aiaa-2009-3671

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Uncertainty Quantification in

Fluid-Structure Interaction Simulations Using a

Simplex Elements Stochastic Collocation Approach

Jeroen A.S. Witteveen, Hester Bijl

Faculty of Aerospace Engineering, Delft University of Technology,

Kluyverweg 1, 2629HS Delft, The Netherlands

An efficient uncertainty quantification method for unsteady problems is presented in

order to achieve a constant accuracy in time for a constant number of samples. The

approach is applied to the aeroelastic problems of a transonic airfoil flutter system and the

AGARD 445.6 wing benchmark with uncertainties in the flow and the structure.

I. Introduction

Numerical errors in multi-physics simulations start to reach acceptable engineering accuracy levels due tothe increasing availability of computational resources. Nowadays, uncertainties in modeling multi-scale phe-nomena in fluid-structure, fluid-thermal, and aero-acoustic interactions have a larger effect on the accuracyof computational predictions than discretization errors. It is therefore especially important in multi-physicsapplications to systematically quantify the effect of physical variations on a routine basis. Furthermore,unsteady fluid-structure interaction applications are practical aeronautical examples of dynamical systemswhich are known to amplify initial variations with time. In these problems, natural irreducible input vari-ability can trigger the earlier onset of unstable flutter behavior, which can lead to unexpected fatigue damage

and structural failure.Polynomial Chaos uncertainty quantification methods1,2,7,8,12,13,17,20,21,29 however usually result in afast increasing number of samples with time to resolve the effect of random parameters in dynamical systemswith a constant accuracy.16 Resolving the asymptotic stochastic effect, which is of practical interest in post-flutter analysis, can in these long time integration problems lead to thousands of required samples. Theincreasing number of samples is caused by the increasing nonlinearity of the response surface for increasingintegration times. This effect is especially profound in problems with oscillatory solutions in which thefrequency of the response is affected by the random parameters. The frequency differences between therealizations lead to increasing phase differences with time, which in turn result in an increasingly oscillatoryresponse surface and more required samples.

In order to enable efficient uncertainty quantification in time-dependent simulations, a special uncer-tainty quantification methodology for unsteady oscillatory problems is developed. The approach based ontime-independent parameterization of oscillatory samples achieves a constant uncertainty quantification in-

terpolation accuracy in time with a constant number of samples.22 A parameterization in terms of thetime-independent functionals frequency, relative phase, amplitude, a reference value, and the normalizedperiod shape is used for period-1 responses. The extension with a damping factor and an algorithm foridentifying higher-period shape functions is also applicable to more complex and non-periodic realizations.23

A second uncertainty quantification formulation for achieving a constant accuracy in time with a constantnumber of samples is developed based on interpolation of oscillatory samples at constant phase instead ofat constant time.24 The scaling of the samples with their phase eliminates the effect of the increasing phase

Postdoctoral researcher, Member AIAA, +31(0)15 2782046, [email protected], http://www.jeroenwitteveen.com.Full Professor, Member AIAA, +31(0)15 2785373.

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American Institute of Aeronautics and Astronautics
mailto:[email protected]:[email protected]://www.jeroenwitteveen.com/http://www.jeroenwitteveen.com/http://www.jeroenwitteveen.com/mailto:[email protected]

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differences in the response, which usually leads to the fast increasing number of samples with time. Theresulting formulation is not subject to a parameterization error and it can resolve time-dependent functionalsthat occur for example in transient behavior. It is also proven that phase-scaled uncertainty quantificationresults in a bounded error as function of the phase for periodic responses and under certain conditions alsoin a bounded error in time.26 The unsteady approaches are independent of the employed non-intrusiveuncertainty quantification to perform the actual interpolation of the oscillatory samples. Here the SimplexElements Stochastic Collocation (SESC) method based on Newton-Cotes quadrature in simplex elements isemployed.25 Since the piecewise polynomial approximation of the response of SESC satisfies the extrema

diminishing robustness criterion extended to probability space, it enables us to resolve also bifurcationphenomena reliably.26 The formulation is also extended to multi-frequency responses of continuous structuresby using a wavelet decomposition preprocessing step in order to treat the different frequency componentsseparately.28

After the introduction of the mathematical statement of the uncertainty quantification problem in sec-tion II, the efficient uncertainty quantification method for unsteady problems is presented in section III. Thedeveloped approach is applied to the unsteady fluid-structure interaction test cases of a two-dimensional air-foil flutter problem and the three-dimensional aeroelastic AGARD 445.6 wing. In section IV the stochasticpost-flutter analysis of a two-degree-of-freedom rigid airfoil in Euler flow with nonlinear structural stiffness inpitch and plunge with uncertainty in a combination of randomness in two system parameters is considered.The effect of free stream velocity fluctuation is analyzed in section V for the AGARD aeroelastic wing in thetransonic flow regime 5% below the deterministic flutter speed. Additional applications to other aeroelasticsystems are reported in.18,19,27 Finally, the conclusions are summarized in section VI.

II. Mathematical formulation of the uncertainty quantification problem

Consider a dynamical system subject to na uncorrelated second-order random input parameters a() ={a1(), . . ,ana()} A with parameter space A R

na , which governs an oscillatory response u(x, t, a)

L(x, t, a; u(x, t, a)) = S(x, t, a), (1)

with operator L and source term S defined on domain D T A, and appropriate initial and boundaryconditions. The spatial and temporal dimensions are defined as x D and t T, respectively, with D Rd,d = {1, 2, 3}, and T = [0, tmax]. A realization of the set of outcomes of the probability space (, F, P) isdenoted by = [0, 1]na, with F 2 the -algebra of events and P a probability measure.

Here we consider a non-intrusive uncertainty quantification method l which constructs a weighted ap-

proximation w(x, t, a) of response surface u(x, t, a) based on ns deterministic solutions vk(x, t) u(x, t, ak)of (1) for different parameter values ak a(k) for k = 1, . . ,ns. The samples vk(x, t) can be obtained bysolving the deterministic problem

L(x, t, ak; vk(x, t)) = S(x, t, ak), (2)

for k = 1, . . ,ns, using standard spatial discretization methods and time marching schemes. A non-intrusiveuncertainty quantification method l is then a combination of a sampling method g and an interpolationmethod h. Sampling method g defines the ns sampling points {ak}

nsk=1 and returns the deterministic samples

v(x, t) = {v1(x, t), . . ,vns (x, t)}. Interpolation method h constructs an interpolation surface w(x, t, a) throughthe ns samples v(x, t) as an approximation ofu(x, t, a). We are eventually interested in an approximation ofthe probability distribution and statistical moments ui(x, t) of the output u(x, t, a), which can be obtainedby sorting and weighted integration of w(x, t, a)

ui(x, t) wi(x, t) =A

w(x, t, a)ifa(a)da. (3)

This information can be used for reducing design safety factors and robust design optimization, in contrastto reliability analysis in which the probability of failure is determined.14

III. An efficient uncertainty quantification method for unsteady problems

The efficient uncertainty quantification formulation for oscillatory responses based on interpolation ofscaled samples at constant phase is developed in section B. Scaling the samples with their phase is proven

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to result in a bounded error for non-periodic responses in section C. The robust extrema diminishinguncertainty quantification method based on Newton-Cotes quadrature in simplex elements employed in theunsteady approach is first presented in the next section.

A. Robust extrema diminishing uncertainty quantification

A multi-element uncertainty quantification method l evaluates integral (3) by dividing a bounded parameterspace A into ne non-overlapping simplex elements Aj A

wi(x, t) =

nej=1

Aj

w(x, t, a)ifa(a)da. (4)

Here we consider a multi-element Polynomial Chaos method based on Newton-Cotes quadrature pointsand simplex elements.25 A piecewise polynomial approximation w(x, t, a) is then constructed based on nsdeterministic solutions vj,k(x, t) = u(x, t, aj,k) for the values of the random parameters aj,k that correspondto the ns Newton-Cotes quadrature points of degree d in the elements Aj

wi(x, t) =

nej=1

nsk=1

cj,kvj,k(x, t)i, (5)

where cj,k is the weighted integral of the Lagrange interpolation polynomial Lj,k(a) through Newton-Cotes

quadrature point k in element Aj

cj,k =

Aj

Lj,k(a)fa(a)da, (6)

for j = 1, . . ,ne and k = 1,.., ns. Here, second degree Newton-Cotes quadrature with d = 2 is consideredin combination with adaptive mesh refinement in probability space, since low order approximations aremore effective for approximating response response surfaces with singularities. The initial discretizationof parameter space A for the adaptive scheme consists of the minimum of neini = na! simplex elementsand nsini = 3

na samples, see Figure 1. The example of Figure 1 for two random input parameters cangeometrically be extended to higher dimensional probability spaces. The elements Aj are adaptively refinedusing a refinement measure j based on the largest absolute eigenvalue of the Hessian Hj , as measure of thecurvature of the response surface approximation in the elements, weighted by the probability fj containedby the elements

fj =Aj

fa(a)da, (7)

withne

j=1 fj = 1. The stochastic grid refinement is terminated when convergence measure ne is smaller

than a threshold value ne < where

ne = max

wne/2(x, t) wne (x, t)

wne (x, t) ,

wne/2(x, t) wne (x, t)

wne (x, t)

, (8)

with w(x, t) and w(x, t) the mean and standard deviation of w(x, t , ), or when a maximum number ofsamples ns is reached. Convergence measure ne can be extended to include also higher statistical momentsof the output.

In elements where the quadratic second degree interpolation results in an extremum other than in aquadrature point, the element is subdivided into ne = 2

na subelements with a linear first degree Newton-

Cotes approximation of the response without performing additional deterministic solves. It is proven in26that the resulting approach satisfies the extrema diminishing (ED) robustness concept in probability space

minA

(w(a)) minA

(u(a)) maxA

(w(a)) maxA

(u(a)) u(a), (9)

where the arguments x and t are omitted for simplicity of the notation. The ED property leads to theadvantage that no non-zero probabilities of unphysical realizations can be predicted due to overshoots orundershoots at discontinuities in the response. Due to the location of the Newton-Cotes quadrature pointsthe deterministic samples are also reused in successive refinements and the samples are used in approximatingthe response in multiple elements.

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(a) Element (b) Initial grid (c) Adapted grid

Figure 1. Discretization of two-dimensional parameter space A using 2-simplex elements and second-degreeNewton-Cotes quadrature points given by the dots.

B. Efficient uncertainty quantification interpolation at constant phase

Polynomial Chaos methods usually require a fast increasing number of samples with time to maintain aconstant accuracy. Performing the uncertainty quantification interpolation of oscillatory samples at constantphase instead of at constant time results, however, in a constant accuracy with a constant number of samples.Assume, therefore, that solving equation (2) for realizations of the random parameters ak results in oscillatorysamples vk(t) = u(ak), of which the phase vk(t) = (t, ak) is a well-defined monotonically increasing functionof time t for k = 1, . . ,ns.

In order to interpolate the samples v(t) = {v1(t), . . ,vna(t)} at constant phase, first, their phase as functionof time v(t) = {v1(t), . . . , vna (t)} is extracted from the deterministic solves v(t). Second, the time seriesfor the phase v(t) are used to transform the samples v(t) into functions of their phase v(v(t)) accordingto

vk(vk(t)) = vk(t), (10)

for k = 1, . . ,ns, see Figure 2. And, third, the sampled phases v(t) are interpolated to the function w(t, a)

w(t, a) = h(v(t)), (11)

as approximation of (t, a). Finally, the transformed samples v(v(t)) are interpolated at a constant phase

w(t, a) to w(, a) = h(v()). (12)

Repeating the latter interpolation for all phases w(t, a) results in the function w(w(t, a), a). Theinterpolation w(w(t, a), a) is then transformed back to an approximation in the time domain w(t, a) asfollows

w(t, a) = w(w(t, a), a). (13)

The resulting function w(t, a) is an approximation of the unknown response surface u(t, a) as function oftime t and the random parameters a(). The actual sampling g and interpolation h is performed usingthe extrema diminishing uncertainty quantification method l based on Newton-Cotes quadrature in simplexelements described in the previous section.

The phases v(t) are extracted from the samples based on the local extrema of the time series v(t).A trial and error procedure identifies a cycle of oscillation based on two or more successive local maxima.

The selected cycle is accepted if the maximal error of its extrapolation in time with respect to the actualsample is smaller than a threshold value k for at least one additional cycle length. The functions for thephases v(t) in the whole time domain T are constructed by identifying all successive cycles of v(t) andlinear extrapolation to t = 0 and t = tmax before and after the first and last complete cycle, respectively.The phase is normalized to zero at the start of the first cycle and a user defined parameter determineswhether the sample is assumed to attain a local extremum at t = 0. The interpolation at constant phase isrestricted to the time domain that corresponds to the range of phases that is reached by all samples in eachof the elements. If the phase vk (t) cannot be extracted from one of the samples vk(t) for k = 1, . . ,ns, thenuncertainty quantification interpolation h is directly applied to the time-dependent samples v(t).

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kv

time t

(a) samples vk(t)

kv^

phase

(b) samples vk()

Figure 2. Oscillatory samples as function of time and phase.

C. Error bound for non-periodic responses

It is shown below that the uncertainty quantification interpolation of non-periodic samples at constant phaseresults in a bounded relative error. Let u(t, a) be a non-periodic response as function of time t for t Rgiven by

u(t, a) = A(t, a)up(t, a), (14)with amplitude A(t, a) > 0 t R and up(t, a) a periodic function in time

up(t + zT(a), a) = up(t, a), for all z Z and a A, (15)

with T(a) = 1/f(a) > 0 the period length and f(a) the frequency affected by the random input a(). Thephase (t, a) of the periodic part up(t, a) is given by

(t, a) = 0(a) +t

T(a), (16)

with 0(a) = (0, a). Consider non-periodic response u(t, a) that can then be written as function of (t, a)as follows

u(t, a) = u((t, a), a) =

A((t, a))up((t, a), a), (17)with A((t, a)) = A(t, a) and up((t, a), a) = up(t, a). Consider uncertainty quantification method l whichresults in an approximation w(t, a) ofu(t, a) based on applying interpolation method h at constant phase tons samples v(t) = {v1(t), . . . , vns} for parameter values ak for k = 1, . . . , ns resulted from sampling methodg. Let h be a linear interpolation method in the sense that

ch(v) = h(cv), (18)

with c a constant, which holds for virtually all interpolation techniques including the Simplex ElementsStochastic Collocation approach.

Theorem 1. Relative error (, a) in approximation w(, a) with respect to non-periodic response surfaceu(, a) given by

(, a) = |w(, a) u(, a)|A()

, (19)

with A((t, a)) = A(t, a), as results from uncertainty quantification methodl applied at constant phase isbounded for all R and a A by for which holds

(, a) < , for all [0, 1] and a A. (20)

Proof. For the periodic part up(t, a) of the non-periodic response u(t, a) = A(t, a)up(t, a) holds

up(t, a) = up((t, a), a), (21)

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and

up( + z, a) = up

0(a) +

t + zT(a)

T(a), a

= up(t + zT(a), a)

= up(t, a) = up(, a), for all z Z and a A. (22)

As function of the phase also holds

u(, a) = A()up(, a), (23)

with A() = A(t, a). The samples vk(t) resulting from sampling method g

vk(t) = gk(u(t, a)) = u(t, ak), (24)

for k = 1, . . . , ns can also be written in terms of a periodic part vpk(t)

vk(t) = A(t, ak)vpk (t), (25)

with vpk (t) = gk(up(t, a)) = up(t, ak). The signals vpk(t) are periodic with period length vTk = T(ak), sinceusing (15)

vpk(t + zvTk) = up(t + zT(ak), ak) = up(t, ak) = vpk(t) for all z Z, (26)

for k = 1, . . . , ns. The phase vk(t) = (t, ak) of the signals vpk (t) is then in correspondence with (16) givenby

vk (t) = v0k +t

vTk, (27)

for k = 1, . . . , ns, with v0k = vk (0). Scaling the functions vpk(t) with their phase vk(t) results in

vpk(t) = vpk (vk(t)), (28)

for k = 1, . . . , ns. Periodicity of vpk(t) gives

vpk(vk(t) + z) = vpk

v0k +

t + zvTkvTk

= vpk(t + zvTk) = vpk(t) = vpk(vk(t)), for all z Z, (29)

for k = 1, . . . , ns. In terms of the phase vk(t), (25) becomes

vk(vk(t)) = A(vk(t))vpk(vk(t)), (30)

with vk(t) = vk(vk(t)) and A(t, ak) = A(vk(t)). Uncertainty quantification method l results in approxi-mation w(, a) by applying interpolation method h of the samples v() at a constant phase

w(, a) = h(v()) = h(v1(), . . . , vns()). (31)

Introduce also the following notations

wp(, a) = h(vp1(), . . . , vpns ()), (32)

andw(, a) = A() wp(, a). (33)

The relative error (, a) as function of phase in approximation w(, a) with respect to u(, a) is definedby (19) as

(, a) =|w(, a) u(, a)|

A(). (34)

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Then holds for ( + z, a) the following

( + z, a) =|w( + z, a) u( + z, a)|

A( + z)

=

h(v1( + z), . . . , vns( + z))

A( + z)

A( + z)up( + z, a)

A( + z)

=h A( + z)vp1( + z)A( + z) , . . . ,

A( + z)vpns ( + z)

A( + z)

up( + z, a)

= |h(vp1(), . . . , vpns ()) up(, a)|

=|A() wp(, a) A()up(, a)|

A()

=|w(, a) u(, a)|

A()= (, a), for all z Z and R and a A. (35)

Error (, a) is, therefore, a periodic function of . Define for which holds (20)

(, a) < , for all [0, 1] and a A,

then holds

( + z, a) = (, a) < , for all z Z and [0, 1] and a A, (36)and

(, a) < , for all R and a A, (37)

Relative error (, a) in approximation w(, a) is, therefore, bounded by for all R and a A.

Notice that the proof of Theorem 1 is independent of uncertainty quantification method l, samplingmethod g, and interpolation method h, as long as the interpolation method satisfies (18), ch(v) = h(cv).For the special case of periodic samples with A(t, a) a constant, the error bound (, a) < holds for theabsolute error (, a) = |w(, a) u(, a)|.26

For non-periodic responses which cannot be written as u(t, a) = A(t, a)up(t, a), with A(t, a) = A((t, a)),the relative error (, a) is not strictly bounded by . However, scaling the resulting samples vk(t) with theirphase still eliminates the effect of the increasing phase differences on the increase of the number of required

samples. Also for these responses, performing the uncertainty quantification interpolation at constant phaseresults practically in a constant accuracy with a constant number of samples.The bounded error (, a) as function of phase also results in a bounded error (t, a) in time for the

Simplex Elements Stochastic Collocation method l, if initial phase 0(a) and frequency f(a) of periodicfunction up(t, a) depend linearly on a, with u(t, a) = A(t, a)up(t, a). Let initial phase 0(a), therefore,depend linearly on the random parameters a

0(a) = c0,0 + c0,1 a. (38)

where denotes the vector inner product, with c0,0 constant and c0,1 a vector containing na constants.And let frequency f(a) also depend linearly on a()

f(a) = cf,0 + cf,1 a, (39)

with cf,0 constant and cf,1 an na-dimensional constant vector. Further is used that Simplex Elements

Stochastic Collocation exactly interpolates linear functions.

Theorem 2. Relative error (t, a) in approximation w(t, a) with respect to non-periodic response surfaceu(t, a) given by

(t, a) =|w(t, a) u(t, a)|

A(t, a), (40)

with A(t, a) = A((t, a)), as resulted from Simplex Elements Stochastic Collocation l applied at constantphase is bounded for all t R and a A by for which holds (20)

(, a) < , for all [0, 1] and a A,

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if initial phase 0(a) and frequency f(a) of periodic function up(t, a) depend linearly on a, with u(t, a) =A(t, a)up(t, a).

Proof. The phase (t, a) of the periodic part up(t, a) of non-periodic response u(t, a) is given by (16)

(t, a) = 0(a) +t

T(a)= 0(a) + f(a)t.

The linear dependence of 0(a) and f(a) on a given by (38) and (39) results in

(t, a) = c0,0 + cf,0t + (c0,1 + cf,1t) a (41)

The ns sampled phases v(t) = {v1(t), . . . , vns (t)} resulting from sampling method g are, therefore,

vk(t) = c0,0 + cf,0t + (c0,1 + cf,1t) ak, (42)

for k = 1, . . . , ns. The resulting w(t, a) of Simplex Elements Stochastic Collocation interpolation h of thesamples v(t) then exactly reconstructs the function (t, a)

w(t, a) = h(v(t)) = c0,0 + cf,0t + (c0,1 + cf,1t) a = (t, a). (43)

Therefore, error ((t, a), a) in the approximation w(w(t, a), a) of response u((t, a), a) becomes

((t, a), a) =|w(w(t, a), a) u((t, a), a)|

A((t, a))

=|w((t, a), a) u((t, a), a)|

A((t, a))

=|w(t, a) u(t, a)|

A(t, a)

= (t, a) < for all R and a A, (44)

according to Theorem 1. Using (16) gives

(t, a) < for all t R and a A. (45)

IV. Transonic airfoil flutter

The combined effect of independent randomness in the ratio of natural frequencies () and the freestream velocity U() on the post-flutter behavior of an elastically mounted airfoil is analyzed. The struc-tural model of the pitch-plunge airfoil with cubic nonlinear spring stiffness is given by: 6,11

+ x +

U

2( +

3) = 1

Cl(), (46)

xr2

+ +1

U2( +

3) =2

r2Cm(), (47)

where = 0m2 and = 300rad2 are the cubic spring parameters, () = h/b is the non-dimensionalplunge displacement of the elastic axis, see Figure 3, () is the pitch angle, and () denotes differentiationwith respect to non-dimensional time = Ut/b, with half-chord length b = c/2 = 0.5m. The radius ofgyration around the elastic axis is rb = 0.25m, bifurcation parameter U is defined as U = U/(b),and the airfoil-air mass ratio is = m/b2 = 100, with m the airfoil mass. The elastic axis is locatedat a distance ahb = 0.25m from the mid-chord position and the mass center is located at a distancexb = 0.125m from the elastic axis. The ratio of natural frequencies is defined as () = /, with and the natural frequencies of the airfoil in pitch and plunge, respectively. The randomness in () isdescribed by a uniform distribution around mean value = 0.25 with a coefficient of variation of 10%. The

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ahb x

b

h

midchordelastic axis

centre of mass

reference position

cb

Figure 3. The elastically mounted pitch-plunge airfoil model.

free stream velocity U() is subject to a symmetric unimodal beta distribution with 1 = 2 = 2 with acoefficient of variation of 1% around mean U = 276.27m/s, which corresponds to M = 0.8.

The non-dimensional aerodynamic lift and moment coefficients, Cl() and Cm(), are determined bysolving the unsteady Euler equations. The two-dimensional flow domain is discretized by an unstructured

hexahedral mesh of 12 103

cells, which was selected based on a grid convergence study. The governingequations are discretized using a second order central finite volume discretization stabilized with artificialdissipation. An Arbitrary Lagrangian-Eulerian formulation is employed to couple the fluid mesh with themovement of the structure. The fluid mesh is deformed using radial basis function interpolation of theboundary displacements.4 Time integration is performed using the second order BDF-2 method until t = 3with time step t = 0.002, which was established after a time step refinement study. Initially the airfoil is atrest at a deflection of (0) = 0.1deg and (0) = 0 from its equilibrium position. In order to study the post-bifurcation behavior, the bifurcation parameter U is fixed at 130% of the deterministic linear bifurcationpoint for the mean values of the random parameters. The stochastic behavior of the angle of attack (t, )is resolved as indicator for the post-flutter airfoil behavior.

The Unsteady Adaptive Stochastic Finite Elements response surface approximation of the angle of attack(t, ) as function of the random parameters () and U() at t = {0.5; 1.5; 2.5} given in Figure 4 showsan increasingly oscillatory response surface with time. The 10% variation in () has a larger effect on

the frequency of the response than U() with 1% variation. Both parameters have a small effect onthe amplitude of the oscillation of (t, ) of approximately 3o. At t = 0.5 the airfoil exhibits transientbehavior from its initial perturbation of (0) = 0.1o, which is indicated by the smaller amplitude of theresponse surface variations of approximately 2o. These results are obtained using the time-independentgrid in probability space shown in Figure 4d with ns = 9 samples, ne = 2 elements, and nesub = 4096post-processing subelements.

The resulting UASFE approximation of the mean (t) and standard deviation (t) of the angle ofattack (t, ) in Figure 5 shows two frequency signals due to the effect of the two random parameterson the frequency of the response. The mean (t) exhibits initially an increasing oscillation caused bythe deterministic transient of the samples, after which it develops a decaying oscillation due to the effectof the random parameters on the frequency of the response. The large effect of the random parameterson the dynamical system is illustrated by the fast initial increase of the standard deviation (t) fromits deterministic initial condition. Although the deterministic post-flutter behavior is highly unsteady, the

stochastic response reaches a steady asymptotic behavior with a standard deviation of = 1.6o, which is afactor 16 larger than the initial angle of attack (0) = 0.1o. The discretizations with ns = {9, 13, 25} samplesand ne = {2, 4, 8} uniformly refined elements, respectively, indicate that the results are uniformly convergedin time. The approximation with ns = 25 is converged up to ne = 6.210

3, where ne is defined by (8). Thelocal convergence for (t) and (t) at t = {0.5; 1.0; 1.5; 2.0; 2.5} given in Tables 1 and 2 for ns = {13, 25}shows no clear increase of convergence measure with time. This illustrates that the convergence and theaccuracy of the UASFE approximation are in practice constant in time.

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(a) t = 0.5 (b) t = 1.5

(c) t = 2.5 (d) Stochastic grid

Figure 4. Response surface of angle of attack () as function of random natural frequency ratio () and free

stream velocity U() for transonic airfoil flutter.

0 0.5 1 1.5 2 2.5 3

0.6

0.4

0.2

0

0.2

0.4

0.6

time t [s]

mean[

deg]

ns=9

ns=13

ns=25

(a) Mean

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

time t [s]

standarddeviation[

deg]

ns=9

ns=13

ns=25

(b) Standard deviation

Figure 5. Mean and standard deviation of angle of attack () for transonic airfoil flutter with random naturalfrequency ratio () and free stream velocity U().

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Table 1. Convergence measure ne for mean angle of attack (t, ) for transonic airfoil flutter with randomnatural frequency ratio () and free stream velocity U().

ns ne t = 0.5 t = 1.0 t = 1.5 t = 2.0 t = 2.5

13 4 0.640 103 3.712 103 4.426 103 7.207 103 4.331 103

25 8 0.268 103 2.455 103 3.138 103 4.422 103 2.684 103

Table 2. Convergence measure ne for the standard deviation of angle of attack (t, ) for transonic airfoilflutter with random natural frequency ratio () and free stream velocity U().

ns ne t = 0.5 t = 1.0 t = 1.5 t = 2.0 t = 2.5

13 4 2.943 103 3.275 103 7.500 103 5.378 103 3.896 103

25 8 0.973 103 4.388 103 0.859 103 2.194 103 3.344 103

V. Three-dimensional transonic wing

The transonic AGARD 445.6 wing30 is a standard benchmark case for the fluid-structure interaction ofa three-dimensional continuous structure. The discretization of the aeroelastic configuration is described insection A. In section B randomness is introduced in the free stream velocity. The stochastic response of thesystem and the flutter probability are determined.

A. AGARD 445.6 wing benchmark problem

The AGARD aeroelastic wing configuration number 330 known as the weakened model is considered herewith a NACA 65A004 symmetric airfoil, taper ratio of 0.66, 45o quarter-chord sweep angle, and a 2.5-footsemi-span subject to an inviscid flow. The structure is described by a nodal discretization using an undampedlinear finite element model in the Matlab finite element toolbox OpenFEM.15 The discretization contains inthe chordal and spanwise direction 6 6 brick-elements with 20 nodes and 60 degrees-of-freedom, and atthe leading and trailing edge 2 6 pentahedral elements with 15 nodes and 45 degrees-of-freedom as in.33

The orthotropic material properties are obtained from3 and the fiber orientation is taken parallel to thequarter-chord line.

The Euler equations for inviscid flow5 are solved using a second-order central finite volume discretizationon a 60 15 30m domain using an unstructured hexahedral mesh. The free stream conditions for thedensity = 0.099468kg/m3 and pressure p = 7704.05Pa are taken from.30 Time integration of thesamples is performed using a third-order implicit Runge-Kutta scheme9 until t = 1.25s to determine thestochastic solution until t = 1s. The first bending mode with a vertical tip displacement of ytip = 0.01m isused as initial condition for the structure, see Figure 6.

The coupled fluid-structure interaction system is solved using a partitioned IMEX scheme31,32 with ex-plicit treatment of the coupling terms without sub-iterations. An Arbitrary Lagrangian-Eulerian formulationis employed to couple the fluid mesh with the movement of the structure. The flow forces and the structuraldisplacements are imposed on the structure and the flow using nearest neighbor and radial basis functioninterpolation,33 respectively. The fluid mesh is also deformed using radial basis function interpolation ofthe boundary displacements.4 A convergence study has been performed to determine a suitable flow mesh

discretization and time step size. Deterministic results for the selected flow mesh with 3.1 10

4

volumes andtime step of t = 2.5 103s agree well with experimental and computational results in the literature.10,30,33

The deterministic flutter velocity is found to be Uflut = 313m/s, which corresponds to a Mach number ofM = 0.951.

B. Randomness causes non-zero flutter probability

In the following, the effect of randomness in the free stream velocity U() is studied. The mean freestream velocity is chosen 5% below the actual deterministic flutter velocity, U = 0.95Uflut, to assessthe effectiveness of a realistic design safety factor. The coefficient of variation of the assumed unimodal

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Figure 6. Initial condition and grid for the AGARD 445.6 wing for mean free stream velocity U.

beta distribution is set to cvU = 3.5%. The outputs of interest are the lift L(t, ) and the vertical tip

displacement of the tip-node ytip(t, ).The first Ns = 3 sampled time series of the lift Li(t, ) of the UASFE discretization with Ne = 1 element

show in Figure 7a that the first bending mode is the dominant mode in the system response. A second modewhich is initially present in the response, damps out quickly, such that a wavelet decomposition pre-processingstep is in this case not necessary to obtain the stochastic solution using UASFE. The samples illustrate thatthe free stream velocity has a significant effect on the frequency and the damping of the system response,which results in a diverging oscillation for i = 3, and decaying oscillations for i = 1 and mean value Uat i = 2. The same conclusions can be drawn from Figure 7b in which the response surface approximationof the lift L(t, ) at t = 1 is given for Ne = 5 elements and Ns = 11 samples. The response surface hasan oscillatory character due to the effect of the random U() on the frequency of the lift oscillation andconsequently on the phase differences in L(t, ) at t = 1. The adaptive UASFE grid refinement resultsautomatically in a gradually finer mesh in the region of large lift amplitudes at large values of U().

0 0.2 0.4 0.6 0.8 1250

200

150

100

50

0

50

100

150

200

250

time t

lift[N]

UASFE (Ne=1,N

s=3)

i=1 i=2 i=3

(a) lift samples Li(t)

0.88 0.9 0.92 0.94 0.96 0.98 1 1.0250

0

50

100

150

200

250

U/Uflut

liftL[N]

t=1UASFE (N

e=5)

Ns=11 samples

(b) lift response surface L(t, ) at t = 1

Figure 7. Results for the AGARD 445.6 wing with random free stream velocity U().

Results for the time evolution of the mean L(t) and the standard deviation L(t) of the lift are givenin Figure 8 for Ne = 4 and Ne = 5 elements. The two approximations are converged with respect to eachother up to 5 103. The time history for the mean lift L(t) shows a decaying oscillation up to t = 0.4s

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from the initial value of L = 23.9N. This behavior can be explained by the decaying lift oscillation fora large range of U() values and the effect of U() on the increasing phase differences with time. Fort > 0.4 the decay is approximately balanced by the exponentially increasing amplitude of the unstable partof the U() parameter domain. In contrast, the standard deviation shows an oscillatory increase from theinitial L = 2.46N up to a local maximum of L = 18.3N at t = 0.31s due to the increasing phase differenceswith time. For t > 0.31 the standard deviation slightly decreases due to the decreasing lift amplitude in partof the parameter domain. Eventually, the unstable realizations result in an increasing standard deviationwhich reaches at t = 1 values between L = 14 and L = 19, which corresponds to an amplification of the

initial standard deviation with a factor 6 to 8.

0 0.2 0.4 0.6 0.8 140

30

20

10

0

10

20

30

40

time t [s]

meanliftL

[N]

UASFE (Ne=5,N

s=11)

UASFE (Ne=4,N

s=9)

(a) mean L

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

time t [s]

standarddeviationliftL

[N]

UASFE (Ne=5,N

s=11)

UASFE (Ne=4,N

s=9)

(b) standard deviation L


The nodal description of the structure directly returns the vertical tip-node displacement ytip(t, ). Theapproximations of the mean ytip(t) and standard deviation ytip(t) of ytip(t, ) show in Figure 9 a quali-tatively similar behavior as the lift L(t, ). The standard deviation ytip(t) vanishes, however, initially dueto the deterministic initial condition for the structure in contrast with the non-zero L(t) at t = 0. The

standard deviation reaches values between ytip = 4.2 10

3

m and ytip = 5.6 10

3

m at t = 1, whichcorresponds to a standard deviation equal to 42% and 56% of the deterministic initial vertical tip deflection.

0 0.2 0.4 0.6 0.8 10.015

0.01

0.005

0

0.005

0.01

0.015

time t [s]

me

antipdisplacementy,t

ip[m]

UASFE (Ne=5,N

s=11)

UASFE (Ne=4,N

s=9)

(a) mean ytip

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7x 10

3

time t [s]

standard

deviationtipdisplacementy,t

ip[m]

UASFE (Ne=5,N

s=11)

UASFE (Ne=4,N

s=9)

(b) standard deviation ytip


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The probability of flutter can be determined by constructing the probability distribution of the dampingfactor of the system given in Figure 10. The damping factor is here extracted from the last period ofoscillation of the sampled vertical tip node displacements. Positive and negative damping factors denoteunstable and damped oscillatory responses, respectively. Even though the mean free stream velocity Uis fixed at a safety margin of 5% below the deterministic flutter velocity Uflut, the non-zero probability ofpositive damping indicates a non-zero flutter probability. The 3.5% variation in U() results actually ina probability of flutter of 6.19%. Taking physical uncertainties into account in numerical predictions is,therefore, a more reliable approach than using safety margins in combination with deterministic simulation

results.

4 3 2 1 0 1 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

damping factor

probabilitydistribution

UASFE (Ne=5,N

s=11)

UASFE (Ne

=4,Ns

=9)


VI. Conclusions

An uncertainty quantification formulation for achieving a constant accuracy in time with a constantnumber of samples in multi-physics aeroelastic problems is developed based on interpolation of oscillatorysamples at constant phase instead of at constant time. The scaling of the samples with their phase eliminates

the effect of the increasing phase differences in the response, which usually leads to the fast increasing numberof samples with time. The resulting formulation is not subject to a parameterization error and it canresolve time-dependent functionals that occur for example in transient behavior. Phase-scaled uncertaintyquantification results in a bounded error as function of the phase for periodic responses and under certainconditions also in a bounded error in time.

The developed approach is applied to the unsteady fluid-structure interaction test cases of a two-dimensional airfoil flutter problem and the three-dimensional aeroelastic AGARD 445.6 wing. In the stochas-tic post-flutter analysis of a two-degree-of-freedom rigid airfoil in Euler flow with nonlinear structural stiffnessin pitch and plunge, a combination of randomness in two system parameters is considered. The uncertaintyin the ratio of natural frequencies of the structure is described by uniform distribution around mean valueof 0.25 with a coefficient of variation of 10%. The free stream velocity is subject to a symmetric unimodalbeta distribution with a coefficient of variation of 1% around a mean of 276.27m/s, which corresponds to aMach number of 0.8. In order to study the post-bifurcation behavior, the bifurcation parameter is fixed at130% of the deterministic linear bifurcation point for the mean values of the random parameters.

The behavior of the mean, standard deviation, and response surface of the angle of attack is resolvedas indicator for the post-flutter airfoil behavior. The results show that the frequency of the increasinglyoscillatory response surface with time is affected more by the frequency ratio than the free stream velocity.Both parameters have a small effect on the amplitude of the oscillation. The asymptotic stochastic behaviorof the time-dependent problem is steady with a standard deviation of 1.6 degrees, which is a factor 16larger than the deterministic initial angle of attack. Convergence results for discretizations with increasingnumber of samples indicate that the applied uncertainty quantification methodology results in practice in atime-independent accuracy with a constant number of samples.

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The AGARD aeroelastic wing configuration number 3 known as the weakened model is also consideredwith a NACA 65A004 symmetric airfoil, taper ratio of 0.66, 45 degrees quarter-chord sweep angle, and a2.5-foot semi-span subject to an inviscid flow. The free stream conditions for the density and pressure are0.099468 kg/m3 and 7704.05 Pa, respectively. The first bending mode with a vertical tip displacement of0.01 m is used as initial condition for the structure. The deterministic flutter velocity for this configurationis found to be 313 m/s, which corresponds to a Mach number of 0.951.

The effect of randomness in the free stream velocity is studied. The mean free stream velocity is chosen5% below the actual deterministic flutter velocity to assess the effectiveness of a realistic design safety

factor. The coefficient of variation of the assumed unimodal beta distribution is set to 3.5%. The outputsof interest are the mean and standard deviation of the lift and the vertical tip displacement of the tip-node,the probability distribution of the damping coefficient, and the probability of flutter.

Even though the mean free stream velocity is fixed at a safety margin of 5% below the deterministic fluttervelocity, a resulting non-zero probability of positive damping indicates a non-zero flutter probability. The3.5% variation in free stream velocity results actually in a probability of flutter of 6.19%. Taking physicaluncertainties into account in numerical predictions is, therefore, a more reliable approach than using safetymargins in combination with deterministic simulation results.

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