Une approche de l’identification en dynamique des structures combinant l’erreur en relation de comportement et le filtrage de Kalman These` present´ ee´ pour l’obtention du titre de DOCTEUR DE L’ECOLE´ POLYTECHNIQUE Specialit´ e´ : Mecanique´ par Albert Alarcon´ Cot Soutenue le 4 juin 2012 devant le jury compose´ de President´ : Claude Blanze´ Professeur, CNAM, France Rapporteurs : Laurent Champaney Professeur, ENS-Cachan, France Geert Lombaert Professeur, KU Leuven, Belgique Examinateur : Andrew W. Smyth Professeur, Columbia University, USA Directeur : Marc Bonnet Directeur de recherche, CNRS, France Encadrant : Charles Bodel Ingenieur´ de recherche, EDF, France Laboratoire de Mecanique´ des Solides LaMSID Ecole´ Polytechnique, France UMR EDF - CNRS 2832, France b a` Carole a la meva fam´ılia d Acknowledgments I would like to express my most sincere gratitude to my advisors, Marc Bonnet and Charles Bodel, not only for the quality of their guidance but also for the confidence they offered me from the very beginning and the patience and help they showed all along this period. I also owe all my gratitude to all the members of my defense committee: Claude Blanze´ for accepting being the committee chairman, Geert Lombaert and Laurent Champaney for their effort in reviewing this dissertation, their questions, comments, and useful feedback on this work. And obviously, a very special thanks to Andrew W. Smyth for having accepted to be part of my defense committee, which honors me, but also for warmly hosting me at Columbia University during a summer internship, where I had the chance to share rich and inspiring interactions with him. During this work, I spent most of my time at the Mechanics and Acoustics research department of EDF, where I always felt pleased to work. I would therefore like to sincerely thank Laurent Billet, Sebastien Caillaud and Franc¸ois Weackel, not only for having accepted me as a Ph.D student, but also for their permanent trust that made possible new, challenging, projects such as giving me the possibility work at Columbia University or establishing a cooperation with the LMT laboratory of ENS-Cachan. In the cooperation with the LMT, I had he chance to work with Frederic Ragueneau and the LMT laboratory fellows in the construction of a nontrivial test setup, and I would like to acknowledge them for their invaluable contribution. Of course, this research wouldn’t had been possible without Mathieu Corus and Jean-Philippe Argaud from EDF R&D, with whom I had discussion of priceless help from both technical and scientific point of view. I cannot forget either the nice experience I had supervising Ma¨ılys Pache during her final-year undergraduate internship. Her commitment and diligence considerably helped me to both improve this work and acquire a deeper insight of the involved techniques. And, most importantly, I want to thank, warmly, all those who have have been there for all what is not written hereafter: colleagues, new and old friends, and all the people I love. The list will be too long, but let me at least mention a few, starting from the guys from the Laboratoire de Mecanique´ des Solides of the Ecole´ Polytechnique, with a special big up to Eva and Nico, whose ability to unconditionally make me laugh goes far beyond expectations. I also truly want to thank all the colleagues from T61 without exception, for their permanent good vibes. And specially “la bande de joyeux connards” with Laurent, Thibaud, Charles, Lise, John ’Blue Eyes’, Nicolas, JC and Emeric, for having shared so many “romantiK“ moments together and inspired me day after day. The moments I spent in NYC will also last e f as an unforgettable experience and I wish to specially thank Mike, Adrian, and particularly Aude for her contagious energy. I won’t forget to broadly thank all my friends who have continuously been by my side, for their continuous encouragement and interest that helped me carry on, and all those little moments that makes a strong whole. Special thoughts go to Laura, Esteban and Paola without whom anything would be as it is today, so many ”calle treces, haches intercaladas, loquitos-por-ti, peches, penestines, guayavez, vueltas imposibles, vinitos y trasnochadas“ to regret. For sure, I would also wish to express all my gratitude to my beloved family, from Barcelona to Caen, from the youngest (2!) to the elder (99!) whose unconditional love made this thesis possible. Finally, and above all, my deepest gratitude and love goes to Carole for her never ending encourage- ment, support, patience and love during all these years. Abstract Throughout its industrial activity, and particularly in the field of structural vibrations, French electricity producer EDF faces dimensioning, monitoring and diagnosis problems. Experimental information is often combined with numerical simulations to complete the a priori knowledge of structural behavior needed to address industrial issues. Vibration expertise is thus required in a broad range of fields such as health monitoring, structural modification assessment and boundary conditions identification. This work aims to find a method to combine experimental and numerical information for model- updating purposes and thus improve their predictive power. More specifically, the problem of structures with evolutionary mechanical properties is addressed. To this end, this thesis proposes a combined use of the Error in Constitutive Relation (ECR) and Kalman filtering (KF) techniques. In structural dynamics, the ECR is an energy-based approach to solve inverse problems. ECR func- tionals measure the model error by evaluating the difference between kinematically and dynamically admissible fields using an energy norm. This technique presents interesting features such as good ability to spatially localize erroneously modeled regions, strong robustness in presence of noisy data, and good regularity properties of cost functions. On the other hand, the Kalman filtering techniques are prediction- correction algorithms for recursive system estimation. The Kalman filtering is particularly suitable for studying evolutionary systems embedding noisy data from both model and observation. The main part of this work is devoted to establish and evaluate a general-purpose identification ap- proach using ECR and KF. In order to achieve this goal, the ECR is initially used to improve the a priori knowledge of model errors. Furthermore, ECR functionals are introduced in a state-space description of the identification problem. Its resolution is performed by means of the Unscented Kalman Filter (UKF), a second-order, reduced-cost, Kalman filter. The adequacy of the ECR-UKF approach to address problems of industrial relevance is shown through different numerical examples, such as structural time-varying damage assessment of a com- plex structures, boundary conditions identification of in-operation structures and field reconstruction problems. Moreover, these examples are used to improve the performance of the ECR-UKF algorithm, particularly the introduction of algebraic constraints in the ECR-UKF algorithm and the influence of error covariance matrix design. Finally, this approach is evaluated in more complex problems such as the identification of boundary impedances from an experimental campaign and the damage assessment in a complex civil structure subjected to seismic loads. g h Contents List of Figures iii Notations vii Introduction and general overview ix EDF’s industrial need....................................... ix Considered methods........................................ xi Overview of the thesis....................................... xii I Introduction to Error in Constitutive Relation and Data Assimilation methods1 1 Identification methods and Error in Constitutive Relation3 1.1 Reference Problem.....................................3 1.2 Energy-based functionals. Introduction to the Error in Constitutive Relation.......6 1.3 Conclusions......................................... 14 2 Data Assimilation 15 2.1 Introduction......................................... 15 2.2 Concepts and classic notation in data assimilation..................... 16 2.3 Sequential and variational formalisms: Kalman filter and 4D-Var............. 17 2.3.1 Variational formalism: 4D-Var........................... 18 2.3.2 Sequential formalism: The Kalman filter...................... 20 2.4 Example of nonlinear identification by means of the Unscented KF............ 26 2.5 Conclusions......................................... 31 II Towards a combined use of Kalman filtering and Error in Constitutive Relation 33 3 A Kalman filter and ECR strategy for structural dynamics model identification 35 3.1 Purpose........................................... 35 3.2 Improving a priori knowledge with the ECR........................ 36 3.3 Introducing the ECR functionals into Kalman Filtering.................. 39 3.4 Solving the identification problem by using ECR - UKF coupled method......... 45 3.5 Numerical example of structural parameter identification................. 45 3.6 Conclusions......................................... 53 i ii CONTENTS 4 ECR and UKF for model enhancement in problems of industrial relevance 55 4.1 Damage identification through the ECR-UKF strategy for high DOF models....... 55 4.1.1 Case of evolving parameters............................ 59 4.2 Identifying incorrect modelling of boundary conditions.................. 62 4.2.1 A time-domain approach for the identification of mis-modeled boundaries.... 75 4.3 Comparison of ECR and BLUE methods for structural field