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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 18 (2009) 035014 (13pp) doi:10.1088/0964-1726/18/3/035014

Determinate structures for wing cambercontrolD Baker and M I Friswell

Department of Aerospace Engineering, Queens Building, University of Bristol,Bristol BS8 1TR, UK

E-mail: [email protected]

Received 7 July 2008, in final form 29 December 2008Published 3 February 2009Online at stacks.iop.org/SMS/18/035014

AbstractAn investigation of truss structures for the purpose of creating a continuously variable cambertrailing edge device for an aircraft wing is presented. By creating structures that are bothstatically and kinematically determinate and then substituting truss elements for actuators, it ispossible to impose structural deflection without inducing member stress.

A limited number of actuators with limited strain capabilities are located within thestructure in order to achieve a target deflected shape starting from an initially symmetric profile.Two objective functions are used to achieve this: a geometric objective for which the targetdisplacement is fixed and a shape objective for which the target displacement is dependent onthe surface shape of the targeted aerofoil. The proposed shape objective function is able to offerimprovements over the geometric objective by removing some of the constraints applied to thetargeted structure joint locations.

Four methods for selecting the location of a set of actuators are compared, namelyexhaustive search, a genetic algorithm, stepwise forward selection (SFS) and incrementalforward selection (IFS). Both SFS and IFS are variations of regression methods for subsetselection; in each case an approach has been created to allow the imposing of upper and lowerbounds on the search space. It is shown that the genetic algorithm is well suited to addressingthe problem of optimally locating a set of actuators; however, regression methods, particularlyIFS, can provide a rapid tool suitable for addressing large selection problems.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The design of conventional fixed wing aircraft is constrainedby the conflicting requirements of multiple objectives.Mechanisms such as deployable flaps provide the currentstandard of adaptive aerofoil geometry, reducing the problemto a few degrees of freedom. This solution places notablelimitations on, amongst others, manoeuvrability and efficiencythus providing a design that is non-optimal at many flightregimes.

Variable camber control was demonstrated by the missionadaptive wing (MAW) programme. Here variable camberleading and trailing edge flaps enabled a smooth, variablecamber by the use of hydraulic motors and mechanicalactuators. The goal of the system was to permit L/Doptimization, roll control, manoeuvre load control, increasedlift without increased angle of attack and gust load alleviation.Secondary benefits included terrain following improvements as

a result of lift alteration at constant angle of attack, reducedradar cross section and parasitic drag reduction [1]. Additionalbenefits lie with cruise camber control in maximizing vehicleefficiency during straight and level flight. Based on a studyof a subsonic transport aircraft at low speed and cruise flightregimes the benefits of camber optimization are discussed withreference to the L/D ratio. The return from such systemsappeared to be more apparent at increased lift coefficients (CL)where increased camber is used to attain maximal L/D as CLincreases [2].

The DARPA/AFRL/NASA Smart Wing programmeinvestigated the application of smart materials to aircraftcontrol surface design. The first phase of the investigationincorporated shape memory alloy (SMA) based smoothlycontoured trailing edge control surfaces and wing twistmechanisms [3]. Phase 2 focused on the use of higherbandwidth piezoelectric based actuation systems integratedinto a larger scale full span model [4]. Wind tunnel test

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Smart Mater. Struct. 18 (2009) 035014 D Baker and M I Friswell

results demonstrated roll moment improvements of the smoothcontour trailing edge flap when compared with a discrete flapsystem [4]. An analytical investigation of conformal controlsurfaces for aerodynamic control compared with a discretearticulation recorded improvements for CL at a range of flaplength to chord ratios [5].

The mission adaptive compliant wing (MACW) makesuse of compliant structures to configure a variable geometrytrailing edge flap. Wind tunnel testing of a long enduranceaerofoil with a conventional trailing edge flap demonstratedincreased drag due to flow separation in the aft portion of theaerofoil. This region of flow separation was induced by thesharp change to the curvature of the aerofoil surface as theflap was rotated. Wind tunnel data for an adaptive complianttrailing edge flap developed by Flexsys Inc. was observed tohave consistently low drag as the flap was deflected, achievedby removal of sharp changes in the surface curvature of theaerofoil [6]. Flight testing of the Flexsys concept has reportedpreliminary results that suggest reduced fuel consumption,reduced weight and possible gust load alleviation benefits [7].

The structural systems used to create smooth variablecamber have ranged from a mechanistic approach, mostprominently demonstrated by the MAW but ranging througha number of patented designs [8, 9], to the more recentapplication of compliant and flexible structures. Saggereand Kota [10] described a method of providing small shapechanges in structures comprising flexible beam elements.Here the emphasis was placed on the achievement of smoothcurved forms by a compliant structure in a two stageprocedure: topological synthesis and dimensional synthesis.The problem of topology generation was addressed byspecification of a series of output points on the 2D structuralsurface from which a topological design forms the basisof the dimensional synthesis problem. The dimensionalsynthesis was undertaken as a structural optimization schemeencompassing the dimensions of the beam elements thatcomprised the structures topology, in addition to the locationand magnitude of the input actuation.

Anusonti-Inthra et al [11] applied a continuous optimiza-tion method to optimize the distribution of an array of piezo-electric elements forming the trailing edge of a rotor aerofoil.Here a starting topology was subjected to topology and geo-metric optimization in which each of the internal elements tookthe form of a piezoelectric actuator. Tip deflections resulted inan effective flap angle of 4.24. Gandhi et al [12] developeda methodology for the design of a smart materials based con-formable aerofoil for a rotor blade with the aim of providingvibration control and reduced aerodynamic drag properties. Apre-determined frame topology was envisaged with integratedpiezoelectric active elements. A continuous optimization pro-cedure performed optimal dimensional sizing of the passivestructural elements allowing the formation of compliant hingeswithin the structure. In this case an effective flap angle of 4.6was achievable.

Bart-Smith and Risseeuw [13] created a variable camberaerofoil structure employing the principles of staticallyand kinematically determinate structures. In this case anexperimental model featuring an SMA based actuation system

was created. Wind tunnel tests at 0 angle of attack(AoA) were completed up to velocities of 38 ms1 at whichpoint the aeroelastic response of the model became unstable.Ramrkahyani et al [14] also investigated the application ofstatically determinate structures for aircraft structural control.Here a network of truss and cable elements were envisagedto form the wing structure, by varying the length of cableelements structural deflection could be induced.

With regard to the problem of locating action systemswithin space structures for geometric control Haftka et al[15] noted that when the representation of sites available foractuator location placement forms a continuous problem thenconventional gradient based optimization methods may beemployed. For those problems where the actuator locationis described by a discrete variable the location selectionbecomes an integer programming problem that is typicallymore computationally expensive to solve than a continuousproblem. In this case two iterative improvement techniqueswere applied, worst-out-best-in (WOBI) and exhaustive singlepoint substitution (ESPS). The WOBI algorithm began with aninitial configuration containing the desired number of actuatorsand removed a number of actuators considered to be worst.The process continued until no further performance benefit wasfound. The ESPS algorithm moved each actuator in the initialconfiguration to each redundant location in turn, analysing theperformance at each trial step. The configuration where nofurther performance increase could be found was finally settledupon.

Chen et al [16] addressed the issue of entrapment in localminima by the use of the simulated annealing (SA) technique.The premise of SA when selecting a number of locationsfrom a larger system is that non-improving solutions mustoccasionally be accepted in order to avoid local optima [17].This is in contrast to iterative improvement approaches whereevery improved location set was selected.

Santos e Lucato et al [18] investigated various algorithmsto select actuator locations that provide maximum structuraldeflection. The chosen procedures provided a locationset for the actuators from the discrete selection, then foreach configuration a local optimization process selected therespective force input for each actuator location. Due to thelarge unordered search space it was determined that methodsallowing inferior cost moves such as SA and genetic algorithms(GAs) are most effective for the global configuration selection.Such methods allowed escape from local optima and in theexample cases both SA and GA methods produced results ofequal quality.

This paper investigates 3 statically and kinematicallydeterminate truss structures as candidates for the formation of aconformal trailing edge. The promise of these structures is thatby the introduction of a number of active actuated elementsin place of passive truss elements then the structure shalldeform without storage of elastic energy. An approximationof predefined surface form is created, differentiated fromthe un-deflected form by variation of the aerofoil mean line.Two methods of comparing the target and deflected surfaceforms are presented defined as geometry comparison and shapecomparison methods. In addition a number of methods for

2


locating the optimal placement of the active elements areemployed that may be categorized as exhaustive, heuristicand iterative improvement methods. The performance of theselection methods with varying actuator strain constraints,actuator quantity and truss topology is discussed.

2. Determinate structures

The performance of a pin jointed truss structure has beenutilized to provide a measure of the performance of a framestructure with elastic joints [19, 20]. The stability of a truss ischaracterized by the presence or not of mechanisms within thestructure, that is the appearance of modes of nodal deflectionsdefined by a lack of extension in any of the componentmembers. A structure possessing mechanisms is referred toas kinematically indeterminate. Conversely in the example ofa structure in which the defining equations of equilibrium areinsufficient to determine the internal forces and reactions onthe structure then the truss is said to be statically indeterminate.The physical interpretation of this result is that the number andconfiguration of members within the structural system resultsin member redundancy, or states of self-stress.

Pellegrino and Calladine [19] introduced the rank (rB) ofthe structures equilibrium matrix for the characterization ofthe degree of static and kinematic determinacy. Accordingto this analysis the number of states of self-stress (s) and thenumber of mechanisms (v) may be determined as

s = e rB (1)

v = kgg kc rB (2)where kg is the number of degrees of freedom afforded toeach joint, kc is the number of constrained degrees of freedomprovided by the support system, e is the number of structureelements and g is the total number of joints.

For a truss fulfilling the criteria of static and kinematicdeterminacy then the locations of each joint, and byextension the structures geometry, is determined solely bythe constituent member lengths. Thus by substituting amember with an element able to change its length the structuralgeometry may be altered with no storage of elastic energywithin the system. This property of determinate structurespermits limited geometry control without incurring resistancefrom the structure in the form of induced member deformation.In addition such structures are able to support applied loads dueto the satisfaction of the conditions of kinematic determinacy.

3. Problem formulation

For a structure forming the basis for a morphing wingthe internal forces simulating actuator inclusion fact and theapplied aerodynamic load faero are related to the structuraldisplacements q using the stiffness matrix K

fact + faero = Kq (3)

fact = Bu (4)

fact is defined as the vector of forces applied to the structure inorder to simulate actuator inclusion. Therefore the i th elementof u(ui ) defines the force that is transformed to applied loadsin the global coordinate system by matrix B.

By satisfaction of the conditions for kinematic and staticdeterminacy then K is non-singular and so may be inverted andby substitution of equation (4) into equation (3) the vector ofnodal displacements is obtained as

q = K1Bu + K1faero. (5)A reduced selection of structural displacements (qs) are ofinterest for the creation of a conformal trailing edge, definedas

qs = Cq (6)where C is a [ks k] binary matrix that selects the ks controlleddegrees of freedom from q. It is now possible to formulate thelinear forcedisplacement system as

qs CK1faero = CK1Bu. (7)In order that the stiffness matrix shall remain determinate theactuator elements are assigned stiffness Ea and cross sectionalarea Aa. Therefore the actuator simulation force may becorrected to overcome the included stiffness of the activeelement

ui = ui Ea Aai (8)where ui is the corrected force and i is the strain of the activeelement with no aerodynamic loads due to ui .

3.1. Geometry objective

Two displacement control objectives are considered, denotedas a geometry objective and a shape objective. Considering aninitial NACA 0012 aerofoil profile, the trailing edge portion ofthe profile is defined as the region 0.6cc, where c is the chordlength which in this symmetric case is equal to the mean linelength. The mean line in this region is redefined according to aparticular function. In this case the quadratic function

ym = C1x2m + C2xm + C3 (9)where xm and ym define the coordinates of the aerofoil meanline is applied for simplicity and computational efficiency. Thedefinition of a conformal trailing edge initiated at 0.6c requiresym = 0, and y m = 0 at xm = 0.6c so that,

ym = C1(xm 0.6c)2. (10)The length of the mean line in the target and initial aerofoilconfigurations remains constant at c. In addition the profilethickness at each point on the length of the mean line remainsequivalent in the initial and target profiles. The vector of targetdisplacements yg may now be defined as those displacementsrequired to transform the joints of the aerofoil at the surfaceto the target profile defined by a non-zero value of C1. Thegeometry objective is thus quantified by calculation of the sumof squares error (SSEg)

SSEg = yg qs2. (11)3


3.2. Shape objective

The shape objective forms the target profile in a similarmanner to the geometry objective, however in this case thenodal positions that define the target profile are dependentnot just on the mean line function but also the location ofthe surface nodes of the deflected structure. In this casethe target nodal displacement is defined as the displacementrequired to transform a surface node to the nearest point on thecorresponding upper or lower surface of the target profile. Ifpoint P defines a surface node with coordinates (xP , yP) thenan approximation of point R on the target profile that is closestto P is made by first calculating the location of point Q. Q isthe point on the target mean line through which a normal to themean line would also intersect P . The coordinates of Q maybe defined as

xQ = D13D3

D3 D13

(12)

yQ = C1x2Q + C2xQ + C3 (13)where,

D1 = D5 D24

3

D2 = D6 + 2D34 9D4 D5

27

D3 =[

D22

(

D224

+ D31

27

)1/2]1/3

D4 = 3C22C1

D5 = C22 2C1 yP + 2C1C3 + 1

2C21

D6 = C2C3 C2 yP xP2C21

.

Point R is then defined by application of the NACA thicknessfunction at Q. Figure 1 illustrates the method of defining pointQ with respect to the target mean line and point P . The shapeobjective can now be quantified by calculation of the sum ofsquares error (SSEs)

SSEs = ys qs2 (14)where ys is the target deflection vector for the shape objective,which is obtained from the displacement required to transformpoint P to point R for all surface joints. The vector P R is notnormal to the surface of the target aerofoil, however where thecurvatures of the target mean line and target aerofoil surface aresimilar then this approximation provides satisfactory results. Itis possible to determine the precise location of the point on thetarget aerofoil closest to point P using iterative methods at theexpense of increased computational burden.

4. Subset selection

For this study subset selection is the process of selecting asubset of size n defined by vectors t1, t2, . . . , tn from a largerset of m basis vectors that provided the best approximation toa target vector y. From equation (7) the potential basis vectorsare the m columns of T where

T = CK1B. (15)

Figure 1. The derivation of shape finding objective, defined asminimization of length PR for all surface nodes.

A vector ti was selected by the inclusion of a non-zero value forui and for all example problems the influence of faero was notconsidered due to the assumption of high structural stiffness.

Four methods were applied that either allow searching ofthe entire model space, searching of a fraction of the totalmodel space or alternatively provide a systematic method toexamine a path through the model space.

For problems in which the number of possible solutioncombinations is suitably small it is possible to implementan exhaustive search (ES) procedure for the evaluation ofall possible actuator location combinations. Within theconfines of each possible subset, a gradient based constrainedoptimization routine is employed to evaluate the optimumvalues of u1, u2, . . . , un . If the search space within allpossible subsets is convex then this procedure will reach aglobal optimum at the expense of high computational cost.

A genetic algorithm (GA) is a search method thatseeks to improve the selection by applying the principles ofnatural selection [21]. For similar problems in which theuse of an exhaustive search was not feasible then heuristicmethods that permit inferior cost moves, such as a GA,were recommended [18]. The limiting features of a GAare the requirement for a large number of objective functionevaluations and typically the rapid location of optimal regionswithin the search space but with slow convergence to anoptimum value.

Stepwise forward selection (SFS) attempts to provide apath through the search space whereby variables are addedat each step, keeping previously selected variables untila particular termination criteria is met. This may be ifinsufficient improvement to the model is made by the additionof the variable or the maximum size of subset is reached. Thevector t j that provides the closest correlation to the currentresidual vector is selected in turn and is that which givesthe minimum value to [22]

S = tju j2. (16)The regression coefficient that minimizes S for variable t j isgiven by

u j =tTj

tTj t j. (17)

At each step is updated as

= y Tu. (18)4


Figure 2. The incremental forward selection (IFS) method.

The applied force that simulates the effect of an actuatorinclusion was limited to a range of values dependent upon thereal actuator constraints, the maximum and minimum availablestrains maxj and

minj giving

umaxj = E j A jmaxi uminj = E j A jmini (19)where umaxj and u

minj are the upper and lower bounds of the

regression coefficients calculated in equation (17). Given theselimitations is not necessarily orthogonal to y after the firststep as is the case with conventional SFS methods. Of noteis the omission of the influence of faero due to the assumptionof high structural stiffness with respect to faero, as a result nocorrection is made to umaxj and u

minj within this investigation.

The process of applying the maximum possible regressioncoefficient or that which results in an orthogonal residualimplies a greedy selection process that may overlook possibleimproved combinations of actuator locations and strains. Anumber of well established methods are able to provide animproved solution compared to SFS methods. Backwardselection begins with the full model and sequentially removesvariables from the starting set until a subset of the requiredsize is reached, whilst methods such as WOBI begin with thedesired number of variables and sequentially add and subtractvariables from the subset. For both methods the applicationof upper and lower bounds to the regression coefficients isdifficult to implement.

Less aggressive forward selection techniques includingthe Lasso [23] and incremental forward selection (IFS)algorithms [24] have been developed more recently. IFS is a

Figure 3. The 14 element trailing edge in its initial configurationwith element numbering, and the target geometry and approximatetarget shape.

max

SS

Eg

Figure 4. Geometry objective (SSEg) against maximum absoluteactuation strain (max) for a 14 element truss with 3 actuatorsubstitutions.

more cautious version of SFS and is described by the flowchartin figure 2. At each step the coefficient of the variable mostclosely correlated with the current residuals is incremented by, with the sign determined by the sign of the calculatedcorrelation. By reducing the step size, improvements in thecorrelation between the actual and target structure responsemay be achieved at the cost of an increased computationalburden.

5. A 14 element truss structure

A simple 14 element truss forming the trailing edge fraction0.6c to c(c = 100 mm) of a NACA 0012 aerofoil is illustratedby figure 3. Each of the 14 elements provide a possible locationfor actuator substitution with the goal of replicating either theillustrated shape or geometry objective forms. Both of theobjectives are defined by C1 = 4.7 103 resulting ina trailing edge deflection that would require 10 equivalentflap angle [11] in order to achieve a similar deflection with adiscrete hinge device.

5.1. Geometry objective results

Table 1 and figure 4 show the effect of varying themaximum and minimum allowable actuation strain on thegeometry objective (SSEg) defined in equation (11). Threeactuators were located in the structure using 4 methods:

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Table 1. Geometry objective SSE error with respect to max for the14 element truss with 3 actuator substitutions.

Selectionmethod max SSEg

Actuatorlocation i

ES 0.01 65.873 05 3, 5, 7 0.01, 0.01,0.010.025 24.600 51 3, 5, 7 0.025, 0.025,0.0250.05 0.847 64 3, 5, 7 0.05, 0.042 68,0.050.1 0.575 63 3, 5, 11 0.072 72, 0.037 29,0.042 850.15 0.575 63 3, 5, 11 0.072 72, 0.037 29,0.042 85

GA 0.01 65.873 05 3, 5, 7 0.01, 0.01,0.010.025 24.600 51 3, 5, 7 0.025, 0.025,0.0250.05 0.847 64 3, 5, 7 0.05, 0.042 68,0.050.1 0.575 63 3, 5, 11 0.072 72, 0.037 29,0.042 850.15 0.575 63 3, 5, 11 0.072 72, 0.037 29,0.042 85

SFS 0.01 65.873 05 3, 5, 7 0.01, 0.01,0.010.025 24.600 51 3, 5, 7 0.025, 0.025,0.0250.05 0.847 64 3, 5, 7 0.05, 0.042 68,0.050.1 1.403 34 3, 4, 9 0.1,0.019 06, 0.044 750.15 2.197 72 3, 4, 14 0.124 77,0.052 01,0.048 63

IFS 0.01 65.873 05 3, 5, 7 0.01, 0.01,0.010.025 24.600 51 3, 5, 7 0.025, 0.025,0.0250.05 0.849 63 3, 5, 7 0.05, 0.041 75,0.050.1 0.765 63 3, 5, 7 0.049, 0.033 96,0.062 010.15 0.765 63 3, 5, 7 0.049, 0.033 96,0.062 01

exhaustive search (ES), genetic algorithm (GA), stepwiseforward selection (SFS) and incremental forward selection(IFS) and for each actuator maxi = mini = max.

The GA provided a match with the optimal ES solutionfor all max. For max = 0.01, 0.025, 0.05 the search spacewas constrained by limitations on the maximum absoluteactuation strain, however at max 0.072 72 table 1 reveals thesearch space was constrained only by the number of actuatorsubstitutions. In this instance the minimum obtainable SSEgwas 0.575 63, a result illustrated by figure 5, which gives thetarget node distribution.

SFS provided the optimum actuator locations and strainswhen the search space was limited to max 0.072 72. Ifmax was increased beyond this the method became overlygreedy, resulting in a first step of 3 = 0.124 77 givingan orthogonal residual. This aggressive selection methodbypassed improved selections, resulting in high applied strainsin the first selection step instead of a more even actuationdistribution that characterizes optimal designs. As the searchspace became larger the results in this case illustrate theworsening performance of the SFS method when compareddirectly to results obtained with reduced max.

IFS with = 0.5, Ei = 1600 and Ai = 1, in the absenceof max constraints, advanced the applied strain initially at site3 until 3 = 0.0265. At this point sites 3 and 7 were equallycorrelated with the target vector and progression was sharedbetween the two locations. At 3 = 0.04 and 7 = 0.0148site 5 had equal correlation, so progression was shared betweenthe 3 locations until the result in table 1 where max 0.1was obtained. If max = 0.05 then the strain progressionoccurred as described previously until the point was reachedwhere 7 = 0.05. At this point the max constraint becameactive and the reduced search space resulted in 3 reaching thebounding max value, unlike the example of the unrestricted

Figure 5. 14 element truss with 3 actuators located using ES and thegeometry objective, together with the target node geometry. Theelement shade refers to the applied strain.

Table 2. Shape objective error (SSEs) with respect to max for the 14element truss with 3 actuator substitutions.

Selectionmethod max SSEs

Actuatorlocation i

ES 0.01 61.191 95 3, 5, 7 0.01, 0.01,0.010.025 23.264 01 3, 5, 7 0.025, 0.025,0.0250.05 0.523 97 3, 5, 7 0.05, 0.045 55,0.050.1 0.040 19 3, 7, 11 0.072 32,0.051 11,0.027 190.15 0.040 19 3, 7, 11 0.072 32,0.051 11,0.027 19

GA 0.01 61.191 95 3, 5, 7 0.01, 0.01,0.010.025 23.264 01 3, 5, 7 0.025, 0.025,0.0250.05 0.523 97 3, 5, 7 0.05, 0.045 55,0.050.1 0.040 19 3, 7, 11 0.072 32,0.051 11,0.027 190.15 0.040 19 3, 7, 11 0.072 32,0.051 11,0.027 19

SFS 0.01 61.191 95 3, 5, 7 0.01, 0.01,0.010.025 23.264 01 3, 5, 7 0.025, 0.025,0.0250.05 1.321 00 3, 7, 11 0.05,0.05,0.050.1 0.329 43 2, 3, 7 0.018 65,0.045 13,0.10.15 0.424 60 4, 6, 7 0.078 95,0.018 05,0.146 14

IFS 0.01 61.191 95 3, 5, 7 0.01, 0.01,0.010.025 23.264 01 3, 5, 7 0.025, 0.025,0.0250.05 1.342 47 3, 7, 11 0.05,0.05,0.0490.1 0.041 11 3, 7, 11 0.072 87,0.048 12,0.029 380.15 0.041 11 3, 7, 11 0.072 87,0.048 12,0.029 38

search. The conservative progression of the IFS method did notpermit the selection of the optimal locations; for max = 0.05 itwas the worst performing method. However as the size of thesearch space was increased IFS did not suffer the performancedeterioration that afflicted the SFS procedure.

5.2. Shape objective results

Table 2 and figure 6 show the results for the shape objectiveSSEs defined in equation (14) for the 14 element truss withrespect to varying max. Three actuator substitutions wereagain selected using the ES, GA, SFS and IFS methods. Aswith the geometry objective, the GA results matched thoseobtained with ES and thus they may be considered to beglobally optimal. In the cases where max < 0.072 32 thesearch space was constrained by max leading to selection oflocations 3, 5 and 7 when max 0.05. This provided closeagreement with the geometry objective solutions. However

6


max

Figure 6. Shape objective (SSEs) against maximum absoluteactuation strain (max) for a 14 element truss with 3 actuatorsubstitutions.

Figure 7. 14 element truss with 3 actuators located using ES and theshape objective, together with the target node geometry (dashed) andthe location of surface objective points R for the pictured deflection.The element shade refers to the applied strain.

at max 0.072 32 the location selections switched to 3, 7and 11, identified as the elements forming the lower facingsurface of the aerofoil illustrated by figure 7. Comparisonof these shape objective results with those derived using thegeometric objective reveals a similar trend, whereby in theregion 0 max 0.05 large reductions in the SSEg andSSEs were achievable with small increases in max. As max

neared the point at which it no longer constrained the selectionproblem, then increased max had a reduced effect on the SSEin the case of both objective measures.

The SFS method at max 0.025 provided the optimalselection however, as with the geometry objective, as the searchspace was increased the greedy selection method fell in to anon-optimal path. In each case taking an excessively large firststep resulted in highly localized deflections of the structure.This effect is revealed in table 2 by the large strains appliedat location 7 for the SFS method at max 0.1, with respectto the optimal solutions found by the ES and GA methodssubjected to the same constraints. The use of a shape objectivefunction that varied the target vector ys as a function of thestructure deflection at each selection step, coupled with thegreedy selection characterizing SFS, resulted in the selectionof a non-optimal path. Thus as max was increased beyond0.025 subsequent actuator selection was applied to correct theoverly greedy selections of previous steps. As was the case

Figure 8. 1752 element truss structures with identical geometry anddiffering topology and referred to as structures A and B.

Table 3. Number of iterations and CPU time required to reach theoptimal solution using the shape objective error (SSEs) for the 14element truss with 3 actuator substitutions and max = 0.1.

Selectionmethod

Number ofiterations CPU time

ES 364 35.89GA 2940 9.01SFS 3 0.10IFS 650 0.96

with the geometry objective the enlargement of the searchspace resulted in overly large initial selection steps resultingin reduced performance of the method at max 0.1.

The IFS method obtained optimal actuator selectionswhen max 0.025, however as max 0.1 the methodprovided optimum locations with non-optimum regressioncoefficient values. At max = 0.05 the limitations of theIFS procedure were highlighted by the greedy selection of7 = 0.05 which led to the subsequent selection of location11 in place of optimal location 5.

Table 3 lists the number of iterations and CPU timeto complete the actuator selection analysis for 3 actuatorsubstitutions in the 14 element truss using ES, GA, SFS andIFS selection methods. All of the analyses were implementedwith MATLAB v7.2 using a 2.8 GHz CPU. For the ESmethod the computational speed could be reduced by analyticalformulation of the objective function sensitivities, rather thanthe finite difference method employed. For the GA the totalnumber of function evaluations is listed, this was achieved over147 generations with a population size of 20.

6. Two 1752 element truss structures

Figure 8 illustrates two statically and kinematically determi-nate truss structures comprising 1752 elements and referred toas A and B. Each structure is geometrically identical andtopologically similar; the difference being the connectivity ofelements adjacent to the surface. As with the 14 element trussboth form the trailing 0.4c fraction of a NACA 0012 aerofoil.

7


max

SS

Eg

Figure 9. Geometry objective (SSEg) against maximum absoluteactuation strain (max) for trusses A and B with 88 actuatorsubstitutions.

Table 4. Geometry and shape objective SSE error with respect tomax for the 1752 element truss with 88 actuator substitutions.

Truss A Truss B

Selectionmethod max

GeometrySSEg

ShapeSSEs

GeometrySSEg

ShapeSSEs

GA 0.01 109.560 98 98.724 06 124.551 44 115.044 250.025 8.383 63 6.673 65 16.461 45 13.395 240.05 0.546 01 0.031 81 0.510 00 0.015 810.1 0.546 01 0.018 17 0.271 42 0.008 480.15 0.536 77 0.018 17 0.271 42 0.008 48

SFS 0.01 147.696 34 132.135 95 154.670 79 140.511 440.025 48.124 39 46.803 27 48.141 31 44.024 650.05 5.565 88 6.071 91 1.723 53 1.288 410.1 1.065 48 0.565 86 1.250 71 1.717 840.15 1.023 47 0.409 57 1.214 97 1.652 18

IFS 0.01 150.850 47 130.561 71 157.971 06 140.286 890.025 51.545 61 35.011 10 48.113 35 42.004 420.05 3.294 13 1.331 78 1.364 24 0.247 760.1 1.137 95 0.128 39 1.152 74 0.096 800.15 0.810 81 0.071 99 1.152 74 0.096 80

Table 4 and figures 9 and 10 provide the results of locating88 actuators within trusses A and B using the GA, SFS and IFSmethods. The target form was defined by C1 = 4.7 103.

Beginning with truss A, the GA provided the bestperformance in terms of minimizing the SSE for both thegeometry and shape objective functions. In the case of thegeometry objective function a near minimum value of SSEgwas realized at values of max 0.05. In the case of the shapeobjective function little improvement in the SSEs value wasobserved at values of max 0.1, when using the GA searchmethod. The SFS method applied with the geometry objectiveprovided similar performance to the IFS method suggestingboth selection procedures followed non-optimal paths whencompared to the GA. Considering the shape objective, onceagain the GA outperformed both the SFS and IFS procedures,however here a notable improvement of the IFS method overthe SFS method was observed. This is due to the alteration ofthe target vector as the actuator selection process is progressed.

max

SS

Es

Figure 10. Shape objective (SSEs) against maximum absoluteactuation strain (max) for trusses A and B with 88 actuatorsubstitutions.

As each actuator is added to the subset the structure is subjectedto a displacement that, in the case of the shape objective,alters the target displacement. By taking large steps towardsthe target vector the SFS method oversteps possible improvedselections that may appear as the target alters with the valuesof the regression coefficients.

Truss B exhibited similar characteristics in the selectionof optimal locations as for truss A, whereby the GA provedto be the best selection method in terms of reducing theSSEg and SSEs in all cases. As with truss A, the SFS andIFS methods have similar performances when applied to thegeometry objective. Improved results were obtained using theIFS method with the shape objective when compared with theSFS method.

Comparison of the results between the two truss patternsfor the GA selections revealed an improved performance oftruss A at max 0.025 for both the shape and geometryobjective functions. Figure 11 illustrates the deflectedstructures for both A and B with 88 actuator substitutionsand max = 0.1. Some common features of the two resultsare evident, including concentrations of actuator substitutionsalong the entire upper surfaces and between 0.6c and 0.75con the lower surfaces of the trusses. The deflected result oftruss A experienced large rotations of the elements in the uppersurface region between 0.8c and 0.85c leading to a numberof overlapping elements. In addition truss B required reducedstrain in the upper surface element in the region 0.9c to c due tothe increased deflection of the structure forward of this point.

6.1. Selection filtering for regression based subset selection

Figure 12 illustrates the deflected A and B trusses as a resultof a series of actuators located using the IFS method with theshape objective and max = 0.1. Of note is the concentrationof actuation elements on the upper and lower surfaces ofeach truss together with a second concentration between 0.85cand 0.9c. Here the active elements are prescribed regressioncoefficients equivalent to ui = Ei Aimax. Each of thoseelements located in the second concentration had a highcorrelation with the target, however the required regression

8


Figure 11. Trusses A and B (A above) with 88 actuators locatedusing the GA and the shape objective. The element shade refers tothe applied strain.

Figure 12. Trusses A and B (A above) with 88 actuators locatedusing IFS and the shape objective. The element shade refers to theapplied strain.

coefficient in order to reach a point orthogonal to the targetis such that selection of these locations may be considerednon-optimal. This conclusion is confirmed by observation thatthe GA search ignored those regions. In order to improvethe selection procedure using the SFS and IFS methods itwas proposed to exclude those variables from the search forwhich u j > , where u j is the absolute of the j thregression coefficient calculated using equation (17) and is a constant value dictating the limit at which a variable isdeemed unsuited for inclusion in the active set. Due to thelarge number of iterations for IFS, the calculation of u j for

Table 5. Geometry and shape objective SSE error with respect tomax for the 1752 element truss with 88 actuator substitutionsincluding element filtering.

Truss A Truss B

Selectionmethod max

GeometrySSEg

ShapeSSEs

GeometrySSEg

ShapeSSEs

SFS filteredselection

0.01 130.428 89 120.661 81 129.926 06 124.638 220.025 29.434 26 32.748 20 19.379 63 16.227 660.05 1.146 94 0.683 89 0.834 06 0.926 660.1 0.783 07 0.303 31 0.842 46 0.893 370.15 0.699 19 0.186 31 0.665 94 0.864 38

IFS filteredselection

0.01 132.046 10 122.698 36 124.638 22 117.159 980.025 31.123 39 31.059 41 16.227 66 14.403 170.05 1.763 78 0.168 33 0.926 66 0.012 350.1 0.816 96 0.030 63 0.893 37 0.006 170.15 0.760 73 0.013 67 0.864 38 0.005 43

all variables at every iteration was prohibitively expensive,negating many of the benefits of the method when comparedto the GA. Thus the regression coefficients of all variableswere calculated and screened before undertaking the selectionanalysis using the IFS method. This system does not allow forchanges experienced by as the selection progresses, howeverby taking a suitable large value of this was assumed to haveno effect.

Table 5 and figures 13 and 14 illustrate the SSEg andSSEs results for trusses A and B with a selection filter valueof = 1 103 E j A j . In all cases 88 actuators were locatedwithin the structures using the SFS and IFS methods with atarget form defined by C1 = 4.7 103. For both objectivetypes the filter process eliminated the same variables, howeverfor both truss A and truss B the shape objective was found toexperience the greatest improvement due to selection filtering.Figure 15 illustrates the revised actuator locations for trussesA and B where max = 0.1. Of note is the relocation of thefiltered selection between 0.85c and 0.9c that was observed infigure 12.

6.2. Shape and geometry objective comparison

In order to compare the performance of the two objectivefunctions, the effectiveness of the geometry objective infulfilling the requirement of replicating pre-determined surfaceform was evaluated. Table 6 lists SSEsg, the shape objectivevalue for the actuator selections obtained using the geometryobjective value. The results reveal that the use of a geometricobjective for the requirement of forming a target shape is aviable option as the trends between SSEsg and SSEg withrespect to max are consistent.

Figure 16 illustrates the ratio SSEsg/SSEs with respect tomax, a value SSEsg/SSEs < 1 for any point implies that theuse of the geometric objective function provides an improvedmatch with the target shape than the shape objective function.For SSEsg/SSEs > 1 the shape objective function may beconsidered an improvement over the geometry objective forthe purpose of replicating a target shape. In the case of theGA and IFS selection methods for trusses A and B the shapeobjective function provided the best method for selection of

9


max

SS

Eg

max

SS

Es

Figure 13. SSEg (left) and SSEs (right) against maximum absolute actuation strain max for truss A with SFS and IFS selection methods, bothwith and without element filtering.

max

SS

Eg

max

SS

Es

Figure 14. SSEg (left) and SSEs (right) against maximum absolute actuation strain max for truss B with SFS and IFS selection methods, bothwith and without element filtering.

optimal actuator locations for the replication of a target shape.As the search space was expanded by increased max thenthis improvement of the shape objective over the geometryobjective became more pronounced. In the case of the SFSmethod, for both truss A and truss B the geometry objectiveproved to be marginally improved compared to the shapeobjective for the purpose of replicating a target shape. Thisresult was due to the non-constant ys and the greedy selectionsteps of SFS.

6.3. Actuator fraction analysis

Consideration has thus far only been given to the effectof variation of max on the SSEg and SSEs. The numberof actuator substitutions was fixed at 88 corresponding toapproximately 5% of the number of elements, m. Thefollowing section details the result of fixing max = 0.01 andvarying the number of actuators, n.

Table 7 summarizes the results of attempting to optimallylocate a set of n = 18, 44, 88, 175, 263, 438, 613 and 876

actuators within trusses A and B, corresponding to an actuatorfraction of fa = 0.01, 0.025, 0.05, 0.1, 0.15, 0.25, 0.35 and0.5 respectively where fa = n/m. The results using boththe SFS and IFS methods proved to be similar in all cases,indicating that the previous difference between the results ofthe two methods was primarily due to large max.

Figure 17 plots the SSE for both shape and geometryobjectives with respect to varying actuator fraction when theactuator locations were selected using the IFS method. TrussB experienced a rapid reduction in the SSE value as fa wasincreased from 0 to 0.05 and in this region provided improvedperformance when compared with truss A. If fa > 0.05 trussA provided much improved performance with respect to trussB irrespective of the objective type and this pattern is repeatedwith the results using the SFS method.

Figure 18 illustrates the deflected form of truss A and trussB with fact = 0.025 and max = 0.01 and actuator locationselections made using the IFS method. The concentration ofactuator substitutions at the upper and lower surface is evidentfor both structures A and B due to the requirement of structural

10


Figure 15. Trusses A and B (A above) with 88 actuators locatedusing IFS, the shape objective and element location filtering. Theelement shade refers to the applied strain.

Table 6. Shape objective value (SSEsg) calculated using selectionsobtained with the geometry objective, with respect to max for the1752 element truss with 88 actuator substitutions without elementfiltering.

Selectionmethod max

Truss ASSEsg

Truss BSSEsg

GA 0.01 98.667 42 112.837 430.025 6.620 21 14.382 040.05 0.293 05 0.208 500.1 0.293 05 0.100 520.15 0.284 37 0.100 52

SFS 0.01 131.281 93 138.910 690.025 42.880 65 43.788 310.05 4.933 98 0.878 560.1 0.474 41 0.780 930.15 0.393 56 0.567 55

IFS 0.01 135.025 65 142.138 860.025 46.307 68 43.909 010.05 2.595 13 0.505 630.1 0.672 95 0.398 260.15 0.329 06 0.398 26

elongation and contraction of the upper and lower surfacesrespectively. Figure 19 illustrates the topology of half of asymmetric single cell that may be repeated to form the upperand lower surfaces of structures A and B. If all elements of theunit cells were replaced by actuators then for truss A, the strainof the cell parallel to the x-axis is formulated as

x = 1cos

[(1 + 3)

1 sin

2

(1 + 3)2

+

1 sin 2 (1 + 2)2]

2 (20)

where 2 and 3 are the strains of elements with length 2L2and L3 respectively. In the case of truss B x = 1. Thus byalteration of the geometry and the substitution of all elements

max

SS

Esg

/SS

Es

Figure 16. Ratio of the shape objective value from a solution derivedusing the geometry objective function (SSEsg), to the shape objectivevalue derived using the shape objective function (SSEs), with respectto max.

Table 7. Geometry and shape objective SSE error with respect tomax for the 1752 element truss with 88 actuator substitutionsincluding element filtering.

Truss A Truss B

Selectionmethod

Actuatorfractionfa

GeometrySSEg

ShapeSSEs

GeometrySSEg

ShapeSSEs

SFS 0.01 227.856 99 202.446 37 181.032 37 184.133 860.025 189.197 55 166.289 19 134.318 94 134.582 730.05 130.428 89 120.661 81 129.926 06 117.996 390.1 60.463 14 55.717 60 120.404 45 109.158 440.15 33.218 85 30.536 69 117.222 77 106.034 020.25 21.801 90 20.062 73 110.088 12 99.018 010.35 17.669 77 15.766 83 104.254 33 93.263 080.5 14.374 92 12.290 47 96.816 71 85.925 72

IFS 0.01 229.620 86 204.732 94 201.697 69 182.276 860.025 189.770 60 166.358 42 149.398 49 134.288 340.05 132.046 10 122.698 36 124.638 22 117.159 980.1 64.962 68 59.618 10 120.404 45 109.157 890.15 33.151 85 30.570 01 117.269 57 106.095 120.25 21.620 99 19.924 66 110.322 38 99.253 020.35 17.743 91 15.882 73 104.601 39 93.616 020.5 14.486 34 12.443 36 97.422 88 86.993 08

connected to a surface facing node it is possible to achievex >

max with the surface topology of truss A. In contrastfor truss B the surface extension/contraction is approximatelyequal to max.

7. Conclusions

The use of a statically and kinematically determinate truss wasproposed to act as a host for an integrated actuation system thatis able to support an externally applied load. The consideredtrusses were 2D networks of articulated elements and in thisform offer no resistance to actuation loads; however, for therealization of a practical structure the use of compliant jointsbetween relatively stiff elements must be considered to reducecomplexity. In this case the cross section of the elementis varied in order to obtain localized deflection at the jointlocations whist maintaining a monolithic type construction.

11

Smart Mater. Struct. 18 (2009) 035014 D Baker and M I FriswellS

SE

Actuator Fraction fa

Figure 17. Sum of squares error (SSE) for both the shape objective(SSEs) and the geometry objective (SSEg) with respect to theactuator fraction ( fa). For all results max = 0.01.

Figure 18. Deflected trusses A and B (A above) with fa = 0.25 andmax = 0.01, of note is the increased deflection of truss A. Theelement shade refers to the applied strain.

The actuation system within the proposed structures hasa structural role to fulfil and so a system is required todeliver moderate actuation strain coupled with an ability toact as a structurally integral element with minimal energyconsumption. Possible smart actuation systems may includehigh strain piezoelectric based actuators, SMA actuatorsand magnetorheological actuation systems in addition toconventional hydraulic and electric actuators. The practicalimplementation of a large array of actuators in a dense trusstopology would present considerable challenges includingphysically locating the system within the truss, connectionof the actuators to the surrounding truss and the routing ofservices to the actuators.

Periodic truss type structures have a load bearingperformance greatly superior to stochastic foams andcomparable to honeycombs when used as the core material forsandwich panels [25]. Using techniques such as investmentcasting, deformation forming and additive layer manufactureintricate trusses can be generated to exacting tolerances atacceptable costs. The topologies investigated for the purpose of

A B

Figure 19. Unit cell that when repeated approximates the surfacetopology of trusses A (left) and B.

this investigation were selected for their determinate propertiesand as an illustration of the selection methods to both largeand small scale problems, using similar methods the selectiontechniques may also be applied to three-dimensional trussarrangements. By selection of the appropriate truss topologyand geometry it may be possible to provide an efficient loadbearing structural solution with integrated morphing controlusing a periodic truss system. Additional functional benefitsof an open cell system may be realized by integrated liquidstorage or by heat transfer into a coolant fluid.

In order to find appropriate locations in which to makean actuator substitution in a previously passive element twoobjective functions were defined. A geometric type objectivedefined a set of nodal deflections based on the perturbation ofthe aerofoil mean line, this target deflection remained constantthroughout the selection process. A shape type objective wasalso proposed whereby the target deflection was a functionof the location of the structure at that particular phase inthe selection process. This was achieved by formulating anapproximation of the minimum distance between a structuraljoint on the surface and the target surface. This approximationoffers a direct solution without the requirement of iterative rootfinding analysis. It was found that, except for those cases inwhich the SFS method was chosen, the shape objective offeredimproved performance.

Four methods to locate actuators within a truss usingthe two objective functions were investigated. An exhaustivesearch provided the optimum solution at the expense of highcomputational requirements and thus was only applicable toproblems of limited size. The GA permitted the search of afraction of the search space, however the number of functionevaluations remained high and there was no guarantee oflocating a global optimum. In isolated cases for the largerstructures it was outperformed by IFS. IFS employs forwardselection methods, however in order to reduce the problemsthat such methods have in large problems the regressioncoefficient is only incremented by a small amount at each step.In contrast SFS uses the computationally efficient method oftaking greedy selection steps, either within the actuator strainlimits or to a point orthogonal to the target vector. It was foundthat the IFS method offered significant advantages over SFSwhen selecting actuator locations using the shape objective,particularly when coupled to a problem in which the searchspace was enlarged by increased max.

12


A simple 14 element truss was first used to provide insightin to the relative merits of the selection schemes followed bytwo 1752 element trusses. The two larger trusses were similarexcept for the connectivity of the elements near the surface. Forexamples in which the maximum actuator strain was limited to0.01, the structure that permitted greater extension/contractionof the geometry by combining the effects of a greater numberof actuators could outperform a structure in which the surfaceforming joints were directly connected. However, in the samelimited strain investigation if the number of actuation elementswas reduced then the structure with direct connections betweenthe surface joints could provide better performance.

By removing those selections that provide good correla-tion with the target deflection yet require large regression coef-ficients, far beyond practical constraints, it was possible to sig-nificantly improve the SFS and IFS selection processes. Thishad the twin effect of removing erroneous selection choiceswhilst also reducing the search space size, making SFS andIFS more competitive with the GA. In most cases where com-putation time of a single function evaluation was sufficientlysmall the GA provided the best choice for actuator location se-lection. However, it has been shown that regression methods,particularly IFS, can provide good, but sub-optimal, selectionchoices.

Acknowledgments

The authors acknowledge funding from the EuropeanCommission through the Marie Curie Excellence GrantMEXT-CT-2003-002690 and from the EPSRC.

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http://dx.doi.org/10.2514/3.57533http://dx.doi.org/10.1117/12.351561http://dx.doi.org/10.1177/1045389X04042796http://dx.doi.org/10.2514/2.3062http://dx.doi.org/10.1117/12.483869http://dx.doi.org/10.2514/2.775http://dx.doi.org/10.2514/1.1519http://dx.doi.org/10.2514/1.24476http://dx.doi.org/10.1016/0045-7949(85)90105-1http://dx.doi.org/10.2514/3.10739http://dx.doi.org/10.1126/science.220.4598.671http://dx.doi.org/10.1088/0964-1726/14/4/047http://dx.doi.org/10.1016/0020-7683(86)90014-4http://dx.doi.org/10.1016/j.jmps.2005.02.011http://dx.doi.org/10.1214/07-EJS004http://dx.doi.org/10.1016/S0266-3538(03)00266-5

1. Introduction2. Determinate structures3. Problem formulation3.1. Geometry objective3.2. Shape objective

4. Subset selection5. A 14 element truss structure5.1. Geometry objective results5.2. Shape objective results

6. Two 1752 element truss structures6.1. Selection filtering for regression based subset selection6.2. Shape and geometry objective comparison6.3. Actuator fraction analysis

7. ConclusionsAcknowledgmentsReferences

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