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 dynamics 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 surrogate 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 solution with the accurate numerical solvers confirms that the solution is feasible.
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Agte JS (2005) A tool for application of bi-level integrated system synthesis (BLISS) to Multidisciplinary design optimization problems. Institute of Aeroelasticity, German Aerospace Center, Bunsenstraße vol 10. Göttingen, pp 37073
Alexandrov NM, Lewis RM (1999) Comparative properties of collaborative optimization and other approaches to MDO. In: Proceedings of the first ASMO UK/ISSMO conference on engineering design optimization, Bradford, 8–9 July 1999
Alexandrov NM, Lewis RM (2000) Analytical and computational properties of distributed approaches to MDO. In: 8th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis & optimization, Long Beach, CA, USA. AIAA Paper, 6–8 September 2000, pp 2000–4718
Anderson JD (1995) Computation fluid dynamics—The basics with applications. McGraw-Hill, New York, 547 pp
Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25:539–575
Braun RD, Kroo IM (1997) Development and application of the collaborative optimization architecture in a multidisciplinary design environment. In: Multidisciplinary design optimization state of the art, SIAM series: Proceedings in applied mathematics, vol 80. SIAM, Philadelphia, pp 98–116
Breitkopf P, Rassineux A, Villon P (2002) An introduction to moving least squares meshfree methods. Rev Europ Élém Finis 11/7–8:825–867
Bui-Thanh T, Damodaran M, Willcox K (2003) Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics. In: 21th AIAA applied aerodynamics conference, AIAA Paper, Orlando, 23–26 June 2003, pp 2003–4213
Cramer EJ, Denis JE, Frank PD, Lewis RM, Shubin GR (1994) Problem formulation for multidisciplinary optimization. SIAM J Optim 4:754–776
Dulong J-L, Druesne F, Villon P (2007) A model reduction approach for real-time part deformation with nonlinear mechanical behavior. Int J Interact Des Manuf 1(4):1955–2513
Duvigneau R (2006) Adaptive parameterization using free-form deformation for aerodynamic shape optimization. INRIA Research report RR-5949, July
Filomeno Coelho R, Breitkopf P, Knopf-Lenoir C (2008) Model reduction for multidisciplinary optimization—application to a 2D wing. Struct Multidisc Optim. doi:10.1007/s00158-007-0212-5
Giassi A, Bennis F, Maisonneuve JJ (2004) Multidisciplinary design optimisation and robust design approaches applied to concurrent design. Struct Multidisc Optim 28:356–371
Giunta AA, Watson LT, Koehler J (1998) A comparison of approximation modeling techniques: polynomial versus interpolating models. In: 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis & optimization, AIAA paper, St. Louis, 2–4 September 1998, pp 1998–4758
Hulme KF, Bloebaum CL (1999) A comparison of formal and heuristic strategies for iterative convergence of a coupled multidisciplinary analysis. In: 3rd world congress on structural and multidisciplinary optimization, Buffalo, 17–21 May 1999
Kodiyalam S, Sobieszczanski-Sobieski J (1999) Bi-level integrated system synthesis with response surfaces. In: 40th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference. AIAA Paper 1999–1306-wip, St. Louis, 12–15 April 1999
LeGresley PA, Alonso JJ (2004) Improving the performance of design decomposition methods with POD. In: 10th AIAA/ISSMO multidisciplinary analysis and optimization conference. AIAA Paper, Albany, 30 August–1 September 2004, pp 2004–4465
LeGresley PA (2005) Application of proper orthogonal decomposition (POD) to design decomposition methods. PhD thesis, Department of Aeronautics and Astronautics, Stanford University
Masmoudi M, Parte Y (2006) Disciplinary interaction variable elimination (DIVE) approach for multi-disciplinary optimization. In: Wesseling P, Oñate E , Périaux J (eds) ECCOMAS CFD-2006, Egmond aan Zee, 5–8 September 2006
Massin P, Al Mikdad M (2005) Eléments de coques volumiques en non linéaire géométrique. Code_Aster User’s manual, 56 pp
Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318
Newman A (1996) Model reduction via the Karhunen-Loève expansion. Part I: An exposition. Technical report TR96–32, Inst. Systems Research, April 1996
Powel MJD (1994) A direct search optimization method that models the objective and constraint functions by linear interpolation. In: Advances in optimization and numerical analysis. Kluwer, Dordrecht, pp 51–67
Samareh JA (2004) Aerodynamic shape optimization based on free-form deformation. In: 10th AIAA/ISSMO multidisciplinary analysis and optimization conference. AIAA paper, Albany, 30 August–1 September 2004, pp 2004–4465
Schenk CA, Pradlwarter HJ, Schuller GI (2005) Non-stationary response of large, non-linear finite element systems under stochastic loading. Comput Struct 83(14):1086–1102
Simpson TW, Korte JJ, Mauery TM, Mistree F (1998) Comparison of response surface and kriging models for multidisciplinary design optimization. In: 7th AIAA/USAF/NASA/ ISSMO symposium on multidisciplinary analysis & optimization, AIAA Paper, St. Louis, 2–4 September 1998, pp 1998–4755
Sobieszczanski-Sobieski J, Agte J, Sandusky R (1998) Bi-level integrated system synthesis (BLISS). Langley Research Center, Hampton, Virginia, NASA Technical Report TM-1998-208715
Tedford N, Martins J (2006) On the common structure of MDO problems: a comparison of architectures. In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, 6–8 September 2006
Zadeh PM, Toropov VV, Wood AS (2005) Use of moving least squares method in collaborative optimization. In: 6th world congress on structural and multidisciplinary optimization, Rio de Janeiro, 30 May–3 June 2005
Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics (6th edn). Elsevier, Oxford, 631 pp
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This paper is an extended version of a study presented at the EngOpt conference held at Rio, Brazil (June 1–5, 2008).
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Filomeno Coelho, R., Breitkopf, P., Knopf-Lenoir, C. et al. Bi-level model reduction for coupled problems. Struct Multidisc Optim 39, 401–418 (2009). https://doi.org/10.1007/s00158-008-0335-3
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DOI: https://doi.org/10.1007/s00158-008-0335-3