Struct Multidisc Optim (2009) 39:401–418 DOI 10.1007/s00158-008-0335-3 RESEARCH PAPER Bi-level model reduction for coupled problems Application to a 3D wing Rajan Filomeno Coelho · Piotr Breitkopf · Catherine Knopf-Lenoir · Pierre Villon Received: 3 July 2008 / Revised: 10 October 2008 / Accepted: 19 October 2008 / Published online: 11 November 2008 © Springer-Verlag 2008 Abstract In this work a methodology is proposed for the optimization of coupled problems, and applied to a 3D flexible wing. First, a computational fluid dy- namics code coupled with a structural model is run to obtain the pressures and displacements for different wing geometries controlled by the design variables. Secondly, the data are reduced by Proper Orthogonal Decomposition (POD), allowing to expand any field as a linear combination of specific modes; finally, a sur- rogate model based on Moving Least Squares (MLS) is built to express the POD coefficients directly as functions of the design variables. After the validation of this bi-level model reduction strategy, the approximate models are used for the multidisciplinary optimization of the wing. The proposed method leads to a reduction of the weight by 6.6%, and the verification of the This paper is an extended version of a study presented at the EngOpt conference held at Rio, Brazil (June 1–5, 2008). R. Filomeno Coelho (B ) Laboratoire Roberval, UTC-CNRS, UMR 6253, Centre de Recherches de Royallieu, Université de Technologie de Compiègne, BP 20529 – 60205 Compiègne, France e-mail: rajan.filomeno-coelho@utc.fr P. Breitkopf · C. Knopf-Lenoir · P. Villon Laboratoire Roberval, Université de Technologie de Compiègne, Compiègne, France e-mail: piotr.breitkopf@utc.fr C. Knopf-Lenoir e-mail: cklv@utc.fr P. Villon e-mail: pierre.villon@utc.fr solution with the accurate numerical solvers confirms that the solution is feasible. Keywords Multidisciplinary optimization (MDO) · Model reduction · Proper orthogonal decomposition (POD) · Moving least squares (MLS) · 3D wing · Fluid–structure coupling 1 Introduction In recent years, more and more applications in struc- tural engineering take different disciplines into consid- eration. Indeed, in various cases, the structural analysis is tightly coupled to one or more disciplines (typically fluid and/or thermal). On the other hand, the trend nowadays is to use independent and dedicated compu- tational codes for each discipline. In this context, the aim of multidisciplinary analysis (= MDA) is to develop mathematical and numerical methods in order to guarantee the coherence of the physical variables involved. Several approaches have been proposed in the literature to reach this goal, as the fixed-point method (Alexandrov and Lewis 2000), the minimization of the discrepancy between the coupling variables from the different disciplines (Tedford and Martins 2006) and the CASCADE method (Hulme and Bloebaum 1999). However, beside the specific features of each MDA technique, all of them require the models to exchange information interactively, which has greatly hindered the systematic use of MDA in industrial applications. Indeed, since the responses (e.g. the mass, the max- imum von Mises stress, the displacements at given nodes) are generally post-processed results computed