Computers and Structures 201 (2018) 26–36

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Computers and Structures

journal homepage: www.elsevier .com/locate/compstruc

A harmonic balance method for nonlinear fluid structure interactionproblems

https://doi.org/10.1016/j.compstruc.2018.02.0030045-7949/� 2018 The Authors. Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

⇑ Corresponding author at: Department of Mechanical Sciences, University ofSurrey, Guildford GU2 7XH, UK.

E-mail addresses: s.marques@surrey.ac.uk, s.marques@qub.ac.uk (S. Marques).

W. Yao, S. Marques ⇑School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast BT9 5AH, UK

a r t i c l e i n f o

Article history:Received 2 September 2017Accepted 12 February 2018

Keywords:Limit cycle oscillationsVortex-induced vibrationsAeroelasticityCFDHarmonic balanceNonlinear

a b s t r a c t

Over the past decade the Harmonic-Balance technique has been established as a viable alternative todirect time integration methods to predict periodic aeroelastic instabilities. This article reports the pro-gress made in using a frequency updating procedure, based on a coupled fluid-structural solver using theHarmonic-Balance formulation. In particular, this paper presents an efficient implicit time-integrator thataccelerates the convergence of the structural equations of motion to the final solution. To demonstratethe proposed approached, the paper includes a detailed investigation of the impact of input parametersand exercises the method for two types of fluid-structural nonlinear instabilities: transonic limit-cycleoscillations and vortex-induced vibrations.� 2018 The Authors. Published by Elsevier Ltd. This is an openaccess article under the CCBY license (http://

creativecommons.org/licenses/by/4.0/).

1. Introduction

The ever growing capability of computing hardware and soft-ware, enabled high fidelity computational fluid dynamics (CFD)to become the primary tool for the study of fluid physics. With sim-ilar advancements in computational structural dynamics (CSD) andcoupling algorithms, CFD has also been extensively applied tofluid-structure interaction problems where flow nonlinearitiessuch as shock waves or flow separation play a dominant role.The time dependency of this type of analysis means it usuallyrequires additional computational resources. Nevertheless, fluid-structure interactions play a safety critical role in several applica-tions such as aircraft flutter or vortex induced vibration (VIV) fre-quency lock-in. Therefore, the efficient prediction of these andassociated phenomena attracts much attention from the numericalmethods research community.

Several alternatives to full time-domain simulations have beenproposed to investigate the loss of linear stability of an aeroelasticsystem. Badcock et al. avoid conducting time-domain analysis bytracking the critical eigenvalue of the coupled system Jacobianmatrix [1–3], at an equivalent cost of a few steady-state calcula-tions; if the Jacobian of the coupled system is not available, linearreduced-order models (ROM) can be built from prescribed time

domain simulations [4,5] and used to infer the system’s stabilityat a lower cost than using the full-order model.

If nonlinearities are present, additional instabilities have beenobserved which pose further challenges to the development of effi-cient simulation tools [6]. In particular, nonlinearities such asshocks, vortex shedding or free-play can cause periodic oscilla-tions, generally referred to as limit cycle oscillations (LCO) thatare not captured by linear ROMs. Therefore, the development ofnonlinear ROMs is a popular topic of research [7–10]. The system’snonlinear response can be approximated based on input-outputrelations using a recursive nonlinear interpolator in lieu of the fullorder aerodynamic model [11–13]; Yao and Marques applied aninput-output technique employing a radial basis function (RBF)neural network together with a discrete empirical interpolationmethod to reconstruct the complete flow field for the predictionof LCO [14].

The aforementioned nonlinear methods rely on performingtime domain simulations a priori, under specific conditions thatenable a suitable nonlinear model for the problem to be built.Alternative methods have been proposed that preserve the under-lying physics of the problem and allow the direct computation ofthe nonlinear system. Following the stability analysis describedin Refs. [1,3], LCO can be predicted by model order reduction usingthe critical eigenbasis of the Jacobian of the coupled system.Another alternative in this class of methods is to exploit the peri-odicity of the problem and solve the fluid-structural system inthe frequency domain, these strategies are commonly known asTime-Spectral method or high-dimensional Harmonic Balance

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W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36 27

(HB) approach. The HB method transfers the time dependent prob-lem into a steady solution based on Fourier expansions. Originallyapplied in the context of CFD for turbomachinery problems [15,16],the HB method has been used in several diverse applications,including the prediction of dynamic derivatives [17,18], periodicflows found in rotorcraft [19] or wind-turbines [20]. Progress hasalso been made in applying the HB methodology to fluid structureinteraction. For the majority of such cases, it is worth noting thatthe fundamental frequency of the oscillation is not known a prioriand is an additional unknown. Thomas et al. developed the HB/LCOmethod to predict LCO based on a Newton–Raphson approach,where a HB formulation solves the fluid problem and amplitudeand frequency are determined simultaneously [21]. Blanc et al.proposed a fully coupled aero-structural Harmonic Balance solverfor forced motions. Ekici and Hall also developed a fully coupledfluid-structure coupled HB strategy capable of predicting LCOsfor one degree-of-freedom (DOF) turbomachinery components[22]. Yao and Marques, building upon these approaches, proposedan Aeroelastic-HB (A-HB) to analyze fixed wing nonlinear aeroelas-tics [23] and this was further adapted to VIV lock-in by Yao and Jai-man [24].

The A-HB approach and its version developed for VIVemploys an explicit scheme to resolve the structural equationsof motion, together with a relaxation approach to update thefundamental frequency of the oscillation. Both these strategieslimit the efficiency of the iterative scheme. This paperaddresses these limitations by reformulating the A-HB methodusing an approximate exponential time differencing schemeand subsequent modified strategy to update the frequency.The following sections will describe the numerical formulationfor the flow and structural models, this is followed by theintroduction of the new coupling procedure; the final two sec-tions of the paper present a diverse range of test cases used tocritique the new A-HB method and the conclusions obtainedfrom this work.

2. Aeroelastic - harmonic balance formulation

2.1. Fluid governing equation

The governing fluid equations used in this work are the com-pressible Euler or laminar Navier-Stokes equations. For time-dependent problems with moving boundaries, the system of equa-tions is solved using an arbitrary Lagrangian-Eulerian (ALE) formu-lation as follows:

dwdt

¼ �RðwÞ ð1Þ

RðwÞ ¼ r � Fi � Fv� �

; ð2Þ

the integral form of these equations for a control volume V j withsurface dS can be written as:

ddt

ZVjðtÞ

wdV þI@VjðtÞ

Q � n dS ¼ 0 ð3Þ

and

Q ¼ ðFi � Fv �wvgÞ; ð4Þ

where t is the physical time, w ¼ q;qu;qE½ �T is the vector of con-served variables, the over-bar denotes the control volume averagequantities; q is the density and E is the energy, u and vg are theCartesian flow and grid velocities vectors, respectively and n is

the outward unit normal of every cell face. Fi and Fv correspondto the inviscid and viscous fluxes, respectively and can be writtenin compact form as:

Fi ¼qu

quiuþ pdijðqEþ pÞu� pvg

264

375; Fv ¼

0sij

siju� qi

264

375; i ¼ 1;2;3: ð5Þ

The inviscid flux is calculated by the AUSMþ � up flux function[25], together with the Van Albada limiter to achieve 2nd orderspatial accuracy, details of this implementation can be found inRef. [26]. The viscous stress tensor sij and the heat flux qi aregiven by:

sij ¼ l @ui@xj

þ @uj@xi

� 23@uk@xk

dij

� �ð6Þ

qi ¼ �lcpPr

@T@xi

ð7Þ

where l is the coefficient for laminar viscosity, cp is the specific heatratio for a perfect gas and Pr ¼ 0:72 is the laminar Prandtl number.The pressure is given by the ideal gas law:

p ¼ ðc� 1Þq E� 12ðu � uÞ

� �ð8Þ

where c ¼ 1:4 and represents the ratio of specific heats for a dia-tomic gas.

Following Hall et al. [16], Eq. (1) can be written in the highdimensional HB format as

xDwhb þ Rhb ¼ 0; ð9Þwhere x is the system’s fundamental frequency, whb and Rhb arethe fluid and residual solution at equally spaced time intervalsDt ¼ T=ð2NH þ 1Þ where T ¼ 2p=x and NH corresponds to thenumber of harmonics selected. The matrix D is given by:

Di;k ¼ 22NH þ 1XNHn¼1

n sin2pnðk� iÞ2NH þ 1

� �: ð10Þ

Full details of the HB implementation and derivation of matrix Dcan be found in Refs. [23,27]. The Eq. (9) is solved by introducinga pseudo time variable, s:

dwhbds

þxDwhb þ Rhb ¼ 0: ð11Þ

This means that the time dependency of the fluid equation isconverted into a steady problem. An LU-SGS [28] scheme isemployed to March the Eq. (11) forward in pseudo-time.

2.2. Structural governing equations

The equations of motion for a linear structural model can bedescribed in general terms as

M€gþ f _gþ Kg ¼ f ; ð12Þhere M; f; K are the mass, damping and stiffness matrices of thestructure, g and f are the displacement and external force appliedto the structure. The latter two quantities also correspond to theoutput and input to the structural system. The structural equationsin state space format can be described as,

_n ¼ Asnþ Bsf ð13Þwhere

As ¼0 I

�M�1K �M�1f

� �; Bs ¼

0M�1

� �; n ¼ g

_g

� �:

Similar to the fluid equations, the same HB operator D can beapplied to Eq. (13), resulting in

ðxD� AsÞghb ¼ Bsf hb: ð14Þ

28 W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36

The structural residual L2 norm is then defined

L2 ¼ 12RTsRs ð15Þ

and

Rs ¼ ðxD� AsÞghb � Bsf hb ð16Þ

Table 1Pitch/plunge aerofoil parameters.

Static unbalance, xa 0.25Radius of gyration about elastic axis, r2a 0.75Frequency ratio, xh=xa 0.5Mass ratio, . ¼ 4m= pq1c2

� 75

2.3. Harmonic-balance fluid structure coupling

The basic idea in the original A-HBmethod proposed by Yao andMarques [23], is to transform the fluid structure interaction prob-lem in the frequency domain with a fixed point algorithm, which isrepeated in Algorithm 1.

Algorithm 1. A-HB

In the original A-HB approach, a direct solver with a relaxationfactor, Eq. (17), was adopted for step 2 in Algorithm 1, which canbe considered as a one step explicit algorithm. However, the relax-ation factor k restricts the convergence rate.

gnhb ¼ gn�1hb þ kðĝhb � gn�1hb Þ; ð17Þwhere ĝhb ¼ ðxD� AsÞ�1Bsf nhb. An alternative time integrator for thestructural system is necessary to lift up this restriction. FollowingEkici and Hall [22] and similar to Eq. (11), a pseudo time steppingis introduced in Eq. (14) to integrate in time the HB form of thestructural equations:

_ghb ¼ Asdghb þ Bsf hb ð18Þwhere Asd ¼ �ðxD� AsÞ.

The exact discretization of Eq. (18) is given by [29]

gnhb ¼ eAsdDsgn�1hb þZ nDsðn�1ÞDs

eAsdðnDs�rÞBsf hbðrÞdr: ð19Þ

The integral term in Eq. (19) can be simplified by definingt ¼ nDs� r, and f hbðrÞ ¼ 1=2ðf nhb þ f n�1hb Þ, resulting in

gnhb ¼ eAsdDsgn�1hb þ12A�1sd ðeAsdDs � IÞBsðf nhb þ f n�1hb Þ: ð20Þ

The practicality of this result depends on the evaluation of thematrix exponential. To improve robustness, the matrix exponential

is approximated as eAsdDs � I� DsAsdð Þ�1, this corresponds to a firstorder Padé type approximation [30]. Additionally Eq. (20) can besimplified by noting that

A�1sd eAsdDs � I� � A�1sd I� DsAsdð Þ�1 � Ih i

¼ A�1sd I� I� DsAsdð Þ½ � I� DsAsdð Þ�1

¼ Ds I� DsAsdð Þ�1; ð21Þwhich leads to:

gnhb ¼ ðI� AsdDsÞ�1 gn�1hb þ12DsBsðf nhb þ f n�1hb Þ

� �: ð22Þ

This exponential time integrator is the same as a first order linearlyimplicit one, in which the force term f hb is approximated by theTrapezoidal rule. A standard implicit Euler method can be recoveredby assuming constant f hb or f ðrÞ ¼ f nhb in the exact discretization Eq.(19). The results shown in the next section suggest that the conver-gence rate of Eq. (22) improves significantly with respect to thestandard implicit method. Therefore, the Eq. (17) in step 2 of Algo-rithm 1 is replaced by Eq. (22).

Unlike the time accurate fluid and structure interaction compu-tation, Algorithm 1 is typically a steady state searching process,where only the converged solution matters. The frequency x isupdated by minimizing the system’s residual L2 norm [31,22],defined in Eq. (15), by using the gradient with respect to the fre-quency, which in this work is given by:

@L2@x

¼ Dghb � Bs@f hb@x

� �TðxD� AsÞg� Bsf hb½ � ð23Þ

Previously, the authors demonstrate that @f hb@x , which is computed by

finite differencing, improves the frequency updating process signif-icantly, however this is dependent on the step size used. By intro-ducing the implicit method in step 2 in the A-HB method theimpact of the force derivative becomes less significant to the overallefficiency of the algorithm (results indicate up to 10% reduction incomputing time).

The modified A-HB method is referred to A-HB+ method. Thecontrolling parameters for the A-HB+ method are defined as½Ds;NH;Ni;x0;g0�, where Ds;NH;Ni;x0;g0 are the pseudo timestep, number of harmonics, number of sub-iterations, initial fre-quency and displacement, respectively.

In the next section, the sensitivity of the A-HB+ method withrespect to these parameters is investigated using a pitch/plungetransonic aerofoil system, undergoing an LCO; following this initialproblem, the proposed method is tested to predict LCOs for theGoland wing and the versatility of the approach is further demon-strated by predicting oscillations due to vortex shedding.

3. Results

3.1. Aerofoil aeroelastic system

To assess the A-HB+ method, a NACA 64A010 aerofoil with pitch(a) and plunge (h) DOF is adopted. The non-dimensional structuralgoverning equation, Eq. (24), and parameters given in Table 1follow the descriptions found in Refs. [21,23]:

M€gþ 1V2

Kg ¼ 4p.

f ð24Þ

where

M ¼ 1 xaxa r2a

� �; K ¼

xhxa

� �20

0 r2a

24

35; f ¼ �Cl

2Cm

� �;

g ¼2hc

a

" #; V ¼ U1

xac:

W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36 29

U1 represents the freestream velocity, Cl is the lift coefficient andCm is the pitching moment coefficient. Critical conditions are setusing the velocity index is defined as Vs ¼ 2U1=ðxac ffiffiffi.p Þ, and thenapplying the procedure given by Algorithm 1.

The fluid domain is descritized using an O-type grid with121� 41 points in the circumferential and radial direction, respec-tively; the solution of the fluid problem is obtained by solving theEuler equations. The Mach number and initial angle of attack are,respectively, ½M1;a0� ¼ ½0:8;0��. The CFL number is set to 1 andkept the same for all the LCO calculations.

First, the number of harmonics required to convergethis problem with the time integrator from Eq. (22) is investi-gated. Conditions are set as: ½Ds;Ni;j0� ¼ ½50;100;0:105�,g0 ¼ ½0:2;0:02�T , Vs ¼ 0:8. The A-HB+ method starts with the ini-tial flow solution computed with ½j0;g0�, where j ¼ 2xU1c is thereduced frequency and j0 is near to the flutter onset frequency(jflutter � 0:109) [23]. Fig. 1 shows the fluid and structure conver-gence history, which indicates both fluid and structural residual

Iteration

L2 n

orm

of

resi

du

al

50 100 150 200 25010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100 HB1HB3HB5

Fig. 1. L2 norm of residual of (a) fluid and (b) str

h

h

-0.4 -0.2 0 0.2 0.4-0.1

-0.05

0

0.05

0.1

0.15 TDcycle_HB3HB1HB3HB5

Fig. 2. LCO position-velocity diagram retaining different number of harmonics: (a) plungiA-HB+ method.

converge rapidly. The stopping criteria is based on the structuralresidual reducing 15 orders of magnitude. As shown in Fig. 2,both plunging and pitching amplitudes converge when usingthree harmonics (NH ¼ 3), the fully reconstructed cycle usingthree harmonics (cycle HB3), shows good agreement with theconventional time-domain (TD) result. Fig. 3(a) shows that theLCO solution (½j;g�) converges after approximately 100 iterationswith the corresponding structural residual near 10�10. The shockmotion drives the aerodynamic forces and Fig. 3(b) shows theaerofoil motion at the maximum plunging position, while pitch-ing nose down.

After the NH convergence study, the time integrator from Eq.(22) is compared against the standard implicit temporal discretiza-

tion with ½Ds;NH;Ni;j0� ¼ ½50;3;100;0:105�, g0 ¼ ½0:2; 0:02�T . Fig. 4shows that Eq. (22) improves the convergence rate significantlywith respect to the standard first order implicit method. Therefore,the solution of Eq. (22) is the default setting in A-HB+ method andadopted for the remainder of the calculations.

Iteration

L2 n

orm

of

resi

du

al

50 100 150 200 25010-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100 HB1HB3HB5

ucture with different number of harmonics.

α

. α

-0.04 -0.02 0 0.02 0.04

-0.01

0

0.01

0.02 TDcycle_HB3HB1HB3HB5

ng and (b) pitching. The large symbols correspond to the time intervals solved by the

Iteration

κ

L2 n

orm

of

dis

pla

cem

ent

50 100 1500.09

0.095

0.1

0.105

0.11

0.05

0.1

0.15

0.2

0.25

0.3

κL2(η)

X

Y

0 0.5 1

-0.5

0

0.5

P

1.64831.50421.36011.21601.07190.92780.78370.6396

Fig. 3. (a) Displacement L2 norm and reduced frequency convergence history and (b) pressure contour near hmax .

Fig. 4. L2 norm of the structural residual using the proposed approximateexponential integrator, Eq. (22), and the standard implicit Euler method.

Iteration

L2

no

rm o

f re

sid

ual

100 101 102 103

10-14

10-12

10-10

10-8

10-6

10-4

10-2

10 0

Δτ=10Δτ=25Δτ=50Δτ=100Δτ=1000

Fig. 5. L2 norm of structural residual with different time step, Ds.

30 W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36

The sensitivity of the time step size Ds is performedby using ½NH;Ni;j0� ¼ ½3;100;0:105�, g0 ¼ ½0:2;0:02�T , andDs ¼ ½10;25;50;100;1000�. Fig. 5 suggests that values between10 6 Ds 6 100 produce adequate convergence rates, with the opti-mum being near Ds ¼ 50. The results indicate that the time stepsize if enlarged arbitrarily, deteriorates the convergence rate, butthe system remained stable up to Ds ¼ 1000; it is worth notingthat the standard implicit formulation convergence stalled forDsP 100 at a value of 10�8.

The initial guess for the frequency j0 is normally chosen nearthe flutter onset frequency. The flutter onset can be predicted byseveral approaches as described in the introduction. Nevertheless,the sensitivity to this parameter is carried out by using½Ds;NH;Ni� ¼ ½50;3;100�, g0 ¼ ½0:2;0:02�T and j ¼ ½0:08;0:15�.Fig. 6 shows the initial frequency and updating strategy impacton the convergence rate. The inclusion of the force derivative inEq. (23) has a stronger influence on the convergence for poorer ini-tial frequency estimates. However, and unlike the original A-HB

method, this typically yields less than 10% computational savings,both in the number of iterations and in time. Overall, the LCO solu-tion converges within 160–220 iterations, even for the case wherethe initial frequency deviated approximately 50% from final solu-tion. The solution to this problem using the original A-HB methodtook nearly 1 h, the current method reduces the computationalelapsed time by approximately a factor of two.

The A-HB+ coupling process is a steady state computation,therefore, it is not necessary to drive the fluid residual to a verylow level for each sub-iteration, unlike time domain loosely cou-pled, aero-structural solvers. However, the number of iterations(Ni) has a significant impact on the convergence rate as shown inFig. 7; the inputs used are ½Ds;NH;j0� ¼ ½50;3;0:105� andg0 ¼ ½0:2;0:02�T . There is a marginal speed up in convergence whenincreasing Ni from 100 to 150, whereas there is a deterioration for

Iteration

Wall clock [s]

L2 n

orm

of

resi

du

al

50 100 150 200

0 500 1000 1500

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100κ=0.08κ=0.15

Iteration

Wall clock [s]

κ

50 100 150 200

0 500 1000 1500

0.08

0.1

0.12

0.14

0.16

κ=0.08κ=0.15

Fig. 6. (a) L2 norm of structural residual and (b) reduced frequency convergence with different initial frequencies (the symbols denote results that include the force derivativein the frequency updating procedure).

Iteration

L2 n

orm

of

resi

du

al

100 200 30010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

subit=50subit=100subit=150

Iteration

L2 n

orm

of

resi

du

al

100 200 30010-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100 subit=50subit=100subit=150

Fig. 7. L2 norm of (a) fluid and (b) structural residuals using different number of sub-iterations.

W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36 31

Ni ¼ 50. Therefore, Ni ¼ 100 is considered sufficient for the 2Daerofoil case.

The parameter sensitivity study shows that the method isrobust and accurate and provides a guideline to chose adequateparameter combinations.

3.2. Goland wing aeroelastic system

The second test case is the Goland+ wing structure (the +denotes the heavy version of the wing) with a tip store. The pur-pose of this problem is to investigate and exercise the A-HB+method for 3D fixed-wing LCO problems. This case has beenreported in the literature to exhibit aeroelastic instabilities, bothflutter and LCOs, driven by the presence and oscillation of shock-waves [32–34,1,35,36]. Results reported in the cited literature,show the presence of a transonic flutter bucket near M1 ¼ 0:92

where both modes 1 and 3 are lightly damped, driven by the pres-ence of a shock-wave near the trailing edge of the wing, which canresult in a LCO.

The wing is of rectangular shape, with a semi-span and chordlength (c) of 6.096 m, 1.8288 m, respectively, and the thickness-to-chord-ratio (s) is 0.04. The aerofoil profile consists of a parabolicarc, constant along the span, given by:

zc¼ �2s 1� x

c

� � xc

� �ð25Þ

A linear finite element model (FEM), shown here in Fig. 8(a), wasbuilt following the details provided by Beran et al. [33], but thepresence of the tip store is not reflected in the CFD mesh, as illus-trated in Fig. 8(b). The CFD mesh follows an O-topology and has atotal of 69,741 cells, with 81 points along the semi-span and 41along the chord direction. The structural mode shapes and natural

X

Z

Y

X

Z

Y

Fig. 8. Goland+ (a) FEM model and (b) CFD mesh.

Fig. 9. Structural normal modes projected onto the CFD grid: (a) Mode 1 (1.71 Hz), (b) Mode 2 (3.05 Hz), (c) Mode 3 (9.20 Hz), and (d) Mode 4 (10.90 Hz).

Iteration

L2 n

orm

of

resi

du

al

100 200 30010-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Iteration

κ

L2 n

orm

of

dis

pla

cem

ent

100 200 300

0.06

0.065

0.07

5

10

15

κL2 (η)

Fig. 10. L2 norm of (a) fluid and (b) structure residuals, U1 ¼ 137:16 m s�1.

32 W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36

frequencies are obtained by using the software MSC/NASTRAN. Thefirst four normal modes, depicted in Fig. 9, are retained for the LCOanalysis.

The problem is initially set up with the knowledge of the flutterconditions from the reported literature. The input parameters forthe A-HB+ method are ½M1;a� ¼ ½0:92; 0��; ½Ds;NH;Ni� ¼ ½25;3;50�,g0 ¼ ½2;1;0:1;0:1�T and j ¼ 0:06; in addition, all results employstandard sea-level air density. A reference LCO is sought atU1 ¼ 137:16 m s�1 which is about 15% above the flutter onsetcondition reported by Snyder et al. [32]. For these conditions, thestructural residual reaches 10�10 after approximately 200

iterations as shown in Fig. 10(a). The LCO frequency and amplitudeconverge as depicted in Fig. 10(b).

Next, further LCO conditions were computed for freestreamvelocities below and above 137:16 m s�1, which show the develop-ment of a supercritical LCO branch, Fig. 11(a), with the amplitudeincreasing rapidly beyond U1 ¼ 116 m s�1. The flutter and LCOconditions result from the interaction between mode 1 (first bend-ing) and mode 2 (first torsion). Results are consistent with thosereported by Beran et al. [33] using conventional time-domainmethods. There is a small change in the frequency between116 m s�1 and 125 m s�1, beyond this point the frequency

U [ms-1]

Gen

eral

ized

Co

ord

inat

e

110 115 120 125 130 135 1400

0.5

1

1.5

2

2.5

3 Mode 1Mode 2

U [ms-1]

Fre

qu

ency

[H

z]

110 115 120 125 130 135 1402.5

2.55

2.6

2.65

2.7

2.75

2.8

Fig. 11. LCO development with velocity: (a) amplitude and (b) frequency.

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

X Y

Z

0.6211 0.7431 0.8651 0.9872 1.1092

Fig. 12. Surface pressure flow field at HB time slices 2, 4 and 6; (a) U1 ¼ 116 m s�1, (b) U1 ¼ 120 m s�1, and (c) U1 ¼ 125 m s�1.

W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36 33

increases at an accelerating rate. The corresponding shock motionat U1 ¼ ½116;120;125�m s�1 is represented by the pressure fieldat three time-slices from the HB calculation in Fig. 12. AtU1 ¼ 116 m s�1, the shock exhibits small fluctuations, but itsposition remains largely unchanged. At U1 ¼ 120 m s�1, the wingtwist at the tip increases, the shock excursion at the wing root is

significant and during the cycle, a strong region of lower pressureextends towards the wing tip; overall significant changes in sur-face pressure occur both at the wing’s tip and root. At speeds ofU1 ¼ 125 m s�1, the shock at the root reaches the trailing edgeand it weakens as it moves forward, at U1 ¼ 137:16 m s�1 all butvanishes. For speeds above U1 ¼ 125 m s�1, the frequency starts

X

Y

-2 0 2

-2

0

2

Fig. 13. Mesh around cylinder for VIV problem.

X

Y

0.09 0.1 0.11 0.12 0.13-0.6

-0.4

-0.2

0

0.2

0.4

0.6TDCycle_HB5HB5

Fig. 15. Typical ‘‘8” trajectory of VIV system from TD and A-HB+ simulations.

34 W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36

to increase and the low pressure region observed at the wing tipleading edge, extends further inboard, originating large pressuredifferentials, sustaining higher amplitudes.

3.3. Circular cylinder VIV system

This case presents a different type of self sustained oscillations,i.e. driven by vortex shedding. For elastic immersed structures, thevortex shedding frequency can depart from the Strouhal frequencyand become synchronized, or locked in, with the frequency of thestructure’s motion; this nonlinear coupling between the vortexshedding and structure results in self limiting oscillations.Recently, Yao and Jaiman demonstrated how the lock-in frequencyand motion amplitude can be determined using the A-HB approachin Algorithm 1. In this section, the A-HB+ formulation is evaluatedfor VIV problems.

Consider the nondimensional structural equation for a vibratingcylinder with 2-DOF:

M€gþ f _gþ Kg ¼ f ð26Þ

Iteration

L2 n

orm

of

resi

du

al

500 1000 1500 200010-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

κ

Fig. 14. The convergence of (a) structu

where

M ¼ 1 00 1

� �; f ¼ �41pFs 0

0 �41pFs

� �; K ¼ �ð2pFsÞ

2 0

0 �ð2pFsÞ2" #

f ¼ 2pm

ClCd

� �; g ¼ x

y

� �;

here the x and y are the horizontal and vertical displacements; Cland Cd are the lift and drag coefficient, m; 1 are the mass ratioand damping coefficient, respectively. Following convention, thereduced natural frequency is defined as Fs ¼ 1=Ur , where Ur is thereduced velocity. The mass ratio m is defined as the ratio betweenthe vibrating structure and the mass of displaced fluid.

The fluid domain is discretized with 171� 65 points in the cir-cumferential and radial directions respectively, with the outerboundary located 50 diameter lengths away from the cylinder’scentre; the resultant mesh and point distribution near the surface

Iteration

L2 n

orm

of

dis

pla

cem

ent

500 1000 1500 20000.4

0.45

0.5

0.55

0.5

1

1.5

2

2.5

κL2 (η)

ral residual and (b) VIV response.

X

Y

0 2 4

-2

0

2

U

1.50831.34401.17981.01550.85120.68690.52270.35840.19410.0299

-0.1344

X

Y

0 2 4

-2

0

2

U

1.50831.34401.17981.01550.85120.68690.52270.35840.19410.0299

-0.1344

X

Y

0 2 4

-2

0

2

U

1.50831.34401.17981.01550.85120.68690.52270.35840.19410.0299

-0.1344

X

Y

0 2 4

-2

0

2

U

1.50831.34401.17981.01550.85120.68690.52270.35840.19410.0299

-0.1344

Fig. 16. Snapshots of the flow field horizontal velocity component: (a) upstroke; (b) maximum heave; (c) downstroke; and (d) minimum heave.

W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36 35

is illustrated in Fig. 13. The fluid problem is solved assuminglaminar flow.

The inputs to the A-HB+ solver are: ½m;Ur ; 1� ¼ ½10;6;0:0�,½M1;Re� ¼ ½0:3;100�, where Re corresponds to the Reynolds num-ber, ½Ds;NH;Ni� ¼ ½25;5;50� and with the initial displacement ofg0 ¼ ½0:1;0:5�T . The initial frequency f 0 ¼ 1=Ur is used for the VIVcomputation. Hence, for a given Reynolds number and remainderinput conditions, the A-HB+ solver determines the correspondinglocked-in frequency and amplitude, if they exist. Based on the find-ings by Yao and Jaiman [26], for 2-DOF vortex shedding problems,five harmonics are retained to fully capture the nonlinear motionand flow characteristics. This results in several more iterationsthan what was observed in the previous two cases, as shown inFig. 14. The trajectory obtained for the cylinder is shown inFig. 15; agreement with the time domain result is excellent. Illus-tration of the velocity field at four distinct points is shown inFig. 16, detailing the cylinder’s extreme heave positions andspeeds. The alternating vortex shedding on opposite sides of themotion cycle is clearly captured.

4. Conclusions

A new HB methodology to predict periodic, nonlinear fluid-structures interaction problems, was proposed and demonstrated.By implementing a particular implicit structural time integrator,that largely removes the restrictions on the scheme’s time step,the convergence of the HB equations of motion to the finalamplitude-frequency combination is accelerated. The necessity toapproximate force derivatives to converge on the final frequencyremains advantageous but is now secondary.

The proposed method was demonstrated for three test cases.The first case analyzed was a 2-DOF aerofoil undergoing an LCOthat was the focus of the original A-HB method study; a detailedparametric investigation of the new approach showed an improvedrobustness and consistent results over a wide range of conditions.The Goland+ wing allowed exercising the method for a fixed wingundergoing LCO at a transonic Mach number and over a range ofdifferent dynamic pressures. The last test case featured a fluid-structure instability driven by vortex-shedding exciting a self

36 W. Yao, S. Marques / Computers and Structures 201 (2018) 26–36

sustained oscillation. The ability of the A-HB+ to determine the socalled locked-in conditions is demonstrated for a 2-DOF vibratingcylinder. For all test cases, the A-HB+ provided detailed informa-tion on the periodic oscillatory flow characteristics and structuralmotion. Results indicate the computational effort was reduced bya factor of two with respect to the original A-HB method.

Acknowledgements

This work was sponsored by the United Kingdom Engineeringand Physical Sciences Research Council (Grant No. EP/P025692/1). The authors gratefully acknowledged this support.

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A harmonic balance method for nonlinear fluid structure interaction problems1 Introduction2 Aeroelastic - harmonic balance formulation2.1 Fluid governing equation2.2 Structural governing equations2.3 Harmonic-balance fluid structure coupling

3 Results3.1 Aerofoil aeroelastic system3.2 Goland wing aeroelastic system3.3 Circular cylinder VIV system

4 ConclusionsAcknowledgementsReferences