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Preface

P.1. The book series

This book constitutes the first volume of the set of books entitled “IsoGeometric Analysis (IGA) tools for optimization applications in structural mechanics”. The objective of the series is to present, in a comprehensive and detailed manner, some advanced modeling and numerical strategies, recently developed in the context of IGA, that provide strong benefits not only for direct simulations but also for the resolution of optimization problems in structural mechanics. The IGA paradigm was originally introduced by Hughes and co-workers in 2005 (Hughes et al. 2005) to reunify the fields of computer-aided design (CAD) and finite element analysis (FEA). The core idea is to resort to the same higher-order and smooth spline bases for the representation of the geometry in CAD as well as for the approximation of solutions fields in FEA. The use of such families of functions quickly made IGA highly attractive for two main reasons: first, a common geometrical model can be used by both the designers and analysts; second, an increased per-degree-of-freedom (per-DOF) accuracy can be reached in comparison to standard finite element methods (FEM). This technology is thus often seen as a high-performance computational tool.

Beyond its undeniable superior analysis properties, IGA is also highly relevant for addressing the higher-level problems of optimization (also characterized as inverse problems) that enable us to question and adapt a numerical model with regard to key features. Indeed, IGA provides a natural regularization framework for such problems since it allows us to look for the solution in more regular approximation subspaces. Following this mindset, many progresses can be reported in the current literature, with the common goal of further consolidating IGA by demonstrating its performance for optimization-type applications. This book series is meant to be part of this attempt.

P.2. The present volume

This first volume tackles design optimization that is essential to reduce the usual long trial-and-error learning process during the product development in industry. More precisely, we focus on structural shape optimization, which aims at finding the optimal shape of a given topology of a structure with respect to a certain objective, such as minimal mass and maximal rigidity. Shape optimization undoubtedly constitutes one of the most valuable applications for IGA as the latter combines high-quality geometric representations and efficient analysis capabilities into one single model. In this book, we seek to develop a full analysis-to-optimization framework that is applicable to complex structures in order to tend to real-world (industrial) applications.

From an analysis point of view, this forces us to face two very important challenges encountered in the field today, namely (i) the simple local enrichment of rigid tensor-product spline models (which may also go with trimming when introducing an arbitrary local region within a spline patch) and (ii) the efficient analysis of large multipatch structures. The methods proposed to answer these issues are quite original for the domain and are based on developments in which the French community is particularly active. More precisely, the strategies rely on domain coupling and, in particular: (i) global/local non-invasive coupling algorithms that enable us to locally enrich a global model without altering its corresponding numerical operators and (ii) parallel domain decomposition solvers that allow us to separate the computational resources between several subdomains.

In a next step, the developed direct solvers are finely integrated into the shape optimization loop for full robustness and efficiency, and a geometrical modeling, based on embedded entities, is formulated to arrive at what we may refer to as optimization-suitable models, i.e. models that are ready for natural isogeometric shape optimization regardless of their complexity. The resulting framework is very robust and generic; it merges all the attractive features of IGA (design–analysis link, numerical efficiency, natural regularization, etc.) and thus brings the possibility of exploring new types of design. The performance of the methodology is demonstrated in the context of elastic solid and thin shell structures (with the Kirchhoff–Love theory). In particular, we consider as a guiding application the optimal design of stiffened structures, which are ubiquitous in aeronautics. Today, designs of wings and fuselages contain a lot of straight and aligned stiffeners uniformly distributed. However, it seems that defining curvilinear stiffeners instead of straight ones can further improve the mechanical behavior of the overall stiffened structures. Figure P.1 highlights what could be, in the future, the aerostructures based on free-form stiffeners. The developed framework contributes to reach such innovative designs.

Figure P.1. Innovative stiffened structures with curvilinear stiffeners: (a) from Renard (2018) and (b) from Havens et al. (2011). For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

P.2.1. Intended audience

Overall, this book provides a contemporary vision of IGA by first discussing the current challenges to achieve a true CAD-analysis bridge, then it proposes original solutions to answer the related issues from an analysis point of view and eventually goes up to the shape optimization of structures, which is one of the greatest applications of IGA. This book is intended to be sufficiently self-contained so as to not require that much background in IGA. We wish this book to be accessible for computational scientists, with a good background on finite element analysis and structural mechanics, and to be addressed to people either already part of or willing to join the IGA community. Throughout the chapters, we endeavor to provide detailed information and accurate datasets so that the reader should be able to understand all the essential ideas and to reproduce the numerical experiments. In addition, an exhaustive list of relevant references is provided as it is not possible to address every topic in the full generality and completeness that it deserves. In particular, the first and leading book in IGA by Cottrell et al. (2009), as well as the research papers by the authors upon which most of the content is based (see, for example, Bouclier et al. (2016, 2017), Bouclier and Passieux (2018), Hirschler et al. (2019a, 2019b, 2019c)), obviously constitute relevant additional resources. Finally, conscious effort has been made to present material that is not in research papers and to draw relevant perspectives, which in our opinion allows us to appreciate the full potential of the developed framework.

P.2.2. Organization of the text

Following these opening remarks, this book is organized as follows: first, Chapter 1 re-introduces IGA by highlighting the opportunities and remaining issues for the analysis and shape optimization of complex structures; then, Chapters 2 and 3 are devoted to the derivation of the direct solvers, i.e. the non-invasive coupling scheme for flexible global/local simulation and the family of parallel domain decomposition solvers for efficient multipatch analysis, respectively; finally, the constructed algorithms are integrated into the shape optimization loop and complemented by a sound geometrical modeling to achieve the optimal design of innovative complex structures in Chapter 4. From here on, it should be stressed that the contributions regarding optimization concern the modeling and the involved direct resolutions rather than the optimization algorithm itself.

The first chapter serves as a prerequisite for the others. It does not include innovative developments, but is intended to provide all the necessary information on IGA by discussing in particular the current challenges related to 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