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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright Page
Preface
Chapter 1. IGA: A Projection of FEM onto a Powerful Reduced Basis
1.1. Introduction
1.2. Some necessary elements for B-spline and NURBS-based IGA
1.3. The link between IGA and FEM
1.4. Non-invasive implementation using a global bridge between IGA and FEM
1.5. Numerical experiments
1.6. Summary and discussion
1.7. References
Chapter 2. Non-invasive Global/Local Hybrid IGA/FEM Coupling
2.1. Introduction
2.2. Origin of non-invasiveness: a need for industry
2.3. General formulation of the coupling and iterative solution
2.4. Interest for the local enrichment of isogeometric models
2.5. Fully non-invasive global-IGA/local-FEM analysis
2.6. Summary and discussion
2.7. References
Chapter 3. Non-invasive Spline-based Regularization of FE Digital Image Correlation Problems
3.1. Brief introduction
3.2. An introduction to the general field of FE-DIC from a numerical point of view
3.3. Multilevel and non-invasive CAD-based shape measurement
3.4. A spline FFD-based regularization for FE-DIC
3.5. References
Index
Other titles from iSTE in Numerical Methods in Engineering
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1. Quadratic univariate B-spline and Lagrange functions (plotted with...
Figure 1.2. Examples of B-spline and NURBS geometric entities (the control mes...
Figure 1.3. An example of a spline refinement procedure that maintains a highe...
Figure 1.4. Illustration of the trimming concept on a simple example of a plat...
Figure 1.5. From B-spline to Lagrange: (a) the initial B-spline based discreti...
Figure 1.6. Illustration of the extraction operators involved to go from a Lag...
Figure 1.7. General standard implementation procedure using the Lagrange extra...
Figure 1.8. Approach to link (B-spline and NURBS) IGA to standard FEM. An FE m...
Figure 1.9. The pragmatic approximation in case of a quarter circular beam. Th...
Figure 1.10. Flowchart of the non-invasive and nonlinear implementation in an ...
Figure 1.11. Isogeometric simulation of the curved beam problem: NURBS mesh co...
Figure 1.12. Convergence of the relative energy error for the 2D circular beam...
Figure 1.13. Symbolic illustration of the increased accuracy of IGA: interpret...
Figure 1.14. Description and results for the bi-material problem. The problem ...
Figure 1.15. Geometry and results for the solid horseshoe (non-invasive IGA im...
Figure 1.16. Problem description of the elastoplastic dog-bone sample in tensi...
Figure 1.17. Initial CAD model (top) and refined FE mesh (bottom) for the dog-...
Figure 1.18. Accuracy of the proposed non-invasive isogeometric implementation...
Figure 1.19. Force versus displacement curve. Comparison between Code_Aster FE...
Chapter 2
Figure 2.1. Real fuselage section subjected to a large cut: description of the...
Figure 2.2. Example of a global/local problem. The global model over subdomain...
Figure 2.3. The iterative non-invasive exchange procedure. Starting with a glo...
Figure 2.4. Three different strategies may be applied in IGA to numerically so...
Figure 2.5. Four distinct coupling situations: (a) matching meshes; (b) non-ma...
Figure 2.6. Illustration of the proposed pragmatic strategy based on a transit...
Figure 2.7. Results for the real fuselage section: (a) global horizontal displ...
Figure 2.8. Comparison between the submodeling and the iterative non-invasive ...
Figure 2.9. Illustration of the proposed procedure to build in a simple and au...
Figure 2.10. Communications between the global isogeometric and local FE model...
Figure 2.11. Global/local non-invasive analysis of the linear curved beam prob...
Figure 2.12. Convergence of the relative energy error for the linear elastic 2...
Figure 2.13. Global/local non-invasive analysis of the linear curved beam prob...
Figure 2.14. Convergence of the relative energy error for the linear elastic 2...
Figure 2.15. Non-invasive introduction of holes, cracks and frictional contact...
Figure 2.16. Solution obtained for the 2D curved beam with holes, cracks and c...
Figure 2.17. Convergence of the non-invasive global/local algorithm for the 2D...
Figure 2.18. Description of the 2D plate problem with multiple inclusions and ...
Figure 2.19. Local FE meshes considered for the plate with multiple inclusion ...
Figure 2.20. Bilinear law for the cohesive elements at the inclusion-to-matrix...
Figure 2.21. Convergence of the non-invasive global/local algorithm for the pl...
Figure 2.22. Stress distribution for the plate problem with 16 inclusions at t...
Figure 2.23. Zoom on a local model for the plate problem with 16 inclusions at...
Figure 2.24. Reaction force versus prescribed displacement at each loading ste...
Figure 2.25. 3D mechanical assembly example: description and data of the probl...
Figure 2.26. Meshes for the 3D mechanical assembly example. For a color versio...
Figure 2.27. Convergence of the non-invasive algorithm for the 3D mechanical a...
Figure 2.28. Obtained von Mises stress and deformed configuration (scale facto...
Figure 2.29. Highlighting of the compression zones (i.e. where σxx is negative...
Chapter 3
Figure 3.1. Graylevel conservation problem: find transformation ϕ (or, in othe...
Figure 3.2. Experimental setup for the 2D beam subjected to bending. Yellow ar...
Figure 3.3. Example of FE-DIC on a four-point bending test on a PMMA specimen:...
Figure 3.4. Principle of the formulation of the calibration phase in FE-SDIC. ...
Figure 3.5. Principle of the formulation of the displacement measurement phase...
Figure 3.6. Twisted specimen: shape obtained with unregularized FE-SDIC (ampli...
Figure 3.7. Bending of a 2D beam with FE-DIC and Tikhonov regularization. Evol...
Figure 3.8. Twisted plate: shape obtained using a Tikhonov regularization for ...
Figure 3.9. Shape modification of a two-element quadratic C
1
B-spline curve
Figure 3.10. Shape modification of a hemisphere (made of four quadratic C1 NUR...
Figure 3.11. Going from a coarse spline representation to a fine FE mesh witho...
Figure 3.12. Multilevel design approach: design and analysis spaces describe t...
Figure 3.13. Principle of the proposed geometric regularization: a multilevel ...
Figure 3.14. Overview of the different transformations enabling to communicate...
Figure 3.15. Overview of the proposed regularization scheme included in the fu...
Figure 3.16. Initial CAD parametrization, multilevel NURBS meshes and final fi...
Figure 3.17. Twisted specimen with the proposed geometric regularization (def....
Figure 3.18. Convergence (versus the number of iterations) of the relative res...
Figure 3.19. Convergence speed (versus the dimensionless time) for the monosca...
Figure 3.20. Deformation of an FE mesh using FFD. The blue dots are the contro...
Figure 3.21. Non-influential points in a 2D plate with a hole. Plate of size 6...
Figure 3.22. Bending beam problem: gray and blue meshes represent, respectivel...
Figure 3.23. Zoom on defects that are not part of the displacement field. The ...
Figure 3.24. Measured strain on the bending beam. First line: no regularizatio...
Figure 3.25. Application of the FFD approach to the twisted plate example (def...
Figure 3.26. Spherical cap, considered FE mesh (in yellow), and visible parts ...
Figure 3.27. FE mesh embedded in the FFD morphing box at the finest scale, wit...
Figure 3.28. Shape measurement on the spherical cap. The spherical cap is seen...
Figure 3.29. Symbolic illustration of the measured shape correction
Figure 3.30. Shape measurement obtained for the spherical cap when starting wi...
List of Tables
Chapter 3
Table 3.1. Comparison of the sphere shape measurement with our method and with the laser scan
Guide
Cover
Table of Contents
Title Page
Copyright Page
Preface
Begin Reading
Index
Other titles from iSTE in Numerical Methods in Engineering
End User License Agreement
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Isogeometric Analysis Tools for Optimization Applications in Structural Mechanics Set
coordinated byPiotr Breitkopf
Volume 2
IGA: Non-Invasive Coupling with FEM and Regularization of Digital Image Correlation Problems
Robin Bouclier
Jean-Charles Passieux
First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2023The rights of Robin Bouclier and Jean-Charles Passieux to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023935724
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-825-2
Preface
P.1. The book series
This document constitutes the second 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 has been originally introduced by Hughes and co-workers in 2005 (Hughes et al. 2005), and formalized in the book by Cottrell et al. (2009), 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; then, 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 also appears highly relevant for addressing the higher-level problems of optimization (also characterized as inverse problems), that seek a solution through a certain minimization allowing to meet, at best, given criteria. 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.
The book series is composed, to this date, of two volumes: the first one by Bouclier and Hirschler (2022) and the present one. The first book of this series provides a contemporary vision of IGA and discusses many current challenges in the field, namely the true bridging of CAD and analysis, local enrichment in IGA, the coupling of non-conforming patches and the isogeometric shape optimization of structures.
P.2. The present volume
Unlike the first volume of this series that addresses pure IGA issues, the idea in this second volume is to use IGA jointly with standard FEM to enhance the latter, in a non-invasive fashion, in situations where it is relevant. More precisely, we undertake to shed a new light on IGA in relation to FEM, by showing that it can be implemented on top of an FE code in order to improve the solution of problems requiring more regularity. We thus adopt a rather pragmatic vision in this book: the aim is to develop hybrid IGA/FEM methodologies which draw up the best of each analysis technology. Given the different targets of the two books of this series, the present volume 2 can be read independently from the first one.
More specifically, we start in this book by making a distinction on the relation between IGA and FEM from the viewpoint of the technologies (splines versus Lagrange polynomials) and the viewpoint of the resulting approximation spaces for numerical simulation (isogeometric versus FE discretization space). While spline technologies undeniably outperform Lagrangian polynomials because they offer higher regularity, the resulting IGA approximation subspace is actually included in the FE one, so that IGA can simply be interpreted as a projection of FEM onto a powerful reduced basis. This observation appears original in the field and is central in this book: it constitutes the cornerstone of the developed strategies. First, it appears possible to perform IGA from an available FE software by using the latter as a black box, thereby fostering the access of IGA in industry. As an illustration, this is performed in this book to simply incorporate isogeometric capabilities in the industrial FE code Code_Aster (CodeAster 1989–2017) developed by the EDF R&D company. In particular, such a strategy allows us to benefit from the increased per-DOF accuracy of IGA while reducing to the minimum the human time required for the program development.
Then, the book addresses two problems in which the French FE community is particularly active, namely (i) the coupling of global/local models and associated numerical codes with the aim of being non-invasive, and (ii) the field of digital image correlation and in particular FE digital image correlation which appears ubiquitous nowadays within the context of data assimilation and digital twin in mechanics of structures and of materials.
For problem (i), an automatic global-IGA/local-FEM strategy is developed to combine the efficiency of IGA for capturing global regular responses, and the robustness of FEM to compute strongly non-linear or even singular behaviors. It results in a fully non-invasive approach with respect to FEM, that improves global/local FEM in the sense that it saves many DOF for the same accuracy in global regions; and reciprocally, that enhances global/local IGA from the perspective that it allows the integration of complex local behaviors within IGA by making use of optimized FE routines specifically developed for these purposes.
Problem (ii) consists of an optimization (or inverse) problem: from a set of images taken during an experimental test, the aim is to compute a displacement field through a least-squares minimization allowing to match, at best, the different images. This problem is known to be highly ill-posed in the FEM community due to the roughness of the FE modeling that inevitably leads to wiggly irregular fields. On the contrary, IGA, with its reduced smoother space, appears as an ideal choice to solve such a problem. Benefiting from the established IGA–FEM link, and even extending it to any arbitrary FE mesh with the free-form deformation concept, we thus build two spline-based regularization schemes that are non-invasive in the sense that the implementation still relies on existing FE routines, and that they do not modify the considered FE description.
According to us, the proposed interpretation on IGA along with the success of the different studies performed in this book demonstrate that FEM and IGA may not be opposed in any case, but could be wisely combined to handle challenging and practical applications. We hope that such a book will motivate future research following this mindset.
P.3. Intended audience
This book somehow desacralizes IGA by introducing it from the FEM viewpoint, thereby highlighting its links, advantages and shortcomings with respect to FEM. It is therefore rather aimed at the standard FEM community.
Although it is slightly less exhaustive than the first volume regarding some technical details of IGA, this document is intended to be sufficiently self-contained as to be able to grasp all the concepts without 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, to be addressed especially to people willing to foresee the potential of integrating IGA capabilities in their research. Over all chapters, we endeavor to provide didactic and complete presentations to help the reader who may be unfamiliar with the treated topics to understand the essential ideas and reproduce the numerical experiments. In addition, an exhaustive list of relevant references is provided, as it is not possible to address every notion in the full generality and completeness that it deserves. In particular, the first volume of this book series (Bouclier and Hirschler 2022) may constitute an interesting additional reference for an alternative presentation of IGA, perhaps more contemporary and including more implementation details. Then, obviously, we advise the reader to consult the first and leading book in IGA by Cottrell et al. (2009) for the origins. Finally, the interested reader may also find further relevant information in the research papers by the authors upon which most of the content is based (see, e.g., Bouclier et al. 2016; Tirvaudey 2016–2019; Guinard et al. 2018; Passieux and Bouclier 2019; Colantonio et al. 2020; Chapelier et al. 2021; Lapina et al. 2023).
P.4. Organization of the text
Following these opening remarks, the document is organized as follows: Chapter 1 introduces IGA as a projection of FEM onto a regular reduced basis, and pushes forward the reasoning by proposing a non-invasive implementation of IGA in an available FE software; then, Chapter 2 develops a fully non-invasive hybrid IGA/FEM algorithm for the efficient and robust multiscale global/local simulation of structures; and finally, Chapter 3 builds two non-invasive spline regularization schemes, the first one being based on CAD and the second one on free-form deformation, for the inverse problem of FE digital image correlation.
Although not mandatory to understand the essential ideas of Chapters 2 and 3, we advise the interested reader to first read Chapter 1. Indeed, Chapter 1 offers an original vision of IGA compared to what is found in the current literature, which seems to remain very close to the first book by Cottrell et al. (2009). Furthermore, this chapter may be helpful for the reader unfamiliar with IGA and to understand the spirit of our work. Then, Chapters 2 and 3 consist of different applications, so they can be read independently according to the reader’s interest. In addition, precise introductions to pave the context are provided at the beginning of these two chapters so that they appear self-contained with respect to the literature in the associated fields. Finally, all chapters end with conclusions that summarize our most important points and motivate future research based on the proposed methodologies.
P.5. Acknowledgments
The book series has been prepared under the suggestion of Piotr Breitkopf, director of the ISTE collection Numerical Methods in Mechanics, following the thesis of Robin Bouclier (2020) entitled “Habilitation à diriger des recherches”. Many thanks to Piotr for giving us this opportunity and for helpful comments and advice concerning an initial draft of this book. We would also like to thank our collaborators for the work contained in this volume. It is the fruit of many exciting collaborations that occurred in diverse projects. In particular, the efforts of our PhD and master’s students Mateus Toniolli, Marie Tirvaudey, Guillaume Colantonio and Morgane Chapelier, as well as those of our colleagues Stéphane Guinard and Paul Oumaziz have led to many examples of this book. Your efforts are all greatly appreciated, and we look forward to many fruitful collaborations again in the future. We would also like to single out, for special acknowledgment, Michel Salaün and Jean-Noël Périé, our most favorite druids, who have been and remain very important to us not only for this book but also for our day-to-day working life in Toulouse (France).
May 2023
P.6. References
Bouclier, R. (2020). Some numerical tools based on IGA for optimization applications in strucutral mechanics. HDR Thesis, Université fédérale de Toulouse-Midi-Pyrénées.
Bouclier, R. and Hirschler, T. (2022).
IGA: Non-conforming Coupling and Shape Optimization of Complex Multipatch Structures
. ISTE Ltd., London, and John Wiley & Sons, New York.
Bouclier, R., Passieux, J.C., Salaün, M. (2016). Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture.
Computer Methods in Applied Mechanics and Engineering
, 300, 1–26.
Chapelier, M., Bouclier, R., Passieux, J.C. (2021). Free-form deformation digital image correlation (FFD-DIC): A non-invasive spline regularization for arbitrary finite element measurements.
Computer Methods in Applied Mechanics and Engineering
, 384, 113992.
CodeAster (1989–2017). Finite element CodeAster, analysis of structures and thermomechanics for studies and research. Open source [Online]. Available at:
www.code-aster.org
.
Colantonio, G., Chapelier, M., Bouclier, R., Passieux, J.C., Marenić, E. (2020). Non-invasive multilevel geometric regularization of mesh-based three-dimensional shape measurement.
International Journal for Numerical Methods in Engineering
, 121, 1877–1897.
Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y. (2009).
Isogeometric Analysis: Toward Integration of CAD and FEA
. John Wiley & Sons, Chichester.
Guinard, S., Bouclier, R., Toniolli, M., Passieux, J.C. (2018). Multiscale analysis of complex aeronautical structures using robust non-intrusive coupling.
Advanced Modeling and Simulation in Engineering Sciences
, 5(1), 1–27.
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement.
Computer Methods in Applied Mechanics and Engineering
, 194, 4135–4195.
Lapina, E., Oumaziz, P., Bouclier, R., Passieux, J.C. (2023). A fully non-invasive hybrid IGA/FEM scheme for the analysis of localized non-linear phenomena.
Computational Mechanics
, 71, 213–235.
Passieux, J.C. and Bouclier, R. (2019). Classic and inverse compositional Gauss-Newton in global DIC.
International Journal for Numerical Methods in Engineering
, 119, 453–468.
Tirvaudey, M., Bouclier, R., Passieux, J.C., Chamoin, L. (2019). Non-invasive implementation of nonlinear isogeometric analysis in an industrial FE software.
Engineering Computations
, 37, 237–261.
