the true dialog between CAD and FEA. This chapter may be helpful for the reader who is unfamiliar with IGA and helps explain the motivations of our work. Although Chapters 2 and 3 share many similarities from the conceptual point of view and both start with domain coupling, they can be read independently according to the reader’s interest. Chapter 4 can also more or less be read on its own by people involved in IGA; yet, fully skipping Chapters 2 and 3 would inherently lead to difficulties in clearly understanding the benefits of the non-invasive and domain decomposition solvers to be integrated in shape optimization loops. All chapters, except Chapter 1, start with a precise introduction to pave the context and end with a conclusion that summarizes our most important points and motivates future research based on the proposed methodology.
P.3. Acknowledgments
The book series has been prepared under the suggestion of Piotr Breitkopf, director of the ISTE series “Numerical Methods in Engineering”, following Robin Bouclier’s (2020) thesis, known as a “Habilitation à Diriger des Recherches” (HDR). Many thanks to Piotr for giving the authors this opportunity and for helpful comments and advice concerning an initial draft of this book. The authors would also like to thank their collaborators on the work contained in this volume. In particular, this volume completes many texts and results from the PhD thesis of Thibaut Hirschler (2019); thus, the authors would like to gratefully thank the colleagues from the supervision team of this PhD, starting with Thomas Elguedj and going up to Joseph Morlier without omitting Arnaud Duval. Finally, Robin Bouclier would like to single out for special acknowledgments Jean-Charles Passieux and Michel Salaün who initiated him into the field of domain coupling, specifically in the context of non-invasive global/local simulations, at the early stage of his arrival in Toulouse (France).
Robin BOUCLIER
Thibaut HIRSCHLER
January 2022
1
Introduction to IGA: Key Ingredients for the Analysis and Optimization of Complex Structures
1.1. Brief introduction
IsoGeometric analysis (IGA) was originally introduced by Hughes et al. (2005) and formalized in Cottrell et al. (2009), in order to reunify the fields of geometric modeling in computer-aided design (CAD) and numerical simulation using the finite element method (FEM). The main idea is to resort to the same bases for analysis as the ones used to describe the geometry in CAD. In this framework, the method can be viewed as a generalization of the FEM that considers smooth and higher-order functions, for example, the non-uniform-rational-B-spline (NURBS) functions (Cohen et al. 1980; Piegl and Tiller 1997; Rogers 2000; Farin 2002), to replace typical Lagrange polynomials in the computations. Some other geometric descriptions include T-splines (Bazilevs et al. 2010) and subdivision surfaces (Cirak et al. 2002). Within this work, we only use the NURBS (which constitute the most commonly used technology in CAD) and simpler B-splines. We use the spline and isogeometric terminologies indifferently to denote a NURBS and a B-spline object, respectively.
Now, about 15 years after its birth, there are substantial works in the area of IGA, which makes it a very competitive methodology for the general field of scientific computing. In this chapter, we attempt to introduce IGA by providing a contemporary vision on its interests, limitations and related challenges that still need to be faced in order to meet its full potential. The discussion is performed in accordance with the core objective of this book, which is the analysis and shape optimization of complex structures. More precisely, we insist on the opportunities offered by the underlying spline technologies to represent, modify and mechanically simulate any geometrical shape in structural mechanics. We also outline the remaining issues, and the main research paths currently followed to answer them, to achieve a true dialog between CAD and FE analysis. An effort is made to help readers unfamiliar with IGA to understand all of the necessary key points related to this method. This first chapter serves as a prerequisite for the contributions presented in the next chapters of this book.
1.2. Geometric modeling and simulation with splines
To start with, IGA is introduced from a technical viewpoint by providing the key ingredients regarding the considered spline geometric modeling techniques, namely the B-spline and NURBS variants. Particular care is taken to highlight the ability of these spline tools to describe any geometrical shape and to also control them smoothly. These aspects are of paramount importance for the works presented in this book, which address the general field of computational solid mechanics to the shape optimization of structures. Finally, with the IGA concept now being mature and relatively well known in the scientific computing community, we shortly review its basics from an analytical point of view. In this respect, we recall that the major difference, with respect to standard FEM, is to use the spline-based parameterizations of CAD to build the approximation subspaces when applying the Galerkin’s method. For further details, besides the pioneering contributions (Hughes et al. 2005; Cottrell et al. 2009), refer to the works cited hereafter.
1.2.1. Parametric representation of geometries
The spline formalism offers a natural way to represent geometries in the parametric form. Hence, let us specify the parametric representation of geometries before entering into the details of the B-spline and NURBS technologies. To begin with, we consider univariate geometric entities living in 2D physical spaces. These entities simply consist of planar curves. Describing these curves in the parametric form consists of expressing each coordinate of a point on the curve separately as an explicit function of an independent parameter (Piegl and Tiller 1997):
[1.1]
where
[1.2]
Figure 1.1. Two simple planar curves
Parameter ξ1 takes any real value to represent the complete line or is bounded to represent a segment of the line. Using the same idea, the curve of the circle can be parameterized as follows:
[1.3]
Note that at this stage the parameterization is not unique and parametric forms are obviously not the only way of representing curves. Implicit equations are also commonly used to describe curves.
