aero elasticity of micro vehicles.pdf

Struct Multidisc Optim (2009) 38:301316DOI 10.1007/s00158-008-0292-x

INDUSTRIAL APPLICATION

Aeroelastic topology optimization of membrane structuresfor micro air vehicles

Bret Stanford Peter Ifju

Received: 7 March 2008 / Revised: 2 May 2008 / Accepted: 10 June 2008 / Published online: 19 July 2008 Springer-Verlag 2008

Abstract This work considers the aeroelastic optimiza-tion of a membrane micro air vehicle wing throughtopology optimization. The low aspect ratio wing isdiscretized into panels: a two material formulation onthe wetted surface is used, where each panel can bemembrane (wing skin) or carbon fiber (laminate re-inforcement). An analytical sensitivity analysis of theaeroelastic system is used for the gradient-based op-timization of aerodynamic objective functions. An ex-plicit penalty is added, as needed, to force the structureto a 01 distribution. The dependence of the solu-tion upon initial design, angle of attack, mesh density,and objective function are presented. Deformation andpressure distributions along the wing are studied forvarious load-augmenting and load-alleviating designs(both baseline and optimized), in order to establisha link between stiffness distribution and aerodynamicperformance of membrane micro air vehicle wings. Thework concludes with an experimental validation of thesuperiority of select optimal designs.

Keywords Micro air vehicle Topology optimization Aeroelastic membrane wing

Nomenclature

a adjoint vectorA aeroelastic Jacobian

B. Stanford (B) P. IfjuDepartment of Mechanical and Aerospace Engineering,University of Florida, Gainesville, FL 32611, USAe-mail: [email protected]

angle of attackb wingspan small number to prevent singularityc root chordC vortex lattice influence matrixCD drag coefficientCL lift coefficientCL lift slopeCm pitching moment slopeD carbon fiber flexural rigidityg objective functionG aeroelastic system of equations horseshoe vortex circulationsK reduced global stiffness matrixKe element stiffness matrixKp, Km carbon fiber and membrane stiffness

matricesL vortex lattice source vectorL/D efficiencyn wing outward normalp nonlinear power lawP, Q interpolation matricesr aeroelastic responseR penalty parameterS sensitivity of objective function to vortex

circulationsT in-plane membrane pre-stress resultantu solution to finite element analysisuo, vo, wo induced velocitiesU free-stream velocityw out-of-plane displacementx, y chordwise and spanwise positionX element densitieszo, z rigid and flexible wing shapes

302 B. Stanford, P. Ifju

1 Introduction

This work considers the design of flexible micro airvehicle wings (MAVs, with a wingspan less than 15 cm)through aeroelastic topology optimization. The struc-ture for these vehicles resemble larger hang gliders andsail wings (Ormiston 1971): a flexible membrane sur-face (latex rubber, in this case) strategically reinforcedwith stiffer rib and batten structures (carbon fiber lam-inates). The flight performance of flexible MAVs hasindicated several desirable properties directly attribut-able to the elastic nature of the wing: primarily, passiveshape adaptation (Albertani et al. 2007). Dependingupon the nature of the laminate reinforcement (ofwhich the trailing edge is particularly important), thewing deformation may be able to alleviate the flightloads (which may reduce the drag, delay stall, or pro-vide gust rejection) or augment the flight loads (higherlift, longer static stability margins). Either methodologymay be preferred (or perhaps a compromise betweenthe two), depending upon flight environment (indooror outdoor flight), payload, propulsion, range and du-ration requirements. This problem can be solved bydiscretizing the wing into panels for an aeroelastictopology optimization study, with a two-material for-mulation (each panel can be stiff carbon fiber lami-nate or flexible latex membrane) on the wetted surfaceof the wing. A series of aerodynamic objective func-tions can be considered, including L/D, CL, CD, CL ,and Cm .

The basics of topology optimization are given byZuo et al. (2007): the design domain is discretized,and the relative density of each element can be 0(void) or 1 (solid). Early work in this field optimizesa structure under static loads by minimizing the com-pliance with an equality constraint upon the volumefraction, though applications of topology optimizationhave expanded dramatically in recent years. A reviewof the extensive and disparate applications that havebeen successfully solved with topology optimization isgiven by Eschenauer and Olhoff (2001), Bendse andSigmund (2003), and Bendse et al. (2005).

Solving a topology optimization scheme with strictlydiscrete design variables is rare; Anagnostou et al.(1992) utilize simulated annealing, while Deb and Goel(2001) and Wang et al. (2006) use a genetic algorithm(with the latter paper providing a detailed literaturereview of the topic). Beckers (1999) uses a dual methodto solve the large-scale discrete problem, while recentwork by Stolpe and Stidsen (2006) use a hierarchi-cal technique: the topology optimization problem issolved on successively refined meshes. The optimumobtained from the previous coarse mesh is used as a

good starting point for a local search algorithm, solvedas a convex mixed 01 program. The computationalrequirements of such binary techniques can be severefor even moderately-sized problems, however. A more-commonly utilized strategy lets the density vary contin-uously between 0 and 1 and then penalizes intermediatevalues during the optimization (Bendse and Sigmund2003), and is used for this work.

As noted by Stegmann and Lund (2005a), solid-voidtopology optimization of engineering shell structureshas little practical relevance: holes in the structure candepreciate the ability of the structure to carry mem-brane loads, and may be completely forbidden outof aerodynamic considerations. Two (or more) mate-rial formulations can potentially be used; interpolationschemes for such a model are given by Bendse andSigmund (2003), and a similar interpolation is used inthis work. Alternatively, layered shell models may beused, by constructing the shell from a combination ofsolid layers and artificial layers (which may or may notbe void). As discussed by Lee et al. (2000), the lay-up can use single artificial layers (potentially resultingin holes), two layers for an eccentric stiffness aboutthe mid-surface, or three layers with potential enclosedvoids or holes concentric to the midplane. This work isextended by Stegmann and Lund (2005b), who locatethe optimal distribution of glass/epoxy laminate andpolymeric foam in a shell structure, as well as the fiberangles of the former.

From a literature standpoint, aeronautical applica-tions of topology optimization of structures are rare;aeroelastic applications are rarer still. The literaturecan be roughly divided into three categories. The firstuses an aerodynamic solver to compute the pressuredistribution over the wing (during steady flight, pull-upmaneuvers, etc). This load distribution is then appliedto the structure for optimization: global metrics suchas compliance, weight/volume fraction, and vibrationfrequency or local metrics such as displacement andstress objective functions and constraints can all be con-sidered. The re-distribution of the aerodynamic loadingdue to elastic wing deformation is ignored. The secondcategory explicitly uses the aerodynamic performanceof a flexible wing for objective functions or constraints,and necessarily includes aeroelastic coupling. The thirdcategory is topology optimization of channel flows:changes in the force imparted by a fluid flow are notmade with flexible structures, but by dividing the com-putational domain into either fluid or solid elements.

Literature indicative of the first category can befound in the work of Balabanov and Haftka (1996),who optimize the internal structure of a transport wing,using a ground structure approach (the domain is filled

Aeroelastic topology optimization of membrane structures for micro air vehicles 303

with interconnected trusses, and the cross-sectionalarea of each is a design variable (Bendse and Sigmund2003)) for compliance minimization. Eschenauer andOlhoff (2001) optimize the topology of an internal wingrib under both pull-up load maneuvers and internaltank pressures, using a bubble method. Krog et al.(2004) also optimize the topology of wing box ribs, anddiscuss methods for interpretation of the results to forman engineering design, followed by sizing and shapeoptimization. Luo et al. (2006) compute the optimaltopology of an aerodynamic missile body, consideringboth static loads and natural frequencies.

Aeroelastic topology optimization (second categorydiscussed above) is an under-served method of aircraftdesign, presumably due to the large computational costassociated with such an undertaking. The seminal workin this field is given by Maute and Allen (2004), whoconsider the topological layout of stiffeners within aswept wing, using a three-dimensional Euler solvercoupled to a linear finite element model. Results froman adjoint sensitivity analysis of the coupled aeroelasticsystem (Maute et al. 2002; Martins et al. 2001) are fedinto an augmented Lagrangian optimizer to minimizemass with constraints upon the lift, drag, and wingdisplacement. The authors are able to demonstrate thesuperiority of designs computed with aeroelastic topol-ogy optimization, rather than considering a constantpressure distribution. This research effort is extendedby Maute and Reich (2006) for topology optimizationof a compliant morphing mechanism within an airfoil,considering both passive and active shape deforma-tions. Superior optima are found with this method,as compared to a jig-shape approach: optimizing theaerodynamic shape, and then locating the mechanismthat leads to such a shape. Very recent work is givenby Gomes and Suleman (2008), who use a spectrallevel set method to maximize aileron reversal speedby reinforcing the upper skin of a wing torsion box viatopology optimization.

For the third category listed above, early work inthe topology optimization of fluid flows is given byBorrvall and Petersson (2003), who divide the designdomain into fluid and solid elements. Flow is assumedto be governed by Stokes differential equations forcreeping planar flows. Interpolation between fluid andsolid is obtained by means of the distance between thetwo surfaces that contain the fluid. Gersborg-Hansenet al. (2005) extend this technique to account for fluidinertia in the governing equations, thus utilizing a non-linear flow model. Pingen et al. (2007) find the mini-mum drag profile of submerged bodies, using a latticeBoltzmann method as an approximation to the NavierStokes equations. Drag is minimized by a football shape

(with front and back angles of 90) at low Reynoldsnumbers (where reducing surface area is important),and a symmetric airfoil at higher Reynolds numbers(where streamlining is more important).

The research detailed in this work is solely con-cerned with the second category discussed above:aeroelastic topology optimization of a thin flexible mi-cro air vehicle wing. This work details the locationof holes within a carbon fiber wing shell, holes whichwill then be covered with a thin, taught, rubber mem-brane skin. Numerically, the wing will be discretizedinto a series of panels, wherein each panel can bea carbon fiber laminated shell or an extensible latexrubber skin. The wings performance is determined by astatic aeroelastic solver, and objective functions consistof lift, drag, and pitching moment-based aerodynamicmetrics. Nonlinear interpolation schemes allow eachwing panel to vary continuously between carbon fiberskeleton and rubber membrane skin, necessitating anadjoint sensitivity analysis of the coupled aeroelasticsystem.

The remainder of this work is organized as fol-lows. A brief description of membrane micro air ve-hicle aeroelasticity is given, followed by details of thetopology optimization within the wing structure, andhow the technique fits into the overall design schemerequired for MAVs. This is followed by the compu-tational framework: material interpolation, the staticaeroelastic model, sensitivity analysis, and optimizationprocedures. Results are then given in terms of con-vergence histories, and the dependence of the optimaldesign (upon initial design, flight conditions, mesh den-sity, and objective functions). Lift generation as seen bythe topology optimizer is examined in detail, in orderto elucidate several load-alleviating and -augmentingmechanisms. The work is concluded with an experimen-tal validation of the superiority of select optimal designsover baseline wing structures.

2 Membrane micro air vehicle wings

Two extensively-flight tested membrane wing topolo-gies are given in Fig. 1. On the left of the figureis a batten-reinforced wing design (BR), where threecarbon fiber strips (battens) are imbedded into eachmembrane semi-wing. The trailing edge of this designis permitted to lift up due to an applied pressure load.The nose-down geometric twist of each wing sectionwill alleviate the flight loads, and possibly decreaseCD, and CL (as compared to a rigid wing; Albertaniet al. 2007). Conversely, on the right side of Fig. 1is a perimeter-reinforced wing design (PR). Now the


Fig. 1 Batten-reinforced(left) and perimeter-reinforced (right) membranewing designs

trailing edge is constrained by a laminate perimeter,and the membrane skin will inflate due to an appliedload. This aerodynamic twist of each flexible wing sec-tion will augment the flight loads, and possibly increaseCL and decrease Cm (Albertani et al. 2007).

A rigorous aeroelastic topology optimization of themembrane wings may be able to produce structures thatout-perform these two designs. A fairly fine structuralgrid is needed to resolve topologies on the order ofthose seen in Fig. 1, which will, of course, increase thecomputational cost associated with the optimization.The wing is discretized into a set of quadrilaterals,which represent the density variables: 0 or 1. Thesequadrilaterals are used as panels for the aerodynamicsolver, and broken into two triangles for the finiteelement solver, as shown in Fig. 2. As in Fig. 1, thewing topology at the root, leading edge, and wing tip isfixed as carbon fiber, to maintain some semblance of anaerodynamic shape capable of sustaining lift. The wingtopology in the figure is randomly distributed.

It should also be noted that aeroelastic topologyoptimization is one of many techniques that should betaken into account when designing a micro air vehi-cle: Torres (2002), for example, minimizes a combina-tion of payload, endurance, and agility metrics, withwing/tail planform, aspect ratio, propeller location, andangle of attack variables. This work fixes all of these

variables, and only considers un-constrained single-objective topology optimization of longitudinal aerody-namic performance metrics. The goal of this researchis to provide insight into the complex relationship be-tween stiffness distribution and aerodynamic behaviorof a membrane MAV wing, with the future goal ofutilizing this technique as part of a larger, global designscheme.

3 Computational framework

3.1 Material interpolation

Topology optimization with continuous density vari-ables may require a mechanism to push the final struc-ture to a 01 distribution. An implicit penalty uponintermediate densities can be achieved through anonlinear power law interpolation. This technique isknown as the solid isotropic material with penalizationmethod, or SIMP (Zuo et al. 2007). The power lawseffectiveness as an implicit penalty is predicated upona volume constraint: intermediate densities are unfa-vorable, as their stiffness is small compared to theirvolume (Bendse and Sigmund 2003). For the two-material wing considered above (membrane or carbon

Fig. 2 Sample wing topology(left), aerodynamic mesh(center), and structural mesh(right)


fiber), the stiffness matrix Ke of each finite element inFig. 2 can be computed as:

Ke =(Kp (1 ) Km

) Xpe + Km + Kp (1)where Kp and Km are the plate and membrane ele-ments, respectively (the latter with zeros placed withinrows and columns corresponding to bending degrees offreedom). is a small number used to prevent singular-ity in the pure membrane element (due to the bendingdegrees of freedom), Xe is the density of the element,varying from 0 (membrane) to 1 (carbon fiber), and pis the nonlinear penalization power.

No volume constraint is utilized here, due to an un-certainty upon what this value should be. Furthermore,for aeronautical applications it may be desired to mini-mize the mass of the wing itself, as discussed by Mauteand Allen (2004). Regardless, the nonlinear power lawof SIMP is still useful for the current application, asdemonstrated in Fig. 3. Both linear and nonlinear ma-terial interpolations are given for the lift computation(the aeroelastic model used here is described below),and the wing topology is altered uniformly.

For the linear interpolation (i.e., without SIMP), theaeroelastic response is a weak function of the densityuntil X becomes very small (0.001), when the sys-tem experiences a very sharp change as X is furtherdecreased to 0. This is a result of the large stiffnessimbalance between the carbon fiber laminates and themembrane skin, and the fact that lift is a direct functionof the wings compliance (the inverse of the weightedsum of the two disparate stiffness matrices). The inclu-sion of a nonlinear penalization power (p = 5), spreadsthe response evenly between 0 and 1: at X = 0 and 1,both linear and nonlinear interpolation curves coincide.Aeroelastic topology optimization with linear materialinterpolation experiences convergence difficulties, as

Fig. 3 Effect of linear and nonlinear material interpolationupon lift

the gradient-based technique struggles with the nearly-disjointed design space; a penalization power of 5 isutilized for the remainder of this work. It can also beseen in Fig. 3 that the sensitivity of the aeroelastic re-sponse to element density is zero for a pure membranewing (X = 0), as can be inferred from the interpolationequation above. As such, using a pure membrane wingas an initial guess for optimization will not work, as thedesign wont change.

Two local optima exist in the design space of Fig. 3,which may prevent the gradient-based optimizer fromconverging to a 01 material distribution. To counteractthis problem, an explicit penalty on intermediate densi-ties is added to the objective function (Chen and Wu2008):

R NX

i=1sin (Xi ) (2)

where R is a penalty parameter appropriately sized soas not to overwhelm the aerodynamic performance ofthe wing topology. Several potential issues may appearwith this formulation: the inclusion of a large penaltyearly in the optimization process will upset the compro-mise between improving the aerodynamic performanceand removing intermediate densities. In order to pre-vent such interference, this penalty is only added whenand if the aeroelastic optimizer has converged upon adesign with intermediate densities. R is sized such thatthe penalty is 1015% of the objective function, and isnot increased during the optimization process. Such astrategy is not found to significantly alter the optimaltopology, as will be discussed below.

3.2 Static aeroelastic analysis

Due to the large number of expected function eval-uations (200) needed to converge upon an optimalwing topology, and the required aeroelastic sensitivities(computed with an adjoint method), a lower-fidelityaeroelastic model is used for the current application.The membrane elements within the wing structure areassumed to obey Poissons differential equation fora taught elastic membrane subjected to a transversedistributed pressure (Cook et al. 2002):

2w = p/T (3)where p is the applied pressure, w is the out-of-planedisplacement, and T is the in-plane pre-stress resultant.This linear membrane model is known to be accurate,as long as the plane strains that develop within the elas-tic sheet due to the pressure loading are much smallerthan the original pre-strains. Discretization of the wing


into finite elements leaves one degree of freedom permembrane node: the out-of-plane displacement w. Sim-ilarly, the carbon fiber elements within the wing areassumed to be governed by:

4w = p/D (4)where D is the plates flexural rigidity, and the or-thotropy of the plain weave laminate is ignored. Eachnode of a plate element will have three degrees of free-dom: one out-of-plane displacement, and two rotations.The finite element mesh is seen on the right of Fig. 2.

A linear vortex lattice method is used to computethe aerodynamic pressure distribution over the flexiblewing. The continuous distribution of bound vorticity isapproximated by discretizing the wing into a paneledgrid, and placing a horseshoe vortex upon each panel.Each horseshoe vortex is comprised of a bound vortex(which coincides with the quarter-chord line of eachpanel), and two trailing vortices extending downstream.Each vortex filament creates a velocity whose magni-tude is assumed to be governed by the BiotSavartlaw (Katz and Plotkin 2001). Furthermore, a controlpoint is placed at the three-quarter-chord point of eachpanel. The tangency condition is applied (i.e., the wingbecomes a streamline of the flow) by stipulating thatthe induced flow (from the horseshoe vortices) alongthe outward normal at each control point exactly can-cels with that caused by the free-stream velocity. Thefollowing system of equations results:

C11 C12 C1NC21 C22 C2N...

.... . .

...

CN1 CN2 CNN

12...

N

=

{U cos () , 0,U sin ()} n1{U cos () , 0,U sin ()} n2

...

{U cos () , 0,U sin ()} nN

(5)

The source vector in (5) assumes longitudinal flow (noside-slip): U is the free-stream velocity, the angleof attack of the wing, and ni the outward normal ofthe wing at the ith control point. i is the unknowncirculation strength of each horseshoe vortex, and theinfluence coefficients are given by:

Cij = {uo, vo, wo}ij ni (6)where uo,i j, vo,i j, and wo,i j are the velocities inducedat the ith control point by the jth horseshoe vortex.Further information on the vortex lattice method usedhere can be found by Katz and Plotkin (2001). Pro-peller, fuselage, and stabilizers are not included in the

analysis, and the resulting aerodynamic mesh is given inthe center of Fig. 2.

Drag computations present a problem for this in-viscid modeling technique, which can only provide in-duced drag due to lift (the downwash of the finite spandeflects the local velocity vector at each wing sectiondownward, tilting the lift vector slightly backward toprovide a small drag force): drag due to flow separa-tion and viscous shear is unaccounted for. The latterterms are included by augmenting the drag with a non-zero CDo, estimated from experimental data to be 0.05.The viscous drag terms are not truly constant (flowseparation generally increases with angle of attack, forexample), and so the vortex lattice method is expectedto under-predict the overall drag. Previous work hasshown this method to provide an adequate computa-tion of drag (as well as lift and pitching moments) atmoderate angles of attack, as compared to wind tunneldata, for micro air vehicle wings (Stanford et al. 2007a).It should also be noted here that higher-fidelity aero-dynamic modeling techniques are available: unsteadymethods that model the wing-wake interactions, as wellas panel methods suitable for simulations of entirevehicles.

Aeroelastic coupling is facilitated by considering thesystem as defined by a three field response vector r:

r = {uT zT T }T (7)where u is the solution to the system of finite elementequations (composed of both displacements and rota-tions) at each free node, z is the shape of the flexiblewing, and is the vector of unknown horseshoe vortexcirculations (the solution vector of (5)). The coupledsystem of equations G(r) is then:

G (r) =

K u Q z zo P u

C L

= 0 (8)

The first row of G is the finite element analysis: K is thestiffness matrix assembled from the elemental matrices,and appropriately reduced based upon fixed boundaryconditions along the wing root. Q is an interpolationmatrix that converts the circulation of each horseshoevortex into a pressure, and subsequently into the trans-verse force at each free node. The second row of G isa simple grid regeneration analysis: zo is the original(rigid) wing shape, and P is a second interpolationmatrix that converts the finite element state vector intodisplacements at each free and fixed node along thewing. The third row of G is the vortex lattice method,rewritten from (5): C is the influence matrix dependingsolely on the wing geometry, and L is the source vector


depending on the wings outward normal vectors, theangle of attack, and the free stream velocity. Conver-gence of this system can typically be obtained within 25iterations, when the logarithmic error in the wings liftcoefficient is less than 5.

3.3 Adjoint sensitive analysis

As the number of variables in the aeroelastic system(essentially the density of each element) will alwaysoutnumber the constraints and objective functions, asensitivity analysis can be most effectively carried outwith an adjoint analysis. The sought-after total deriv-ative of the objective function with respect to eachdensity variable is given through the chain rule:

dgdX

= gX

+ gr

T

drdX

(9)

where g is the objective function and r is the aeroelas-tic state vector of (7). The term g/X is the explicitportion of the derivative, while the latter term is theimplicit portion through dependence on the aeroelasticsystem (Haftka and Grdal 1992). Only aerodynamicobjective functions are considered in this work: theexplicit portion is then zero, unless the intermediatedensity penalty (2) is included.

The derivative of the objective function with respectto the aeroelastic state vector is:

gr

= {0 0 ST }T (10)where S is the derivative of the aerodynamic objectivefunction with respect to the vector of horseshoe vortexcirculations. For metrics such as lift and pitching mo-ment, g = ST , though more complex expressions existfor drag-based metrics. The derivative of the aeroelasticstate vector with respect to the element densities isfound by differentiating the coupled system:

dG (X, r)dX

= 0 GX

+ A drdX

= 0 (11)Only the finite element analysis of the aeroelastic sys-tem contains the element densities, and so the deriva-tive of the aeroelastic system with respect to the densityvariables is then:

GX

=

KX u

00

(12)

A is the Jacobian of the aeroelastic system, defined by:

A = Gr

=

K 0 Q

P I 00 dC

/dz L/z C

(13)

The terms from (1013) are inserted into (9); the im-plicit portion of the derivative becomes a triple productwhich is solved with the adjoint method. The adjointvector does not contain the density of each element (X),and only needs to be solved once, as intended:

a = AT gr

(14)

Two three-dimensional tensors are required for theadjoint method. From a computational standpoint bothare relatively sparse, but K/X (12) is fairly inex-pensive to compute (Haftka and Grdal 1992), whiledC/dz (13) is very intensive, and represents a signif-icant portion of the cost associated with the gradientcalculations. The former term is mostly zero: as seen inFig. 2, each quadrilateral wing panel is divided into twotriangles. As such, K/Xi only receives contributionsfrom two element stiffness matrices, and the remainderof the assembled matrix is zero. The dC/dz term isthe derivative of the influence matrix with respect tothe wing shape, and requires a sensitivity analysis of theBiotSavart law. z is defined at the nodes of the meshin Fig. 2, or the corners of each vortex panel. As such,a change in a single zi will the change the geometricalposition of up to four control points and four horseshoevortices (both of which are defined within the panels):dC/dzi is then found to be a sparse matrix with up tofour non-zero rows and columns.

The second derivative of the objective function isrequired if aerodynamic derivative metrics such as CLand Cm are of interest. Two options are available forthis computation. The first involves a similar analyticalapproach to the one described above. This would even-tually necessitate the extremely difficult computationof A/r, which is seldom done in practice (Mauteet al. 2002). For the current work, only C dependson the aeroelastic state vector, while the rest of theJacobian would be zero. Nonetheless, this remainingterm d2C/dz2 is a four-dimensional tensor, and thecomputational complexities and cost associated with itsconstruction is expected to be severe. As an alternative,finite differences are used here:

g2

X 1

(gX

( +) gX

()

)(15)

The term g/ can be computed using anotherfinite difference, or with the adjoint method describedabove, substituting the angle of attack for the elementdensities X.

In order to solve the linear system of the Jaco-bian matrix, a staggered approach is adapted, ratherthan solving the entire system of (un-symmetric sparse)equations as a whole (Maute et al. 2002). Each


sub-problem is solved with the same algorithm usedin the aeroelastic solver (direct sparse solver for thefinite element equations, and an iterative GaussSeidelsolver for the vortex lattice equations), and as such, thecomputational cost and number of iterations neededfor convergence is approximately equal between theaeroelastic solver and the adjoint vector solver. Thesize of the numerical system is moderate: each opti-mization iteration requires 3 min of simulation timeon a Compaq Alpha workstation with a UNIX OSF1V5.0 910 operating system (for the 30 30 grid over thesemi-wing, as seen in Fig. 2), with construction of dC/dzallocated about 50% of this time. The computationalcost can be expected to decrease for stiffer wings.

3.4 Optimization framework

In order to ensure the existence of the optimal wingtopologies, a mesh-independent filter is employed.Such a filter acts as a moving average of the gradi-ents throughout the membrane wing, and limits theminimum size of the imbedded carbon fiber struc-tures (Bendse and Sigmund 2003). As no constraintsare included in the optimization, an unconstrainedFletcherReeves conjugate gradient algorithm (Haftkaand Grdal 1992) is employed. Step size is kept con-stant, at a reasonably small value to preserve the fi-delity of the sensitivity analysis. In order to increasethe chances of locating a global optimum, each opti-mization is run with three distinct initial designs:Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo = 0.1. A

pure membrane wing (Xo = 0) cannot be considered forreasons discussed above.

4 Results

Five objective functions are considered in this work:two load-augmenting metrics (maximum lift and min-imum Cm), two load-alleviating metrics (minimumdrag, and minimum CL) and efficiency (maximumL/D). The MAV wing under consideration has aZimmerman planform, with a 152 mm wingspan,124 mm root chord, and an aspect ratio of 1.25.7of positive geometric twist (nose up) is built into thewingtip, with 7 of dihedral between 2y/b = 0.4 and thewingtip. Both a reflex airfoil and a singly-curved airfoilare studied (the former mitigates the pitching momentof the wing, and allows for the removal of a horizontalstabilizer to decrease the size of the MAV). Both havea maximum camber at the root of 6.8% (at x/c = 0.22),and the former airfoil has a maximum reflex at the rootof 1.4% (at x/c = 0.86). Flight speed is kept constantat 13 m/s (equating to a Reynolds number of 105), butboth 3 and 12 angles of attack are considered, with a of 1 for finite differences. The flexural rigidity ofthe carbon fiber laminates is 0.5 Nm, and the pre-stressresultant of the latex membrane is 7 N/m. The thicknessof the former is 0.5 mm, and the latter is 0.1 mm.

A typical convergence history of the aeroelastictopology optimizer can be seen in Fig. 4, for a re-flex wing at 3 angle of attack, with a maximum L/Dobjective function. Results are computed with a 30

Fig. 4 Convergence historyfor maximizing L/D, = 3,reflex wing (30 30 grid overthe semi-wing)


30 grid over the semi-wing, and the initial guess is anintermediate density of 0.5. Within four iterations, theoptimizer has removed all of the carbon fiber adjacentto the root of the wing, with the exception of theregion located at three-quarters of the chord, whichcorresponds to the inflection point of the reflex airfoil.The material towards the leading edge and at the wingtip is also removed. Further iterations see topologicalchanges characterized by intersecting threads of mem-brane material that grow across the surface, leaving be-hind islands of carbon fiber. These structures arentconnected to the laminate wing, but are imbeddedwithin the membrane skin.

These results indicate two fundamental differencesbetween the designs in Fig. 1 and those computedvia aeroelastic topology optimization. The first is thepresence of islands; these designs can be built, butthe process is significantly more complicated than witha monolithic wing skeleton. Such structures could beavoided with a manufacturability constraint/objectivefunction (such as discussed by Lyu and Saitou (2005)),but the logistics of such a metric are difficult to formu-late. Furthermore, the aeroelastic advantages of free-floating laminate structures are significant, as will bediscussed below. A second difference is the fact thatthe designs of Fig. 1 are composed entirely from thinstrips of carbon fiber embedded within the membrane,while the topology optimization is apt to utilize two-dimensional laminate structures.

After 112 iterations in Fig. 4, the optimization haslargely converged (with only minimal further improve-ments in L/D), but some material with intermedi-ate densities remains towards the leading edge of thewing. Many techniques exist for effectively interpretinggray level topologies (Hsu and Hsu 2005); the explicitpenalty of (2) is used here. Surprisingly, the L/D seesa further increase with the addition of this penalty,contrary to the conflict between performance and 01convergence reported by Chen and Wu (2008). Theexplicit penalty does not significantly alter the topology,

but merely forces all of the design variables to theirlimits, as intended.

The final wing skeleton has three trailing edge bat-tens (one of which is connected to a triangular structuretowards the center of the membrane skin), and a fourthbatten oriented at 45 to the flow direction. The struc-ture appears to be a topological combination of a BRand a PR wing, with both battens and membrane infla-tion towards the leading edge. The optimized topologyincreases the L/D by 9.5% over the initial design, and(perhaps more relevant, as the initial intermediate-density design does not technically exist) by 10.2% overthe rigid wing.

The affect of mesh density is given in Fig. 5, for areflex wing at 12 angle of attack, with L/D maximiza-tion as the objective function. The 30 30 grid, forexample, indicates that 900 vortex panels (and 1,800finite elements) cover each semi-wing. As the leadingedge, root, and wing tip of each wing are fixed ascarbon fiber, 480 density design variables are left for thetopology optimization. One obvious sign of adequateconvergence is the efficiency of the rigid wing, withonly a 0.44% difference between that computed on thetwo finer grids. The optimal wing topologies in Fig. 5are similar, with three distinct carbon fiber structuresimbedded within the membrane skin. While the 20 20 grid is certainly too coarse to adequately resolvethe geometries of interest, the topology computed onthe 30 30 grid is very similar to that computed on the40 40 grid. The computational cost of each optimiza-tion iteration upon the coarser grid is five times lessthan that seen for a 40 40 grid, and will be used forthe remainder of this work.

The affect of the initial starting design is given inFig. 6, for a reflex wing at 12 angle of attack, with dragminimization as the objective function. As mentionedabove, Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo =0.1 are all considered. The three final optimal topolo-gies are very different, indicating a large dependencyupon the initial guess and no guarantee that a global

Fig. 5 Affect of mesh densityupon optimal L/D topology, = 12, reflex wing


Fig. 6 Affect of initial designupon the optimal CDtopology, = 12, reflex wing

optimum has been located. Nevertheless, the indicatedimprovements in drag are promising, with a potential6.7% decrease from the rigid wing. As expected, thedenser the initial topology, the denser the final opti-mized topology.

All three wing topologies utilize some form of adap-tive washout for load and drag alleviation. The struc-tures must be flexible enough to generate sufficientnose-down rotation of each wing section, but not soflexible that the membrane areas of the wing will inflateand camber, increasing the forces. The wing structure inthe center of Fig. 6 (with Xo = 0.5) strikes the best com-promise between the two deformations, and providesthe lowest drag. When Xo = 1, the structure is too stiff,relying upon a membrane hinge between the carbonfiber wing and root. When Xo = 0.1, the optimizer isunable to fill in enough space with laminates to preventmembrane inflation. Of the three designs, this is theleast tractable from a manufacturing point of view aswell.

The dependency of the optimal topology (maximumlift) upon both angle of attack and airfoil shape aregiven in Fig. 7, for both a reflex (left two plots) anda cambered wing (right two plots). For the wing withtrailing edge reflex, the optimal lift design looks similarto that found in Fig. 4: trailing edge battens that extendno farther up the wing than the half-chord, a spanwisemember that coincides with the inflection point of theairfoil, and unconstrained membrane skin towards the

leading edge, where the forces are largest. The opti-mizer has realized that it can maximize lift by bothcambering the wing through inflation at the leadingedge, and forcing the trailing edge battens downwardfor wash-in.

This latter deformation is only possible due to thereflex (negative camber) in this area, included to offsetthe nose-down pitching moment of the remainder ofthe flying wing, and thus allow for removal of ahorizontal stabilizer due to size restrictions. Increasingthe angle of attack from 3 to 12 shows no significantdifference in the wing topology, slightly increasing thelength of the largest batten. At the lower angle ofattack, up to 22% increase in lift is indicated throughtopology optimization.

For the cambered wing (singly-curved airfoil, righttwo plots of Fig. 7), the lift over the rigid wing is, asexpected, much larger than found in the reflex wings,but adequate stability becomes critical. With the re-moval of the negatively-cambered portion of the airfoil,most of the forces generated over this wing will bepositive, and the topology optimizer can no longer gainadditional lift via wash-in. Imbedding batten structuresin the trailing edge will now result in washout, surelydecreasing the lift. As such, the optimizer producesa trailing edge member that outlines the planformand connects to the root (similar to the perimeter-reinforced wing designs), restraining the motion of thetrailing edge and inducing an aerodynamic twist.

Fig. 7 Affect of angle of attack and airfoil upon the optimal CL topology


Unlike the PR wing, this trailing edge reinforcementdoes not extend continuously from the root to the tip,instead ending at 65% of the semi-span. This is thenfollowed by a trailing edge batten that extends into themembrane skin, similar to the designs seen for the re-flex wing in Fig. 7. Why such a configuration should bepreferred over the PR wing design for lift enhancementwill be discussed below. As before, increasing the angleof attack has little bearing on the optimal topology,again increasing the size of the trailing edge batten. Apotential increase in lift by 15% over the rigid wing isindicated at the lower angle of attack.

Similar results are given in Fig. 8, with L/D max-imization as the topology design metric. Presumablydue to the conflictive nature of the ratio, the wingtopology that maximizes L/D is a strong function ofangle of attack. For the reflex wing at lower angles, theoptimal design resembles topologies used above for liftenhancement (Fig. 7), while at 12 the design is closer intopology to one with minimum drag (Fig. 6). Increasinglift is more important to L/D at lower angles, whiledecreasing drag becomes key at larger angles. Thedrag is very small at low angles of attack (technicallyzero for this inviscid formulation, if not for the inclu-sion of a constant CDo), and insensitive to changes viaaeroelasticity.

This concept is less true for the cambered wing (righttwo plots of Fig. 8), where designs at both 3 and 12angle of attack utilize a structure with trailing edgeadaptive washout. At the lower angle, the topology op-timizer leaves a large triangular structure at the trailingedge (connected to neither the root nor the wing tip),and the leading edge is filled in with carbon fiber. Atthe higher angle of attack, four batten-like structuresare placed within the membrane skin, oriented parallelto the flow, one of which connects to the wing tip. Po-tential improvements are generally smaller than thoseseen above, though a 10% increase in L/D is availablefor the cambered wing at 12.

Wing displacements and pressure distributions aregiven for select wing designs in Fig. 9, for a reflex wingat 12 angle of attack. Corresponding data along thespanwise section 2y/b = 0.58 is given in Fig. 10. Asthe wing is modeled with no thickness in the vortexlattice method, distinct upper and lower pressure distri-butions are not available, only differential terms. Fivetopologies are discussed, beginning with a pure carbonfiber wing. Lift-augmentation designs are representedby a baseline PR wing and the topology optimized formaximum lift. Lift-alleviation designs are representedby a baseline BR wing and the topology optimized forminimum lift slope.

The differential pressure distribution over the mono-lithic carbon fiber wing is defined by leading edgesuction due to flow stagnation, pressure recovery (andpeak lift) over the maximum camber, and negativeforces over the reflex portion of the wing. As ex-pected, the inviscid solver misses the low-pressure cellsat the wingtip (from the vortex swirling system), andthe plateau in the pressure distribution, indicative ofa separation bubble. This aerodynamic loading causesa moderate wash-in of the carbon fiber wing (0.1),resulting in a computed lift coefficient of 0.604.

Computed deformation of the PR wing is muchlarger than the carbon fiber wing, though the deforma-tions are within the range of validity of a linear finiteelement solver. The sudden changes in wing geometryat the membrane/carbon fiber interfaces lead to sharpdownward forces at the leading and trailing edges, thelatter of which exacerbates the effect of the airfoilreflex. Despite this, the membrane inflation increasesthe camber of the wing and thus the lift, by 6.5% overthe carbon fiber wing.

Several disparate deformation mechanisms con-tribute to the high lift of the MAV design located by theaeroelastic topology optimizer (middle column, Fig. 9).First, the membrane inflation towards the leading edgeincreases lift via cambering, similar to the PR wing (the

Fig. 8 Affect of angle of attack and airfoil upon the optimal L/D topology


Fig. 9 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom) for baseline and optimal topologydesigns, = 12, reflex wing

pressure distributions over the two wing structures areidentical through x/c = 0.25). The main trailing edgebatten structure is then depressed downward along thetrailing edge (due to the reflex) for wash-in, increasingthe local angle of attack of each flexible wing section,and thus the lift. It can also be seen (from the left sideof Fig. 10 in particular) that the local bending/twistingof this batten structure is minimal: the deformationalong this structure is largely linear down the wing. Theintersection of this linear trend with the curved inflatedmembrane shape produces a cusp in the airfoil. The

small radius of curvature forces very large velocities,resulting in the lift spike at 46% of the chord.

This combination of wash-in and cambering leads toa design which out-performs the lift of the PR wing by5.6%, but the former effect is troubling. The wash-inessentially removes the reflex from the airfoil (as doesthe aerodynamic twist of the PR wing), an attributeoriginally added to mitigate the nose-down pitchingmoment. This fact leads to two important ideas. First,thorough optimization of a single design metric is ill-advised for micro air vehicle design, as other aspects of

Fig. 10 Deformations andpressures along 2y/b = 0.58for baseline and optimaltopology designs, = 12,reflex wing


the flight performance will surely degrade. Its inclusionhere is only meant to emphasize the relationship be-tween aeroelastic deformation and flight performance,and show the capabilities of the topology optimiza-tion. Secondly, if the design goal is a single-mindedmaximization of lift, a reflex airfoil is a poor choicecompared to a singly-curved airfoil, a shape which thetopology optimizer strives to emulate through aeroelas-tic deformations.

Referring now to the load-alleviating MAV wingstructures of Figs. 9 and 10, the deformation of theBR wing is relatively small, allowing for just 0.1 ofadaptive washout. The BR wing is very sensitive topre-tensions in the span direction; the structure is toostiff. Furthermore, the reflex in the wing pushes thetrailing edge down, limiting the ability of the battens towashout for load reduction. Less than a 2% drop in liftfrom the carbon fiber wing is obtained, and the pressuredistributions of the two wings in Fig. 10 are very similar.

The load alleviating design located by the topologyoptimizer (right column, Fig. 9) is significantly moresuccessful. By filling the design space with patches ofdisconnected carbon fiber structures (dominated by along batten which extends the length of the mem-brane skin, but is not connected to the wings leadingedge), the MAV wing is very flexible, but none of themembrane portions of the wing are large enough tocamber the wing via inflation. Wing deformation is the

same magnitude as that seen in the PR-type wings, butthe motion is located at the trailing edge for adaptivewashout, and lift is decreased by 5%. The local defor-mation within the membrane between the leading edgeand the long batten structure is substantial, and the flowdeceleration over this point sees a further loss in lift, aswith the PR wing.

Similar results are given in Figs. 11 and 12, for acambered wing at 12 angle of attack. The three base-line wings are again shown (carbon fiber wing, PR, andBR), as well as the designs located by the topologyoptimization to maximize lift and minimize lift slope.As the forces are generally larger for the camberedairfoil, the deformations have increased to 5% of theroot chord. The negative forces at the trailing edgeof the airfoil are likewise absent. As before, the PRmembrane wing effectively increases the lift over itscarbon fiber counterpart through adaptive cambering,along with aerodynamic penalties from the shape dis-continuities at the leading and trailing edge of themembrane skin.

There is an appreciable amount of upward defor-mation of the PR wings trailing edge carbon fiberstrip, leading to washout of each flexible wing section,degrading the lift. As such, the aeroelastic topologyoptimizer can maximize lift (Fig. 11, middle column) byadding more material to this strip and negating the mo-tion of the trailing edge. As discussed above, this strip

Fig. 11 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom) for baseline and optimal topologydesigns, = 12, cambered wing


Fig. 12 Deformations andpressures along 2y/b = 0.58for baseline and optimaltopology designs, = 12,cambered wing

does not continue unbroken to the wing tip, but ends at65% of the semispan. The remaining membrane trailingedge is filled with a free-floating carbon fiber batten.Such a configuration can (theoretically) improve the liftin several ways, similar to the trailing edge structureused for lift optimization in Fig. 9.

Placing a flexible membrane skin between two rigidsupports produces a trade-off: the cambering via in-flation increases lift, but this metric is degraded bythe sharp discontinuities in the airfoil shape. Towardsthe inner portion of the MAV wing, this trade-off isfavorable for lift. Towards the wingtip however (eitherdue to the changes in chord or in pressure) this is nolonger true: the total lift can be increased by allowingthis portion of the trailing edge to washout, therebyavoiding the negative pressures seen elsewhere alongthe trailing edge.

The forward portion of this batten structure alsoproduces a cusp in the wing geometry, forcing a verystrong low pressure spike over the upper portion ofthe airfoil to increase the lift, as before. The inviscidsolver probably over-predicts the strength of this spike:the presence of viscosity will attenuate the speed of theflow, and thus both the magnitude of the low pressurespike and its beneficial effect upon lift. The aeroelastictopology optimizer predicts a 3.5% increase in lift overthe PR wing, and 12.5% increase over the carbon fiberwing, though the veracity of these comparisons requiresa viscous solver to ascertain the actual height of thelow-pressure spike at x/c = 0.68.

The batten-reinforced design of Fig. 11 is substan-tially more effective with the cambered wing, largelydue to the positive forces at the trailing edge (ratherthan the negative forces seen over the reflex wing). The

Fig. 13 Experimentally measured forces and moments for baseline and optimal topology designs, reflex wing


1.6 of washout in the cambered BR wing decreasesthe load throughout most of the wing and decreasesthe lift by 8.5% (compared to the carbon fiber wing),but, as before, the load-alleviating design located bythe topology optimizer (right column, Fig. 11) is su-perior. Similar to above, the design utilizes a series ofdisconnected carbon fiber structures, oriented parallelto the flow, and extending to the trailing edge. Thediscontinuous wing surface forces a number of high-pressure spikes on the upper surface, notably at x/c =0.2 and 0.6. This, in combination with the substantialadaptive washout at the trailing edge, decreases the liftby 13.6% over the carbon fiber wing and by 5.6% overthe BR wing.

5 Experimental validation

Due to the low-fidelity nature of the aeroelastic analysisemployed in this work, a certain measure of experimen-tal validation of the superiority of computed optimaover baseline designs is required. This is particularlytrue for drag-based metrics, which, as discussed above,the vortex lattice method with struggle to accuratelycompute. For this purpose, six designs are built forexperimental testing: three baseline designs (mono-lithic carbon fiber wing, PR, and BR wings), and threetopologically-optimized designs (minimum CL , mini-mum drag, and minimum Cm , the third, fourth, andfifth wings shown in Fig. 13). The latter three designsare found by optimizing with a reflex airfoil, at 3 angleof attack. All testing is conducted in a closed-loopwind tunnel, and loads are measured with a strain gagesting balance through an entire -sweep; more detailspertaining to the experimental testing can be found byStanford et al. (2007a).

Experimental results are given in Fig. 13, in termsof CL, CD, and Cm of the three baseline designs andthe relevant optimal design. All three structures locatedby the topology optimizer show marked improvementsover the baseline experimental data, validating the useof a low fidelity aeroelastic model as a surrogate forcomputationally-intensive nonlinear models. With theexception of very low (where deformations are small)and very high angles of attack (where the wing hasstalled), the optimized designs consistently out-performthe baselines. The minimum CL design has a shallowerlift slope than the BR wing, and a flatter stalling curve(left of Fig. 13). The minimum drag design similarlyout-performs the BR wing (middle of Fig. 13). Theminimum Cm design has a steeper pitching momentslope than the PR wing (right of Fig. 13), indicative of alonger static stability margin.

6 Conclusions

Aeroelastic topology optimization has been used to de-sign the laminate reinforcement of a membrane microair vehicle wing, with a two-material formulation ofthe wetted surface. The static aeroelasticity is modeledwith a vortex lattice method coupled to linear mem-brane and shell elements. The sensitivity of the wingsaerodynamic performance to each element density iscomputed with an adjoint analytical sensitivity analysis.The following conclusions can be drawn:

1. Nonlinear power-law interpolation between 0(membrane) and 1 (carbon fiber) ensures a smoothaeroelastic response.

2. An explicit penalty based upon the sine functionadequately pushes the design to a 01 distributionwithout significantly altering topology.

3. The optimal solution is very dependent upon theinitial topology, requiring multiple runs for eachcase.

4. Optimal lift- and drag-based metrics are insensitiveto flight condition, though L/D is not.

5. The optimizer makes use of several disparate phe-nomena for lift augmentation: membrane inflationto camber the wing, depression of the trailing edgefor wash-in, and sharp cusps in surface geometrythat cause the pressure to locally spike.

6. Optimal load-alleviation is obtained with a seriesof disjointed laminate structures: the wing is flexi-ble enough to substantially washout at the trailingedge, but the remaining membrane patches arentlarge enough to locally inflate, which would addcamber and increase the lift.

7. Relevant improvements in aerodynamic perfor-mance can be obtained despite the low-fidelityaeroelastic model employed, as indicated by exper-imental testing.

This work has clearly demonstrated the feasibility ofaeroelastic topology optimization for membrane wings,but future work would need to concentrate on severalareas. First, multi-objective analysis is required to lo-cate designs with adequate performance along a widerange of flight conditions. Secondly, it is hypothesizedhere that minimizing the lift slope will help with gustalleviation as well, but the veracity of this claim needsto be tested with unsteady aeroelastic models and non-homogenous incoming flow. Finally, agility-based met-rics should be considered as well: a MAV wing in alateral roll maneuver, for example.


Acknowledgements This work was jointly supported by the AirForce Research Laboratory and the Air Force Office of ScientificResearch under the MURI program F49620-03-1-0381.

References

Albertani R, Stanford B, Hubner J, Ifju P (2007) Aerodynamiccoefficients and deformation measurements on flexible mi-cro air vehicle wings. Exp Mech 47(5):625635

Anagnostou G, Rnquist E, Patera A (1992) A computationalprocedure for part design. Comput Methods Appl Mech Eng97(1):3348

Balabanov V, Haftka R (1996) Topology optimization of trans-port wing internal structure. J Aircr 33(1):232233

Beckers M (1999) Topology optimization using a dual methodwith discrete variables. Struct Multidisc Optim 17(1):1424

Bendse M, Sigmund O (2003) Topology optimization. Springer,Berlin, Germany

Bendse M, Lund E, Olhoff N, Sigmund O (2005) Topologyoptimization: broadening the areas of application. ControlCybern 34(1):735

Borrvall T, Petersson J (2003) Topology optimization of fluids instokes flow. Int J Numer Methods Fluids 41(1):77107

Chen T, Wu S (2008) Multiobjective optimal topology design ofstructures. Comput Mech 21(6):483492

Cook R, Malkus D, Plesha M, Witt R (2002) Concepts and appli-cations of finite element analysis. Wiley, New York, NY

Deb K, Goel T (2001) A hybrid multi-objective evolutionaryapproach to engineering shape design. In: Internationalconference on evolutionary multi-criterion optimization,79 March, Zurich, Switzerland

Eschenauer H, Olhoff N (2001) Topology optimization of contin-uum structures: a review. Appl Mech Rev 54(4):331390

Gersborg-Hansen A, Sigmund O, Haber R (2005) Topologyoptimization of channel flow problems. Struct MultidiscOptim 30(3):181192

Gomes A, Suleman A (2008) Topology optimization of a re-inforced wing box for enhanced roll maneuvers. AIAA J46(3):548556

Haftka R, Grdal Z (1992) Elements of structural optimization.Kluwer, Dordrecht, The Netherlands

Hsu M, Hsu Y (2005) Interpreting three-dimensional structuraltopology optimization results. Comput Struct 83(4):327337

Katz J, Plotkin A (2001) Low-speed aerodynamics. CambridgeUniversity Press, Cambridge, UK

Krog L, Tucker A, Kemp M (2004) Topology optimizationof aircraft wing box ribs. In: AIAA/ISSMO multidisci-plinary analysis and optimization conference, 30 August1 September, Albany, NY

Lee S, Bae J, Hinton E (2000) Shell topology optimization usingthe layered artificial material model. Int J Numer MethodsEng 47(4):843867

Luo Z, Yang J, Chen L (2006) A new procedure for aero-dynamic missile designs using topological optimizationapproach of continuum structures. Aerosp Sci Technol10(5):364373

Lyu N, Saitou K (2005) Topology optimization of multicompo-nent beam structure via decomposition-based assembly syn-thesis. J Mech Des 127(2):170183

Martins J, Alonso J, Reuther J (2001) Aero-structural wing de-sign optimization using high-fidelity sensitivity analysis. In:Confederation of European aerospace societies conferenceon multidisciplinary analysis and optimization, 2526 June,Cologne, Germany

Maute K, Allen M (2004) Conceptual design of aeroelasticstructures by topology optimization. Struct Multidisc Optim27(1):2742

Maute K, Reich G (2006) Integrated multidisciplinary topol-ogy optimization approach to adaptive wing design. J Aircr43(1):253263

Maute K, Nikbay M, Farhat C (2002) Sensitivity analysis and de-sign optimization of three-dimensional non-linear aeroelas-tic systems by the adjoint method. Int J Numer Methods Eng56(6):911933

Ormiston R (1971) Theoretical and experimental aerodynamicsof the sail wing. J Aircr 8(2):7784

Pingen G, Evgrafov A, Maute K (2007) Topology optimizationof flow domains using the Lattice Boltzmann method. StructMultidisc Optim 34(6):507524

Stanford B, Abdulrahim M, Lind R, Ifju P (2007a) Investigationof membrane actuation for roll control of a micro air vehicle.J Aircr 44(3):741749

Stanford B, Sytsma M, Albertani R, Viieru D, Shyy W, IfjuP (2007b) Static aeroelastic model validation of membranemicro air vehicle wings. AIAA J 45(12):28282837

Stegmann J, Lund E (2005a) Nonlinear topology optimizationof layered shell structures. Struct Multidisc Optim 29(5):349360

Stegmann J, Lund E (2005b) Discrete material optimization ofgeneral composite shell structures. Int J Numer MethodsEng 62(14):20092027

Stolpe M, Stidsen T (2006) A hierarchical method for dis-crete structural topology design problems with local stressand displacement constraints. Int J Numer Methods Eng69(5):10601084

Torres G (2002) Aerodynamics of low aspect ratio wings at lowReynolds numbers with applications to micro air vehicledesign. Ph.D. Dissertation, Department of Aerospace andMechanical Engineering, University of Notre Dame, SouthBend, IN

Wang S, Tai K, Wang M (2006) An enhanced genetic algorithmfor structural topology optimization. Int J Numer MethodsEng 65(1):1844

Zuo K, Chen L, Zhang Y, Yang J (2007) Study of key algo-rithms in topology optimization. Int J Adv Manuf Technol32(7):787796

Aeroelastic topology optimization of membrane structures for micro air vehiclesAbstractIntroductionMembrane micro air vehicle wingsComputational frameworkMaterial interpolationStatic aeroelastic analysisAdjoint sensitive analysisOptimization framework

ResultsExperimental validationConclusionsReferences

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