， 2020， 201： 108376.
18	BENDSOE M P， GUEDES J M， HABER R B， et al. An analytical model to predict optimal material properties in the context of optimal structural design［J］. Journal of Applied Mechanics， 1994， 61（4）： 930-937.
19	FUJII D， CHEN B C， KIKUCHI N. Composite material design of two-dimensional structures using the homogenization design method［J］. International Journal for Numerical Methods in Engineering， 2001， 50（9）： 2031-2051.
20	NEVES M M， SIGMUND O， BENDSØE M P. Topology optimization of periodic microstructures with a penalization of highly localized buckling modes［J］. International Journal for Numerical Methods in Engineering， 2002， 54（6）： 809-834.
21	ROSEN D W. Computer-aided design for additive manufacturing of cellular structures［J］. Computer-Aided Design and Applications， 2007， 4（5）： 585-594.
22	钱令希. 工程结构优化设计［M］. 北京：水利电力出版社， 1983： 1-2.
	QIAN L X. Optimal design of engineering structure［M］. Beijing： Water Resources and Electric Power Press， 1983： 1-2 （in Chinese）.
23	罗云锋. 具有特定几何特征的增材制造结构拓扑优化设计方法［D］. 大连：大连理工大学， 2021： 9-10.
	LUO Y F. Topology optimization of structures with specific geometric features for additive manufacturing［D］. Dalian： Dalian University of Technology， 2021： 9-10 （in Chinese）.
24	高杰. 基于参数化水平集的结构/材料多尺度拓扑优化设计研究［D］. 武汉：华中科技大学， 2019： 5-7.
	GAO J. Research on topology optimization for multiscale design of structure-material based on parametric level set［D］. Wuhan： Huazhong University of Science and Technology， 2019： 5-7 （in Chinese）.
25	都柄晓. 基于移动可变形空腔的结构拓扑优化方法研究［D］. 长沙：国防科技大学， 2019： 5-10.
	DU B X. Research on topology optimisation using moving morphable voids［D］. Changsha： National University of Defense Technology， 2019： 5-10 （in Chinese）.
26	郭中泽，张卫红，陈裕泽. 结构拓扑优化设计综述［J］. 机械设计， 2007， 24（8）： 1-6.
	GUO Z Z， ZHANG W H， CHEN Y Z. An overview on the topological optimization design of structures［J］. Journal of Machine Design， 2007， 24（8）： 1-6 （in Chinese）.
27	文桂林，刘杰，陈梓杰，等. 非线性连续体拓扑优化方法综述［J］. 力学学报， 2022， 54（10）： 2659-2675.
	WEN G L， LIU J， CHEN Z J， et al. A survey of nonlinear continuum topology optimization methods［J］. Chinese Journal of Theoretical and Applied Mechanics， 2022， 54（10）： 2659-2675 （in Chinese）.
28	龙凯，王选，孙鹏文. 连续体结构拓扑优化方法及应用［M］. 北京：中国水利水电出版社， 2022： 1-3.
	LONG K， WANG X， SUN P W. Topology optimization method of continuum structure and its application［M］. Beijing： China Water & Power Press， 2022： 1-3 （in Chinese）.
29	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.
30	BENDSØE M P， SIGMUND O. Material interpolation schemes in topology optimization［J］. Archive of Applied Mechanics， 1999， 69（9）： 635-654.
31	STOLPE M， SVANBERG K. On the trajectories of penalization methods for topology optimization［J］. Structural and Multidisciplinary Optimization， 2001， 21（2）： 128-139.
32	STOLPE M， SVANBERG K. An alternative interpolation scheme for minimum compliance topology optimization［J］. Structural and Multidisciplinary Optimization， 2001， 22（2）： 116-124.
33	XIE Y M， STEVEN G P. A simple evolutionary procedure for structural optimization［J］. Computers & Structures， 1993， 49（5）： 885-896.
34	QUERIN O M， STEVEN G P， XIE Y M. Evolutionary structural optimisation （ESO） using a bidirectional algorithm［J］. Engineering Computations， 1998， 15（8）： 1031-1048.
35	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.
	YANG D Q， LIU Z X， SUI Y K. ICM method for topology optimization design of continuum structures［J］. Journal of Shanghai Jiao Tong University， 1999， 33（6）： 734-736 （in Chinese）.
36	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.
37	ALLAIRE G， JOUVE F， TOADER A M. A level-set method for shape optimization［J］. Comptes Rendus Mathematique， 2002， 334（12）： 1125-1130.
38	ALLAIRE G， JOUVE F， TOADER A M. Structural optimization using sensitivity analysis and a level-set method［J］. Journal of Computational Physics， 2004， 194（1）： 363-393.
39	WANG M Y， ZHOU S W. Phase field： A variational method for structural topology optimization［J］. Computer Modeling in Engineering and Sciences， 2004， 6（6）： 547-566.
40	TAKEZAWA A， NISHIWAKI S， KITAMURA M. Shape and topology optimization based on the phase field method and sensitivity analysis［J］. Journal of Computational Physics， 2010， 229（7）： 2697-2718.
41	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.
42	ZHANG W S， CHEN J S， ZHU X F， et al. Explicit three dimensional topology optimization via moving morphable void （MMV） approach［J］. Computer Methods in Applied Mechanics and Engineering， 2017， 322： 590-614.
43	DU B X， YAO W， ZHAO Y， et al. A moving morphable voids approach for topology optimization with closed B-splines［J］. Journal of Mechanical Design， 2019， 141（8）： 081401.
44	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.
45	张卫红，周莹，酒丽朋，等. 特征驱动的结构拓扑优化方法［J］. 中国科学：技术科学， 2019， 49（10）： 1177-1185.
	ZHANG W H， ZHOU Y， JIU L P， et al. Feature-driven method for structural topology optimization［J］. Scientia Sinica （Technologica）， 2019， 49（10）： 1177-1185 （in Chinese）.
46	ZHANG W H， ZHOU Y， ZHU J H. A comprehensive study of feature definitions with solids and voids for topology optimization［J］. Computer Methods in Applied Mechanics and Engineering， 2017， 325： 289-313.
47	ESCHENAUER H A， KOBELEV V V， SCHUMACHER A. Bubble method for topology and shape optimization of structures［J］. Structural Optimization， 1994， 8（1）： 42-51.
48	SIGMUND O， MAUTE K. Topology optimization approaches［J］. Structural and Multidisciplinary Optimization， 2013， 48（6）： 1031-1055.
49	DEATON J D， GRANDHI R V. A survey of structural and multidisciplinary continuum topology optimization： post 2000［J］. Structural and Multidisciplinary Optimization， 2014， 49（1）： 1-38.
50	VAN DIJK N P， MAUTE K， LANGELAAR M， et al. Level-set methods for structural topology optimization： A review［J］. Structural and Multidisciplinary Optimization， 2013， 48（3）： 437-472.
51	MUNK D J， VIO G A， STEVEN G P. Topology and shape optimization methods using evolutionary algorithms： A review［J］. Structural and Multidisciplinary Optimization， 2015， 52（3）： 613-631.
52	XIA L， XIA Q， HUANG X D， et al. Bi-directional evolutionary structural optimization on advanced structures and materials： A comprehensive review［J］. Archives of Computational Methods in Engineering， 2018， 25（2）： 437-478.
53	WEIN F， DUNNING P D， NORATO J A. A review on feature-mapping methods for structural optimization［J］. Structural and Multidisciplinary Optimization， 2020， 62（4）： 1597-1638.
54	PAVLIOTIS G A， STUART A M. Multiscale methods： Averaging and homogenization［M］. New York： Springer， 2008.
55	杜善义，王彪. 复合材料细观力学［M］. 北京：科学出版社， 1998： 1-6.
	DU S Y， WANG B. Mesomechanics of composite materials［M］. Beijing： Science Press， 1998： 1-6 （in Chinese）.
56	陈玉丽，马勇，潘飞，等. 多尺度复合材料力学研究进展［J］. 固体力学学报， 2018， 39（1）： 1-68.
	CHEN Y L， MA Y， PAN F， et al. Research progress in multi-scale mechanics of composite materials［J］. Chinese Journal of Solid Mechanics， 2018， 39（1）： 1-68 （in Chinese）.
57	江山. 多尺度有限元方法的一些研究及应用［D］. 湘潭：湘潭大学， 2008： 1-6.
	JIANG S. Some studies and applications on the multiscale finite element method［D］. Xiangtan： Xiangtan University， 2008： 1-6 （in Chinese）.
58	黄富华. 周期性复合材料有效性能的均匀化计算［D］. 哈尔滨：哈尔滨工业大学， 2010： 3-18.
	HUANG F H. Homogenizated numeration of effective properties for composite material with periodicity［D］. Harbin： Harbin Institute of Technology， 2010： 3-18 （in Chinese）.
59	吴敬凯. 非均质材料力学性能与失效分析的多尺度有限元法研究［D］. 大连：大连理工大学， 2013： 3-11.
	WU J K. Study on multiscale finite element method for mechanical behavior and failure analyses of heterogeneous materials［D］. Dalian： Dalian University of Technology， 2013： 3-11 （in Chinese）.
60	BABUŠKA I. Homogenization approach in engineering［C］∥ GLOWINSKI R， LIONS J L. Computing Methods in Applied Sciences and Engineering. Berlin： Springer， 1976： 137-153.
61	BENSOUSSAN A， LIONS J L， PAPANICOLAOU G. Asymptotic analysis for periodic structures［M］. Providence： American Mathematical Society， 1978： 1-20.
62	CIORANESCU D， SAINT JEAN PAULIN J. Homogenization in open sets with holes［J］. Journal of Mathematical Analysis and Applications， 1979， 71（2）： 590-607.
63	OLEINIK O A. On homogenization problems［M］∥ Trends and Applications of Pure Mathematics to Mechanics. Berlin： Springer Berlin Heidelberg， 1984： 248-272.
64	GUEDES J， KIKUCHI N. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods［J］. Computer Methods in Applied Mechanics and Engineering， 1990， 83（2）： 143-198.
65	HASSANI B， HINTON E. A review of homogenization and topology optimization I—Homogenization theory for media with periodic structure［J］. Computers & Structures， 1998， 69（6）： 707-717.
66	HASSANI B， HINTON E. A review of homogenization and topology opimization II—Analytical and numerical solution of homogenization equations［J］. Computers & Structures， 1998， 69（6）： 719-738.
67	CHUNG P W， TAMMA K K， NAMBURU R R. Asymptotic expansion homogenization for heterogeneous media： Computational issues and applications［J］. Composites Part A： Applied Science and Manufacturing， 2001， 32（9）： 1291-1301.
68	KALAMKAROV A L， ANDRIANOV I V， DANISHEVS’KYY V V. Asymptotic homogenization of composite materials and structures［J］. Applied Mechanics Reviews， 2009， 62（3）： 669-676.
69	HASSANI B， HINTON E. A review of homogenization and topology optimization III—Topology optimization using optimality criteria［J］. Computers & Structures， 1998， 69（6）： 739-756.
70	SIGMUND O. Materials with prescribed constitutive parameters： An inverse homogenization problem［J］. International Journal of Solids and Structures， 1994， 31（17）： 2313-2329.
71	ANDREASSEN E， ANDREASEN C S. How to determine composite material properties using numerical homogenization［J］. Computational Materials Science， 2014， 83： 488-495.
72	DONG G Y， TANG Y L， ZHAO Y F. A 149 line homogenization code for three-dimensional cellular materials written in Matlab［J］. Journal of Engineering Materials and Technology， 2019， 141（1）： 011005.
73	BELYTSCHKO T， SONG J H. Coarse-graining of multiscale crack propagation［J］. International Journal for Numerical Methods in Engineering， 2010， 81（5）： 537-563.
74	UNGER J F， ECKARDT S. Multiscale modeling of concrete［J］. Archives of Computational Methods in Engineering， 2011， 18（3）： 341-393.
75	LLOBERAS-VALLS O， RIXEN D J， SIMONE A， et al. Multiscale domain decomposition analysis of quasi-brittle heterogeneous materials［J］. International Journal for Numerical Methods in Engineering， 2012， 89（11）： 1337-1366.
76	WU J， SIGMUND O， GROEN J P. Topology optimization of multi-scale structures： A review［J］. Structural and Multidisciplinary Optimization， 2021， 63（3）： 1455-1480.
77	SU W Z， LIU S T. Size-dependent optimal microstructure design based on couple-stress theory［J］. Structural and Multidisciplinary Optimization， 2010， 42（2）： 243-254.
78	HUANG X， ZHOU S W， XIE Y M， et al. Topology optimization of microstructures of cellular materials and composites for macrostructures［J］. Computational Materials Science， 2013， 67： 397-407.
79	KATO J， YACHI D S， TERADA K， et al. Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis［J］. Structural and Multidisciplinary Optimization， 2014， 49（4）： 595-608.
80	KATO J， YACHI D S， KYOYA T， et al. Micro-macro concurrent topology optimization for nonlinear solids with a decoupling multiscale analysis［J］. International Journal for Numerical Methods in Engineering， 2018， 113（8）： 1189-1213.
81	DA D C， YVONNET J， XIA L， et al. Topology optimization of periodic lattice structures taking into account strain gradient［J］. Computers & Structures， 2018， 210： 28-40.
82	MONTEMURRO M， BERTOLINO G， ROINÉ T. A general multi-scale topology optimisation method for lightweight lattice structures obtained through additive manufacturing technology［J］. Composite Structures， 2021， 258： 113360.
83	MOSES E， FUCHS M B， RYVKIN M. Topological design of modular structures under arbitrary loading［J］. Structural and Multidisciplinary Optimization， 2002， 24（6）： 407-417.
84	HUANG X， XIE Y M. Optimal design of periodic structures using evolutionary topology optimization［J］. Structural and Multidisciplinary Optimization， 2008， 36（6）： 597-606.
85	ZUO Z H， XIE Y M， HUANG X D. Optimal topological design of periodic structures for natural frequencies［J］. Journal of Structural Engineering， 2011， 137（10）： 1229-1240.
86	XIE Y M， ZUO Z H， HUANG X D， et al. Convergence of topological patterns of optimal periodic structures under multiple scales［J］. Structural and Multidisciplinary Optimization， 2012， 46（1）： 41-50.
87	THOMAS S， LI Q， STEVEN G. Topology optimization for periodic multi-component structures with stiffness and frequency criteria［J］. Structural and Multidisciplinary Optimization， 2020， 61（6）： 2271-2289.
88	ALMEIDA S R M， PAULINO G H， SILVA E C N. Layout and material gradation in topology optimization of functionally graded structures： A global-local approach［J］. Structural and Multidisciplinary Optimization， 2010， 42（6）： 855-868.
89	THOMAS S， LI Q， STEVEN G. Finite periodic topology optimization with oriented unit-cells［J］. Structural and Multidisciplinary Optimization， 2021， 64（4）： 1765-1779.
90	ZUO Z H， HUANG X D， YANG X Y， et al. Comparing optimal material microstructures with optimal periodic structures［J］. Computational Materials Science， 2013， 69： 137-147.
91	LIU L， YAN J， CHENG G D. Optimum structure with homogeneous optimum truss-like material［J］. Computers & Structures， 2008， 86（13-14）： 1417-1425.
92	YAN J， CHENG G D， LIU L. A uniform optimum material based model for concurrent optimization of thermoelastic structures and materials［J］. International Journal for Simulation and Multidisciplinary Design Optimization， 2008， 2（4）： 259-266.
93	NIU B， YAN J， CHENG G D. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency［J］. Structural and Multidisciplinary Optimization， 2009， 39（2）： 115-132.
94	DENG J D， YAN J， CHENG G D. Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material［J］. Structural and Multidisciplinary Optimization， 2013， 47（4）： 583-597.
95	WADBRO E， NIU B. Multiscale design for additive manufactured structures with solid coating and periodic infill pattern［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 357： 112605.
96	YAN X L， HUANG X D， ZHA Y， et al. Concurrent topology optimization of structures and their composite microstructures［J］. Computers & Structures， 2014， 133： 103-110.
97	VICENTE W M， ZUO Z H， PAVANELLO R， et al. Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures［J］. Computer Methods in Applied Mechanics and Engineering， 2016， 301： 116-136.
98	XIA L A， BREITKOPF P. A reduced multiscale model for nonlinear structural topology optimization［J］. Computer Methods in Applied Mechanics and Engineering， 2014， 280： 117-134.
99	YAN X L， HUANG X D， SUN G Y， et al. Two-scale optimal design of structures with thermal insulation materials［J］. Composite Structures， 2015， 120： 358-365.
100	YAN X L， XU Q W， HUANG D F， et al. Concurrent topology design of structures and materials with optimal material orientation［J］. Composite Structures， 2019， 220： 473-480.
101	YAN X L， XU Q W， HUA H Y， et al. Concurrent optimization of macrostructures and material microstructures and orientations for maximizing natural frequency［J］. Engineering Structures， 2020， 209： 109997.
102	FRITZEN F， XIA L A， LEUSCHNER M， et al. Topology optimization of multiscale elastoviscoplastic structures［J］. International Journal for Numerical Methods in Engineering， 2016， 106（6）： 430-453.
103	YAN J， HU W B， DUAN Z Y. Structure/material concurrent optimization of lattice materials based on extended multiscale finite element method［J］. International Journal for Multiscale Computational Engineering， 2015， 13（1）： 73-90.
104	YAN J， SUI Q Q， FAN Z R， et al. Multi-material and multiscale topology design optimization of thermoelastic lattice structures［J］. Computer Modeling in Enginee-ring & Sciences， 2022， 130（2）： 967-986.
105	CHENG G D， XU L. Two-scale topology design optimization of stiffened or porous plate subject to out-of-plane buckling constraint［J］. Structural and Multidisciplinary Optimization， 2016， 54（5）： 1283-1296.
106	WANG Y Q， WANG M Y， CHEN F F. Structure-material integrated design by level sets［J］. Structural and Multidisciplinary Optimization， 2016， 54（5）： 1145-1156.
107	ROBBINS J， OWEN S J， CLARK B W， et al. An efficient and scalable approach for generating topologically optimized cellular structures for additive manufacturing［J］. Additive Manufacturing， 2016， 12： 296-304.
108	CHEN W J， TONG L Y， LIU S T. Concurrent topology design of structure and material using a two-scale topology optimization［J］. Computers & Structures， 2017， 178： 119-128.
109	DA D C， CUI X Y， LONG K， et al. Concurrent topological design of composite structures and the underlying multi-phase materials［J］. Computers & Structures， 2017， 179： 1-14.
110	DA D C， CUI X Y， LONG K， et al. Multiscale concurrent topology optimization of structures and microscopic multi-phase materials for thermal conductivity［J］. Engineering Computations， 2019， 36（1）： 126-146.
111	DA D C， XIA L A. Design of heterogeneous mesostructures for nonseparated scales and analysis of size effects［J］. International Journal for Numerical Methods in Engineering， 2021， 122（5）： 1333-1351.
112	LONG K， YUAN P F， XU S Q， et al. Concurrent topological design of composite structures and materials containing multiple phases of distinct Poisson’s ratios［J］. Engineering Optimization， 2018， 50（4）： 599-614.
113	LI H， LUO Z， XIAO M， et al. A new multiscale topology optimization method for multiphase composite structures of frequency response with level sets［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 356： 116-144.
114	FU J J， LI H， GAO L， et al. Design of shell-infill structures by a multiscale level set topology optimization method［J］. Computers & Structures， 2019， 212： 162-172.
115	ZHANG W H， SUN S P. Scale-related topology optimization of cellular materials and structures［J］. International Journal for Numerical Methods in Engineering， 2006， 68（9）： 993-1011.
116	TROMME E， KAWAMOTO A， GUEST J K. Topology optimization based on reduction methods with applications to multiscale design and additive manufacturing［J］. Frontiers of Mechanical Engineering， 2020， 15（1）： 151-165.
117	ALEXANDERSEN J， LAZAROV B S. Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner［J］. Computer Methods in Applied Mechanics and Engineering， 2015， 290： 156-182.
118	COELHO P G， RODRIGUES H C. Hierarchical topology optimization addressing material design constraints and application to sandwich-type structures［J］. Structural and Multidisciplinary Optimization， 2015， 52（1）： 91-104.
119	LI H， LUO Z， ZHANG N， et al. Integrated design of cellular composites using a level-set topology optimization method［J］. Computer Methods in Applied Mechanics and Engineering， 2016， 309： 453-475.
120	LI H， LUO Z， GAO L， et al. Topology optimization for functionally graded cellular composites with metamaterials by level sets［J］. Computer Methods in Applied Mechanics and Engineering， 2018， 328： 340-364.
121	FUJIOKA M， SHIMODA M， AL ALI M. Shape optimization of periodic-microstructures for stiffness maximization of a macrostructure［J］. Composite Structures， 2021， 268： 113873.
122	SIVAPURAM R， DUNNING P D， KIM H A. Simultaneous material and structural optimization by multiscale topology optimization［J］. Structural and Multidisciplinary Optimization， 2016， 54（5）： 1267-1281.
123	GUO X， ZHAO X F， ZHANG W S， et al. Multi-scale robust design and optimization considering load uncertainties［J］. Computer Methods in Applied Mechanics and Engineering， 2015， 283： 994-1009.
124	LI H， LUO Z， GAO L， et al. Topology optimization for concurrent design of structures with multi-patch microstructures by level sets［J］. Computer Methods in Applied Mechanics and Engineering， 2018， 331： 536-561.
125	XU L， CHENG G D. Two-scale concurrent topology optimization with multiple micro materials based on principal stress orientation［J］. Structural and Multidisciplinary Optimization， 2018， 57（5）： 2093-2107.
126	LIU K， DETWILER D， TOVAR A. Cluster-based optimization of cellular materials and structures for crashworthiness［J］. Journal of Mechanical Design， 2018， 140（11）： 111412.
127	GAO J， LUO Z， LI H， et al. Topology optimization for multiscale design of porous composites with multi-domain microstructures［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 344： 451-476.
128	GAO J， LUO Z， LI H， et al. Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites［J］. Composite Structures， 2019， 211： 401-417.
129	PIZZOLATO A， SHARMA A， MAUTE K， et al. Multi-scale topology optimization of multi-material structures with controllable geometric complexity—Applications to heat transfer problems［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 357： 112552.
130	WANG Y G， KANG Z. Concurrent two-scale topological design of multiple unit cells and structure using combined velocity field level set and density model［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 347： 340-364.
131	QIU Z， LI Q H， LIU S T， et al. Clustering-based concurrent topology optimization with macrostructure， components， and materials［J］. Structural and Multidisciplinary Optimization， 2021， 63（3）： 1243-1263.
132	JIA J， DA D C， LOH C L， et al. Multiscale topology optimization for non-uniform microstructures with hybrid cellular automata［J］. Structural and Multidisciplinary Optimization， 2020， 62（2）： 757-770.
133	YAN J， SUI Q Q， FAN Z R， et al. Clustering-based multiscale topology optimization of thermo-elastic lattice structures［J］. Computational Mechanics， 2020， 66（4）： 979-1002.
134	XU Z， ZHANG W H， ZHOU Y， et al. Multiscale topology optimization using feature-driven method［J］. Chinese Journal of Aeronautics， 2020， 33（2）： 621-633.
135	LIAO H T. A single variable-based method for concurrent multiscale topology optimization with multiple materials［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 378： 113727.
136	GU X C， HE S M， DONG Y H， et al. An improved ordered SIMP approach for multiscale concurrent topology optimization with multiple microstructures［J］. Composite Structures， 2022， 287： 115363.
137	AL ALI M， SHIMODA M. Investigation of concurrent multiscale topology optimization for designing lightweight macrostructure with high thermal conductivity［J］. International Journal of Thermal Sciences， 2022， 179： 107653.
138	PANTZ O， TRABELSI K. A post-treatment of the homogenization method for shape optimization［J］. SIAM Journal on Control and Optimization， 2008， 47（3）： 1380-1398.
139	ALLAIRE G， GEOFFROY-DONDERS P， PANTZ O. Topology optimization of modulated and oriented periodic microstructures by the homogenization method［J］. Computers & Mathematics with Applications， 2019， 78（7）： 2197-2229.
140	GEOFFROY-DONDERS P， ALLAIRE G， PANTZ O. 3-D topology optimization of modulated and oriented periodic microstructures by the homogenization method［J］. Journal of Computational Physics， 2020， 401： 108994.
141	GROEN J P， SIGMUND O. Homogenization-based topology optimization for high-resolution manufacturable microstructures［J］. International Journal for Numerical Methods in Engineering， 2018， 113（8）： 1148-1163.
142	GROEN J P， STUTZ F C， AAGE N， et al. De-homogenization of optimal multi-scale 3D topologies［J］. Computer Methods in Applied Mechanics and Engineering， 2020， 364： 112979.
143	TAMIJANI A Y， VELASCO S P， ALACOQUE L. Topological and morphological design of additively-manufacturable spatially-varying periodic cellular solids［J］. Materials & Design， 2020， 196： 109155.
144	KUMAR T， SURESH K. A density-and-strain-based K-clustering approach to microstructural topology optimization［J］. Structural and Multidisciplinary Optimization， 2020， 61（4）： 1399-1415.
145	LEE J， KWON C， YOO J， et al. Design of spatially-varying orthotropic infill structures using multiscale topology optimization and explicit de-homogenization［J］. Additive Manufacturing， 2021， 40： 101920.
146	WU J， WANG W M， GAO X F. Design and optimization of conforming lattice structures［J］. IEEE Transactions on Visualization and Computer Graphics， 2021， 27（1）： 43-56.
147	JUNG T， LEE J， NOMURA T， et al. Inverse design of three-dimensional fiber reinforced composites with spatially-varying fiber size and orientation using multiscale topology optimization［J］. Composite Structures， 2022， 279： 114768.
148	XU L， QIAN Z H. Topology optimization and de-homogenization of graded lattice structures based on asymptotic homogenization［J］. Composite Structures， 2021， 277： 114633.
149	BURBLIES A， BUSSE M. Computer based porosity design by multi phase topology optimization［C］∥ AIP Conference Proceedings. AIP Conference Proceedings， 2008， 973（1）： 285-290.
150	BRACKETT D， ASHCROFT I， HAGUE R. Topology optimization for additive manufacturing［C］∥ 2011 International Solid Freeform Fabrication Symposium. Austin： University of Texas， 2011.
151	ARABNEJAD KHANOKI S， PASINI D. Multiscale design and multiobjective optimization of orthopedic hip implants with functionally graded cellular material［J］. Journal of Biomechanical Engineering， 2012， 134（3）： 031004.
152	WANG Y J， ARABNEJAD S， TANZER M， et al. Hip implant design with three-dimensional porous architecture of optimized graded density［J］. Journal of Mechanical Design， 2018， 140（11）： 111406.
153	ALZAHRANI M， CHOI S K， ROSEN D W. Design of truss-like cellular structures using relative density mapping method［J］. Materials & Design， 2015， 85： 349-360.
154	CHENG L， ZHANG P， BIYIKLI E， et al. Integration of topology optimization with efficient design of additive manufactured cellular structures［C］∥ 2015 International Solid Freeform Fabrication Symposium. Austin： University of Texas， 2015.
155	CHENG L， ZHANG P， BIYIKLI E， et al. Efficient design optimization of variable-density cellular structures for additive manufacturing： Theory and experimental validation［J］. Rapid Prototyping Journal， 2017， 23： 660-677.
156	ZHANG P， TOMAN J， YU Y Q， et al. Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing： Theory and validation［J］. Journal of Manufacturing Science and Engineering， 2015， 137（2）： 021004.
157	CHENG L， LIANG X A， BELSKI E， et al. Natural frequency optimization of variable-density additive manufactured lattice structure： Theory and experimental validation［J］. Journal of Manufacturing Science and Engineering， 2018， 140（10）： 105002.
158	WANG X， ZHANG P， LUDWICK S， et al. Natural frequency optimization of 3D printed variable-density honeycomb structure via a homogenization-based approach［J］. Additive Manufacturing， 2018， 20： 189-198.
159	CHENG L， LIU J K， TO A C. Concurrent lattice infill with feature evolution optimization for additive manufactured heat conduction design［J］. Structural and Multidisciplinary Optimization， 2018， 58（2）： 511-535.
160	CHENG L， LIU J K， LIANG X， et al. Coupling lattice structure topology optimization with design-dependent feature evolution for additive manufactured heat conduction design［J］. Computer Methods in Applied Mechanics and Engineering， 2018， 332： 408-439.
161	CHENG L， BAI J X， TO A C. Functionally graded lattice structure topology optimization for the design of additive manufactured components with stress constraints［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 344： 334-359.
162	CLAUSEN A， AAGE N， SIGMUND O. Exploiting additive manufacturing infill in topology optimization for improved buckling load［J］. Engineering， 2016， 2（2）： 250-257.
163	WU T， LIU K， TOVAR A. Multiphase topology optimization of lattice injection molds［J］. Computers & Structures， 2017， 192： 71-82.
164	WANG Y J， XU H， PASINI D. Multiscale isogeometric topology optimization for lattice materials［J］. Computer Methods in Applied Mechanics and Engineering， 2017， 316： 568-585.
165	PANESAR A， ABDI M， HICKMAN D， et al. Strategies for functionally graded lattice structures derived using topology optimisation for additive manufacturing［J］. Additive Manufacturing， 2018， 19： 81-94.
166	TANG Y L， ZHAO Y F. Multifunctional design of heterogeneous cellular structures［J］. Structural and Multidisciplinary Optimization， 2018， 58（3）： 1121-1138.
167	WU Z J， XIA L， WANG S T， et al. Topology optimization of hierarchical lattice structures with substructuring［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 345： 602-617.
168	WU Z J， FAN F， XIAO R B， et al. The substructuring-based topology optimization for maximizing the first eigenvalue of hierarchical lattice structure［J］. International Journal for Numerical Methods in Engineering， 2020， 121（13）： 2964-2978.
169	LIU Z， XIA L， XIA Q， et al. Data-driven design approach to hierarchical hybrid structures with multiple lattice configurations［J］. Structural and Multidisciplinary Optimization， 2020， 61（6）： 2227-2235.
170	ZHU Y， ZHAO J Q， ZHANG M， et al. An improved density-based design method of additive manufacturing fabricated inhomogeneous cellular-solid structures［J］. International Journal of Precision Engineering and Manufacturing， 2020， 21（1）： 103-116.
171	WATTS S， ARRIGHI W， KUDO J， et al. Simple， accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design［J］. Structural and Multidisciplinary Optimization， 2019， 60（5）： 1887-1920.
172	JANSEN M， PIERARD O. A hybrid density/level set formulation for topology optimization of functionally graded lattice structures［J］. Computers & Structures， 2020， 231： 106205.
173	KIM D， LEE J， NOMURA T， et al. Topology optimization of functionally graded anisotropic composite structures using homogenization design method［J］. Computer Methods in Applied Mechanics and Engineering， 2020， 369： 113220.
174	KIM J E， PARK K. Multiscale topology optimization combining density-based optimization and lattice enhancement for additive manufacturing［J］. International Journal of Precision Engineering and Manufacturing-Green Technology， 2021， 8（4）： 1197-1208.
175	KUMAR T， SRIDHARA S， PRABHUNE B， et al. Spectral decomposition for graded multi-scale topology optimization［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 377： 113670.
176	ZHOU M D， GENG D. Multi-scale and multi-material topology optimization of channel-cooling cellular structures for thermomechanical behaviors［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 383： 113896.
177	ZHOU H， ZHU J H， WANG C， et al. Hierarchical structure optimization with parameterized lattice and multiscale finite element method［J］. Structural and Multidisciplinary Optimization， 2022， 65（1）： 39.
178	CRAMER A D， CHALLIS V J， ROBERTS A P. Microstructure interpolation for macroscopic design［J］. Structural and Multidisciplinary Optimization， 2016， 53（3）： 489-500.
179	LI D W， LIAO W H， DAI N， et al. Optimal design and modeling of gyroid-based functionally graded cellular structures for additive manufacturing［J］. Computer-Aided Design， 2018， 104： 87-99.
180	LI D W， DAI N， TANG Y L， et al. Design and optimization of graded cellular structures with triply periodic level surface-based topological shapes［J］. Journal of Mechanical Design， 2019， 141（7）： 071402.
181	HU J Q， LI M， GAO S M. Texture-guided generative structural designs under local control［J］. Computer-Aided Design， 2019， 108： 1-11.
182	DENG H， TO A C. Projection-based implicit modeling method （PIMM） for functionally graded lattice optimization［J］. JOM， 2021， 73（7）： 2012-2021.
183	LI M， ZHU L C， LI J Z， et al. Design optimization of interconnected porous structures using extended triply periodic minimal surfaces［J］. Journal of Computational Physics， 2021， 425： 109909.
184	NGUYEN C H P， CHOI Y. Multiscale design of functionally graded cellular structures for additive manufacturing using level-set descriptions［J］. Structural and Multidisciplinary Optimization， 2021， 64（4）： 1983-1995.
185	NGUYEN C H P， KIM Y， DO Q T， et al. Implicit-based computer-aided design for additively manufactured functionally graded cellular structures［J］. Journal of Computational Design and Engineering， 2021， 8（3）： 813-823.
186	MONTEMURRO M， REFAI K， CATAPANO A. Thermal design of graded architected cellular materials through a CAD-compatible topology optimisation method［J］. Composite Structures， 2022， 280： 114862.
187	LIU H， CHEN L X， JIANG Y， et al. Multiscale optimization of additively manufactured graded non-stochastic and stochastic lattice structures［J］. Composite Structures， 2023， 305： 116546.
188	WANG Y Q， CHEN F F， WANG M Y. Concurrent design with connectable graded microstructures［J］. Computer Methods in Applied Mechanics and Engineering， 2017， 317： 84-101.
189	WANG Y Q， ZHANG L， DAYNES S， et al. Design of graded lattice structure with optimized mesostructures for additive manufacturing［J］. Materials & Design， 2018， 142： 114-123.
190	ZONG H M， LIU H， MA Q P， et al. VCUT level set method for topology optimization of functionally graded cellular structures［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 354： 487-505.
191	LIU H， ZONG H M， SHI T L， et al. M-VCUT level set method for optimizing cellular structures［J］. Computer Methods in Applied Mechanics and Engineering， 2020， 367： 113154.
192	LIU H， CHEN L X， SHI T L， et al. M-VCUT level set method for the layout and shape optimization of stiffeners in plate［J］. Composite Structures， 2022， 293： 115614.
193	LIU H， CHEN L X， BIAN H. Data-driven M-VCUT topology optimization method for heat conduction problem of cellular structure with multiple microstructure prototypes［J］. International Journal of Heat and Mass Transfer， 2022， 198： 123421.
194	XIA Q， ZONG H M， SHI T L， et al. Optimizing cellular structures through the M-VCUT level set method with microstructure mapping and high order cutting［J］. Composite Structures， 2021， 261： 113298.
195	ZHOU Y， GAO L， LI H. Graded infill design within free-form surfaces by conformal mapping［J］. International Journal of Mechanical Sciences， 2022， 224： 107307.
196	HUO S L， DU B X， ZHAO Y， et al. Thermal design of functionally graded cellular structures with multiple microstructure configurations through topology optimization［J］. Composite Structures， 2023， 313： 116922.
197	SCHURY F， STINGL M， WEIN F. Efficient two-scale optimization of manufacturable graded structures［J］. SIAM Journal on Scientific Computing， 2012， 34（6）： B711-B733.
198	LIU C， DU Z L， ZHANG W S， et al. Additive manufacturing-oriented design of graded lattice structures through explicit topology optimization［J］. Journal of Applied Mechanics， 2017， 84（8）： 081008.
199	ZHU Y C， LI S S， DU Z L， et al. A novel asymptotic-analysis-based homogenisation approach towards fast design of infill graded microstructures［J］. Journal of the Mechanics and Physics of Solids， 2019， 124： 612-633.
200	WANG C， ZHU J H， ZHANG W H， et al. Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures［J］. Structural and Multidisciplinary Optimization， 2018， 58（1）： 35-50.
201	WANG C， GU X J， ZHU J H， et al. Concurrent design of hierarchical structures with three-dimensional parameterized lattice microstructures for additive manufacturing［J］. Structural and Multidisciplinary Optimization， 2020， 61（3）： 869-894.
202	ZHANG Y， XIAO M， LI H， et al. Multiscale concurrent topology optimization for cellular structures with multiple microstructures based on ordered SIMP interpolation［J］. Computational Materials Science， 2018， 155： 74-91.
203	ZHANG Y， GAO L， XIAO M. Maximizing natural frequencies of inhomogeneous cellular structures by Kriging-assisted multiscale topology optimization［J］. Compu-ters & Structures， 2020， 230： 106197.
204	ZHANG Y， LI H， XIAO M， et al. Concurrent topology optimization for cellular structures with nonuniform microstructures based on the kriging metamodel［J］. Structural and Multidisciplinary Optimization， 2019， 59（4）： 1273-1299.
205	HAN Y F， LU W F. A novel design method for nonuniform lattice structures based on topology optimization［J］. Journal of Mechanical Design， 2018， 140（9）： 091403.
206	IMEDIEGWU C， MURPHY R， HEWSON R， et al. Multiscale structural optimization towards three-dimensional printable structures［J］. Structural and Multidisciplinary Optimization， 2019， 60（2）： 513-525.
207	IMEDIEGWU C， MURPHY R， HEWSON R， et al. Multiscale thermal and thermo-structural optimization of three-dimensional lattice structures［J］. Structural and Multidisciplinary Optimization， 2022， 65（1）： 13.
208	KANG D， PARK S， SON Y， et al. Multi-lattice inner structures for high-strength and light-weight in metal selective laser melting process［J］. Materials & Design， 2019， 175： 107786.
209	LIU H， ZONG H M， TIAN Y， et al. A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design［J］. Structural and Multidisciplinary Optimization， 2019， 60（6）： 2221-2247.
210	LI Q H， XU R， LIU J， et al. Topology optimization design of multi-scale structures with alterable microstructural length-width ratios［J］. Composite Structures， 2019， 230： 111454.
211	ZHAO J Q， ZHANG M， ZHU Y， et al. A novel optimization design method of additive manufacturing oriented porous structures and experimental validation［J］. Materials & Design， 2019， 163： 107550.
212	XU M M， XIA L， WANG S T， et al. An isogeometric approach to topology optimization of spatially graded hierarchical structures［J］. Composite Structures， 2019， 225： 111171.
213	DAS S， SUTRADHAR A. Multi-physics topology optimization of functionally graded controllable porous structures： Application to heat dissipating problems［J］. Materials & Design， 2020， 193： 108775.
214	DONG G Y， TANG Y L， LI D W， et al. Design and optimization of solid lattice hybrid structures fabricated by additive manufacturing［J］. Additive Manufacturing， 2020， 33： 101116.
215	JIANG L， GUO Y， CHEN S K， et al. Concurrent optimization of structural topology and infill properties with a CBF-based level set method［J］. Frontiers of Mechanical Engineering， 2019， 14（2）： 171-189.
216	LI D W， LIAO W H， DAI N， et al. Anisotropic design and optimization of conformal gradient lattice structures［J］. Computer-Aided Design， 2020， 119： 102787.
217	WU T Y， LI S. An efficient multiscale optimization method for conformal lattice materials［J］. Structural and Multidisciplinary Optimization， 2021， 63（3）： 1063-1083.
218	YU H C， HUANG J Q， ZOU B， et al. Stress-constrained shell-lattice infill structural optimisation for additive manufacturing［J］. Virtual and Physical Prototyping， 2020， 15（1）： 35-48.
219	AGRAWAL Y， ANANTHASURESH G K. Towards optimal heterogeneity in lattice structures［J］. Structural and Multidisciplinary Optimization， 2021， 64（4）： 2489-2512.
220	GANGWAR T， SCHILLINGER D. Concurrent material and structure optimization of multiphase hierarchical systems within a continuum micromechanics framework［J］. Structural and Multidisciplinary Optimization， 2021， 64（3）： 1175-1197.
221	JIANG L， GU X D， CHEN S K. Generative design of bionic structures via concurrent multiscale topology optimization and conformal geometry method［J］. Journal of Mechanical Design， 2021， 143（1）： 011701.
222	LI Q H， XU R， WU Q B， et al. Topology optimization design of quasi-periodic cellular structures based on erode-dilate operators［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 377： 113720.
223	LUO Y F， HU J Y， LIU S T. Self-connected multi-domain topology optimization of structures with multiple dissimilar microstructures［J］. Structural and Multidisciplinary Optimization， 2021， 64（1）： 125-140.
224	MOUSSA A， MELANCON D， ELMI A EL， et al. Topology optimization of imperfect lattice materials built with process-induced defects via Powder Bed Fusion［J］. Additive Manufacturing， 2021， 37： 101608.
225	YPSILANTIS K I， KAZAKIS G， LAGAROS N D. An efficient 3D homogenization-based topology optimization methodology［J］. Computational Mechanics， 2021， 67（2）： 481-496.
226	ZHANG J W， YANAGIMOTO J. Topology optimization of microlattice dome with enhanced stiffness and energy absorption for additive manufacturing［J］. Composite Structures， 2021， 255： 112889.
227	BRUGGI M， ISMAIL H， LÓGÓ J， et al. Lightweight design with displacement constraints using graded porous microstructures［J］. Computers & Structures， 2022， 272： 106873.
228	LIU X L， GAO L， XIAO M， et al. Kriging-assisted design of functionally graded cellular structures with smoothly-varying lattice unit cells［J］. Computer Methods in Applied Mechanics and Engineering， 2022， 390： 114466.
229	COSTA M R， SOHOULI A， SULEMAN A. Multi-scale and multi-material topology optimization of gradient lattice structures using surrogate models［J］. Composite Structures， 2022， 289： 115402.
230	YU C， WANG Q F， XIA Z H， et al. Multiscale topology optimization for graded cellular structures based on level set surface cutting［J］. Structural and Multidisciplinary Optimization， 2022， 65（1）： 32.
231	WU J， AAGE N， WESTERMANN R， et al. Infill optimization for additive manufacturing-approaching bone-like porous structures［J］. IEEE Transactions on Visualization and Computer Graphics， 2018， 24（2）： 1127-1140.
232	SCHMIDT M P， PEDERSEN C B W， GOUT C. On structural topology optimization using graded porosity control［J］. Structural and Multidisciplinary Optimization， 2019， 60（4）： 1437-1453.
233	GARAIGORDOBIL A， ANSOLA R， QUERIN O M， et al. Infill topology optimization of porous structures with discrete variables by the sequential element rejection and admission method［J］. Engineering Optimization， 2021， 55（3）： 457-475.
234	LONG K， CHEN Z， ZHANG C W， et al. An aggregation-free local volume fraction formulation for topological design of porous structure［J］. Materials， 2021， 14（19）： 5726.
235	ZOWE J， KOČVARA M， BENDSØE M P. Free material optimization via mathematical programming［J］. Mathematical Programming， 1997， 79（1）： 445-466.
236	RODRIGUES H， GUEDES J M， BENDSOE M P. Hierarchical optimization of material and structure［J］. Structural and Multidisciplinary Optimization， 2002， 24（1）： 1-10.
237	GUEDES J M， LUBRANO E， RODRIGUES H C， et al. Hierarchical optimization of material and structure for thermal transient problems［J］. Solid Mechanics and its Applications， 2006， 137： 527-536.
238	COELHO P G， FERNANDES P R， GUEDES J M， et al. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures［J］. Structural and Multidisciplinary Optimization， 2008， 35（2）： 107-115.
239	BARBAROSIE C， TOADER A M. Optimization of bodies with locally periodic microstructure［J］. Mechanics of Advanced Materials and Structures， 2012， 19（4）： 290-301.
240	BARBAROSIE C， TOADER A M. Optimization of bodies with locally periodic microstructure by varying the periodicity pattern［J］. Networks & Heterogeneous Media， 2014， 9（3）： 433-451.
241	XIA L A， BREITKOPF P. Concurrent topology optimization design of material and structure within FE² nonlinear multiscale analysis framework［J］. Computer Methods in Applied Mechanics and Engineering， 2014， 278： 524-542.
242	XIA L A， BREITKOPF P. Multiscale structural topology optimization with an approximate constitutive model for local material microstructure［J］. Computer Methods in Applied Mechanics and Engineering， 2015， 286： 147-167.
243	FERRER A， OLIVER J， CANTE J C， et al. Vademecum-based approach to multi-scale topological material design［J］. Advanced Modeling and Simulation in Engineering Sciences， 2016， 3（1）： 1-22.
244	FERRER A， CANTE J C， HERNÁNDEZ J A， et al. Two-scale topology optimization in computational material design： An integrated approach［J］. International Journal for Numerical Methods in Engineering， 2018， 114（3）： 232-254.
245	LIU X C， SHAPIRO V. Sample-based synthesis of two-scale structures with anisotropy［J］. Computer-Aided Design， 2017， 90： 199-209.
246	ZHU B， SKOURAS M， CHEN D S， et al. Two-scale topology optimization with microstructures［J］. ACM Transactions on Graphics， 36（5）： 164.
247	SEO M， PARK H， MIN S. Heat flux manipulation by using a single-variable formulated multi-scale topology optimization method［J］. International Communications in Heat and Mass Transfer， 2020， 118： 104873.
248	SHA W， XIAO M， ZHANG J H， et al. Robustly printable freeform thermal metamaterials［J］. Nature Communications， 2021， 12： 7228.
249	廉艳平，王潘丁，高杰，等. 金属增材制造若干关键力学问题研究进展［J］. 力学进展， 2021， 51（3）： 648-701.
	LIAN Y P， WANG P D， GAO J， et al. Fundamental mechanics problems in metal additive manufacturing： A state-of-art review［J］. Advances in Mechanics， 2021， 51（3）： 648-701 （in Chinese）.
250	ZHOU S W， LI Q. Design of graded two-phase microstructures for tailored elasticity gradients［J］. Journal of Materials Science， 2008， 43（15）： 5157-5167.
251	ZHOU S W， LI Q. Microstructural design of connective base cells for functionally graded materials［J］. Materials Letters， 2008， 62（24）： 4022-4024.
252	DURIEZ E， MORLIER J， CHARLOTTE M， et al. A well connected， locally-oriented and efficient multi-scale topology optimization （EMTO） strategy［J］. Structural and Multidisciplinary Optimization， 2021， 64（6）： 3705-3728.
253	LIU P， KANG Z， LUO Y J. Two-scale concurrent topology optimization of lattice structures with connectable microstructures［J］. Additive Manufacturing， 2020， 36： 101427.
254	RADMAN A， HUANG X， XIE Y M. Topology optimization of functionally graded cellular materials［J］. Journal of Materials Science， 2013， 48（4）： 1503-1510.
255	DU Z L， ZHOU X Y， PICELLI R， et al. Connecting microstructures for multiscale topology optimization with connectivity index constraints［J］. Journal of Mechanical Design， 2018， 140（11）： 111417.
256	GARNER E， KOLKEN H M A， WANG C C L， et al. Compatibility in microstructural optimization for additive manufacturing［J］. Additive Manufacturing， 2019， 26： 65-75.
257	SANDERS E D， PEREIRA A， PAULINO G H. Optimal and continuous multilattice embedding［J］. Science Advances， 2021， 7（16）： eabf4838.
258	NAKSHATRALA P B， TORTORELLI D A， NAKSHATRALA K B. Nonlinear structural design using multiscale topology optimization. Part I： Static formulation［J］. Computer Methods in Applied Mechanics and Engineering， 2013， 261-262： 167-176.
259	FU J J， XIA L A， GAO L A， et al. Topology optimization of periodic structures with substructuring［J］. Journal of Mechanical Design， 2019， 141（7）： 071403.
260	ZHANG Z Y， LI Y， ZHOU W E， et al. TONR： An exploration for a novel way combining neural network with topology optimization［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 386： 114083.
261	WHITE D A， ARRIGHI W J， KUDO J， et al. Multiscale topology optimization using neural network surrogate models［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 346： 1118-1135.
262	KIM C， LEE J， YOO J. Machine learning-combined topology optimization for functionary graded composite structure design［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 387： 114158.
263	GUO Y L， YING HSI J FUH， FENG L W. Multiscale topology optimisation with nonparametric microstructures using three-dimensional convolutional neural network （3D-CNN） models［J］. Virtual and Physical Prototyping， 2021， 16（3）： 306-317.
264	ZHENG L， KUMAR S， KOCHMANN D M. Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy［J］. Computer Methods in Applied Mechanics and Engineering， 2021， 383： 113894.
265	BLACK N， NAJAFI A R. Deep neural networks for parameterized homogenization in concurrent multiscale structural optimization［J］. Structural and Multidisciplinary Optimization， 2023， 66（1）： 20.
266	CHAN Y C， DA D C， WANG L W， et al. Remixing functionally graded structures： Data-driven topology optimization with multiclass shape blending［J］. Structural and Multidisciplinary Optimization， 2022， 65（5）： 135.
267	TAN R K， ZHANG N L， YE W J. A deep learning-based method for the design of microstructural materials［J］. Structural and Multidisciplinary Optimization， 2020， 61（4）： 1417-1438.
268	WANG L W， CHAN Y C， AHMED F， et al. Deep generative modeling for mechanistic-based learning and design of metamaterial systems［J］. Computer Methods in Applied Mechanics and Engineering， 2020， 372： 113377.
269	PATEL D， BIELECKI D， RAI R， et al. Improving connectivity and accelerating multiscale topology optimization using deep neural network techniques［J］. Structural and Multidisciplinary Optimization， 2022， 65（4）： 1-19.
270	CHANDRASEKHAR A， SURESH K. TOuNN： Topology optimization using neural networks［J］. Structural and Multidisciplinary Optimization， 2021， 63（3）： 1135-1149.
271	CHANDRASEKHAR A， SRIDHARA S， SURESH K. Graded multiscale topology optimization using neural networks［J］. Advances in Engineering Software， 2023， 175： 103359.
272	CHANDRASEKHAR A， MIRZENDEHDEL A， BEHANDISH M， et al. FRC-TOuNN： Topology optimization of continuous fiber reinforced composites using neural network［J］. Computer-Aided Design， 2023， 156： 103449.
273	RASTEGARZADEH S， WANG J， HUANG J D. Neural network-assisted design： A study of multiscale topology optimization with smoothly graded cellular structures［J］. Journal of Mechanical Design， 2023， 145（1）： 011701.
274	SRIDHARA S， CHANDRASEKHAR A， SURESH K. A generalized framework for microstructural optimization using neural networks［J］. Materials & Design， 2022， 223： 111213.
275	ELINGAARD M O， AAGE N， BÆRENTZEN J A， et al. De-homogenization using convolutional neural networks［J］. Computer Methods in Applied Mechanics and Engineering， 2022， 388： 114197.

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类型	结构柔度		相对误差/%
类型	均匀化	全尺度分析	相对误差/%
	4.763×10⁶	4.860×10⁶	-2.00
	6.286×10⁶	6.464×10⁶	-2.75
	4.653×10⁶	5.248×10⁶	-11.34
	8.420×10⁷	4.963×10⁹	-98.30
	6.538×10⁷	8.322×10¹³	-100.00

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多尺度结构拓扑优化设计方法综述

A review of topology optimization design methods for multi-scale structures

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