Isogeometric Topology Optimization

1 Introduction 11.1 Topology optimization (Top-opt) 11.2 IsoGeometric Analysis (IGA) 31.3 Isogeometric Topology Optimization (ITO) 51.3.1 Density-based ITO methods 51.3.2 Level-set-based ITO methods 91.3.3 MMC/Vs-based ITO methods 111.4 Applications of topology optimization 121.4.1 Multi-material structures 121.4.2 Stress-related designs 141.4.3 Piezoelectric structures 161.4.4 Architected materials 181.4.5 Auxetic meta-materials/composites 201.5 Implementations of topology optimization 211.6 The main focus of the current monograph 222 The Density-based ITO method 252.1 NURBS-based IGA for numerical analysis model 252.1.1 NURBS basis functions 252.1.2 Galerkin's Formulation for elastostatics 272.2 Density Distribution Function (DDF) for topology description model 282.2.1 NURBS for structural geometry 282.2.2 Density Distribution Function ( 302.2.3 Material interpolation model 332.3 The ITO formulations for two problems 342.3.1 ITO formulation for the stiffness-maximization 342.3.2 ITO formulation for compliant mechanism design 352.4 Numerical implementations 362.5 Numerical examples 372.5.1 Several numerical examples in 2D 382.5.2 Several numerical examples in 3D 442.5.3 Discussions on the smoothing mechanism 472.6 Discussions on the indispensability of the ITO method 48II2.6.1 Problems in the classic SIMP method ........................................................................................................................................ 482.6.2 The extension of the DDF ............................................................................................................................................................................ 512.6.3 Comparisons between the ITO and the FEM-based three-field SIMP...................................... 522.6.4 Numerical examples .............................................................................................................................................................................................. 532.7 Appendix for sensitivity analysis .................................................................................................................................................................... 622.8 Summary .............................................................................................................................................................................................................................................. 643 The Multi-material ITO (M-ITO) method ............................................................................................................................................................ 663.1 NURBS-based Multi-Material Interpolation (N-MMI) ................................................................................................ 663.1.1 The Field of Design Variables (DVF) ........................................................................................................................................ 663.1.2 The Field of Topology Variables (TVF) .................................................................................................................................. 663.1.3 Multi-material interpolation model .............................................................................................................................................. 673.2 Multi-material Isogeometric Topology Optimization (M-ITO) ........................................................................ 693.3 Design se