IGA: Non-conforming Coupling and Shape Optimization of Complex Multipatch Structures, Volume 1

Contents

Preface ix

Chapter 1: Introduction to IGA: Key Ingredients for the Analysis and Optimization of Complex Structures 1

1.1 Brief introduction 1

1.2 Geometric modeling and simulation with splines 2

1.2.1.Parametric representationof geometries 2

1.2.2 B-spline and NURBS technologies 5

1.2.3 Design features and shape parameterization 15

1.2.4 Spline-based finite element analysis: isogeometric principle 21

1.3 Improved CAD-CAE integration for robust optimization 23

1.3.1 Returning to the original motivations behind IGA 23

1.3.2 An ideal framework for parametric shape optimization 25

1.4.The analysis-suitablemodel issue 27

1.4.1.The trimmingconcept 28

1.4.2 Non-conformingmultipatch parameterization 30

1.4.3 Imposingshape variation 33

1.5 Computation of non-conforming interfaces: a brief overview of usual weak coupling methods 35

1.5.1.Governingequations 36

1.5.2.Penalty coupling 38

1.5.3.Mortar coupling 39

1.5.4.Nitsche coupling 41

Chapter 2: Non-invasive Coupling for Flexible Global/Local IGA 45

2.1 Brief introduction 45

2.2 The standard non-invasive strategy 46

2.2.1.Origin 46

2.2.2 Non-invasive resolution of the coupling problem 47

2.3 Interest in the field of IGA 54

2.3.1 Global/local modeling in IGA 55

2.3.2.Challenges 57

2.4 A robust algorithm for non-conforming global/local IGA 59

2.4.1 Reference formulation: non-symmetric Nitsche coupling 59

2.4.2 A Nitsche-based non-invasive algorithm 64

2.4.3.Validation 72

2.5.Summaryand discussion 84

Chapter 3: Domain Decomposition Solvers for Efficient Multipatch IGA 87

3.1 Introduction 87

3.2 Benefiting from the additional Lagrange multiplier field for multipatch analysis 89

3.3 Case of multipatch Kirchhoff–Love shell analysis 91

3.3.1 Kirchhoff–Love shell formulation: basics 92

3.3.2 Formulation of the coupled problem 97

3.3.3 Preliminary results: monolithic resolution 100

3.4 On the construction of dual domain decomposition solvers 103

3.4.1 Formulation of the interface problem 104

3.4.2.Solvingthe interfaceproblem 106

3.4.3 Null space and pseudo-inverse 110

3.4.4.Preconditioning 111

3.5 Numerical investigation of the developed algorithms 115

3.5.1.Standardsolid elasticity 116

3.5.2 Heterogeneous plate bending 121

3.5.3.Scordelis–Loroof 123

3.5.4.Stiffenedpanel 125

3.6.Summaryand discussion 128

Chapter 4: Isogeometric Shape Optimization of Multipatch and Complex Structures 131

4.1 Introduction 131

4.2 Isogeometric shape optimization framework 133

4.2.1.Optimizationflowchart 133

4.2.2 Multilevel design 133

4.2.3.Design variables 135

4.2.4.Formulationandresolution 136

4.3 Unify the DD approach and multipatch optimization: towards ultimate efficiency 145

4.3.1 DD computation of the response functions 145

4.3.2 DD computation of the sensitivities 146

4.3.3.Non-designparts 147

4.3.4.Fast re-analysis 148

4.3.5.Optimizationalgorithm 149

4.4 Innovative design of multipatch structures: focus on aeronautical stiffenedpanels 149

4.4.1 Geometric modeling: embedded entities 150

4.4.2 Analysis: an embedded Kirchhoff–Love shell element 156

4.4.3.Two preliminary examples to illustrate the design capabilities 159

4.5 Application to solid structures and first interests 168

4.5.1.Simple extensionof themethod 169

4.5.2.Atest case in 2D 171

4.6 Advanced numerical optimization examples 175

4.6.1 Global shell optimization: stiffened roof 176

4.6.2.Local shell optimization: curvedwall 179

4.6.3.Designingan aircraftwing-box 185

4.7 Towards the optimal design of structural details within isogeometric patches 192

4.7.1.Asimple but instructive test case 192

4.7.2 Unify the non-invasive global/local approach and the optimizationof local details 193

4.7.3.Preliminaryresults and perspectives 197

4.8.Summaryand discussion 198

References 201

Index 229

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