Transonic aeroelastic simulation for instability searches and uncertainty analysis K.J. Badcock à , S. Timme, S. Marques, H. Khodaparast, M. Prandina, J.E. Mottershead, A. Swift, A. Da Ronch, M.A. Woodgate School of Engineering, University of Liverpool, Liverpool L69 3GH, UK article info Available online 29 June 2011 Keywords: Aeroelasticity Computational fluid dynamics Model reduction Non-deterministic analysis Limit cycle oscillation abstract In this paper the use of eigenvalue stability analysis of very large dimension aeroelastic numerical models arising from the exploitation of computational fluid dynamics is reviewed. A formulation based on a block reduction of the system Jacobian proves powerful to allow various numerical algorithms to be exploited, including frequency domain solvers, reconstruction of a term describing the fluid– structure interaction from the sparse data which incurs the main computational cost, and sampling to place the expensive samples where they are most needed. The stability formulation also allows non- deterministic analysis to be carried out very efficiently through the use of an approximate Newton solver. Finally, the system eigenvectors are exploited to produce nonlinear and parameterised reduced order models for computing limit cycle responses. The performance of the methods is illustrated with results from a number of academic and large dimension aircraft test cases. & 2011 Elsevier Ltd. All rights reserved. Contents 1. Introduction ...................................................................................................... 393 2. Formulation of fluid–structure interaction simulation ..................................................................... 393 2.1. CFD code................................................................................................... 393 2.2. Modal structural model ....................................................................................... 394 2.3. Inter-grid transformation...................................................................................... 395 2.4. Mesh movement and solution sequencing ........................................................................ 395 3. Test cases ........................................................................................................ 395 3.1. NACA 0012 ................................................................................................. 395 3.2. Isogai pitch-plunge aerofoil .................................................................................... 395 3.3. Goland wing ................................................................................................ 395 3.4. MDO wing ................................................................................................. 396 3.5. Open source fighter (OSF) ..................................................................................... 397 4. Stability calculation ................................................................................................ 397 4.1. Formulation ................................................................................................ 397 4.2. Schur eigenvalue solver ....................................................................................... 398 4.3. Frequency domain fluid solvers ................................................................................. 398 4.4. Jacobian matrices for CFD models ............................................................................... 400 4.5. Eigenvalues and augmented systems ............................................................................ 401 5. Searching for instability using a hierarchy of models ..................................................................... 402 5.1. The approximation problem ................................................................................... 402 5.2. Series and kriging approximations .............................................................................. 402 5.3. Sampling................................................................................................... 405 5.4. Results .................................................................................................... 408 Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/paerosci Progress in Aerospace Sciences 0376-0421/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.paerosci.2011.05.002 à Corresponding author. E-mail address: K.J.Badcock@liv.ac.uk (K.J. Badcock). Progress in Aerospace Sciences 47 (2011) 392–423