Applied Mathematical Modelling 28 (2004) 211–239

www.elsevier.com/locate/apm

A finite volume unstructured mesh approach to dynamicfluid–structure interaction: an assessment of the challenge

of predicting the onset of flutter

A.K. Slone *, K. Pericleous, C. Bailey, M. Cross, C. Bennett

Centre for Numerical Modelling and Process Analysis, University of Greenwich, The Old Royal Naval College,

Park Row, London SE10 9LS, UK

Received 2 April 2002; received in revised form 9 June 2003; accepted 31 July 2003

Abstract

Computational modelling of dynamic fluid–structure interaction (DFSI) is a considerable challenge. Our

approach to this class of problems involves the use of a single software framework for all the phenomena

involved, employing finite volume methods on unstructured meshes in three dimensions. This methodenables time and space accurate calculations in a consistent manner. One key application of DFSI simu-

lation is the analysis of the onset of flutter in aircraft wings, where the work of Yates et al. [Measured and

Calculated Subsonic and Transonic Flutter Characteristics of a 45� degree Sweptback Wing Planform in

Air and Freon-12 in the Langley Transonic Dynamic Tunnel. NASA Technical Note D-1616, 1963] on the

AGARD 445.6 wing planform still provides the most comprehensive benchmark data available. This paper

presents the results of a significant effort to model the onset of flutter for the AGARD 445.6 wing planform

geometry. A series of key issues needs to be addressed for this computational approach.

• The advantage of using a single mesh, in order to eliminate numerical problems when applying boundary

conditions at the fluid-structure interface, is counteracted by the challenge of generating a suitably high

quality mesh in both the fluid and structural domains.

• The computational effort for this DFSI procedure, in terms of run time and memory requirements, is

very significant. Practical simulations require even finer meshes and shorter time steps, requiring parallelimplementation for operation on large, high performance parallel systems.

• The consistency and completeness of the AGARD data in the public domain is inadequate for use in the

validation of DFSI codes when predicting the onset of flutter.

� 2003 Elsevier Inc. All rights reserved.

* Corresponding author. Tel.: +44-(0) 208-331-8554; fax: +44-(0) 208-331-8695.

E-mail address: a.slone@gre.ac.uk (A.K. Slone).

0307-904X/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0307-904X(03)00142-2

212 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

Keywords: Fluid–structure interaction; Finite volume; Transient structural dynamics; Geometric conservation law;

Newmark algorithm

1. Introduction

Fluid–structure interaction, as applied to flexible structures, has wide application in diverseareas such as flutter in aircraft, wind response of buildings and structures, flows in elastic pipesand blood vessels. A comprehensive computational model of these phenomena involving closelycoupled fluid flow and structural deformation, is a considerable challenge and, until recently,work in this area typically focused on one phenomenon and represented the behaviour of the othermore simply. However, strategies are now emerging for solving the full coupling between the fluidand solid mechanics behaviour. Solving dynamic fluid–structure interaction (DFSI) problems in ageneric time and space accurate fashion without making any simplifying assumptions in either thefluid or structural sub-domains has acquired an increasing degree of attention in the last few years[2–5]. Hughes et al. developed a family of FE based procedures with the SPECTRUM [2] code formodelling DFSI problems in the context of a more general �multi-physics� analysis framework[6,7]. Wall and Ramm [3] have pursued the FE method, with some considerable success, for two-dimensional dynamic fluid–structure problems, including Navier–Stokes flows, following theoverall arbitrary Lagrangian Eulerian (ALE) method of Farhat et al. [4], except that Farhats� teamemployed a mixed FE–FV formulation. The authors and their co-workers have pursued the de-velopment of multi-physics modelling and software tools [8–10] based on finite volume unstruc-tured mesh (FV-UM) techniques for two- and three-dimensional configurations. The low orderelements supported are constant strain triangular elements and bi-linear quadrilateral elements fortwo-dimensional analysis plus linear tetrahedral, bi-linear pentahedral and tri-linear hexahedralelements for three-dimensional analysis. The arising multi-physics simulation toolkit PHYSICA[9], has been used as the environment for modelling a range of DFSI process [11–15].The development of each of the above multi-physics software environments exhibits some

common features:

• Compatible methods of spatial discretisation, where either a FE or FV approach is used foreach of the physical phenomena.

• Similar solution procedures for all phenomena.• A single mesh encompassing all sub-domains.• A single database for the entire solution domain.

In the full spectrum of multi-physics modelling, the above features enable the implementationof volume sources representing the impact of one phenomena on another (for example, electro-magnetic fields coupled to fluid flow in magnetohydrodynamics (MHD) [16]) and in the imple-mentation of internal boundary conditions between sub-domains, each with its own particularphenomena (for example, in metal forming processes [17], which involves interaction between afree surface fluid and a structural domain). The compatibility of the mesh and the discretisationprocedure enables the exchange of volume sources and internal boundary conditions to be de-livered in a straightforward fashion in a space and time accurate manner. For the application of

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 213

multi-physics tools to DFSI simulation [2–5,11–15] there are a number of common features thatmay be identified, i.e., a single family of numerical procedures is employed for both the fluid andstructural sub-domains, where the structural deformation is calculated from the solution of thedynamic equilibrium equations. The calculated deformation subsequently may be used to com-pute the strain and stress throughout the structural sub-domain. Also the flow field is solved aseither an Euler or a Navier–Stokes fluid for a dynamically changing mesh representing the dy-namically changing geometrical domain. The procedure ensures that the mesh in the fluid sub-domain obeys the geometric conservation law and the coupling between the flow and structuralsub-domains is explicit [11–13,15].One application of these DFSI procedures is flow induced vibration of flexible elastic structures

and one classical example, in the context of aeroelasticity, is the consideration of the onset offlutter. Aeroelastic simulation has been the subject of a large research investment over many years[4,5,18–30]. The conventional approach to aeroelasticity simulation involves:

• The calculation of the natural frequencies and mass normalised mode shapes of the structuregeometry, (typically the wing planform) as a free undamped vibration problem [31].

• The re-formulation of the structural dynamic equilibrium equation in a normalised form, wherethe newly defined principal co-ordinates de-couple the structural dynamic equilibrium equa-tion.

• The load from the fluid on the structure, which for an Euler fluid is given by the fluid pressure,is normalised within the uncoupled equation of motion.

• The degrees of freedom may be reduced to retain only the vibrational modes of interest.

The impact of the structural deformation is coupled to the flow field by one of two methods.Either through a transpiration procedure [21,32], where the flow mesh is static and the effect of thedynamic structural deformation is accounted for at the fluid-structure interface by a velocityboundary condition or by directly reflecting the modification of the flow domain by deformingthe mesh [4,5]. Here, themeshmovement is accounted for using the static equilibrium equation witha pseudo-stiffness matrix, where the mesh is modelled using a linear spring approximation [18,33].Typically, the calculations are based upon the experimentally determined flutter conditions.

These calculations attempt to ensure that the model is consistent with the experimental data bypredicting that the oscillation will damp, be self-sustaining or grow, according to the flow con-ditions. This approach is able to represent the whole dynamic envelope, but its predictive capa-bility requires experimental data concerning the flutter conditions. Thus, its genuine predictivecapability is limited, with respect to the onset of flutter.The authors have been involved in the modelling of DFSI problems for a number of years [11–

15] and the techniques have been embedded within an open object based multi-physics modellingsoftware toolkit, PHYSICA [9]. The broad approach to DFSI considered here is conventional inthat the fluid and structure are solved in an iterative staggered manner [13,14]. The pressure andviscous stresses from the flow field provide load conditions for the structure, whilst, at the fluid–structure interface, the deformed structure provides boundary conditions from the structure to thefluid. The structure algorithm also provides the necessary mesh adaption for the flow field, theeffect of which is accounted for in the flow algorithm. The dynamic form of the solid equations issolved using a predictor–corrector version of the Newmark algorithm [14,15]. The mesh is moved

214 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

within the solid algorithm using the static equilibrium equation with a pseudo-stiffness matrix,which unlike the linear spring based methods [33,34], is achieved by defining separate values ofYoung�s modulus for each of the deforming regions of the mesh.This more generic multi-physics DFSI scheme does not require any experimental data con-

cerning the flutter envelope to predict the oscillatory behaviour of the structure, it only requiresthe flow conditions together with the geometric and material properties of the structure. Thus, thisscheme may have the potential to provide a genuine prediction of the onset of flutter without apriori knowledge of the experimentally measured flutter conditions. One set of experimental datathat has been extensively used in the verification of computational aeroelastic procedures is basedon the AGARD 445.6 wing planform carried out by Yates [1] at NASA in the early 1960s.Although the AGARD 445.6 flutter problem has been studied by a number of researchers, this

has been with varying degrees of success. Typically, the method involves the solution of the fluidflow using either the Euler or Navier–Stokes equations and a more simplistic representation of thewing planform dynamics. This approach has been taken by Lee-Rausch and Batina [22,23], whereupwind-spatial differencing is used to solve the three-dimensional time dependent Euler equationsand thin layer Navier–Stokes equations for fluid flow. The structural analysis is performed using atransient aeroelastic procedure, where the aeroelastic equations of motion are formed from a finitemodal series of free-vibration modes. Melville and Morton [35], employed a similar strategy toLee-Rausch and Batina, where the aeroelastic equations were solved on six overlapped sub-gridsfor parallel implementation. Raveh [29] employed a Reduced Order Model of the aerodynamicsystem, where the modal impulses from the first four elastic modes were introduced into the flowanalysis.Farhat et al. [4,5] employed a three-field ALE finite volume/finite element approach to the

solution of large scale three-dimensional nonlinear aeroelastic problems on high performancecomputational platforms. The Euler flow equations are solved using the monotonic upwindingscheme for conservation laws (MUSCL) on unstructured grids. The structural dynamic equilib-rium equations are solved using the finite element tearing and interconnecting (FETI) method.By use of the mid-point rule, both the flow continuity equations and the grid conservation laware enforced. Mesh movement is achieved using the Batina [33,34] method of a linear springsystem.Rifai et al. [30] employed the commercial code SPECTRUM [2] for the AGARD 445.6 flutter

case, where the finite element treatment of both the fluid and the structure was based on the Hu-Washizu variational principle. The computational domain was divided into three distinct regions,i.e., the fluid region, the solid region and an interface between the fluid and solid, thus allowingvariable mesh densities and element topologies in the three regions. The interface was used toenforce the coupling constraint between the fluid and the solid and the wing was perturbed with acombination of the first two vibration modes applied as initial velocities.

2. Governing equations

The coupled DFSI problem may be considered as a three field problem, i.e., fluid flow,structural deformation and the moving mesh [4,36], see Fig. 1. The governing equations for thethree-field problem are:

Fig. 1. Dynamic fluid–structure interaction.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 215

• The Navier–Stokes equation for compressible flow given by:

o

otðquiÞ þ r � quui ¼ �rp þr � lrui þ F fðtÞ;

and the associated continuity equation given by:

oqot

þr � qu ¼ 0; ð1Þ

where u is velocity, q is density, l is dynamic viscosity, p is pressure, ui is the Cartesian com-ponent of velocity u corresponding to direction xi and F fðtÞ is transient load vector defined forthe fluid. For dynamic meshes, the velocity is expressed relative to the movement of the mesh,u ¼ ufluid � umesh.

• The equation governing the dynamic response of a structure or medium may be written in ma-trix form [37] as:

M€dd þ C _dd þ Kd ¼ FsðtÞ; ð2Þ

where d is the displacement vector,M is the mass matrix, C is damping matrix, K is the stiffnessmatrix and FsðtÞ is the transient load vector defined for the solid, where Fs ¼ Fext þ F int. In thecontext of fluid–structure interaction, the external load vector, Fext is composed of the tractionboundary condition imposed by the fluid and any other external applied loads on the structure.F int is the internal force and is composed of the internal stresses within the structure.Damping within the structure is approximated by Rayleigh damping, where the damping

matrix C in Eq. (2) is formed as a linear combination of the stiffness and mass matrices as:

C ¼ arM þ brK ; ð3Þ

where ar and br are the mass and stiffness proportional damping constants.

• Mesh movement may be modelled as a pseudo-structural problem with its own dynamics, witha spring based mesh movement [34], governed by:

Kmdm ¼ fmðtÞ; ð4Þ

216 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

where Km is a pseudo-structural stiffness matrix which is defined for the whole domain and dmis displacement of the mesh, which in the context of fluid–structure interaction, is the dis-placement of the structure in the structural sub-domain.

• In the absence of any other boundary conditions, the method for integrating the flow equationsshould preserve the solution of the Navier–Stokes and continuity equations. This condition issatisfied only when the method for solving the flow equations and the algorithm for updatingthe displacement and velocity of the mesh obey the geometric conservation law [24,36].

The solution algorithm has been implemented using FV-UM techniques within the PHYSICAcode [8,9] and is shown in Fig. 2, from which it may be seen that differential time stepping betweenthe fluid and the structure is allowed. This algorithm is described in detail elsewhere together withits evaluation on a simple geometry cantilever [11–15]. Hence, we simply provide an outline of thescheme here to inform the discussion below when considering its use in modelling the behaviour ofthe AGARD 445.6 wing planform.

3. Navier–Stokes fluid flow solution procedure

The transient fluid flow is governed by the Navier–Stokes momentum and continuity equations.The method of discretisation for the flow equations ensures that the scheme is fully conservativefor domains of arbitrary shape which move in any time-varying fashion [38]. The resultingequations are integrated over each control volume such that the solution sets each of the integralsto zero. By making assumptions concerning the required inter-nodal values, a linear equation isobtained for each control volume in terms of the unknown scalar quantity at the nodes and for thenodes on the adjacent control volumes. For fluid flow and heat transfer the standard boundaryconditions allowable in PHYSICA are those for an inlet, fixed pressure boundary, wall andsymmetry plane. Information in terms of external velocity components, pressure or temperature,is introduced into the equations for those control volumes that have a face coincident with adomain boundary. The set of linear equations for each control volume are assembled into a globalsystem matrix for each of the solved fluid variables, i.e temperature and velocity, using standardcomputational finite volume, unstructured mesh techniques [39]. Based on the continuity equationa pressure correction formulation is applied and the resulting discretised equations are solvedusing an algorithm based on the semi-implicit method for pressure linked equations (SIMPLE) ofPatankar and Spalding [40].

3.1. Flow equations for a moving mesh

For a moving reference frame, the equations governing fluid flow are expressed in terms of thefluid variables relative to the mesh movement i.e. u ¼ uf � um where uf is the velocity of the fluidand um is the velocity of the mesh. Consider the momentum conservation equation for fluid flowand associated continuity equation. For a moving reference frame the appropriate equations are:

o

otðquiÞ þ r � ðqui½uf � umÞ ¼ r � ðlruiÞ �

opoxi

þ Sui ; ð5Þ

Fig. 2. Solution algorithm.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 217

with the associated continuity equation:

oqot

þr � ðq½uf � umÞ ¼ 0; ð6Þ

where l is the dynamic viscosity, q is the density, p is the pressure, u is the resultant face velocity,ui is the Cartesian component of velocity u corresponding to direction xi and Sui is the source term

218 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

for the ith component. At the fluid–solid interface the velocity of the fluid is that of the movingmesh thus imposing a further boundary condition for the fluid:

ubnd ¼ um on Cs: ð7Þ

In addition, a typical DFSI simulation is also subject to the standard boundary conditions forfluid flow, i.e., a mass inflow fixed velocity boundary, a fixed value pressure boundary, a no-slipwall boundary and a symmetry plane, or in other words, an inlet, a free-boundary, symmetryplanes and a wall, see Fig. 4.By comparison with the standard Navier–Stokes momentum and continuity equations, the

moving reference frame has introduced a further flux term of �r � ðquiumÞ to the momentumconservation equation and of �r � ðqumÞ to the continuity equation.It is essential that the calculation of the mesh velocity obeys the geometric conservation law

[4,41], which may be stated as: the change in volume (area) of each control volume between time tn

and tnþ1 must equal the volume (area) swept by the cell boundary during rt ¼ tnþ1 � tn. Thus:

oVot

þr � um ¼ 0; ð8Þ

where V is the volume of the control element. Integrating Eq. (8) over an arbitrary control volumeand applying Gauss� divergence theorem gives the surface mesh flux as [41]:

I

Cum � ndC ¼ DV

Dt: ð9Þ

Eqs. (5) and (6) are integrated over an arbitrary control volume and are discretised using thestandard control volume unstructured mesh techniques [42] to give:

Z

cv

oquiot

dX þI

Ccv

qui½uf � um � ndC ¼I

Ccv

lðrui � nÞdC þZ

Xcv

SuidX; ð10Þ

where n is the unit normal to the face of the control volume. Integration of the continuity equation(6) gives:

Z

Xcv

oqotdX þ

ICcv

q½uf � um � ndC ¼ 0; ð11Þ

where the face velocities are adjusted for mesh movement using Eq. (9).

4. Dynamic structural deformation solution procedure

The equilibrium equations of structural mechanics may derived from Cauchy�s equation ofmotion, which is the governing equation concerning the conservation of momentum for either asolid or a fluid in motion and are given by:

LTr þ b� qs€dd ¼ 0 in Xs; ð12Þ

where b is the body force, €dd is the acceleration and qs is the density of the structure. The dis-placement formulation is derived [43] by use of a finite set of weighting functions Wi to give:

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 219

�Z

Xs

½LWiT½DLd�D�0dX þZ

Xs

WTi bdX �

ZXs

WTi qs€dd dX þ

ZCd

½TTWiT½DLd�D�0dC

þZ

Ct

WTi tp dC; ð13Þ

for i ¼ 1; n where D is the constitutive matrix, L and T are respectively the linear differential andthe outward unit normal operators. r, � respectively, are the stress and strain vectors. Boundaryconditions on the surface Cs of the structural domain Xs are described in terms of prescribedtractions tp on the boundary Ct and prescribed displacements dp on the boundary of the structureCd, where Cs ¼ Cd [ Ct [13]. The essential difference between the finite element method (FEM) andthe finite volume method (FVM) is that for FEM the weighting functions for a node are equal tothe shape functions for that node, i.e., Ni, whereas for cell-vertex FVM the weighting functionsare unity within the control volume and zero elsewhere, i.e. Wi ¼ I within the cv and Wi;¼ 0elsewhere [13].The standard boundary conditions in the structural algorithm are in terms of a specified dis-

placements, concentrated loads and nodal constraints. At the fluid–structure interface thestructure experiences a surface traction due to the fluid, hence the appropriate boundary conditionfor the structure at the fluid–structure interface is expressed as:

tp ¼ �pdij þ louioxj

�þ oui

oxi

�� 2

3lr � udij on Cs: ð14Þ

4.1. Temporal discretisation

The structural algorithm employs a Newmark scheme for temporal discretisation [13,14,44].The Newmark scheme is one of the most popular algorithms in structural dynamics. It is atruncated Taylor series collocation algorithm with quadratic expansion, which is the minimumrequirement for second-order problems. The one-dimensional Newmark scheme is given below:

dnþ1 ¼ dn þ Dt _ddn þDt2

2½ð1� 2bsÞ€ddn þ 2bs€ddnþ1; ð15Þ

_ddnþ1 ¼ _ddn þ Dt½ð1� csÞ€ddn þ cs€ddnþ1; ð16Þ

where bn and cn are specific to the Newmark scheme and are chosen to control stability andaccuracy. The scheme is implicit for bn > 0 and is unconditionally stable for 2bn P cn wherecn P 1

2. For bn ¼ 1

4and cn ¼ 1

2, i.e. a constant-average-acceleration or trapezoidal scheme, it has

zero dissipation for all choices of time step, hence bn ¼ 14and cn ¼ 1

2have been chosen with

consideration of stability, accuracy and dissipation.Applying linear interpolation to Eq. (2) to advance from time tn to time tnþ1 ¼ tn þ dts, where dts

is the structural time step, gives the dynamic solution at time tnþ1 as:

M€ddnþ1 þ C _ddnþ1 þ K Dd ¼ Fextnþ1 þ F int

n ; ð17Þ

where Fext

nþ1 is the external force at tnþ1 which in the context of fluid–structure interaction iscomposed of the traction boundary condition imposed by the fluid and any other external appliedloads, F int

n is the internal force at tn and is composed of the internal stresses within the structure.

220 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

Substituting the Newmark equations (15) and (16) in Eq. (17) gives:

K

�þ 1

bDt2M þ c

bDtC

�Dd0 ¼ F 0

nþ1; ð18Þ

where

F0nþ1 ¼ Fext

nþ1 þ F intn þM� þ C�; ð19Þ

M� ¼ M1

bDt_ddn

�þ 1

b1

2

�� b

�€ddn

�; ð20Þ

C� ¼ Ccb_ddn

�þ c

b1

2

�� b

�Dt€ddn

�: ð21Þ

Solving Eq. (18) for the incremental displacement Dd0, and rearranging Eq. (15), the nodalacceleration, velocity and displacement are updated using standard Newmark methods [45],yielding a first approximation to the acceleration, velocity and displacement for the currenttime step, which is then updated until convergence is achieved using the equilibrium iterationscheme:

K

�þ 1

bDt2M þ c

bDtC

�Ddi ¼ F i

nþ1; ð22Þ

where

F inþ1 ¼ Fext

nþ1 þ F int;inþ1 þMi€dd i

nþ1 þ C _dd inþ1: ð23Þ

The convergence criterion used is based on the displacement Euclidean norm

kDdikdmax

6 toldis and on the energy normðDdiÞTF i

nþ1

ðDd0ÞTF0nþ1

6 tolenergy;

where kDdik is the Euclidean norm of the incremental displacement for the ith equilibrium iter-ation; dmax is the maximum total displacement up to and including the current iteration; toldis andtolenergy are the tolerances set for each case, typically of the order 10

�3 and 10�2, respectively.Divergence is defined in terms of an increasing residual energy norm kF0

nþ1k6 kF iþ1nþ1k and, if the

solution is divergent or is failing to converge within a specified number of equilibrium iterations,the stiffness matrix is reformed using the updated geometry and the equilibrium iterations arecontinued, thus forming a predictor–corrector variant of the standard Newmark scheme [13].

5. Dynamic mesh movement

Mesh movement is governed by the static displacement equation as given by Eq. (4) and may bemodelled as a pseudo-structural problem with its own dynamics. At each time step the followingstatic equilibrium equations are solved:

KmnðtÞ ¼ f ðdðtÞÞ; ð24Þ

Fig. 3. Mesh movement regions.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 221

where Km is a pseudo-structural stiffness matrix which is defined for the entire fluid–structuredomain, nðtÞ is the mesh displacement at time t, the function f ðdðtÞÞ is the given displacementfunction at time t, which for fluid–structure interaction is governed by the movement of thestructure.The objective is to move the mesh close to the structure such that the integrity of the mesh is

preserved and to feed this mesh movement out smoothly to the remainder of the domain such thatfar from the structure mesh movement is negligible. However, moving the entire mesh for a DFSIproblem would be a significant overhead in terms of run time and in practice it is only necessary tomove the mesh in the region close to the structure. This is achieved by defining four regions of thepseudo-stiffness matrix Km, i.e. the structure, the boundary region, a flexible region, subject todisplacement but no additional stress, and a fixed region, see Fig. 3.To ensure that the boundary region moves in the same manner as the structure it is given the

same Young�s modulus as the structure. The flexible region is given a Young�s modulus that is asmall proportion of the Young�s modulus of the structure so as to allow mesh movement such thatthe mesh integrity is maintained and the deformation at the interface with the boundary region isspread smoothly throughout. Typically, the flexible region is given a Young�s modulus 1% of thestructure, since this generally allows the mesh to deform without collapsing. The fixed region istreated as a fixed boundary with zero displacement where dm ¼ d on Xs and dm ¼ 0 on Xfixed.

6. Test cases

The ultimate goal for this research is to investigate the oscillatory characteristics of complexflexible structures subject to fluid loadings using a finite volume approach, where the onset ofaircraft flutter serves as a good example. A simple representation of an aircraft wing is a fixed-freecantilever and a standard problem in structural mechanics is the fixed-free cantilever supportingan applied load at the free end. The structural algorithm used here has been shown to be very

222 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

accurate in predicting the fundamental natural frequency of simple structures with respect to boththe amplitude and period of oscillation [14]. This simple problem was then extended to one of athree-dimensional loaded fixed-free cantilever in fluid flow [13,15] and a sample of the results isincluded below.

6.1. Simple geometry configurations

The three-dimensional loaded cantilever in fluid flow is shown in Fig. 4 and the boundaryconditions are a mass inflow fixed velocity boundary at the high y boundary (inlet), a fixed pressureboundary at the low y boundary (free-boundary), a wall boundary at the low x boundary wherethe structure has a zero displacement boundary condition and symmetry planes for the remainingdomain boundaries. The problem was meshed using 76,844 brick elements and the overallproblem dimensions were: depth 20.0 m, breadth 36.0 m and length 51.0 m. The dimensions of thecantilever were: depth d ¼ 2:0 m, breadth b ¼ 2:0 m and length l ¼ 20:0 m and the cantilever wassubjected to an applied ramped load of 1.05 MN at the free end, which was subsequently re-moved. The material properties of the cantilever were: Young�s modulus 21.0 GPa, Poisson�s ratio0.3 and density 2600 kgm�3.The analytic solution to the structural problem in the absence of fluid is given by standard text

[46,47] as a period of oscillation of 0.4356 s and an amplitude of 0.1 m. To capture the sinusoidalmotion of the cantilever, the time step was taken as 0.022 s, which was expected to give ap-proximately 20 time steps per cycle. Two Reynolds number cases of Re ¼ 200 and Re ¼ 2:0 105

were studied, where Re ¼ Uindm and Uin ¼ 100:0 m s�1 is the fluid inlet velocity, m is kinematic vis-

cosity. Four cases were studied for m ¼ 1.0 and 1 · 10�3 m2 s�1 and fluid density q ¼ 1:0 and 10.0kgm�3. The displacement at the tip of the cantilever in the neutral z plane is shown in Fig. 5,where the effect of the fluid traction load on the cantilever may be seen for the differing Reynoldsnumber cases. There is little difference in the displacement of the tip for fluid density of 1.0 kgm�3

but there is a significant difference in the displacement for the two Reynolds number cases at fluid

Fig. 4. Schematic three-dimensional cantilever in fluid flow.

Fig. 5. Comparison of cantilever tip displacement.

Fig. 6. (a) Re ¼ 200 and (b) Re ¼ 2:0 105.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 223

density 10.0 kgm�3. The shear xy stress for these two cases may be seen in Fig. 6. The greaterdisplacement of the tip for Re ¼ 200 is reflected in the increased shear stresses in comparison withthe Re ¼ 2:0 105 case. The flow field at Re ¼ 200 shows little difference for the two density cases.The flow field for Re ¼ 2 105 in the yz plane at 0.75 cantilever lengths is shown in Figs. 7 and 8.The above analysis of the simple cantilever illustrates that the FV-UM based DFSI procedure

can reliably represent the oscillatory behaviour of the structure, the flow field and their mutualinteraction.

6.2. Complex geometry case––the AGARD 445.6 wing planform

In real, three-dimensional, engineering applications the structure will have several degrees offreedom, each of which will be subject to a traction load from the fluid, which will influence the

Fig. 7. Re ¼ 2 105, velocity vectors, in yz plane at 0.75 cantilever lengths.

Fig. 8. Re ¼ 2 105, pressure contours in yz plane at 0.75 cantilever lengths.

224 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

corresponding mode of vibration. Hence, the structure is unlikely to be in phase with the fluid forall modes of vibration. It is phase difference, rather than amplitude, which governs the onset offlutter [48] and, if the phase difference between the fluid force and the structure velocity exceeds90�, flutter will occur. Models based on a finite volume fluid analysis and a separate finite elementstructural analysis, may introduce spurious phase mismatching between the fluid and the structure[49], which may eliminated by using an integrated (finite volume) technique for both domains.One of the most frequently used benchmarks in aeroelasticity is the AGARD 445.6 wing. This

test case has a panel aspect ratio of 4.0, a quarter-chord sweep of 45� and a taper ratio of 0.6, i.e.,445.6, and a NACA 65A004 aerofoil section. The semi-span of this model was 2.5 feet and theroot chord was 1.833 feet. In the experimental test case, the wall mounted wing was constructed oflaminated mahogany. In order to obtain flutter data throughout the tunnel Mach number anddensity ranges, the stiffness of the wing was reduced by drilling holes, normal to the chord plane,

Fig. 9. AGARD 445.6, weakened model 3.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 225

see Fig. 9, to enable a total of six weakened models to be studied for various panel masses. Theflutter characteristics of this wing planform in air and freon-12 were investigated at the Langleytransonic dynamics tunnel, by Yates et al. in 1963 [1], for Mach numbers in the range 0.338–1.141.The first four coupled vibrational modes had been previously evaluated by Jones and Unangst [50]in 1956, where they describe a method for uncoupling these modes. It is unclear in the Yates et al.paper of 1963 whether the uncoupled vibration frequencies quoted were measured by them orwere based on the results of Jones and Unangst. The distributions of bending and torsionalstiffness, EI and GI were measured and flutter characteristics were calculated using the method ofmodified strip analysis.Some 24 years later, based on his 1963 paper, Yates [51] calculated the flutter characteristics,

using finite element analysis, where the Young�s modulus of the aerofoil was calculated so as toensure that the calculated vibration modes agreed with those calculated/measured in his earlierpaper. Thus, the wing material properties were: density 4.15· 102 kgm�3, Young�s modulus3.2456 GPa and Poisson�s ratio 0.3. In the subsequent flutter calculations, the material propertiesfor the weakened models were treated as homogeneous and the primary purpose of this work wasto provide aerofoil boundary co-ordinates for subsequent aeroelastic flow calculations.Typically in aeroelastic simulations, the modal frequencies are taken as known boundary

conditions to the flow domain and Yates findings of 1963 and 1987 have been used for thispurpose to investigate the flutter characteristics of the AGARD 445.6 wing [22,23]. Thus this typeof simulation of the AGARD 445.6 wing flutter characteristics is guaranteed to observe the samemodal frequencies as Yates. But for present day researchers, who are seeking to perform a fullDFSI simulation of the AGARD 445.6 wing flutter characteristics and only have access to dataheld in the public domain, the data presented by Yates is a source of uncertainty. The effectivevalue of Young�s modulus is dependent on Yates� calculations of 1987, which are themselvesdependent on data collected in 1955. There is uncertainty about the uncoupled modal frequencies,since it is unclear whether they have been calculated or measured at some previous time, more-over, the method of drilling holes in the mahogany must have implications for the homogeneity ofthe aerofoil.

226 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

The work presented here sought only to capture the first mode of vibration and the time stepwas calculated with this goal in mind. The first uncoupled bending frequency for the AGARD445.6 wing, as given by Yates [1,51] for weakened model 3 was 9.6 Hz, hence the time step was setat 0.005 s, yielding approximately 20 time steps per cycle and this time step was also applied to thefluid flow calculations.The test presented is for incompressible flow at Mach number, M1 ¼ 0:338. The fluid material

properties were: density 1.225 kgm�3 and kinematic viscosity 1.461· 10�3 m s�2 and the wingmaterial properties were as given by Yates [51] for weakened model 3, i.e. Young�s modulus3.2456 GPa and Poisson�s ratio 0.3. The mesh for this problem was generated using the com-mercial package GRIDPRO [52] with additional PHYSICA specific software [12,13] and may beseen in Fig. 10. For the test case reported here, the wing is at a 5� angle of attack, with 25 nodesfrom its root to its span-wise boundary and 146 nodes wrapped around the surface. The mesh forthe entire domain had 89,958 hexahedral elements with 95,064 nodes. As such it was near to themaximum memory (l Gb) of a Dec Alpha 466 MHz processor of 100 K elements for singleprocessor implementation. The wing mesh had 5088 hexahedral elements and 7100 nodes. Theboundary layer region of the mesh, where the mesh movement is tightly coupled to the movementof the wing, had 19,806 elements, the moving region of the mesh had 42,186 elements and the fixedregion, where no mesh movement is allowed, had 22,878 elements.The boundary condition follow those for the schematic three-dimensional loaded cantilever in

fluid flow of Fig. 4, but in a different orientation. There is a mass inflow fixed velocity boundary atthe low x boundary (inlet), a fixed pressure boundary at the high x boundary (free-boundary), astationary wall boundary for the entire domain at the low z boundary, where the structure hasfixed zero displacement at its root and symmetry planes for the remaining domain boundaries. Atthe start of the full DFSI simulation, the wing planform was subject only to the fixed displacementcondition at its root.The full DFSI problem was run from the static flow only solution for 30 time steps, making a

total simulation time of 0.155 s and the average run time on a Dec Alpha processor was in excessfor 460 h per cycle of oscillation. The maximum displacements are at the trailing edge of the wing

Fig. 10. Fluid and wing mesh.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 227

tip and are shown in Fig. 11, the shear xy stresses in the wing at 0.105 s are shown in Fig. 12. Thesimulation was stopped at 0.55 s as the aerofoil displacements seemed to have reached an equi-librium position of 0.4791 cm by 0.09 s and the solution was not significantly changing, as may beseen in Fig. 11a. The pressure and velocity of the fluid at the wing tip at 0.105 s are shown in Figs.13–15.It became apparent in the flow simulation that the mesh was not sufficiently fine to capture the

physics of the flow field at the trailing edge of the wing and poor aspect ratios were observed in theflow domain in the wake region, just downstream from the trailing edge of the wing tip [13]. Thesepoor aspect ratios give a CFL of approximately 3· 104 based on the time step for the fundamentalfrequency of the wing. A more detailed flow field may have implications for the oscillatory be-haviour of the wing and when considering the portion of the mesh for the AGARD 445.6 wing inisolation, it also became apparent that the quality of the mesh, in terms of aspect ratios andorthogonality, was an issue.

Fig. 11. Mach 0.338, wing tip displacement at trailing edge.

Fig. 12. Shear XY stress, AGARD 445.6 wing.

Fig. 13. Mach 0.338, flow field at tip of AGARD 445.6 wing, 0.105 s.

Fig. 14. Close-up of Fig. 13 at leading edge, 0.105 s.

228 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

6.3. Torsionally loaded wing

For the case presented above, the fluid solution was not changing appreciably with time andwas providing an almost constant force to the aerofoil, so that the aerofoil eventually came to anequilibrium position, as seen in Figs. 11a and b, and the aerofoil is not displaying flutter char-acteristics.For comparison purposes, it was desirable to use the data given by Yates [51] but one inter-

pretation of the displacements shown in Fig. 11 is that the time step is too large. The time step hadbeen set in accordance with the first coupled bending mode of 9.6 Hz, as calculated by Yates [51],however, the fluid load on the aerofoil is bending and torsional. Thus, it was decided to maintainthe material properties as given by Yates and to investigate the behaviour of the aerofoil inisolation from the flow, i.e. subjected to a physical load but no fluid flow. The aerofoil was treated

Fig. 15. Close-up of Fig. 13 at trailing edge, 0.105 s.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 229

in isolation as a loaded cantilever subjected to an applied torsional load and the time step wasdecreased.The swept-back aerofoil was subjected to a variety of applied torsional loads at the centre of the

wing tip. After several static runs it was found that an applied load of Fs ¼ ð35iþ 35jÞN resultedin a static displacement of 1.11 cm, which is close to the displacement of 0.5 in s or 1.27 cmobserved by Yates [1,51]. Several time steps were investigated and it was found that sinusoidalmotion was exhibited for a time step Dt ¼ 7:878 10�4 s. Lee-Rausch [22,23] employed numericaldamping to aid stability and it was found necessary to employ a stiffness damping parameter of0.011% to ensure no amplitude growth. Using these parameters the problem was run for 80 timesteps, making a total simulation time of 0.063 s. The maximum amplitude of 1.11 cm is again closeto the displacement of 0.5 in s or 1.27 cm observed by Yates [1,51]. The displacements at thetrailing edge of the tip of the torsionally loaded wing are shown in Fig. 16. The displacements

Fig. 16. Torsional loaded aerofoil, wing tip displacement at trailing edge.

230 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

show no amplitude growth and after the first 0.0126 s the frequency of the aerofoil is unaffected bythe initial loading conditions and the period is virtually constant at 0.016 s.A full DFSI simulation was then performed for the torsionally loaded AGARD 445.6, a ¼ 5:0�,

in Mach 0.338 flow with a time step of 7.878 · 10�4, for both the fluid and the aerofoil, but theflow failed to converge. Hence, a further case with a 10:1 differential time stepping between thefluid and that solid was performed, but again the flow failed to converge and relatively large massfluxes due to mesh movement were observed, indicating that some faces in the flow domain wereundergoing significant movement relative to their size. As the mesh moves, the volume of anindividual cell may change appreciably, even though mesh integrity may still be maintained. If thedimensions in a cell are significantly decreased in the direction of the flow, the CFL condition maybe violated and the flow algorithm may fail to converge.The default value of the pseudo-Young�s modulus for the movable region of the mesh was 1.0%

of the structure and this value was increased to 10% of the aerofoil. This had the effect of ensuringthat close to the aerofoil, where the mesh was fine, the mesh movement was almost a simpletranslation, thus reducing the volume change in the movable region, which in turn limited themass flux due to mesh movement. To aid the flow solution the time step was decreased toDt ¼ 7:878 10�5, which gave 200 time steps per cycle for the torsionally loaded wing and a CFLfor the flow of approximately 450. The problem was run as a full DFSI simulation for 0.015756 s,i.e. 200 time steps, on a 466 MHz Dec Alpha processor where the total run time was 460.58 h. Thetorsional load was linearly applied for the first 80 time steps and then released.The maximum displacements are at the leading edge of the tip of the wing and are shown in Fig.

17, which also shows the maximum displacements for the torsionally loaded wing with no fluidflow. The fluid is providing positive lift to the trailing edge of the wing tip and has increased themaximum displacement from the equilibrium position from 1.11 cm at 0.007878 s to 1.27 cm at0.009296 s. By the end of the simulation, at 0.015756 s, the difference in the displacement betweenthe two cases is 0.7776 cm. Fig. 17 shows the maximum amplitude at the tip of the wing occurring

Fig. 17. Comparison of displacement for torsional loaded aerofoil.

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 231

at 0.0093 s, where it may also be seen that the fluid load has increased the period of vibration to aprojected figure of 0.025 s for the first cycle of oscillation.The maximum shear xy stresses occur close to the root of the wing near the trailing edge and the

minimum occurs near the leading edge close to the mid-point of the span. Fig. 18 shows themaximum and minimum shear rxy stresses on the wing at 0.0093 s, when the tip is at its maximumdeflection. These figures also display the relative position of the root and mid-span of the wing.Again, the fluid is providing positive lift to the wing so that mid-way along its span the wing hasbeen lifted, but the leading edge displacements are smaller than those at the trailing edge, thusdecreasing the angle of attack. Figs. 19 and 20 show the fluid pressures and velocity at the wing tipfor 0.0093 and 0.0158 s.From examination of Fig. 17, it is possible that the fluid loaded wing will oscillate around a

non-zero equilibrium position with the fluid eventually damping the motion, but, in view of therun time of over 460 h per cycle, it was not practical to perform the significantly longer runsrequired to establish the full extent of the nature of the oscillatory behaviour.

Fig. 18. AGARD 445.6 wing, shear stress rxz at 0.0093 s.

Fig. 19. Flow field at tip of torsionally loaded wing 0.0093 s.

Fig. 20. Flow field at tip of torsionally loaded wing 0.0158 s.

232 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

7. Issues in modelling complex cases

DFSI problems generally require a finer mesh in the structure and surrounding boundary layerthan in the remainder of the mesh which for even moderately complex geometries may lead to arapid increase in the size of the problem and run times. There are few, if any, mesh generatorscapable of providing a single unstructured mesh suitable for DFSI problems for anything otherthan the simplest geometries.For the dynamic structural mechanics problem, there are a number of analytic benchmark cases

for simple geometries, but to date no such standard benchmarks have been found for DFSI. How-ever, for this work three main areas of concern in modelling DFSI problems have been identified:

• Mesh quality for complex cases.• Excessive run times associated with such cases.• Availability of experimental data.

These issues are addressed below.

7.1. Mesh quality for AGARD 445.6

The two main criteria for assessing the quality of the mesh are the aspect ratio and orthogo-nality of the mesh. Poor aspect ratios decouple the solution in the direction of the smallest side forboth the fluid and the structure and may mask convergence in that the solution is converged onlyin the direction of the largest side. Non-orthogonality in the mesh requires additional correctionterms in the flow equations, increasing the complexity of the calculation for fluid flow.Skewness of the mesh, in terms of non-orthogonality, has implications for fluid flow. When

integrating the Navier–Stokes equations for fluid flow, the cell-centred finite volume discretisationrequires an estimate of the gradient of the face velocity, ou

on, where n is the normal vector to the face.For a structured mesh the line connecting the cell centroids either side of the face is parallel to theface normal so that:

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 233

ouon

¼ uA � uPd

; ð25Þ

where d is the distance form the cell centroid P to the neighbouring centroid A, sharing the face.However, if the face is non-orthogonal, the line connecting the nodes P and A will not be parallelto the face normal, see Fig. 21 so that:

ouov

¼ uA � uPd

; ð26Þ

where v is the vector from node P to node A. Hence, the non-orthogonality of the mesh requires anadditional source term, which is a function of the tangential gradient o

ot, where t is the tangent to theface. At the start of any simulation the velocity gradients may be artificially high and inclusion ofthe orthogonality correction terms would result in reducing the contribution to the system matrixand introducing large source terms. Convergence may be slow and the solution may even diverge.Despite much effort in generating the mesh, the maximum angle between the face normal and

the line connecting the adjacent cell centroids for the mesh in the fluid domain was found to be71.5�, which occurs four cells from the tip of the wing in the span-wise direction i.e., in themovable region, however all the face angles in the same position along the span of the wing are inexcess of 69.0�.The minimum aspect ratio of the wing is 4.19 and is near the root. The maximum aspect ratio of

the wing is 2790 and is close to the tip on the trailing edge. Large aspect ratios will compromisethe accuracy of the solution for any structure and typically finite element analysis codes warn foraspect ratios in excess of 20. The large aspect ratios in the AGARD 445.6 case were a directconsequence of the need to avoid skewness in the mesh while preserving the geometry of theaerofoil and to keep within single processor memory limits. Thus it was felt that the mesh for thiscomplex case was of as good quality as was possible within the limits of available mesh generationtechnology and those imposed by single processor implementation.Using the frequencies and material data claimed by the experimental data of Yates, the

AGARD 445.6 test case did not exhibit flutter, which may have been influenced by inadequaciesin the data and the resolution of the mesh. The block structured nature of GRIDPRO limited the

Fig. 21. Two-dimensional representation of orthogonality.

234 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

way in which the fine grid in the aerofoil could be expanded in the fluid domain, hence, the finegrid was carried far into the fluid domain. There are two possible ways forward for addressing theproblem of mesh generation for complex geometries:

• Using two meshes i.e. one for the fluid and one for the structure, as used by Farhat et al. [4].• Employment of high quality mesh generators.

The use of two, discontinuous meshes would eliminate the problem of feeding a detailed meshfor the structure into the fluid domain and would allow an optimum mesh in each sub-domain.However, using it would also require special effort in terms of an accurate interpolation betweenthe two sub-domains and the two-way exchange of information at the fluid-structure interface. Toensure the accuracy of the particular DFSI boundary conditions, for those cases where the dis-placement of the structure is small in comparison with the machine accuracy, it would be nec-essary to execute the entire software in double precision. In addition, the transfer of data from onesub-domain to the other would require maintaining a data map between the two sub-domains andwould increase the run time and the amount of storage required.Obviously, the problems of large aspect ratios and non-orthogonality of the mesh may be re-

duced by increasing the size of the mesh, which is possible in the context of parallel implementa-tion. But even in this case, the block structured mesh generators would carry the fine hexahedralmesh of the structure into areas where such mesh density is not required. An alternative could havebeen to use tetrahedral meshes which do not suffer in the same way and are capable of locallyrefining the mesh, but it is frequently argued that tetrahedral meshes do not capture boundary layereffects as well as hexahedral ones in the neighbourhood of the fluid-structure interface.However, for wing structures, the flow in the boundary layer is predominately along the surface

of the wing from the leading to the trailing edge, thus the aspect ratios relative to the wing surfacefrom leading to trailing edge are not of primary importance. Ideally, the mesh normal to thesurface should be sufficiently aligned to capture the complex flow patterns and the faces of the cellsin the flow direction should be normal to the flow. This may be achieved by meshing the boundarylayer using hexahedral elements. Away from the wing, the mesh in the fluid domain may berapidly grown, which may be achieved using tetrahedral elements and the mesh in the wing maybe meshed with any combination of hexahedral, tetrahedral or wedge elements, providing theaspect ratios are less than 20. However, to date there is no knowledge of mesh generators that caneasily transform a hexahedral mesh to a tetrahedral mesh.

7.2. Compute times for complex cases

The AGARD 445.6 wing case had approximately 95 K nodes and the average run time per timestep was approximately 2 h on a 466 MHz Dec Alpha processor. Typically, it is necessary tosimulate 10 complete cycles requiring 200 time steps to capture the cyclical motion of the struc-ture, thus the total run time for a 100 K node problem would be of the order of 400 h. But as hasbeen shown above, the mesh for this complex case needed to be refined and if the mesh densitywas only doubled in each direction, there would be an approximate 10-fold increase in the size ofthe problem i.e. 1 million nodes, which would require about 10 Gb of processor memory. As meshdensity increases, the fluid time step would need to be reduced, in order to comply with the CFL

A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239 235

condition, i.e. the fluid time step would be decreased by a factor of 10, requiring 2000 fluid timesteps for a 10 cycle simulation.Variable time stepping would have to be employed in order to avoid capturing multiple modes of

vibration, where 10 flow time steps are performed for one structural time step. Such a scheme iscurrently allowed, where the traction load on the structure is calculated at each time step and theaverage traction load is applied to the structure during the structural time step. Mesh movement iscalculated at the end of the structural time step and the mesh velocity used to modify the flowequations is applied linearly over the fluid time steps. For such a case, the run time per cycle wouldincrease by a factor of 10, thus giving a total run time of 40,000 h. Even allowing for a five timesprocessor speed up, the total run time would still be of the order of 8000 h. The solution to thememory and run time problems has to involve the implementation DFSI procedure in parallel.

7.3. Availability of experimental data

It is difficult to find reliable benchmarks for three-dimensional DFSI. Even in the case ofaeroelasticity, there are so few benchmarks that the AGARD 445.6 is still widely used for thispurpose. The original wind tunnel tests of the flutter dynamics of the AGARD 445.6 wing werecarried out in 1963 and by the standards of today, the information available in the public domainis severely limited, e.g there is little information regarding boundary conditions and the effect onthe homogeneity of the wing in respect of its stiffness. This is not an issue for the aeroelasticsimulation of the flutter characteristics of the AGARD 445.6 wing, where the modal frequenciesare assumed and are used as boundary conditions to the flow, but it is of increasing importance asmore comprehensive, full DFSI simulations are carried out.Material data is of particular concern, since it is evident that drilling holes in the wing and

replacing the mahogany with rigid foam plastic renders the wing a composite material andcompromises the assumption of homogeneity. The Young�s modulus and density of the wing areessential material properties in Eq. (2) and failure to model these correctly will have seriousimplications for the structural response. The fact that sinusoidal motion was observed for a re-duced time step throws further doubt on the values of Young�s modulus, E, and density, q, forYates et al. weakened model 3 data, since frequency f /

ffiffiEq

q. The authors are not alone in being

unable to induce flutter for this case using the data reported by Yates, see, for example, Raveh [29]and Rifai and Smith [53]. In view of the fact that a number of new techniques are emerging formodelling three-dimensional DFSI, it seems appropriate to consider whether a new set ofbenchmarks should be developed.

8. Closure

The simulation of DFSI is a considerable challenge to the computational modelling commu-nity. The development of robust procedures to enable time and space accurate calculations is asignificant challenge, even for the simplest three-dimensional geometries and where such simu-lations have been attempted either the physics or the geometries have usually been simplified.In this work we have considered the application of the essentials of a full physics simulation, i.e.

Navier–Stokes flow, three-dimensional linear elastic structure, in the context of the AGARD

236 A.K. Slone et al. / Appl. Math. Modelling 28 (2004) 211–239

445.6 wing geometry. This research has highlighted three significant issues that need to be ad-dressed if DFSI analysis is to be applied to �flutter� analysis of practical aerostructures.Mesh structure and quality: the constraints of achieving a high quality mesh representation of

the structure, the surrounding boundary layer and the remainder of the flow domain seem to bemutually incompatible. In this work, we have employed a single hexahedral mesh for the wholedomain, which considerably simplifies the mutual exchange of information at the fluid–structureinterface. Although the use of separate meshes should alleviate the meshing problem, it may noteliminate the generation of poor quality cells in the flow domain at the edge of the boundary layer,unless tetrahedral cells are employed outside the boundary layer. Of course if separate meshes areused, the problem of interpolation between the meshes will have to be overcome.Compute speeds: when full Navier–Stokes flow and three-dimensional structural calculations

are made, even for the simplest linear elastic solids and incompressible flows without a turbulencemodel, the calculation times are essentially impractical for current, conventional, single processorcomputers. Meshes with at least an order of magnitude more cells are required to resolve the flowphysics at the trailing edge of the wing. This leads to a time step for the fluid, which if applied tothe structure would result in capturing the higher order structural response modes and the sub-sequent difficulties of resolving these and isolating any possible numerically spurious oscillations.Adequate experimental data: the most comprehensive sets of experimental data to characterise

�flutter� are now well over 30 years old. This data was not collected to validate the kind of so-phisticated computational DFSI models explored in this work and the information in the publicdomain is not sufficiently detailed to enable the �flutter� characteristics of the flow to be clearly andunambiguously identified. This work is not alone in highlighting these problems, see for examplethe work of Bennett and Edwards [54].Hence, successful modelling of the flutter characteristics of aerostructures requires:

• The development of a meshing strategy that enables a single mesh throughout the entire fluid–structure domain, where the boundary layer is meshed using hexahedral element and the re-mainder of the flow domain is meshed using tetrahedral elements. The structure sub-domaincould be any mix of tetrahedral and hexahedral elements.

• The extension of the model to compressible flows with two equation turbulence models, struc-tures with non-linear material properties and their implementation in parallel with good scala-bility up to 100+ processors.

• Sets of three-dimensional �flutter� data that have sufficient detail and are closed from the per-spective of computational modelling requirements.

The future efforts of this programme will focus on the first two of the above issues to produce aDFSI simulation tool that has the essential functionality to address a broad class of problemsinvolving the onset of flutter.

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