Isogeometric Topology Optimization: Methods, Applications and Implementations (Engineering Applications of Computational Methods, 7)

✍ Scribed by Jie Gao, Liang Gao, Mi Xiao

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 230
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This book provides a systematic description about the development of Isogeometric Topology Optimization (ITO) method using the density, and then addresses the effectiveness and efficiency of the ITO method on several design problems, including multi-material structures, stress-minimization structures, piezoelectric structures and also with the uniform manufacturability, ultra-lightweight architected materials with extreme bulk/shear moduli, auxetic metamaterials and auxetic meta-composites with the NPRs behavior in microstructures. A detailed MATLAB implementation of the ITO method with an in-house code “IgaTop” is also presented.

✦ Table of Contents

Preface
Contents
Acronyms
1 Introduction
1.1 Topology Optimization (Top-Opt)
1.2 Isogeometric Analysis (IGA)
1.3 Isogeometric Topology Optimization (ITO)
1.3.1 Density-Based ITO Methods
1.3.2 Boundary-Based ITO Methods
1.4 Applications of Topology Optimization
1.4.1 Multi-material Structures
1.4.2 Stress-Related Problems
1.4.3 Piezoelectric Structures
1.4.4 Architected Materials
1.4.5 Auxetic Meta-Materials/composites
1.5 Implementations of Topology Optimization
1.6 Main Focus of the Current Monograph
2 Density-Based ITO Method
2.1 NURBS-Based IGA
2.1.1 NURBS Basis Functions
2.1.2 Galerkin’s Formulation for Elastostatics
2.2 Density Distribution Function (DDF) for Material Description Model
2.2.1 NURBS for Structural Geometry
2.2.2 Density Distribution Function (DDF)
2.2.3 Material Interpolation Model
2.3 ITO Formulation for Stiffness-Maximization
2.4 Numerical Implementations
2.5 Numerical Examples
2.5.1 Several Numerical Examples in 2D
2.5.2 Several Numerical Examples in 3D
2.6 Discussions on the Indispensability of ITO
2.6.1 Extension of the DDF
2.6.2 Comparisons Between ITO and FEM-Based Three-Field SIMP
2.6.3 Numerical Examples
2.7 Appendix for Sensitivity Analysis
2.8 Summary
3 Density-Based Multi-material ITO (M-ITO) Method
3.1 NURBS-Based Multi-material Interpolation (N-MMI)
3.1.1 Field of Design Variables (DVF)
3.1.2 Field of Topology Variables (TVF)
3.1.3 Multi-material Interpolation Model
3.2 Multi-material ITO (M-ITO)
3.3 Design Sensitivity Analysis
3.4 Numerical Examples in 2D
3.4.1 Two-Material Design
3.4.2 Three-Material Design
3.4.3 Discussions on the Stiffness-To-Mass Ratio
3.4.4 Quarter Annulus
3.5 Numerical Examples in 3D
3.6 Summary
4 ITO for Structures with Stress-Minimization
4.1 Topology Description Model
4.2 NURBS-Based IGA for Stress Computation
4.3 Induced Aggregation Formulations of p-Norm and KS
4.4 ITO for Stress-Minimization Designs
4.4.1 Stress-Minimization Design Formulation
4.4.2 Design Sensitivity Analysis
4.4.3 Numerical Implementations
4.5 Numerical Examples
4.5.1 Discussions on Aggregation Formulations of p-Norm and the Induced p-Norm
4.5.2 Discussions on Aggregation Formulations of KS and the Induced KS
4.5.3 An Inverse L-Type Structure with the Curved Design Domain
4.5.4 A MBB Beam with One Preexisting Crack Notch
4.5.5 A Half Annulus with a Square Hole
4.6 Summary
5 ITO for Piezoelectric Structures with Manufacturability
5.1 NURBS-Based IGA for Piezoelectric Materials
5.1.1 Piezoelectric Constitutive Relations
5.1.2 IGA Formulation for Piezoelectric Materials
5.2 DDF with the Erode–Dilate Operator
5.3 Piezoelectric Materials Interpolation Schemes
5.4 ITO and RITO for Piezoelectric Actuators
5.4.1 ITO Formulation Without Uniform Manufacturability
5.4.2 RITO Formulation with Uniform Manufacturability
5.5 Design Sensitivity Analysis
5.6 Numerical Examples
5.6.1 Design of Piezoelectric Actuators Using ITO
5.6.2 Design of Piezoelectric Actuators Using RITO
5.7 Summary
6 ITO for Architected Materials
6.1 Numerical Implementations of the Homogenization Using IGA
6.2 ITO for Micro-Architected Materials
6.3 Design Sensitivity Analysis
6.4 Optimality Criteria (OC)
6.5 Numerical Examples
6.5.1 2D Micro-Architected Materials
6.5.2 3D Micro-Architected Materials
6.6 Summary
7 ITO for Auxetic Metamaterials
7.1 ITO Formulation for Auxetic Metamaterials
7.2 Design Sensitivity Analysis
7.3 A Relaxed OC Method
7.4 2D Auxetic Metamaterials
7.5 Discussions of the Weight Parameter
7.6 3D Auxetic Metamaterials
7.7 Summary
8 M-ITO for Auxetic Meta-Composites
8.1 Computational Design Framework
8.2 M-ITO Formulation for Auxetic Meta-Composites
8.3 Design Sensitivity Analysis
8.4 Numerical Implementations
8.5 Numerical Examples
8.5.1 2D Auxetic Meta-Composites with Two Materials
8.5.2 2D Auxetic Meta-Composites with Three Materials
8.5.3 3D Auxetic Meta-Composites with Two Materials
8.6 Summary
9 An In-House MATLAB Code of “IgaTop” for ITO
9.1 GeomMod: Construct Geometrical Model Using NURBS
9.2 PreIGA: Preparation for IGA
9.3 BounCond: Define Dirichlet and Neumann Boundary Conditions
9.4 Initializing Control Densities and DDF at Gauss Quadrature Points
9.5 ShepFun: Define the Smoothing Mechanism
9.6 IGA to Solve Structural Responses
9.6.1 StiffEle2D
9.6.2 StiffAss2D
9.6.3 Solving
9.7 Objective Function and Sensitivity Analysis
9.8 OC: Update Design Variables and DDF
9.9 PlotData and PlotTopy: Representation of Numerical Results
9.10 Demos for Several Examples
9.11 Summary
References