[1] R T Haftka, Z Gürdal. Elements of structural optimization. Berlin: Springer, 2012.
[2] M P Bendsøe, O Sigmund. Topology optimization: Theory, methods and applications. Springer, 2003.
[3] A G M Michell. The limits of economy of material in frame-structures. London, Edinburgh, Dublin Philosophical Magazine and Journal of Science, 1904(8): 589-597.
[4] K-T Cheng, N Olhoff. An investigation concerning optimal design of solid elastic plates. International Journal of Solids and Structures, 1981(17): 305-323.
[5] K-T Cheng, N Olhoff. Regularized formulation for optimal design of axisymmetric plates. International Journal of Solids and Structures, 1982(18): 153-169.
[6] M P Bendsøe, N Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988(71): 197-224.
[7] GIN Rozvany. A critical review of established methods of structural topology optimization. Structural and Multidisciplinary Optimization, 2009(37): 217-237.
[8] X Huang, Y-MM Xie. A further review of ESO type methods for topology optimization. Structural and Multidisciplinary Optimization, 2010(41): 671-683.
[9] O Sigmund, K Maute. Topology optimization approaches. Structural and Multidisciplinary Optimization, 2013(48): 1031-1055.
[10] van Dijk NP, K Maute, M Langelaar et al. Level-set methods for structural topology optimization: a review. Structural and Multidisciplinary Optimization, 2013(48): 437-472.
[11] J D Deaton, R V Grandhi. A survey of structural and multidisciplinary continuum topology optimization: post 2000. Structural and Multidisciplinary Optimization, 2014(49): 1-38.
[12] M Zhou, GIN Rozvany. The COC algorithm, Part Ⅱ: Topological, geometrical and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering, 1991(89): 309-336.
[13] M P Bendsøe, O Sigmund. Material interpolation schemes in topology optimization. Archive of Applied Mechanics, 1999(69): 635-654.
[14] Y M Xie, G P Steven. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993(49): 885-969.
[15] J A Sethian, A Wiegmann. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics, 2000(163): 489-528.
[16] M Y Wang, X Wang, D Guo. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003(192): 227-246.
[17] G Allaire, F Jouve, AM Toader. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004(194): 363-393.
[18] MY Wang, S Zhou. Phase field: a variational method for structural topology optimization. Computer Modeling in Engineering & Sciences, 2004(6): 547-566.
[19] A Takezawa, S Nishiwaki, M Kitamura. Shape and topology optimization based on the phase field method and sensitivity analysis. Journal of Computational Physics, 2010(229): 2697-2718.
[20] X Guo, W Zhang, W Zhong. Doing topology optimization explicitly and geometrically—a mew moving morphable components based framework. Journal of Applied Mechanics, 2014(81): 081009.
[21] X Guo, W Zhang, J Zhang et al. Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Computer Methods in Applied Mechanics and Engineering, 2016(310): 711-748.
[22] W Y Yang, W S Zhang, X Guo. Explicit structural topology optimization via Moving Morphable Voids (MMV) approach. 2016 Asian Congr. Struct. Multidiscip. Optim. Nagasaki, Japan, 2016: 98.
[23] W Zhang, W Yang, J Zhou et al. Structural topology optimization through explicit boundary evolution. Journal of Applied Mechanics, 2017: 84.
[24] H A Eschenauer, V V Kobelev, A Schumacher. Bubble method for topology and shape optimization of structures. Structural Optimization, 1994(8): 42-51.
[25] S Cai, W Zhang. An adaptive bubble method for structural shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 2020(360): 112778.
[26] A R Díaaz, N Kikuchi. Solutions to shape and topology eigenvalue optimization problems using a homogenization method. International Journal for Numerical Methods in Engineering, 1992(35): 1487-1502.
[27] J Du, N Olhoff. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Structural and Multidisciplinary Optimization, 2007(34): 91-110.
[28] J Gao, Z Luo, H Li, et al. Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites. Composite Structures, 2019(211): 401-417.
[29] L Yin, G K Ananthasuresh. Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Structural and Multidisciplinary Optimization, 2001(23): 49-62.
[30] S Chu, L Gao, M Xiao, et al. Stress-based multi-material topology optimization of compliant mechanisms. International Journal for Numerical Methods in Engineering, 2018(113): 1021-1044.
[31] G Allaire, J F ouve. Minimum stress optimal design with the level set method. Engineering Analysis with Boundary Elements, 2008(32): 909-918.
[32] S Chu, L Gao, M Xiao, et al. A new method based on adaptive volume constraint and stress penalty for stress-constrained topology optimization. Structural and Multidisciplinary Optimization, 2018(57): 1163-1185.
[33] K Long, X Wang, H Liu. Stress-constrained topology optimization of continuum structures subjected to harmonic force excitation using sequential quadratic programming. Structural and Multidisciplinary Optimization, 2019(59): 1747-1759.
[34] X Zhang, A Takezawa, Z Kang. Robust topology optimization of vibrating structures considering random diffuse regions via a phase-field method. Computer Methods in Applied Mechanics and Engineering, 2019(344): 766-797.
[35] J Zheng, Z Luo, C Jiang, et al. Robust topology optimization for concurrent design of dynamic structures under hybrid uncertainties. Mechanical Systems and Signal Processing, 2019(120): 540-559.
[36] Y Zheng, M Xiao, L Gao, et al. Robust topology optimization for periodic structures by combining sensitivity averaging with a semianalytical method. International Journal for Numerical Methods in Engineering, 2019(117): 475-497.
[37] J Gao, H Li, L Gao, et al. Topological shape optimization of 3D micro-structured materials using energy-based homogenization method. Advances in Engineering Software, 2018(116): 89-102.
[38] Y Wang, J Gao, Z Luo, et al. Level-set topology optimization for multimaterial and multifunctional mechanical metamaterials. Engineering Optimization, 2017(49): 22-42.
[39] K Long, X Du, S Xu, et al. Maximizing the effective Young's modulus of a composite material by exploiting the Poisson effect. Composite Structures, 2016(153): 593-600.
[40] L Xia, P Breitkopf. Design of materials using topology optimization and energy-based homogenization approach in Matlab. Structural and Multidisciplinary Optimization, 2015(52): 1229-1241.
[41] Y Zhang, H Li, M Xiao, et al. Concurrent topology optimization for cellular structures with nonuniform microstructures based on the kriging metamodel. Structural and Multidisciplinary Optimization, 2019(59): 1273-1299.
[42] L Xia, P Breitkopf. Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework. Computer Methods in Applied Mechanics and Engineering, 2014(278): 524-542.
[43] H Li, Z Luo, N Zhang, et al. Integrated design of cellular composites using a level-set topology optimization method. Computer Methods in Applied Mechanics and Engineering, 2016(309): 453-475.
[44] Y Wang, F Chen, M Y Wang. Concurrent design with connectable graded microstructures. Computer Methods in Applied Mechanics and Engineering, 2017(317): 84-101.
[45] H Li, Z Luo, L Gao, et al. Topology optimization for functionally graded cellular composites with metamaterials by level sets. Computer Methods in Applied Mechanics and Engineering, 2018(328): 340-364.
[46] J Gao, Z Luo, H Li, et al. Topology optimization for multiscale design of porous composites with multi-domain microstructures. Computer Methods in Applied Mechanics and Engineering, 2019(344): 451-476.
[47] J Gao, Z Luo, L Xia, et al. Concurrent topology optimization of multiscale composite structures in Matlab. Structural and Multidisciplinary Optimization, 2019(60): 2621-2651.
[48] Y Zhang, M Xiao, L Gao, et al. Multiscale topology optimization for minimizing frequency responses of cellular composites with connectable graded microstructures. Mechanical Systems and Signal Processing, 2020(135): 106369.
[49] TJR Hughes. The finite element method: linear static and dynamic finite element analysis. Courier Corporation, 2012.
[50] J A Cottrell, T J R Hughes, Y Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. 2009.
[51] T J R Hughes, JA Cottrell, Y Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005(194): 4135-4195.
[52] Y-D Seo, H-J Kim, S-K Youn. Shape optimization and its extension to topological design based on isogeometric analysis. International Journal of Solids and Structures, 2010(47): 1618-1640.
[53] Y-D Seo, H-J Kim, S-K Youn. Isogeometric topology optimization using trimmed spline surfaces. Computer Methods in Applied Mechanics and Engineering, 2010(199): 3270-3296.
[54] Y Wang, Z Wang, Z Xia, et al. Structural design optimization using isogeometric analysis: a comprehensive review. Computer Modeling in Engineering & Sciences, 2018(117): 455-507.
[55] Hongliang Liu, Xuefeng Zhu, Dixiong Yang. Research advances in isogeometric analysis-based optimum design of structure. Chinese Journal of Solid Mechanics, 2018(39): 248-267. (in Chinese)
[56] D Aíaz, S O igmund. Checkerboard patterns in layout optimization. Structural Optimization, 1995(10): 40-45.
[57] O Sigmund, J Petersson. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, 1998(16): 68-75.
[58] O Sigmund. Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 2007(33): 401-424.
[59] K Matsui, K Terada. Continuous approximation of material distribution for topology optimization. International Journal for Numerical Methods in Engineering, 2004(59): 1925-1944.
[60] S F Rahmatalla, S CC. A Q4/Q4 continuum structural topology optimization implementation. Structural and Multidisciplinary Optimization, 2004(27): 130-135.
[61] G H Paulino, C H Le. A modified Q4/Q4 element for topology optimization. Structural and Multidisciplinary Optimization, 2009(37): 255-264.
[62] Z Kang, Y Wang. Structural topology optimization based on non-local Shepard interpolation of density field. Computer Methods in Applied Mechanics and Engineering, 2011(200): 3515-3525.
[63] Z Kang, Y Wang. A nodal variable method of structural topology optimization based on Shepard interpolant. International Journal for Numerical Methods in Engineering, 2012(90): 329-342.
[64] Z Luo, N Zhang, Y Wang, et al. Topology optimization of structures using meshless density variable approximants. International Journal for Numerical Methods in Engineering, 2013(93): 443-464.
[65] F Wang, B S Lazarov, O Sigmund. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011(43): 767-784.
[66] A V Kumar, A Parthasarathy. Topology optimization using B-spline finite elements. Structural and Multidisciplinary Optimization, 2011(44): 471.
[67] B Hassani, M Khanzadi, S M Tavakkoli. An isogeometrical approach to structural topology optimization by optimality criteria. Structural and Multidisciplinary Optimization, 2012(45): 223-233.
[68] X Qian. Topology optimization in B-spline space. Computer Methods in Applied Mechanics and Engineering, 2013(265): 15-35.
[69] J Gao, L Gao, Z Luo, et al. Isogeometric topology optimization for continuum structures using density distribution function. International Journal for Numerical Methods in Engineering, 2019(119): 991-1017.
[70] Q X Lieu, J Lee. Multiresolution topology optimization using isogeometric analysis. International Journal for Numerical Methods in Engineering, 2017(112): 2025-2047.
[71] QX Lieu, J Lee. A multi-resolution approach for multi-material topology optimization based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2017(323): 272-302.
[72] R Tavakoli, S M Mohseni. Alternating active-phase algorithm for multimaterial topology optimization problems: A 115-line MATLAB implementation. Structural and Multidisciplinary Optimization, 2014(49): 621-642.
[73] Y Wang, H Xu, D Pasini. Multiscale isogeometric topology optimization for lattice materials. Computer Methods in Applied Mechanics and Engineering, 2017(316): 568-585.
[74] A H Taheri, K Suresh. An isogeometric approach to topology optimization of multi-material and functionally graded structures. International Journal for Numerical Methods in Engineering, 2017(109): 668-696.
[75] J Stegmann, E Lund. Discrete material optimization of general composite shell structures. International Journal for Numerical Methods in Engineering, 2005(62): 2009-2027.
[76] H Liu, D Yang, P Hao, et al. Isogeometric analysis based topology optimization design with global stress constraint. Computer Methods in Applied Mechanics and Engineering, 2018(342): 625-652.
[77] J Gao, Z Luo, M Xiao, et al. A NURBS-based Multi-Material Interpolation (N-MMI) for isogeometric topology optimization of structures. Applied Mathematical Modelling, 2020(81): 818-843.
[78] J Gao, H Xue, L Gao, et al. Topology optimization for auxetic metamaterials based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2019(352): 211-236.
[79] J Xu, L Gao, M Xiao, et al. Isogeometric topology optimization for rational design of ultra-lightweight architected materials. International Journal of Mechanical Sciences, 2020(166): 105103.
[80] M Xu, L Xia, S Wang, et al. An isogeometric approach to topology optimization of spatially graded hierarchical structures. Composite Structures, 2019(225): 111171.
[81] X Xie, S Wang, Y Wang et al. Truncated hierarchical B-spline-based topology optimization. Structural and Multidisciplinary Optimization, 2020(62): 83-105.
[82] Y Wang, Z Liao, M Ye, et al. An efficient isogeometric topology optimization using multilevel mesh, MGCG and local-update strategy. Advances in Engineering Software, 2020(139): 102733.
[83] G Zhao, Y J ang, W Wang, et al. T-Splines based isogeometric topology optimization with arbitrarily shaped design domains. Computer Modeling in Engineering & Sciences, 2020(123): 1033-1059.
[84] S Shojaee, M Mohamadianb, N Valizadeh. Composition of isogeometric analysis with level set method for structural topology optimization. International Journal of Optimization in Civil Engineering, 2012(2): 47-70.
[85] Y Wang, D J Benson. Isogeometric analysis for parameterized LSM-based structural topology optimization. Computational Mechanics, 2016(57): 19-35.
[86] Y Wang, D J Benson. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers in Mechanical Engineering, 2016(11): 328-343.
[87] Z Xia, Y Wang, Q Wang, et al. GPU parallel strategy for parameterized LSM-based topology optimization using isogeometric analysis. Structural and Multidisciplinary Optimization, 2017: 1-22.
[88] H Ghasemi, H S Park, T Rabczuk. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017(313): 239-258.
[89] H A Jahangiry, S M Tavakkoli. An isogeometrical approach to structural level set topology optimization. Computer Methods in Applied Mechanics and Engineering, 2017(319): 240-257.
[90] H A Jahangiry, A Jahangiri. Combination of Isogeometric analysis and level-set method in topology optimization of heat-conduction systems. Applied Thermal Engineering, 2019(161): 114134.
[91] S-W Lee, M Yoon, S Cho. Isogeometric topological shape optimization using dual evolution with boundary integral equation and level sets. Computer-Aided Design, 2017(82): 88-99.
[92] M Xu, S Wang, X Xie. Level set-based isogeometric topology optimization for maximizing fundamental eigenfrequency. Frontiers in Mechanical Engineering, 2019(14): 222-234.
[93] C Yu, Q Wang, C Mei, et al. Multiscale isogeometric topology optimization with unified structural skeleton. Computer Modeling in Engineering & Sciences, 2020(122): 779-803.
[94] S Nishi, T Yamada, K Izui, et al. Isogeometric topology optimization of anisotropic metamaterials for controlling high-frequency electromagnetic wave. International Journal for Numerical Methods in Engineering. 2020(121): 1218-1247.
[95] W Zhang, J Yuan, J Zhang, et al. A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Structural and Multidisciplinary Optimization, 2016(53): 1243-1260.
[96] W Hou, Y Gai, X Zhu, et al. Explicit isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2017(326): 694-712.
[97] X Xie, S Wang, M Xu, et al. A new isogeometric topology optimization using moving morphable components based on R-functions and collocation schemes. Computer Methods in Applied Mechanics and Engineering, 2018(339): 61-90.
[98] X Xie, S Wang, M Ye, et al. Isogeometric topology optimization based on energy penalization for symmetric structure. Frontiers in Mechanical Engineering, 2020(15): 100-122.
[99] X Xie, S Wang, M Xu, et al. A hierarchical spline based isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2020(360): 112696.
[100] W Zhang, D Li, P Kang et al. Explicit topology optimization using IGA-based moving morphable void (MMV) approach. Computer Methods in Applied Mechanics and Engineering, 2020(360): 112685.
[101] Y Gai, X Zhu, Y J Zhang, et al. Explicit isogeometric topology optimization based on moving morphable voids with closed B-spline boundary curves. Structural and Multidisciplinary Optimization, 2020(61): 963-982.
[102] B Du, Y Zhao, W Yao, et al. Multiresolution isogeometric topology optimisation using moving morphable voids. Computer Modeling in Engineering & Sciences, 2020(122): 1119-1140.
[103] L Dedè, M J Borden, T J R Hughes. Isogeometric analysis for topology optimization with a phase field model. Archives of Computational Methods in Engineering, 2012(19): 427-65.
[104] L Yin, F Zhang, X Deng, et al. Isogeometric bi-directional evolutionary structural optimization. IEEE Access, 2019(7): 91134-91145.
[105] NSS Sahithi, KNV Chandrasekhar, TM Rao. A comparative study on evolutionary algorithms to perform isogeometric topology optimisation of continuum structures using parallel computing. Journal of Aerospace Engineering & Technology, 2018(8): 51-58.
[106] X Zhao, W Zhang, T Gao, et al. A B-spline multi-parameterization method for multi-material topology optimization of thermoelastic structures. Structural and Multidisciplinary Optimization, 2020(61): 923-942.
[107] P Kang, S-K Youn. Isogeometric topology optimization of shell structures using trimmed NURBS surfaces. Finite Elements in Analysis and Design, 2016(120): 18-40.
[108] X Yu, J Zhou, H Liang, et al. Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review. Progress in Materials Science, 2018(94): 114-173.
[109] JMJ Guedes, N Kikuchi. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Computer Methods in Applied Mechanics and Engineering, 1990(83): 143-198.
[110] O Sigmund. Materials with prescribed constitutive parameters: An inverse homogenization problem. International Journal of Solids and Structures, 1994(31): 2313-2329.
[111] M Osanov, J K Guest. Topology optimization for architected materials design. Annual Review of Materials Research, 2016(46): 211-233.
[112] Z-P Wang, L H Poh, J Dirrenberger, et al. Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Computer Methods in Applied Mechanics and Engineering, 2017(323): 250-271.
[113] Z-P Wang, L H Poh. Optimal form and size characterization of planar isotropic petal-shaped auxetics with tunable effective properties using IGA. Composite Structures, 2018(201): 486-502.
[114] J K Lüdeker, O Sigmund, B Kriegesmann. Inverse homogenization using isogeometric shape optimization. Computer Methods in Applied Mechanics and Engineering, 2020(368): 113170.
[115] J Gao, M Xiao, L Gao, et al. Isogeometric topology optimization for computational design of re-entrant and chiral auxetic composites. Computer Methods in Applied Mechanics and Engineering, 2020(362): 112876.
[116] C Nguyen, X Zhuang, L Chamoin, et al. Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction. Computer Methods in Applied Mechanics and Engineering, 2020(371): 113306.
[117] W Zhang, L Zhao, T Gao, et al. Topology optimization with closed B-splines and Boolean operations. Computer Methods in Applied Mechanics and Engineering, 2017(315): 652-670.
[118] T H Nguyen, G H Paulino, J Song, et al. A computational paradigm for multiresolution topology optimization (MTOP). Structural and Multidisciplinary Optimization, 2010(41): 525-539.
[119] E Samaniego, C Anitescu, S Goswami, et al. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020(362): 112790.
[120] Y Wang, Z Liao, S Shi, et al. Data-driven structural design optimization for petal-shaped auxetics using isogeometric analysis. Computer Modeling in Engineering & Sciences, 2020(122): 433-458.
[121] J Liu, A T Gaynor, S Chen, et al. Current and future trends in topology optimization for additive manufacturing. Structural and Multidisciplinary Optimization, 2018(57): 2457-2483.
[122] L Meng, W Zhang, D Quan, et al. From topology optimization design to additive manufacturing: Today's success and tomorrow's roadmap. Archives of Computational Methods in Engineering, 2019: 1-26.
[123] D J Benson, Y Bazilevs, MC Hsu et al. Isogeometric shell analysis: The Reissner-Mindlin shell. Computer Methods in Applied Mechanics and Engineering, 2010(199): 276-289.
[124] DJ Benson, Y Bazilevs, M-C Hsu, et al. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 2011(200): 1367-1378.
[125] H Gomez, TJR Hughes, X Nogueira, et al. Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations. Computer Methods in Applied Mechanics and Engineering, 2010(199): 1828-1840.
[126] Y Bazilevs, VM Calo, TJR Hughes, et al. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Computational Mechanics, 2008(43): 3-37.
[127] H Gómez, V M Calo, Y Bazilevs, et al. Isogeometric analysis of the Cahn-Hilliard phase-field model. Computer Methods in Applied Mechanics and Engineering, 2008(197): 4333-4352.
