development and validation of an investigation tool …

26th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

DEVELOPMENT AND VALIDATION OF AN INVESTIGATIONTOOL FOR NONLINEAR AEROELASTIC ANALYSIS

L. Cavagna , P. Masarati , S. Ricci , P. MantegazzaPolitecnico di Milano, Dipartimento di Ingegneria Aerospaziale

Keywords: Aircraft design, aeroelasticity, flutter, non-linear beams

Abstract

This paper presents a design tool based on com-putational methods for the aero-structural anal-ysis of aircraft layouts at the conceptual designstage. Multi-disciplinary optimization (MDO)pushes the design towards the absolute optima.So far it has become one of the unquestioned pri-mary targets for aircraft design tools, to obtainthe best performances at the lowest possible cost.The structural weight tends to be continuouslyreduced in favor of improved performances. Asa consequence, the analysis of increasingly flex-ible aircraft requires dedicated tools that allowto investigate the effects of details like materialanisotropy and geometrical nonlinearities whencomposites materials and aeroelastic tailoring isexploited.SMARTCAD (Simplified Models forAeroelasticity inConceptualAircraft Design) al-lows the creation of low-order, high fidelity mod-els. They can be designed to take into ac-count most of the higher order/nonlinear effectsand couplings of the aircraft under development.SMARTCAD can be used within an MDO frame-work to drive the optimization tool into the mostappropriate direction; aeroelastic performancesfor non-conventional aircraft can be evaluated,potential couplings can be highlighted and theirpropitious or adverse nature can be investigated.The code has been developed under the researchproject named SimSAC (Simulating Aircraft Sta-bility And Control Characteristics for Use inConceptual Design), partially funded by the ECwithin the 6th European Research Framework.

1 Introduction

SMARTCAD represents a design tool for aero-structural analysis to be used in the conceptualphase of fixed wing aircraft design. It allowsto consider aeroelastic aspects from this earlystage. Solving adverse aeroelastic issues likedivergence, control surfaces reversal, flutter, in-creased drag at cruise speed due to structural de-formability may require considerable changes inthe structural design, limitations in flight enve-lope or weight penalties. The late discovery ofthis type of issues may result in significant costincreases and, in some cases, it may even requireto actually close the project. In order to over-come the insurgence of these issues, the influenceof deformability on flight and handling perfor-mances, on structural weight and on design costsneeds to be taken into account as early as possiblein the design process. Rapid and reliable meth-ods need to be used in the initial steps of concep-tual design, when many parameters have not beenestablished yet. The enhancement of this phasewith explicit multi-level design oriented numer-ical tools having physical basis rather than rely-ing on implicit statistics of existing aircraft, pre-vents the design from being excessively modifiedduring the detailed design phase if deficienciesare found, and guarantees the capability to studyunconventional architectures (joined wings andblended wing-body aircraft) and new attractivetechnologies like composite materials.

SMARTCAD adopts a mixture of semi-empirical, computational and analytical tools(see Fig. 1) which allows, starting from a detailed

1

L. CAVAGNA , P. MASARATI , S. RICCI , P. MANTEGAZZA

Fig. 1 Layout of the proposed aero-structural tool.

description of the external geometry, to:

• determine a first reliable distribution ofstructural stiffness for the complete air-frame satisfying local and global buck-ling, compressive yield strength and ulti-mate tensile strength;

• have a better prediction on a physical ba-sis of the airframe, overcoming the lacks ofstatistical methods which may not be evenavailable for unconventional layouts;

• evaluate the static/dynamic aeroelastic per-formances of the sized airframe by meansof beam models and linear/nonlinear aero-dynamic methods.

Different kind of analysis can be carried out:

• vibration modes calculations and flutteranalysis (see Fig. 5,8);

• linear/non-linear static aeroelastic analysis,trimmed calculation for a free-flying rigidor deformable aircraft (see Fig. 7,10);

• steady and unsteady aerodynamic analysisto extract derivatives for flight mechanicsapplications.

In the following sections, the proposed aeroe-lastic tools are outlined and applied to the Boeing747-100 large transport aircraft.

f1

f2

f3

CI

CII

N1

N2

N3

G1

G2

G3

ξ

p

mI

mII

tI

tII

Fig. 2 Finite Volume three-node beam coupledto a classic lifting surface method.

2 Beam formulation

SMARTCAD adopts a three-node linear/non lin-ear finite-volume beam, originally proposed in[11], which proved to be intrinsically shear-lockfree. The finite-volume approach leads to thecollocated evaluation of internal forces and mo-ments, as opposed to usual variational principleswhich require numerical integration on a one-dimensional domain. As sketched in Fig. 2, eachbeam element is divided in three parts. Each partis related to a reference pointGi : the mid- andthe two endpoints. They are referred to geomet-rical nodesNi by means of offsetsf i . This allowsthe elastic axis of the beam to be offset from thecenter of mass. Every node is characterized by aposition vector and a rotation matrix. A referenceline p describes the position of an arbitrary pointp(ξ) on the beam section. Parabolic shape func-tions are used to interpolate displacements androtation parameters of the generic pointp(ξ) asfunctions of those of the reference nodes. Thederivatives of the displacements and the rotationparameters at the two collocation pointsCj (laidatξ =± 1/

√3 to recover the exact static solution

for a beam loaded at the end points) are used toevaluate the strains and the curvatures. The latterare used to compute the internal forces and mo-ments, which must balance the external forcesmi

and momentst i .

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DEVELOPMENT AND VALIDATION OF AN INVESTIGATION TOOL FOR NON LINEARAEROELASTIC ANALYSIS

3 Structural sizing and stick model genera-tion for aeroelastic analysis

As far as the aero-structural design process isconcerned, SMARTCAD is coupled to a pre-processor sizing tool namedGUESS (GenericUnknownsEstimator forStructuralSizing). It isderived from theAnalytical Fuselage and WingWeight Estimation (AFaWWE)method [2], fur-ther extended to the sizing of horizontal andvertical tail planes. This method results in aweight estimate which is directly driven by mate-rial properties, load conditions, and vehicle sizeand shape. Thus it is not confined to an exist-ing data base, but it is rather independent of clas-sic statistical formulas currently used in aircraftdesign to estimate the weight of airframe com-ponents. GUESS determines the distribution ofstiffness and non-structural mass of the airframesubjected to different types of maneuvers (pull-up at prescribed normal load, landing, bump onirregular runway, rudder maximum deflection). Itis based on beam theory and analytical methodsfor the estimation of aerodynamic loads. GUESSis also linked to a tool named CADAC and to aWeight and Balance module for the estimationof the inertia properties. The former provides ageneral parametric description of the geometricshape of the aircraft under design (wing, fuse-lage, tail). The latter provides a semi-empiricalestimation of all inertia properties, as proposedby Raymer and Torenbeek [12]. The interestedreader is referred to Ref. [5], where a more de-tailed description of the whole design environ-ment is presented.

As soon as the airframe is sized, GUESSautomatically creates the stick model forSMARTCAD in order to run the aeroelasticanalysis. The airframe is actually sized withoutconsidering any static/dynamic aeroelastic ef-fect, thus the necessity to assess the aeroelasticperformances of the aircraft and eventuallyimprove them by means of multi-disciplinaryoptimization procedures.

3.1 Mass distribution

The mass distribution is determined by a dedi-cated internal weight and balance module whichcomputes the structural mesh inertia properties.This is particularly important when the trim con-dition for the free-flying aircraft is sought, sincea detailed description of mass values and theirlocation is of primary importance to correctlydefine inertial loads. Thus, when the beamstick model is automatically generated, all non-structural masses are correctly introduced in thestructural mesh as:

• lumped non-structural masses on meshnodes (engines, landing gears, auxiliarytanks, systems);

• non-structural densities per unit lengthalong the beams (passengers in fuselage,fuel in wings, paint, furniture).

As introduced above, these data are either pro-vided by statistical methods or directly by theuser if available. Lumped masses are easily intro-duced in the model by means of rigid offsets froma reference node. Beams can also be used for thispurpose, but an estimation of their stiffness is re-quired (this may happen in the case of engine py-lons). As far as distributed masses are concerned,for example fuel, the availability of an estimate ofthe fuel volume (see Fig. 3) available in the wing-box allows to determine the mass stored for eachbeam along the wing-span, and thus to estimatethe mass per unit length. The same approach canbe applied to any distributed mass (passengers,furniture and so on), provided their value and po-sition is correctly estimated.

3.2 Thrust modelling

For non-linear aeroelastic applications, it may beimportant to include the effect of the displace-ment and rotation of lumped force applicationpoints. This typically highlights potential cou-plings and allows to check their propitious oradverse nature. Thus, SMARTCAD includes afollower-force element which can be used, for ex-ample, to include thrust effects in the aeroelastictrim solution.

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0 5 10 15 20 25 30 350

50

100

150

200

250

Spanwise distance y [m]

Vf [m

3 ]

Fig. 3 Spanwise fuel volume available in wing-box.

3.3 Control surfaces modelling

Control surfaces are currently represented bytheir aerodynamic contributions, considering theearly design phase SMARTCAD is intended for.As for flutter analysis, for example, a furtherstep would consist in the structural sizing ofthe control surfaces, and include a lumped staticimpedance to model a mechanic control-chain orto approximate the impedance of the actuators.This is currently out of the target the code is de-veloped for and it is left to future developments.

When static aeroelastic trim is sought, theuser can specify arbitrary constraints among thecontrol surfaces, with different gains. For exam-ple, antisymmetric ailerons deflection, symmetricelevators deflection, or wing flaps deflection canbe imposed as needed.

4 Spatial coupling methods

SMARTCAD adopts a staggered approach,where different structural and aerodynamic in-dependent codes, each one optimal for its pur-pose, are used for each field. In order for thecodes to interact, a spatial coupling scheme isrequired. The adoption of a partitioned ap-proach [10] requires the definition of an inter-face scheme to exchange displacements, veloci-ties and loads between the structural grid and theCFD boundary surfaces. The two models are typ-ically discretized in very different, often incom-

Structural node

Offset

Aerodynamic mesh

Aeronode for spatial couplingLumped mass

Beam collocation point

Beam elastic axis

(a) Overview of the stick model and nomenclature.

OffsetStructural node

Aerodynamic collocation pointVortex/Doublet singularity

Aeronodes for spatial coupling

Beam collocation point

(b) Detailed overview for tail planes.

Fig. 4 Stick model for Boeing 747 aircraft.

patible ways. Structural models usually presentcomplex geometries, including many discontinu-ities. They are often based on schematic models,which are of common use in the aerospace indus-try, using elements with very different topologies,like beams and plates, which usually hide the realstructural geometry up to the point where the air-craft external shape partially or completely dis-appears. Despite the computational power avail-able nowadays, these simplified models are stillused, and will probably be used in the futurein aerospace industry because of their efficiencyand effectiveness, especially in the early designstages SMARTCAD is supposed to be adoptedfor. As a consequence, it is extremely importantto be able to cope with them. On the contrary,aerodynamic meshes typically require a more ac-

4


Type Φ(δ) , Φ(r)

Volume Spl. rThin Plate Spl. r log(r)Gaussian e−r

Euclid Hat π ·((1/12r3) − (r2max· r)

+(4/3r3max))

WendlandC0 (1− δ)2

WendlandC2 (1− δ)4 ·(4δ + 1)

WendlandC4 (1− δ)6 ·(35/3δ2 + 18/3δ + 1)

WendlandC6 (1− δ)8 ·(32δ3 + 25δ2 + 8δ + 1)

Table 1Weight functions used as RBF.

curate description of boundary surfaces (espe-cially when Computation Fluid Dynamics solversare used to solve for Euler or Navier-Stokes equa-tions). As a consequence, two radically differ-ent representations of the same aircraft geometrymust be made compatible in order to transfer in-formation between them. This is a well knownproblem, deeply investigated in the literature; forfurther reference, see [21; 3].

An innovative scheme, based on a ‘mesh-free’ Moving Least Square (MLS) method [18],is used to cope with incompatible situations,when the two meshes do not share a commonsurface. A second approach is represented bythe Radial Basis Function (RBF) method [4; 20].Both methods ensure the conservation energytransfer between the fluid and the structure andthey are suitable for the treatment of complexconfigurations. To guarantee the conservation be-tween the two models, the correct strategy con-sists in enforcing the coupling conditions in aweak sense, through the use of simple variationalprinciples. Using the Virtual Works Principle(VWP) the energy exchange can be investigatedas reported in [16].

5 Aerodynamic methods available

Two low-fidelity aerodynamic methods are avail-able in SMARTCAD, depending on whethersteady or unsteady analysis is carried out:

Rigid armAerodynamic box

Structural nodeAeronode

Beam model

Fig. 6 Aeronodes for aeroelastic interpolation

• Vortex Lattice Method (VLM) with cam-ber contribution on normalwash once theairfoil description is provided;

• Doublet Lattice Method (DLM) for theprediction of the generalized forces andflutter analysis in the subsonic regime.

Both methods are based on the same geometricdiscretization of the aerodynamic surfaces as aflat lifting surface with singularities and colloca-tion points located respectively at 1/4 and 3/4of each panel chord. Thus the same mesh gen-erator can be used, with the only exception thatthe VLM allows camber contributions to be in-cluded and requires trailing vortexes for wakemodelling.

5.1 Steady aerodynamics

The available VLM is derived from Tornado[17], which has been enhanced with new func-tionalities to support aeroelastic analysis. Sincethe adopted beam model allows to model largedisplacements and rotations, the aerodynamicmesh is physically deformed to accurately fol-low the deformation of the structural shape. Thesame occurs when control surfaces are deflected.Whenever the VLM mesh is deformed, the newposition of its panel nodes, collocation and singu-larities points and trailing wake are updated usingthe techniques introduced in Section 4. The twospatial coupling methods determine an influencematrix H that maps field data from the structuralto the aerodynamic mesh. To overcome the prob-lem of mapping the one-dimensional beam do-main to a two-dimensional or three-dimensionallifting surfaces and CFD domain, an extra set ofnodes, namedaeronodesas sketched in Fig. 6,

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(a) 1st bending - 0.60 Hz (b) 1st torsional - 0.93 Hz (c) 2nd bending - 1.61 Hz (d) 1st inplane - 1.83 Hz

Fig. 5 Boeing 747 vibration modes for maximum take-off configuration.

are introduced in the model; their displacementsare directly recovered from the real structuralnodes under the hypothesis of rigid beam sec-tion. Finally, a new flow solution is determinedon the deformed lattice, and the correspondingforces are transferred to the structural model bymeans of the same spatial coupling matrixH for anew structural solution. These steps are repeateduntil the displacement field converges under theassumption that the system is not approachingor exceeding static aeroelastic divergence condi-tions.

Deformed structrural mesh

Updated trailing vortexes

Updated VLM mesh

Undeformed structural meshUndeformed VLM mesh

Fig. 7 Deformed configuration for Boeing 747clamped wing,M∞=0.8,α=6deg, z=5000m.

5.2 Harmonic aerodynamics

SMARTCAD adopts a built-in DLM [1] tocompute the aerodynamic transfer matrixHam( jk,M∞) for the generalized forces in thereduced frequency domainjk and Mach number

M∞. Two different approximations are availablefor doublet distributions across the span of thebox bound vortex: parabolic, which restrictsthe box aspect ratio to about 3, and quartic,which relaxes the limitation on box aspect ratioand the number of spanwise divisions requiredin high-frequency analyses [19]. Particularcare was dedicated to test the solver for nearlycoplanar wing/tail configurations as illustrated in[13].

Once the transfer matrix is available, flutterinstabilities in the whole flight envelope are de-termined using the classicp-k method, solvingthe equation

(

s2M +sC+K −q∞Ham( jk,M∞))

q = 0 (1)

whereq, M , C, K are respectively modal am-plitudes, mass, damping, and stiffness matrices,and q∞ is the dynamic pressure. The solutionof Eq.(1) determines flutter velocity, frequencyand modes. A well-known three-dimensionalstandard aeroelastic configuration, the AGARD445.6 weakened wing [22], is considered to val-idate the whole procedure. It was originallytested in the Transonic Dynamics Tunnel (TDT)at NASA Langley. The wing semispan model ismade of laminated mahogany, with the NACA65A004 airfoil, a quarter-chord sweep angle of45 deg, an aspect ratio of 1.65 and a taper ra-tio of 0.66. The structural model consists in thefirst four normal modes. Two flight conditionsare investigated by means of SMARTCAD; re-sults are successfully compared with both exper-imental and NASTRAN results, as summarizedin Table 2. The first flight condition at a MachnumberM∞ = 0.678 is subsonic, while the sec-

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Mach VFExp (m/s) VFNAS (m/s) VFSMA (m/s) ωExp (Hz) ωNAS (Hz) ωSMA (Hz)

0.678 231.37 237.94 235.86 13.89 14.82 14.300.960 309.00 337.07 324.23 17.98 20.53 20.06

Table 2 Comparison of flutter velocityVF and frequencyωF for the AGARD 445.6 aeroelastic benchmark

200 250 300 350 4000

2

4

6Velocity−Frequency

Fre

quen

cy

200 250 300 350 400−4

−2

0

2Velocity−Damping

Dam

ping

Flutter Speed: 314.7995 m/s

0.60236 Hz

0.93461 Hz

1.3966 Hz

1.6124 Hz

2.0041 Hz

2.0842 Hz

2.3822 Hz

3.8207 Hz

4.1697 Hz

4.183 Hz

4.232 Hz

4.6348 Hz

Fig. 8 Flutter diagrams for Boeing 747 in freeflight, M∞=0.9, z=0m.

ond one at a Mach numberM∞ = 0.960 is tran-sonic. This flight condition is investigated de-spite the linearized potential theories are knownto overestimate the flutter velocity in this regime.

Fig. 8 shows flutter results through the DLMfor the Boeing 747 in free flight atM∞=0.9 andzero altitude. All rigid body and deformable nor-mal modes are used for this purpose due to thelack of an adequate frequency separation margin.As it can be seen, a flutter instability associatedto the fist bending mode showed in Fig. 5(a) isdetected and a second instability for the first tor-sional mode showed in Fig. 5(b) occurs at highervelocity.

5.3 High-fidelity aerodynamics

Complex aeroelastic phenomena may appear inthe transonic speed range, where moving shockwaves arise in the flow field caused by unsteadymotions of the aircraft structure. The presenceof shock waves may cause a significant drop influtter velocity: the so-calledtransonic dipef-fect. In these cases the flutter velocity is oftenoverpredicted by classical linear velocity poten-tial methods used to determine unsteady aerody-

namic loads. More complex fluid dynamics mod-els need to be adopted, like those based on Euleror Navier-Stokes equations. Following the sameapproach presented in [6], SMARTCAD can ex-port vibration modes to allow the determinationof the aerodynamics transfer matrix by meansof the Edge flow-solver [9]. The results for theAGARD wing with the inviscid flow model aresummarized in Fig. 9, where the transonic dip iscorrectly captured in the transonic Mach region.To highlight the good quality of the results, thesame pictures also show the ones presented in[6; 15], obtained using classical methods like theDLM and the Harmonic Gradient Method (HGM,[8]: ZONA51) and Navier-Stokes equations.

6 Trim solution for the free-flying aircraft

SMARTCAD allows to determine the trimmedsolution for the rigid/deformable free-flying air-craft in steady maneuver, characterized by steadyaerodynamic and inertial forces in body-axes.The approach is the same presented in [7], withthe exception that the full model is used insteadof a reduced modal one by component mode syn-thesis. The governing idea consists in deter-mining the trim condition in a staggered mannerwhen the flight conditions and some parametersregarding the attitude of the aircraft are imposed.This allows, for example, to determine the angleof attack and the controls deflections that makethe aircraft fly with an imposed linear and angu-lar velocity in the inertial reference frame.

A matrix-free Jacobian Free Newton-KrylovMethod (JFNK) [14] or Newton-Raphsonmethod is applied only to the flight mechanicsequilibrium equations. This approach is equiv-alent to solving the coupled problem given bythe flight mechanics and structural equilibriumequations through a block Gauss-Seidel method.Aerodynamic loads are subsequently transferred

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0.4 0.6 0.8 1 1.2 1.40.2

0.3

0.4

0.5

0.6

0.7

0.8

Mach number

VF

ExperimentEdgeDLM,ZONA51Sadeghi et al.Cavagna et al.

0.4 0.6 0.8 1 1.2 1.40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach number

ωF

ExperimentEdgeDLM,ZONA51

Sadeghi et al.Cavagna et al.

Fig. 9 Flutter speed indexVF and frequencyωF for the AGARD 445.6 wing by Edge solver

1 2 3 4 5 6 7 8 9 10 11 12 13 14140

2

4

6

8

10

12

Iteration n.

Ang

le [d

eg]

Angle of attackElevator deflection

Rigid trim

Deformable trim

Initial guess solution

(a) Trim variables history

1 2 3 4 5 6 7 8 9 10 11 12 13 141410

−8

10−6

10−4

10−2

100

102

104

106

108

Iteration n.

log(

resi

dual

)Deformable trim

Rigid trim

(b) Residual norm for flight mechanics equations

(c) Deformed trimmed solution

Deformed VLM mesh Elevator deflected

Undeformed VLM mesh

(d) Horizontal tail detail

Fig. 10 Trim solution for the free flying deformable aircraft,M∞=0.6, z=5000m.

to the structural model to calculate the newdeformed shape. The latter is then used todetermine the new trimmed solution. The pro-cess is repeated until convergence on structuraldisplacements and rigid-body equilibrium isreached (see Fig. 10(b)). The same process can

be applied when the trim condition is sought forboth the rigid and deformable aircraft. For thislast case, the rigid trim is always determined toobtain a good initial guess solution and improvethe convergence. This is also important to high-light the contribution of structural deformability

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on the trimmed solution.Fig. 10 shows the results obtained by apply-

ing the process illustrated above. In this simplecase, a steady levelled symmetric cruise condi-tion is sought. For this simple case, the user spec-ifies null angular velocities, the deflection of thehorizontal tail, the components of the linear ve-locity in the inertial frame and requires a sym-metric flight by enforcing null sideslip angle. Theprocess then determines the angle of attack, theelevator deflection (see Fig. 10(a)) and the struc-tural displacements (Fig. 10(c)). The remainingparameters such as aileron/rudder deflections andbank angle are null. Fig. 10(d) shows the hori-zontal tail bent down as expected, along with thedeflected elevator in order to produce the requirednegative lift.

7 Conclusions and future developments

The paper outlined the different tools developedin order to enhance the conceptual design phasewith aeroelastic analysis. Particular care is ded-icated to provide hierarchical tools from low tohigh fidelity able to:

• be easily coupled with other codes, as mul-tidisciplinary analyses are required;

• be relatively accurate for the conceptualphase and give correct trend data to let thedesign progress in the correct direction; ;

• be computationally efficient, as severalconfigurations need to be examined;

• give the capability to trade accuracy forspeed to the designer who is left free to de-cide the level of discretization and to ruleaccuracy of modelling by the adoption ofdifferent solvers available in the toolbox;

• require minimal time for model prepara-tion and modification; the development ofautomatic procedures and the exploitationof geometry parametrization that can beused as design variables, guarantees to eas-ily reflect the changes of a design variablein all the numerical models;

• provide sensitivity derivatives of the designvariables.

Future developments will allow to provide aeroe-lastic corrections to flight stability derivatives tobe used in flight dynamic applications. Further-more, SMARTCAD will be used within an MDOenvironment to eventually improve the first guessstructural solution in order to satisfy aeroelasticconstraints like flutter speed, divergence, staticaeroelastic deflections and control effectiveness.

Acknowledgments

The development of SMARTCAD environmentwas funded by the European Union under theSixth Research Framework through the Simulat-ing Aircraft Stability And Control Characteristicsfor Use in Conceptual Design (SimSAC) Con-tract number FP6-030838. The authors acknowl-edge Prof. Giampiero Bindolino for the interest-ing discussions on VLM and DLM.

Copyright Statement

The authors confirm that they, and/or their company orinstitution, hold copyright on all of the original mate-rial included in their paper. They also confirm theyhave obtained permission, from the copyright holderof any third party material included in their paper, topublish it as part of their paper. The authors grant fullpermission for the publication and distribution of theirpaper as part of the ICAS2008 proceedings or as indi-vidual off-prints from the proceedings.

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