CFD Based Aeroelastic Stability Predictions Under ... - CFD4Aircraft

This approach was taken for a model wing in reference. 1 The Goland wing was usedas a test case. The main tool for calculating the flutter speed was the linear methon inthe commercial code NASTRAN. This was computationally efficient since the aerodynamicmatrices could be pre-computed and then used for all normal mode sets arising from thevarying structural parameters. The probability density functions for the flutter speedswere computed, and some critical cases identified which were then recomputed using atransonic small disturbance code in the time domain.Methods have been under development to reduce the cost of computing transonic flutterspeeds using high order CFD methods for the aerodynamics. 2 These methods provide asuitable framework for doing analysis for the effects of stuctural variability since theyoperate on the aeroelastic modes. The modes of interest can be identified and then thesemodes can be analysed for dependence on the uncertain structural parameters. There areseveral ways in which this can be done including the Monte-Carlo method, perturbationmethods and interval analysis.This purpose of this paper is to investigate the practicality of using CFD methods foruncertainty analysis. The approach is described and then results are presented for theMonte-Carlo method, perturbation methods and interval analysis methods applied to theGoland wing and a generic fighter.A full literature review will be included in the final paper.III. Aeroelastic Stability FormulationThe semi-discrete form of the coupled CFD-FEM system is written aswheredwdt= 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 (in this paper µ is altitude) which is independent of w. Anequilibrium w 0(µ) of this system 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 work a linear stability analysis is done based on thecoupled system Jacobian matrix which includes the Jacobian of the CFD residual withrespect to the CFD unknowns and the structural unknowns. The calculation of the JacobianA is most conveniently done by partitioning the matrix as2"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 details of the Jacobian calculation are given in references 3 and. 4In the current work, and as is conventional in aircraft aeroelasticity, the structure ismodelled by a small number of modes, and so the number of the fluid unknowns is farhigher than the structural unknowns. This means that the Jacobian matrix has a large,but sparse, block A ff surrounded by thin strips for A fs and A sf . As described in reference 2the stability calculation is formulated as an eigenvalue problem, focussing on eigenvaluesof the coupled system that originate from the uncoupled block A ss.Write the coupled system eigenvalue problem as"#A ff A fsp = λp (5)A sf A sswhere p and λ are the complex eigenvector and eigenvalue respectively. Partition theeigenvector asp = [p f ,p s]. T (6)2 of 13American Institute of Aeronautics and Astronautics

It can be shown 5 that the eigenvalue λ (assuming it is not an eigenvalue of A ff ) satisfiesthe nonlinear eigenvalue problemS(λ)p s = λp s (7)where S(λ) = A ss − A sf (A ff − λI) −1 A fs .The nonlinear equation 7 is solved using Newton’s method. Each iteration requires theformation of the residual and its Jacobian matrix. The formation of the correction matrix isrequired to form the Jacobian matrix. This can be achieved through 2n solutions of a linearsystem against A ff − λI, one for each column of A fs . These solutions are then multipliedagainst A sf . Now, for each value of the bifurcation parameter, there are n solutions ofthe nonlinear system in equation 7, and so the cost of forming the correction matrix ateach Newton step, for each solution and for a range of parameters becomes too high. Toovercome this the expansion(A ff − λI) −1 = A −1ff + λA −1ff A −1ff + λ 2 A −1ff A −1ff A −1ff + ..... (8)is used where λ must be small for the series to converge. Note that this restriction can beovercome by assuming that the eigenvalue we are calculating is a small change from theeigenvalue λ 0 of A ss. Then λ 0 can be used as a shift to the full system eigenvalue problemby replacing A ff by A ff − λ 0I and A ss by A ss − λ 0. This modifies the nonlinear eigenvalueproblem in equation 7 by redefining S(λ) = (A ss − λ 0I) − A sf (A ff − λ 0I − λI) −1 A fs . Now theseries approximation becomes(A ff − λ 0I − λI) −1 = (A ff − λ 0I) −1 + λ(A ff − λ 0I) −2 + λ 2 (A ff − λ 0I) −3 + ..... (9)where λ is a small change to λ 0. When the nonlinear eigenvalue problem is solved for λ,the eigenvalue of the original system is then λ 0 + λ.Now, to evaluate the influence of uncertainty in the structural model parameters, wewant to solve the aeroelastic eigenvalue problem for multiple sets of normal modes. TheNewton solution can be modified slightly to allow this to be done efficiently. First, thematrices for the series approximation to S are computed for the mean values of the structuralparameters. This matrix is then used in an approximate Newton’s method to driveconvergence. However, to account for different mode sets, the right hand side of Newton’smethod is evaluated asS(λ)p s − λp s = [A ss − λI]p s − A sf (A ff − λI) −1 A fs p s. (10)This requires one linear solve against the coefficient matrix A ff −λI for the right hand sideA fs p s. For this iteration the modified Newton’s method will not converge with quadraticconvergence, but if the mean parameter solution provides a good starting solution thenwe might still expect convergence in a small number of quasi-Newton steps. In this waythe aeroelastic eigenvalue for a modified mode set can be obtained in a small additionalnumber of linear solves once the mean parameter eigenvalue is calculated.A. Monte Carlo SimulationIV. Uncertainty FormulationIn a Monte Carlo process a large number of samples of the uncertain structural parametersθ are generated according to the assumed parameter probability density function. In thecurrent work this leads to a set of different normal mode sets which are then used forthe aeroelastic calculations. The respective response values, the real and imaginary partsof the aeroelastic eigenvalues, γ i (which denotes either the real or imaginary part of theith eigenvalue) are evaluated using the Schur method. The mean values and standarddeviation of the eigenvalues can be directly evaluated from the scatter of the computedvalues. The continuous probability density function cannot be calculated directly fromdiscrete response samples. Considering the Kernel density function 7 leads to an estimateof the probability density function from a discrete set of samples. Kernel density estimationcan be constructed as a weighted sum of Gaussian PDFs centred around these samples.3 of 13American Institute of Aeronautics and Astronautics

B. Statistical moments using theory of quadratic formsThe real and imaginary parts of eigenvalues obtained by Schur method at a fixed altitudecan be expanded about the mean value of the uncertain structural parameters θ ∈ R m asγ i = γ i(θ) + g(θ − θ) + G(θ − θ) 2 + ...... (11)whereis the first Jacobian, andis the second Jacobian. Here,g = [∂γ i/∂θ]G = [∂ 2 γ i/∂θ 2 ]G(θ − θ) 2 = Σ m i=1Σ m ∂ 2 γ ij=1 (θ j − θ j )(θ k − θ k ),∂θ j ∂θ kγ i denotes either the real or imaginary part of the eigenvalue and the partial derivativesare evaluated at the mean structural parameters.According to quadratic theory, 8 the statistical moments of γ i can be obtained asandm 1 i = γ j(θ) + 1 2 Trace(Gγ i(θ)Cov(θ, θ)) (12)m r i = r!2 gγ i (θ)T [G γi (θ)Cov(θ, θ)] r−2 Cov(θ, θ)g γi (θ) +(r − 1)!Trace(G γi (θ)Cov(θ, θ)) r (13)2if r ≥ 2. The mean ˆγ j can be obtained for r = 1 (i.e. from equation 12) and the varianceV ar(γ j) for r = 2 asV ar(γ j) = g γi (θ) T Cov(θ, θ)g γi (θ) + 1 2 Trace(Gγ i (θ)Cov(θ, θ))2 . (14)If only first order terms are retained then ˆγ j = γ j(θ) and V ar(γ j) = g γi (θ) T Cov(θ, θ)]g γi (θ).If the statistical moments are known then the probability density function can be derivedby the maximum entropy method 9,10 as1P γi (x) = √2πΨ(γj /σ exp(−(x − γ j )2) (15)j)where σ j is the standard deviation derived from the variance.C. Evaluation of Jacobian MatricesThe first and second Jacobian matrices need to be evaluated at the means structuralparameters to calculate the statistical moments. This is done in two stages. First, normalmode shapes and frequencies are calculated at perturbed values of the structural modelparameters using NASTRAN. The perturbations were chosen to allow a matching withthe sensitivies of normal mode shapes and frequencies to structural parameters alreadyavailable in NASTRAN. There are several orders of magnitude of values for the incrementsthat allow this matching.Then, the Schur method is used to evaluate the aeroelastic eigenvalues at the perturbedparameter values. This is done using the Schur matrix already computed at the meanparameter values to drive the convergence of the quasi-Newton iterations for the perturbedmode shapes and frequencies (i.e. the residual function is evaluated for the perturbedmode shapes). For the first Jacobian one nonlinear system needs to be solved for eachstructural parameter to obtain the Jacobian matrices for the real and imaginary parts ofeach aeroelastic eigenvalue. For the second Jacobian m(m − 1) nonlinear systems must besolved if there are m structural parameters.2σ 2 j4 of 13American Institute of Aeronautics and Astronautics

Figure 1. α-level strategy with 4 α-levels, for a function of two triangular fuzzy parameters (from reference 11 ).D. Interval AnalysisFuzzy finite element methods consist of the application of the α-sublevel strategy on thenumerical procedure of the deterministic aeroelastic analysis. Figure 1, taken from reference,11 illustrates the application of interval analysis for the calculation of a membershipfunction for the aerodynamic damping (represented by fuzzy output) using fuzzy propagationmethods. This figure shows the procedure for a function using two triangular fuzzyvariables with four sublevels.Although the eigenvalue bounds may be obtained from the parameter vertex solution 12but this condition is met when the relation between all uncertain parameters and outputquantity is monotonic. The eigenvalue problem of flutter analysis is nonlinear and it istherefore necessary to apply global optimisation procedure in search of maximum andminimum response values. The interval flutter analysis therefore may be expressed as,[γ i, γ i ] = [min(γ i), max(γ i)] (16)subject to (A − λI)p = 0 for θ ≤ θ ≤ θ where the index i here indicates the mode thatflutters and the under and over bars indicate the lower and upper bounds of the variablerespectively. This represents an optimisation problem that can be solved by gradient basedmethods.A. Goland WingV. ResultsThe Goland wing, shown in figure 2, has a chord of 6 feet and a span of 20 feet. It isa rectangular cantilevered wing with a 4% thick parabolic section. The structural modelfollows the description given in reference, 13 and is shown in figure 2. The CFD grid is blockstructured and uses an O-O topology. This allows points to be focussed in the tip regionwhich is most critical for the aerodynamic contribution to the aeroelastic response. Thefine grid has 236 thousand points and a coarse level was extracted from this grid, which has35 thousand points. Views of the fine grid are shown in figure 2. Four mode shapes wereretained for the aeroelastic simulation. The Schur eigenvalue formulation was evaluatedfor the Goland wing test case in reference. 2The different structural components of the wing are shown in figure 3. It is composedof upper and lower skins, three spars with caps, eleven ribs with caps and 33 posts. Theparameters which define the geometry of the structural model are the thicknesses of theskins, the areas of the spar and rib caps, the thicknesses of the spars and ribs and theareas of the posts. The mean values of these parameters are given in table 1. It should benoted that the density of the structural elements was taken to be negligible and the inertialproperties are modelled as lumped mass elements shown in figure 3(a). As a consequence5 of 13American Institute of Aeronautics and Astronautics

Summary of Geometrysurface CFD gridslice from volume CFD gridFigure 2. Views of geometry and CFD Grids for the Goland Wing.6 of 13American Institute of Aeronautics and Astronautics

the mass and stiffness properties of the wing are decoupled.Parameter Thickness, ft Parameter Area ft2Upper and lower wing skins 0.0155 Leading and trailing edge spar caps 0.0416Leading and trailing edge spars 0.0006 Centre spar cap 0.1496Centre spar 0.0889 Rib caps 0.0422Ribs 0.0347 Posts 0.0008.Table 1. Mean values of thicknesses and areas of finite element parameters for Goland wingThe sensitivity of the flutter speed to each of the structural parameters was calculatedusing the linear method available in Nastran. This was done to indicate which structuralparameters should be considered for the nonlinear analysis. The sensitivies of the aerodynamicdamping at Mach 0.5 and sea level with respect to each parameter is shown infigure 4. This shows that there are 7 parameters which are key to determining the flutterspeed, namely the ticknesses of the leading and trailing edge spars, the thicknesses of theupper and lower skins, and the areas of the leading edge, trailing edge and centre spar caps.Note that the seneitivies here are with respect to the parameters normalised by their meanvalue.The flutter response was next calculated for the mean structural parameters. This wasdone at Mach 0.5 for matched conditions. The Goland wing is very flexible and matchedcalculations at higher values of Mach number predicted flutter at every value of altitude.At Mach 0.5, an interaction between the wing first bending and torsion modes gave flutterbetween ground level and 10000 ft. Although no transonic effects are present at thisMach number the CFD based Schur method was used to allow the feasibility (from acomputational cost viewpoint) of the evaluation of the sensitivity of the flutter speed tostructural parameters to be made.The seven identified structural parameters were randomised by taking a coefficient ofvariation of 0.05 about the mean value indicated in table 1, and a set of 1000 normal modeswas generated. The Schur matrix was calculated at the mean parameter values, at a cost of64 linear solves, and this matrix was then used to drive convergence of the quasi-Newtonmethod for all the other random parameter combinations. The four aeroelastic eigenvalueswere then computed for the 1000 samples. In each case the eigenvalues converged in 3-4quasi-Newton steps, meaning that the computational cost in each case was 3-4 linear solves.The 1000 sets of 4 aeroelastic eigenvalues (both real and imaginary parts) were used toassess how many samples are needed to convergence the probability density function forthe aeroelastic eigenvalues.The aeroelastic eigenvalues are shown in figure 5 as a function of altitude for the 1000samples. It is clear that there is a significant variation in the values of the real part formodes 1 and 2 once they start to interact strongly. The imaginary part shows a similarvariation for all modes, whether interacting or not. This set of results was used to assessthe number of samples required to construct a probability density function (PDF) for thevalue of the real part of the eigenvalue. The PDF is shown in figure 6 for the first bendingmode at an altitude of 2500 ft (i.e. just after flutter onset). The plot shows that 200samples give a good approximation to the shape of the distribution, although 800 samplesis possibly needed to capture the tails.The linear perturbation method was used to predict the PDF and the comparison withthe Monte Carlo results are shown in figure 7. The distributions are in good agreement,with the perturbation results requiring around 7 linear solves compared with around 800linear solves if 200 samples are used in the Monte Carlo generated distribution.B. Generic Fighter ModelHaving demonstrated the approach to evaluating sensitivity on a model wing, a second casewill be computed to show feasibility on a realistically sized aircraft model. The intentionhere is to show that the method can scale to models of the size required for the analysisof real aircraft. The generic fighter was built on data publically available for the F-16aircraft, since this has been the subject of much interest from an aeroelastic viewpoint.7 of 13American Institute of Aeronautics and Astronautics

FE ModelSkinsSparsRibsSpar CapsRib CapsPostsFigure 3. Views of the structural model for the clean Goland wing.8 of 13American Institute of Aeronautics and Astronautics

Figure 4. Sensitivity with respect to the structural parameters of the Aeroelastic Damping at Mach 0.5 andsea level computed by NASTRAN.0.020.010Mode 1 - γMode 2 - γMode 1 (MMode 2 (Mγ+2σγ-2σ0.25Mode 1 - γ (Pb.)Mode 2 - γ (Pb.)Mode 1 (MC)Mode 2 (MC)γ - 2σγ + 2σReal Part-0.01-0.02Imaginary Part0.2-0.03-0.040.15-0.050 10000 20000Altitude [ft]real0 10000 20000 30000Altitude [ft]imaginaryFigure 5. Interaction and variation of the aeroelastic eigenvalues as a function of altitude at Mach 0.5.9 of 13American Institute of Aeronautics and Astronautics

10.810 Samples50 Samples200 Samples400 Samples600 Samples800 SamplesNormalised PDF0.60.40.200 0.005 0.01 0.015Real PartFigure 6. Probability distribution function for the aeroelastic real part of the first wing bending mode computedusing Monte Carlo simulation with varying numbers of modes at Mach 0.5 and 2500 ft.10 of 13American Institute of Aeronautics and Astronautics

1PbMC - 800 Samples0.8Normalised PDF0.60.40.200 0.005 0.01 0.015 0.02Real PartFigure 7. Comparison of Monte Carlo and first order perturbation predictions of the PDF for the aeroelasticreal part of the first wing bending mode at Mach 0,5 and 2500 ft.11 of 13American Institute of Aeronautics and Astronautics

The approach was to establish a test case which is recognisable as an aircraft, and itobviously cannot be representative of the actual behaviour of the F-16.The geometry is summarised in figure 8 and is an extension of that described in reference.2 The wing has a NACA64A204 section, and the twist distribution was chosen tomatch published experimental pressure coefficient measurements. A block structured gridwas used which has 890 thousand points and 240 blocks for the full configuration. Thefull span grid was obtained by reflecting a half span grid. The surface grid is shown infigure 8 and shows that the points are concentrated on the wing which contributes mostto the aeroelastic response. The current structural model follows the description of reference.2 An exercise is underway to match the published mode shapes and frequenciesagainst published values for the F-16.Summary of GeometryCFD surface gridFigure 8. Generic fighter geometry and surface CFD and FEM grids.VI. Results for Final PaperResults are shown in this abstract to demonstrate that a methodology has been establishedto allow the uncertainty in aeroelastic eigenvalue distributions arising from uncertainstructural parameters to be calculated when the aerodynamics is modelled using CFD. Itis the intention to significantly expand the results included in the final paper to include• a full evaluation of the PDF’s for the Goland wing modes pre and post flutter• the inclusion of a tip store in the Goland wing model to modify the behaviour to allowfor significant transonic flow effects• results for the generic fighter configurationIn addition the approaches to calculating the eigenvalue uncertainty will be expandedto include results from the interval and second order perturbation approaches.VII. AcknowledgementsThis work is funded by the European Union for the Marie Curie Excellence TeamECERTA under contract MEXT-CT-2006 042383.12 of 13American Institute of Aeronautics and Astronautics

References1 Kurdi M, Lindsley N and Beran P, Uncertainty quantification of the Goland wing’s flutterboundary, AIAA Paper 2007-6309, August 20-22.2 Badcock, K.J. and Woodgate, M.A., Prediction of Bifurcation Onset of Large Order AeroelasticModels, 49th Structural Dynamics and Materials Conference, Schaumberg, Illinois, May, 2008,AIAA-2008-1820.3 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, AIAAJournal, 42(5), 883-892, May, 2004.4 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, 2005.5 Bekas, K. and Saad, Y., Computation of Smallest Eigenvalues using Spectral Schur Complements,SIAM Journal of Scientific Computing, 27(2), 2005, 458-481.6 Saad, Y and Sosonkina, M, pARMS : A package for the parallel iterative solution of generallarge sparse linear systems user’s guide. Report UMSI2004-8, Minnesota Supercomputer Institute,University of Minnesota, Minneapolis, MN, 2004.7 Bowman, A. W., and A. Azzalini, Applied Smoothing Techniques for Data Analysis, OxfordUniversity Press,1997.8 Mathai A.M., and Provost, S.B., Quadratic Forms in Random Variables: Theory and Applications.Marcel Dekker, New York, NY, 1992.9 Kapur JN, Kesavan HK. Entropy Optimization Principles With Applications. Academic Press:San Diego, CA, 1992.10 Adhikari S. and Friswell M, Random matrix eigenvalue problems in structural dynamics.International journal for Numerical Methods in Engineering, 69, 2007, 562-591.11 Moens D, Vandepitte D. A fuzzy finite element procedure for the calculation of uncertainfrequency response functions of damped structures: Part 1 - procedure. Journal of Sound andVibration 2005; 288(3):431-462.12 Qiu Z., Wang X., Friswell M., Eigenvalue bounds of structures with uncertain but boundedparameters, Journal of Sound and Vibration 2005; 282 :297-312.13 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 of 13American Institute of Aeronautics and Astronautics