Jie Gao is a faculty of Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China. He received PhD degree from joint program of HUST and University of Technology Sydney, Australia in 2019. He is the guest editor of the special issue of the International Journal of Symmetry on "New Advance of Methods and Applications in Topology Optimization and Symmetry" in 2021-2022. He is a member of International Society for Structural and Multidisciplinary Optimization (ISSMO), a member of Mechanics Society of Hubei Province and also a member of Mechanics Society of China. His main research area is the topology optimization, mainly including the developments of topology optimization methods and applications in many design problems of multiscale composite structures, architected materials, auxetic metamaterials, etc. Over the last 5 years, he published nearly 20 SCI-indexed journal papers.

Liang Gao is a Professor at the School of Mechanical Science and Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China. He is an Associate Director of the State Key Laboratory of Digital Manufacturing Equipment and Technology (DMET). He is a member of the Operations Research Society of China, Artificial Intelligence Society of Hubei, Operations Research Society of Hubei and Mechanical Engineering Society of Hubei. He was also the guest co-editor of the special issues of International Journal of Advancements in Computing Technology on "Particle Swarm Optimization and Applications" in 2011 and International Journal of Advanced Manufacturing Technology on "Process Planning and Production Scheduling in Sustainable Manufacturing" in 2012. He was a referee for the International Journal of Production Research, International Journal of Computer Integrated Manufacturing and other related international journals. His main research areas are the modern optimization method and its applications in mechanical design and manufacturing. Over the last 5 years, he has published 2 books and 53 papers in respected journals. He has received a number of awards, including the Ministry of Education Natural Science Award first prize in 2013 and the Chinese Mechanical Engineering Society Youth Science and Technology Achievement Award in 2013. He was selected for the Program for New Century Excellent Talents in University by the Ministry of Education in 2008.

Mi Xiao received his PhD degree in Industrial Engineering from Huazhong University of Science and Technology (HUST), Wuhan, China in 2012. He is currently an Associate Professor in School of Mechanical Science and Engineering, HUST. He is an ASME/IEEE member. He was the guest co-editor of the special issue of International Journal of Materials and Product Technology on "New Advances on Topology Optimization" in 2019, and is currently an Associate Editor of IET Collaborative Intelligent Manufacturing. His research interest is structural optimization design, mainly including topology optimization. He has published more than 70 SCI-indexed journal papers, in journals such as Nature Communications, Computer Methods in Applied Mechanics and Engineering, Reliability Engineering & System Safety, and Structural and Multidisciplinary Optimization.

1.1 Topology optimization (Top-opt) 1

1.2 IsoGeometric Analysis (IGA) 3

1.3 Isogeometric Topology Optimization (ITO) 5

1.3.1 Density-based ITO methods 5

1.3.2 Level-set-based ITO methods 9

1.3.3 MMC/Vs-based ITO methods 11

1.4 Applications of topology optimization 12

1.4.1 Multi-material structures 12

1.4.2 Stress-related designs 14

1.4.3 Piezoelectric structures 16

1.4.4 Architected materials 18

1.4.5 Auxetic meta-materials/composites 20

1.5 Implementations of topology optimization 21

1.6 The main focus of the current monograph 22

2 The Density-based ITO method 25

2.1 NURBS-based IGA for numerical analysis model 25

2.1.1 NURBS basis functions 25

2.1.2 Galerkin's Formulation for elastostatics 27

2.2 Density Distribution Function (DDF) for topology description model 28

2.2.1 NURBS for structural geometry 28

2.2.2 Density Distribution Function ( 30

2.2.3 Material interpolation model 33

2.3 The ITO formulations for two problems 34

2.3.1 ITO formulation for the stiffness-maximization 34

2.3.2 ITO formulation for compliant mechanism design 35

2.4 Numerical implementations 36

2.5 Numerical examples 37

2.5.1 Several numerical examples in 2D 38

2.5.2 Several numerical examples in 3D 44

2.5.3 Discussions on the smoothing mechanism 47

2.6 Discussions on the indispensability of the ITO method 48

II

2.6.1 Problems in the classic SIMP method ........................................................................................................................................ 48

2.6.2 The extension of the DDF ............................................................................................................................................................................ 51

2.6.3 Comparisons between the ITO and the FEM-based three-field SIMP...................................... 52

2.6.4 Numerical examples .............................................................................................................................................................................................. 53

2.7 Appendix for sensitivity analysis .................................................................................................................................................................... 62

2.8 Summary .............................................................................................................................................................................................................................................. 64

3 The Multi-material ITO (M-ITO) method ............................................................................................................................................................ 66

3.1 NURBS-based Multi-Material Interpolation (N-MMI) ................................................................................................ 66

3.1.1 The Field of Design Variables (DVF) ........................................................................................................................................ 66

3.1.2 The Field of Topology Variables (TVF) .................................................................................................................................. 66

3.1.3 Multi-material interpolation model .............................................................................................................................................. 67

3.2 Multi-material Isogeometric Topology Optimization (M-ITO) .…