Isogeometric analysis (IGA) consists of using the same higher-order and smooth spline functions for the representation of geometry in Computer Aided Design as for the approximation of solution fields in Finite Element Analysis. Now, almost twenty years after its creation, substantial works are being reported in IGA, making it very competitive in scientific computing.
This book proposes to use IGA jointly with standard finite element methods (FEM), presenting IGA as a projection of FEM on a more regular reduced basis. By shedding new light on how IGA relates to FEM, we can see how IGA can be implemented on top of an FE code in order to improve the solution of problems that require more regularity. This is illustrated by using IGA with FEM in a non-invasive fashion to perform efficient and robust multiscale global/local simulations in solid mechanics. Furthermore, we show that IGA can regularize the inverse problem of FE digital image correlation in experimental mechanics.
Table of Contents
Preface ix
Chapter 1 IGA: A Projection of FEM onto a Powerful Reduced Basis 1
1.1 Introduction 1
1.2 Some necessary elements for B-spline and NURBS-based IGA 4
1.2.1 B-spline and NURBS basics 4
1.2.2 k-refinement: increasing both the polynomial degree and the regularity 8
1.2.3 The trimming concept and analysis-suitable model issue 12
1.3 The link between IGA and FEM 14
1.3.1 The Bézier extraction 14
1.3.2 The Lagrange extraction 17
1.3.3 The extraction in case of NURBS 19
1.4 Non-invasive implementation using a global bridge between IGA and FEM 20
1.4.1 The common practice 21
1.4.2 A fully non-invasive implementation scheme 23
1.5 Numerical experiments 31
1.5.1 Simple but illustrative examples 31
1.5.2 An example of non-invasive nonlinear isogeometric analysis 37
1.6 Summary and discussion 41
1.7 References 43
Chapter 2 Non-invasive Global/Local Hybrid IGA/FEM Coupling 53
2.1 Introduction 53
2.2 Origin of non-invasiveness: a need for industry 56
2.2.1 Several scales of interest 56
2.2.2 Typical coupling techniques in the industry 57
2.2.3 A non-invasive approach as a remedy 59
2.3 General formulation of the coupling and iterative solution 60
2.3.1 Governing equations 60
2.3.2 Weak form and monolithic approach 62
2.3.3 Non-invasive iterative approach 65
2.4 Interest for the local enrichment of isogeometric models 71
2.4.1 General global-IGA/local-FEM modeling 71
2.4.2 Challenges and implementation issues 73
2.5 Fully non-invasive global-IGA/local-FEM analysis 76
2.5.1 Foundation: non-invasive, non-conforming global/local FEM 76
2.5.2 Extension for the non-invasive hybrid global-IGA/local-FEM coupling 80
2.6 Summary and discussion 103
2.7 References 105
Chapter 3 Non-invasive Spline-based Regularization of FE Digital Image Correlation Problems 115
3.1 Brief introduction 115
3.2 An introduction to the general field of FE-DIC from a numerical point of view 116
3.2.1 FE-DIC: towards an intimate coupling between measurements and simulations 117
3.2.2 Formulation of DIC: a nonlinear least-squares problem 119
3.2.3 Solution of DIC: descent algorithms 124
3.2.4 Extension to stereo-DIC 133
3.2.5 Standard regularization in FE-DIC 146
3.3 Multilevel and non-invasive CAD-based shape measurement 150
3.3.1 Inspiration: structural shape optimization 150
3.3.2 The proposed multilevel geometric and non-invasive scheme 155
3.3.3 Validation through a real example 161
3.3.4 Summary and discussion 165
3.4 A spline FFD-based regularization for FE-DIC 166
3.4.1 The FFD-DIC methodology 167
3.4.2 Application for the displacement measurement of a 2D beam 177
3.4.3 Application to mesh-based shape measurement 180
3.4.4 Summary and discussion 189
3.5 References 191
Index 205
