Abstract submitted to AIAA SDM conference, April 2008 - CFD4Aircraft

Prediction of Bifurcation Onset of Large OrderAeroelastic ModelsK.J. Badcock ∗ and M.A.Woodgate †Computational Fluid Dynamics Laboratory,Flight Sciences and Technology,Department of Engineering,University of Liverpool,L69 3BX, United Kingdom.I. IntroductionThe system derived from coupling a Computational Fluid Dynamicis (CFD) code witha Finite Element Method (FEM) can be used in different ways to simulate the aeroelasticbehaviour of an aircraft. The advancement of the coupled system in time can give informationabout the response to an initial disturbance, and hence the system stability. 1 Thisapproach however must carry the cost of the unsteady CFD simulation and is unattractivefor this reason, particularly when a large parameter space must be investigated. An alternativeis to use the coupled system to generate response information for forced inputs.The response information is then used to setup a data driven model. 2 The data drivenmodel is then used to investigate the system response at much lower cost than the fullorder system. However the data driven models are only applicable within the support ofthe data used to generate them. They are often formulated to be linearisations about amean state, which will more than likely depend on the system parameters as well. If thisis taken into account then the calculations required to generate the model can be quiteextensive, driving up the total cost of the method.An alternative is to stick with the full order system but to exploit some property of thedynamics to reduce the computational cost. One example is the Harmonic Balance methodwhich assumes that the unstable response of the system is periodic. 3 This periodicity isused to reduce the cost of calculations to the equivalent of a few steady state calculations.The approach used in the current paper is to view the problem of computing flutteronset as a stability problem for a steady state of the coupled system. Assuming a HopfBifurcation, stability is lost when the system Jacobian matrix has a pair of eigenvalues whichcross the imaginary axis. The idea is to compute when this happens at a computationalcost much less than time domain analysis.This was first attempted in 4 where a symmetric aerofoil free to move in pitch andplunge was investigated. The coupled system of equations was augmented by an equationfor a purely imaginary eigenvalue. The augmented system was solved using Newton’smethod. Approximations were made to make the direct solution of the linear system forthe updates more tractable. Unfortunately these approximations degraded the convergenceand robustness of the method.These problems were overcome in 5 where an iterative linear solver allowed the unsimplifiedNewton’s method to be used. The method was extended to three dimensional casesin. 6 A problem with the augmented system is that the Newton iterations require the secondJacobian in the coefficient matrix. It is unfeasible to form this matrix explicitly. For∗ Professor, corresponding author. Tel. : +44(0)151 794 4847 , Fax. : +44(0)151 794 6841, Email :K.J.Badcock@liverpool.ac.uk, member AIAA† Research Assistant1 of 9American Institute of Aeronautics and Astronautics

symmetric problems like those tackled in 4–6 the calculation of the steady state is independentof the dynamic pressure and the equilibrium can be calculated separately from thebehaviour of the eigensystem. In this case the second Jacobian is not needed. However,in the general case the steady state and the eigensystem must be calculated in a coupledmanner.To overcome this problem, the classical shifted inverse power method (IPM) was adaptedto track eigenvalues as the dynamic pressure changes. 7 This method solves a system verysimilar to the augmented system, but the steady state can be updated for each value ofthe dynamic pressure before the eigenvalue is calculated. The output of the method is alocus of the aeroelastic system eigenvalues. In a manner identical to classical linear flutteranalysis, the frequency and damping values of the normal modes can be tracked, and thevalue of dynamic pressure where the damping is lost for any mode located.There were two limitations with the method presented in reference. 7 First, a parallelimplementation was needed to deal with realistically sized models. Secondly, the modetracking was not sufficiently robust. These problems are addressed in the current paperwhere a reformulation of the IPM is given, and its parallel implementation described. Theperformance of the method is illustrated on several test cases, including the Goland wing,the buzz onset of an SST configuration and the flutter onset of a fighter trainer.II. FormulationThe semi-discrete form of the coupled CFD-FEM system iswheredwdt= R(w, µ) (1)w = [w f ,w s] T (2)is a vector containing the fluid unknowns w f and the structural unknowns w s, andR = [R f ,R s] T (3)is a vector containing the fluid residual R f and the structural residual R s. The residualalso depends on a parameter µ which is independent of w. An equilibrium of this systemw 0(µ) satisfies R(w 0, µ) = 0.The stability of equilibria of equation 1 is determined by eigenvalues of the Jacobianmatrix A = ∂R/∂w. In the current paper stability is lost through a Hopf Bifurcation wherea pair of eigenvalues of A crosses the imaginary axis.The calculation of the Jacobian A is most conveniently done by partitioning the matrixas2"A = 4∂R f∂w f∂R s∂w f3∂R f∂w s∂R s∂w s5 =#A ff A fsA sf A ss. (4)The contribution A ff is a large sparse matrix derived by differentiating the CFD residual(built up from fluxes calculated using Osher’s method) with respect to the CFD cell values.This non-trivial task has been done analytically. The term A fs arises from the dependenceof the CFD residual on the mesh motion and speeds. These terms are currently evaluatedfrom finite differences. The term A sf is due to the dependence of the generalised forces onthe surface pressures. Finally, A ss is the Jacobian of the structural equations with respectto the structural unknowns.In the current work the structure is modelled by a small number of modes, and so thedimension of the CFD unknowns is far higher. This means that the Jacobian matrix has avery large, but sparse, block A ff surrounded by thin strips for A fs and A sf .The Power Method 8 is an algorithm for calculating the dominant eigenvalue/eigenvectorpair of any given diagonalizable matrix A. Its extension to the shifted inverse powermethod is practical for finding any eigenvalue provided that a good initial approximation isknown. Assume that the n × n matrix A has distinct eigenvalues ω 1, ω 2, . . . , ω n and considerthe eigenvalue ω j. Then a constant λ can be chosen so that 1/(ω j − λ) is the dominanteigenvalue of the matrix (A − λI) −1 . The shifted inverse power method (IPM) uses aniteration to converge to this eigenvalue.Variations on this approach are possible and are formulated in detail in reference. 9However, a disadvantage of the IPM is that it converges to the eigenvalue closest to the shift2 of 9American Institute of Aeronautics and Astronautics

chosen. This can cause problems in tracking modes as the system parameter is changed.Also, if two eigenvalues are close together in the complex plane then the convergence ofthe IPM is reduced.For these reasons a reformulation of the method was made. Let us assume that thesystem parameter is dynamic pressure, although other choices are possible within theframework presented. For low values of the dynamic pressure the important eigenvalues ofA will approach the undamped frequencies of the normal modes of the structure. Therefore,the IPM can be used, with a very good guess at the shift, to calculate initial values for ωand its corresponding eigenvector p. These can be calculated corresponding to each normalmode. Next, the values of ω can be tracked for increasing values of the dynamic pressureby solving the system"#(A − Iω)pR N(ω,p) == 0. (5)q T s p − iThis system, which we will refer to as the Newton Eigenvalue (NEV) system is solved atthe current value of dynamic pressure (updating the Jacobian matrix A to have the correctstatic solution at this parameter value) by Newton’s method. The values of ω and p areused as the initial guess and encourage Newton’s method to converge to the right solution.This is how the tracking problem of the IPM is resolved. The second row in the residualof equation 5 is a scaling to remove the non-uniqueness coming from the eigenvector. Thevector q s is a constant vector.III. Parallel ImplementationThe system R N(ω,p) = 0 is solved using Newton’s method with one step written as"# " # "#(A − Iω) −p δp (A − Iω)p= −. (6)q T s 0 δω q T s p − iThis equation is a large sparse linear system written in complex variables.One of the aims of the current work is to develop a parallel version of the method. Thedecomposition of the data between processors is illustrated in figure 1. 10 The example in thefigure is for 3 processors but the generalisation to more is clear. The same decompositionapplied to the vector w is used for all vectors, including the real and imaginary parts of p.The two noteworthy points from the figure are that first, the components corresponding tothe CFD part of the vectors are assigned grid block by grid block to processors. Secondly,all of the structural unknowns are kept on each processor. This leads to the storage ofthe Jacobian matrix A as indicated in the figure. The formation of A requires halo cellinformation to be communicated for block faces which border on blocks stored on otherprocessors. This communication is standard within a CFD code.The sparse linear system is solved using the Generalized Conjugate Residual (GCR)algorithm. 11 The crucial aspect of the solution is the preconditioning. The options areformulation of this is discussed more fully in. 10 In summary, we use the Block IncompleteUpper Lower (BILU) factorisation of the matrix"#(A ∗ − Iω) 0. (7)0 Iwhere A ∗ is the Jacobian matrix derived when using the first order spatial discretisation forthe CFD residual. Note that all the terms in equation 6 are included for the matrix-vectorproducts, and so the converged solution of the CGR iterations will not be polluted byapproximations. The approximations made in the preconditioning will only influence therate of convergence. In addition, the matrix A ∗ is made local on each processor by droppingterms which couple it to blocks on other processors. This means that communication duringthe CGR iterations is only from the matrix-vector products and the inner products.A. OverviewIV. ResultsThe main purpose of the current paper is to demonstrate that the NEV method can performrobustly for mode tracking and can be used in parallel for systems of large dimension.3 of 9American Institute of Aeronautics and Astronautics

Figure 1. Schematic of the Data decomposition, for a 3 processor calculation, of the Jacobian and right handside vector 10This section presents preliminary results for three test cases. The final paper will presentcomprehensive results for these cases, and the intention is to include an aircraft case inaddition.B. Goland WingThe Goland wing, shown in figure 2, has a chord of 1.8288m and a span of 6.096m. It isa rectangular cantilevered wing with a 4% thick parabolic section. The structural modelfollows the description given in reference. 12 The case used here has a tip store in thestructural model, but not in the aerodynamic model. This case is interesting because itleads to a bucket of shock induced limit cycle oscillation for a freestream Mach number ofaround 0.92. This was investigated in, 7 which also included a review of previous studiesand the prediction of the LCO amplitude using a model reduction based on the criticaleigenvector.In the current section the interest is on solving for the instability onset for CFD grids ofa larger size than was previously possible. The grid used in reference 7 had 27,000 points,with most concentrated in the wing tip region. In the current paper a finer grid of 191kpoints is used.Calculations were made for the previous coarse grid and the new fine grid. Four modesare retained in the structural model. The calculations were unmatched to allow comparisonwith previous results. The density was fixed at the sea level value, the Mach number wasfixed and the freestream velocity was increased from 200 ft/sec to 1100 ft/sec in steps of 100ft/sec. The density and velocity are specified in an input file and can be made consistentwith the chosen Mach number to allow matched calculations.The IPM was used with the normal mode frequency as a shift at the lowest freestreamvelocity. Then, the NEV method was used at each value of the velocity to trace the valueof the eigenvalue. This was done for each of the four normal modes on the coarse grid toidentify the critical mode at each Mach number. To calculate 40 eigenvalues on a single 3GHz P4 processor took about 15 minutes. With the information about the likely criticalmode, this mode was tracked on the fine grid. The calculation this time was made on 8processors. The tracking of the eigenvalues is shown for 2 Mach numbers in figure 3. Theactual flutter speed is different on the fine grid but the motion of the eigenvalues in thecomplex plane is similar at each Mach number.Finally, the summary of the results from the range of Mach numbers computed is shownin figure 4. The dip into the LCO bucket around Mach 0.92 is clear. The flutter speeds at4 of 9American Institute of Aeronautics and Astronautics

Figure 2. Torsion mode of Goland wing.Mach 0.85 Mach 0.92Figure 3. Root locus at 2 Mach numbers for the Goland wing. The red symbols and lines are on the coarsegrid, the blue symbols and lines are on the fine grid.5 of 9American Institute of Aeronautics and Astronautics

nine Mach numbers were computed without problems with the mode tracking.Figure 4. Flutter boundary for the Goland wing. The unstable mode is annotated.C. SST Control Surface BuzzThe second test case involves the prediction of the Mach number for buzz onset for a trailingedge control surface on an SST configuration. A shock wave moves onto the control surfaceto cause the buzz.The SST arrow wing, shown in figure 5, is a cranked double delta with a root chord of2.103 metres. A half model was used in experiments with a semi-span of 1 metre. Thesection profile is a NACA 0003. The inboard delta has a sweep angle of 72.8 ◦ and theoutboard a sweep of 51.6 ◦ . The trailing edge flap starts at 20% half-span and terminatesat 50% half-span. The flap chord is 0.11 metres. Experiments were made for forced flapmotions, 13 including measurements of the wing elastic response. These forced cases werecomputed in reference 14 where a blended control surface mesh treatment was evaluated.Time domain calculations of buzz were reported in. 15The structural model featured the first wing bending (10.4Hz) and torsion (44.24Hz)modes and the flap rotation (16.2Hz) mode. A half model is used for the calculations andthe block structured grid, with spacings suitable for Euler calculations, has 234k points.The approach for this case was to trace the flap rotation mode for increasing freestreamMach number. Twenty-four processors were used for the calculation, and the steady statewas fully converged in around 5 minutes on average. The eigenvalue corresponding tothe flap mode was again computed using the IPM to generate an initial guess, and thenthe NEV method. This time the calculations were matched at each Mach number. Thecalculation of the eigenvalue converged in a similar wall clock time to the steady statecalculation. The real and imaginary parts are shown in figure 6 and from these the buzzonset Mach number is 0.98. This value corresponds closely with the value obtained usingtime domain analysis. 15D. Fighter Trainer Flutter CaseThe third example is based on the BAE SYSTEMS Hawk trainer. A simplified configurationwas computed using time domain analysis in reference. 1 The CFD based resultswere compared with linear potential prediction in the low subsonic regime, and then thetransonic dip was predicted. The wing-body-tail configuration is shown in figure 7 and isrecomputed here. Four structural modes are retained and the grid, relatively coarse, has6 of 9American Institute of Aeronautics and Astronautics

Figure 5. SST configuration.RealImaginaryFigure 6. Real and Imaginary parts of the flap rotation eigenvalue as a function of Mach number.7 of 9American Institute of Aeronautics and Astronautics

128k points. A finer grid, with around 1 million points, will be computed for the finalpaper.Figure 7. Hawk configuration.The method discussed for the Goland wing was used, whereby the IPM was used tocompute a starting solution for each normal mode at a low dynamic pressure, and then theNEV method was used to track the modes as the dynamic pressure was increased. Theresults at Mach 0.8 are shown in figure 8 and show that the second mode goes undamped,with the flutter mechanism being an interaction of the first wing bending and torsionmodes. Note that the predicted flutter is well outside the flight envelope, and is in goodagreement with the time domain predictions. 1RealImaginaryFigure 8. Real and Imaginary parts for the Hawk configuration at Mach 0.8.V. ConclusionsThe final paper will include detailed timings for the test cases, including calculations ona further refined grid. Parallel performance will be evaluated in details.8 of 9American Institute of Aeronautics and Astronautics

VI. AcknowledgementsThis work is funded by the European Union for the Marie Curie Excellence TeamECERTA under contract MEXT-CT-2006 042383.References1 Woodgate, M.A, Badcock, K.J., Rampurawala, A.M., Richards, B.E., Nardini, D and Henshaw,M.J., Aeroelastic Calculations for the Hawk Aircraft Using the Euler Equations, J Aircraft,40(4), 2005, 1005-1012.2 Lucia, D.J., Beran, P.S. and Silva, W.A, Reduced-order modeling: new approaches for computationalphysics, Progress in Aerospace Sciences, 40(1-2), 51-117, 2004.3 Thomas, J.P., Dowell, E.H., Hall, K.C. and Denegri, C.M., An investigation of the sensitivityof F-16 fighter flutter onset and Limit Cycle Oscillations to Uncertainties, AIAA-2006-1847,47th AIAA/ASME/ASCE/AHS/ASC Structures. Structural Dynamics and Materials Conference,Newport, Rhode Island, May, 2006.4 Morton, S.A. and Beran, P.S., Hopf-Bifurcation Analysis of Airfoil Flutter at TransonicSpeeds, J Aircraft, 36, pp 421-429, 1999.5 Badcock, K.J., M.A. Woodgate, M.A. and Richards, B.E., The Application of Sparse MatrixTechniques for the CFD based Aeroelastic Bifurcation Analysis of a Symmetric Aerofoil, AIAA J,42(5), 883-892, May, 2004.6 Badcock, K.J., Woodgate, M.A. and Richards, B.E., Direct Aeroelastic Bifurcation Analysisof a Symmetric Wing Based on the Euler Equations, Journal of Aircraft, 42(3),731-737, 2005.7 Woodgate, M.A. and Badcock, K.J., On the fast prediction of Transonic Aeroelastic Stabilityand Limit Cycles, AIAA Journal, 45(6), 2007, 1370-1381.8 Golub, G.H. and Van Loan, C.F., Matrix Computations, The John Hopkins University Press,1996. 9 Badcock, K.J., Woodgate, M.A. and Beran, P.S., CFD Based Wing Rock Stability Analysisusing Time Marching, Reduced Modelling and Hopf Bifurcation Methods, International Forum onAeroelasticity and Structural Dynamics, Munich, July, 2005.10 Badcock, K.J., Woodgate, M.A., Allen, M.R. and Beran, P.S., Wing Rock Limit Cycle OscillationPrediction Based on Computational Fluid Dynamics, to appear in AIAA Journal, 2007.11 Eisenstat, S.C., Elman, H.C. and Schultz, M., Variational Iterative Methods for NonsymmetricSystems of Linear Equations, SIAM Journal of Numerical Analysis, Vol. 20, No. 2, April 1983,345-357.12 Beran, P.S., Knot, N.S., Eastep, F.E., Synder, R.D. and Zweber, J.V., Numerical Analysis ofStore-Induced Limit Cycle Oscillarion, Journal of Aircraft, 41(6), 1315-1326, 2004.13 Matsushita, H., Tamayama, M., Saitoh, K. and Nakamichi, J, NAL SST arrow wing withoscillating flap. In Verification an Validation Data for Computational Unsteady Aerodynamics, pages295–318. RTO Technical Report–26, October 2000.14 Rampurawala, A.M. and Badcock. K.J., Evaluation of a Simplified Grid Treatment for OscillatingTrailing-Edge Control Surfaces, Journal of Aircraft, 44(4), 1177-1188, 2007.15 Rampurawala. A.M., Aeroelastic Analysis of Aircraft with Control Surfaces using CFD, PhDThesis, University of Glasgow, Glasgow, U.K., 2006.9 of 9American Institute of Aeronautics and Astronautics