[American Institute of Aeronautics and Astronautics 49th AIAA/ASME/ASCE/AHS/ASC Structures,...

Nonlinear models for biologically-inspired elastic

membrane wings

Edward W. Swim∗

United States Military Academy, West Point, NY, 10996

Padmanabhan Seshaiyer†

George Mason University, Fairfax, VA, 22030

Nonlinear membrane wing models that mimic the composition and flexibility of bio-logical tissues such as those found in the wings of bats and insects can be valuable inthe development of structural components for micro air vehicles. In this work, we de-velop a framework for building fluid-structure interaction simulations that allow for bothgeometric and material nonlinearity within a membrane model. A nonconforming finiteelement method is used to allow for independent mesh construction and polynomial degreeof approximation within each subdomain. Moreover, an arbitrary Lagrangian-Eulerianformulation is used to manipulate the meshes at each time step. Both steady-state anddynamic models are tested against data from the literature.

Nomenclature

C right Cauchy-Green tensorE Young’s modulus of elasticityIj strain invariantsL membrane lengthL0 initial membrane lengthn unit normal vectorS nominal strain tensorT surface tensionu axial membrane displacementvj fluid velocity componentz transverse membrane displacementε strain of the middle lineκ curvature of the middle lineµ dynamic fluid viscosityρf fluid densityρm membrane densityσ Cauchy stress tensorτ fluid shear stress at fluid-membrane interface

I. Introduction

Micro air vehicles (MAVs), which typically have a wing span of at most six inches, flight speeds of twentyto forty miles per hour, and a total weight less than half a pound,8,13 encounter flight conditions withReynolds numbers similar to that of an insect or bird (103− 105). As a result, stall may occur for moderate

∗Assistant Professor, Department of Mathematical Sciences, Member AIAA.†Associate Professor, Department of Mathematical Sciences, Member AIAA.

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American Institute of Aeronautics and Astronautics

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br> 16t7 - 10 April 2008, Schaumburg, IL

AIAA 2008-2008

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

angles of attack due to flow separation near the leading edge spar.4 This aerodynamic regime results in vastlydifferent flight dynamics than is usually observed in traditional aircraft. Moreover, propulsion is gained usingsmall gasoline engines or electric motors powered by high energy density batteries. Thus, it is essential thatMAVs be able to perform tasks while consuming very little power.

The goal for most applications is to be able to operate the vehicle remotely, either from a nearby locationor using satellite communications, via a “head-down” display. Successful performance in this paradigm issignificantly enhanced whenever the vehicle can auto-correct under adverse conditions such as hazardousclouds, smoke during fire and rescue operations, darkness, and navigating urban areas or even buildinginteriors. Hence, successful simulation of a wide variety of flight conditions is critical to the design andproduction of MAVs that can meet the needs of today’s military and civilian institutions.

Experimentation with various designs indicates that rigid wing MAVs, constructed from high densityfoam or some other light material, perform poorly when wind gusts are present due to flow separation thatfrequently occurs after a rapid increase in the angle of attack.10 One solution to this problem is the use ofthin membrane wings, usually supported by a skeletal structure composed of rigid beams. The ability ofa membrane wing to deflect at a large scale relative to its panel length, as opposed to the panel width fornonlinear plates, allows for optimal lift.

As the size of MAVs decreases for usage in urban and interior environments, the ability to mimic flightbehavior of insects and other small creatures is an important part of creating a vehicle that can be easilymaneuvered in such settings and still maintain anonymity when necessary. Stability and responsiveness ofMAVs improves whenever a thin material, similar to that used for sailboats and parachutes, is the primarysurface of the wing.1 Moreover, wings constructed in this way can delay the onset of stall. But, as in nature,the payload and function of a MAV can significantly influence the optimal shape of a membrane wing.19

As a result, it is necessary to construct an efficient numerical scheme which allows for the coupling of aviscous incompressible fluid and a nonlinear membrane in a dynamic fluid-structure interaction simulation.In particular, the membrane solver should allow for a wide variety of material nonlinearity so that manydifferent constitutive models can be explored during the development phase of MAV production.

Although several recent studies have focused on a Mooney-Rivlin framework,3,9 when using fibre-reinforcedmaterials to mimic insect wings, an anisotropic model is preferable. Additionally, much improvement isneeded in the efficiency of current software used to simulate the performance of a membrane wing that hasbeen designed in silico. Current efforts to optimize the shape of a membrane wing involve the use of asurrogate rigid wing prior to evaluating the performance of a corresponding membrane wing.12 However,such an approach may bypass a preferred shape due to errors in the computation of aerodynamic loads onthe upper wing surface. By directly computing the membrane wing deflection as well as the fluid velocityand pressure simultaneously at each time step, we hope to improve the performance of simulations that canbe used in developing MAV wing components.

The purpose of this paper is to present a new nonlinear framework for membrane dynamics in thenumerical simulation of micro air vehicle performance. In Section 2 we present previous models that havebeen used successfully by groups that have correlated computational results with experimental data. Weextend those models to include geometric nonlinearity to accommodate large membrane deflections. We thenpresent finite element approximations to the solutions of the resulting boundary value problems and outlinethe performance of these schemes based on manufactured solutions that mimic deformations experimentallyobserved for flexible wing MAVs. In Section 3 we focus on applying the finite element scheme used forthe steady-state boundary value problems to a fully coupled system of dynamic equations that model bothtransverse and axial displacement. Central differences in time are applied to construct a consistent simulation.Finally, in Section 4 a three-field formulation is presented for the fluid-structure interaction problem thatcombines our membrane model with viscous Navier-Stokes equations. This strategy allows for the use of anonconforming finite element method so that both the discretization and polynomial degree of approximationin each subdomain can be constructed independently without loss of accuracy on the interface between thefluid and the membrane.

II. Background

We begin by considering a two-dimensional steady-state problem, i.e., the motion of a fluid is assumed tobe the same in all planes parallel to the xz-plane, with velocity also parallel to that plane and independentof time. Assume that a thin membrane with thickness h is surrounded by a frictionless, incompressible, and

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irrotational fluid with uniform density ρf and consider any point (x∗, z∗) on the membrane. Define by p+

and p− the pressure on the upper and lower surfaces, respectively, on either side of (x∗, z∗). Similarly, definev+ and v− to be the corresponding flow speeds. Then

∆p = −(p+ − p−)

is the net pressure across the membrane and via Bernoulli’s law,

p(x, z) +12ρf |v(x, z)|2 = K,

where ρf is the density of the fluid, we have

12ρf(v2

+ − v2−) = ∆p.

Treating the membrane-fluid interface as a free surface and assuming that fluid particles at the surface remainattached, the net pressure across the surface must balance with surface tension, i.e.,

−κT = ∆p, (1)

where

κ =d2z

dx2

[1 +

(dz

dx

)2]−3/2

is the curvature of a neutral surface, z = η(x), within the structural domain and T is the coefficient of surfacetension at the interface.18

Modifications to (1) include the incorporation of an inertial term by Lillberg, et al.,10 i.e.,

ρmh∂2z

∂t2= T

∂2z

∂x2

[1 +

(∂z

∂x

)2]−3/2

+ ∆p,

where ρm is the material density. A more recent modification by Isaac7 introduced a damping term and aperiodic, time-dependent forcing function, i.e.,

ρm∂2z

∂t2+ cd

∂z

∂t− T

∂2z

∂x2

[1 +

(∂z

∂x

)2]−3/2

= p0 sin ωt,

where cd is the damping coefficient and ω is the forcing frequency.

II.A. Geometrically Nonlinear Models

Smith and Shyy14 used the formulation from equation (1) for equilibrium in the normal direction coupledwith an equilibrium equation for shear stress. In this approach the tension varies across the surface accordingto

dT

dξ= −τ,

where ξ is the tangential curvilinear coordinate along the surface of the membrane and τ = τ− − τ+ is theshear stress. In practice, their simulations avoid coupling these two equations by computing the tensionaccording to Hooke’s law, i.e.,

T = (0S + Eε)h, (2)

where 0S is the membrane prestress, E is the modulus of elasticity, and a linear representation of membranestrain is given by

ε =L− L0

L,

where L represents the arc length of the sail and L0 is the membrane chord length.

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However, geometric and material nonlinearities can have a much more significant impact on the longterm behavior of a membrane wing and realistic flight conditions are likely to produce aperiodic externalloads. Our model will use a nonlinear formulation for the strain-displacement relationship. The motivationfor such an approximation stems from the need to incorporate models of material nonlinearity, as opposedto using Hooke’s law, when using certain materials in the construction of a MAV wing. In this case, it maybe necessary to include varying degrees of geometric nonlinearity in order for a model to be consistent.

Let ds represent a differential arc length along the path traced by the membrane. Then strain is givenby

ε =ds− ds

ds,

which we assume to be small. For a given point (x, z) on our membrane in the undeformed state, afterdeformation its position is given by

x = X (x) + η(x, z)ξ(x) andz = Z(x) + η(x, z)ζ(x),

where (X (x),Z(x)) represents a corresponding point on the deformed middle line of the membrane, ξ(x)and ζ(x) are the components of the outward unit normal vector n at that point, and η(x, z) represents thedeformation of the cross-section of the membrane in the direction of n.15 We will also incorporate Kirchhoff’shypothesis, i.e., normal lines to the original surface remain normal and lines after deformation. Hence, theshear strain is zero, i.e., γxz = 0.

Next, we let u = x− x and w = z − z. Then

u(x, z) = u(x) + η(x, z)ξ(x) andw(x, z) = w(x) + η(x, z)ζ(x)− z,

where u(x) = X (x)− x and w(x) = Z(x)− 0. Now,

ds =√

(∂xX )2 + (∂xZ)2dx

=

√(1 +

∂u

∂x

)2

+(

∂w

∂x

)2

dx

= (1 + ε(x)) dx,

where ε(x) represents the strain of the middle line. Using a Taylor series expansion for√

1 + f(x), we have

ε(x) =12

[2∂u

∂x+

(∂u

∂x

)2

+(

∂w

∂x

)2]− 1

8

[2∂u

∂x+

(∂u

∂x

)2

+(

∂w

∂x

)2]2

+ · · ·

or, simplifying,

ε(x) =∂u

∂x+

12

(∂u

∂x

)2

+12

(∂w

∂x

)2

− 12

(∂u

∂x

)2

− 12

[(∂u

∂x

)3

+∂u

∂x

(∂w

∂x

)2]− 1

8

[(∂u

∂x

)2

+(

∂w

∂x

)2]2

+ · · ·

Neglecting higher order terms, we thus have

ε(x) ≈ ∂u

∂x+

12

(∂w

∂x

)2

.

Hence, after neglecting membrane prestress in (2), the steady-state model (1) can be written as

−Eh

∂2z

∂x2

[∂u

∂x+

12

(∂z

∂x

)2]

[1 +

(∂z

∂x

)2]3/2

= ∆p.

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II.B. Material Nonlinearity: Stress-Strain Relationships

Given a strain tensor E, the right Cauchy-Green tensor is given by

C = 2E + I,

where I is the identity. C has three well-known invariants,

I1 = tr(C),

I2 =12

[I21 − tr(C2)

], and

I3 = det(C).

Using these coefficients in the characteristic polynomial of C, a strain energy function W = W (I1, I2, I3)is often used to define a constitutive relationship between stress and strain for isotropic materials. To dothis, let σ be the Cauchy stress tensor and n an outward unit normal to the deformed surface. Additionally,assume that C = FT F, where

F =∂x∂X

is the deformation gradient tensor. Here, X is a point in the reference configuration and x is a correspondingpoint in the deformed state. Then S = I3F−1σ is the nominal stress tensor. We assume that there exists aW = W (F) such that

S =∂W

∂F.

Such a strain energy function represents the work done per unit volume at X in changing the deformationgradient from I to F whenever the reference state is a natural configuration, i.e., W (I) = 0 and S(I) = 0.Moreover, we assume that W is independent of superimposed rigid deformations, i.e., W (QF) = W (F) forall rotations Q. The constitutive model is then given by11

σ =1I3

FS.

Thus,dT

dξ= −τxz = −σ12.

Material-specific strain energy functions, e.g., the Mooney-Rivlin model for incompressible and isotropicflexible membranes or anisotropic models for fibre-reinforced materials, can then be utilized to replace theassumption of Hooke’s Law.

II.C. Numerical Framework

We consider the BVP consisting of Equation (1) written as

−d2z

dx2

[1 +

(dz

dx

)2]−3/2

=∆p

T,

with homogeneous Dirichlet boundary conditions, or

−a(z′)d2z

dx2= ∆p,

where a(z′) = T (1 + (z′)2)−3/2. We will then apply a Picard iteration by guessing an initial configurationz0 = z0(x), computing its derivative with respect to the spatial coordinate, and solving the two-point BVPcorresponding to −z′′ = F (x; z, z′), where

F (x; z, z′) =∆p

a(z′).

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We denote the new solution at this first iteration z1 = z1(x) and then continue computing a sequence ofsolutions, zk(x), via

−z′′k+1 = F (x; zk, z′k)

until ||zk+1 − zk||1 < ε for some prescribed tolerance ε.The basis functions we use follow the development of Szabo and Babuska17 and are generated using the

Legendre polynomials Pn(x), defined on the interval (−1, 1) using the Rodrigues formula:

Pn(x) =(−1)n

2nn!

(d

dx

)n

(1− x2)n.

Noting that P0(x) = 1 and P1(x) = x, we will use Bonnet’s recursion formula,

(n + 1)Pn+1(x) = (2n + 1)xPn(x)− nPn−1(x), n = 1, 2, . . .

to define any of the polynomials we will need up to any predefined degree in x. For a computational domaindefined by an interval (a, b), we denote a partition on that mesh by a = x1, x2, . . . , xM = b. The kth

element, defined byΩk = x : xk < x < xk+1,

is mapped to a reference element Ωst, given by

Ωst = ξ : −1 < ξ < 1,using the relation x = Qk(ξ), where

Qk(ξ) =1− ξ

2xk +

1 + ξ

2xk+1.

The inverse mapping is given by

ξ = Q−1k =

2x− xk − xk+1

xk+1 − xk.

Denote the space of polynomials of degree p on Ωst by Sp. We use basis functions Nj(ξ)p+1j=1 for Sp, where

N1(ξ) =1− ξ

2, N2(ξ) =

1 + ξ

2,

andNj(ξ) = φj−1(ξ), j = 3, 4, . . . , p + 1,

where

φj(ξ) =

√2j − 1

2

∫ ξ

−1

Pj−1(t) dt, j = 2, 3, . . . , p. (3)

Using the recursion formula

(2n + 1)Pn(x) =d

dxPn+1(x)− d

dxPn−1(x), n = 1, 2, . . . (4)

φj can be computed as

φj(ξ) =1√

2(2j − 1)(Pj(ξ)− Pj−2(ξ)).

Our finite element basis functions will be composed of piecewise polynomials from this space, i.e., η is a finiteelement basis function provided η belongs to the appropriate energy space and η(Qk(ξ)) ∈ Sp. Moreover,the first derivatives can be calculated automatically using equation (3) and Leibniz’s rule:

d

dξφj(ξ) =

√2j − 1

2Pj−1(ξ).

And via equation (4), we can then compute any derivative, since

d

dxPn+1(x) =

d

dxPn−1(x) + (2n + 1)Pn(x).

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For example,d2

dξ2φj(ξ) =

√2j − 1

2

(d

dξPj−3(ξ) + (2j − 3)Pj−2(ξ)

).

This allows to include an arbitrary degree of geometric nonlinearity from our Taylor series expansion for thestrain, ε(x). Figure 1 illustrates the performance of this method using piecewise cubic polynomials and aconstant tension T . Here the prescribed tolerance is ε = 10−4.

0.125 0.25 0.375 0.5 0.625 0.75 0.875 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

k=1k=2k=3k=4k=10

Figure 1. Convergence of a steady-state membrane deflection: constant tension.

III. Dynamic Models

Allowing the membrane to move over time, a dynamic model is given by

ρmh∂2z

∂t2−

Eh∂2z

∂x2

[∂u

∂x+

12

(∂z

∂x

)2]

[1 +

(∂z

∂x

)2]3/2

= ∆p, (5)

where axial displacement satisfies

ρmh∂2u

∂t2− Eh

∂

∂x

[∂u

∂x+

12

(∂z

∂x

)2]

= f(t, x). (6)

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Here f represents external forces in the axial direction. For simplicity, we will assume homogeneous Dirichletboundary conditions at both ends for the transverse displacement z and at the left endpoint for the axialdisplacement u. At the right endpoint, the remaining boundary condition is given by

[∂u

∂x+

12

(∂z

∂x

)2]∣∣∣∣∣

x=1

= r0,

where r0 is a prescribed value for the membrane strain. We assume that the membrane is initially flat, i.e.,z(0, x) = 0, 0 ≤ x ≤ 1.

Next, we consider the dynamic model given by the coupled system of equations (5) and (6). We discretizethe time derivatives using central differences and reformulate the resulting semi-discrete equations as

−∂2xZn+1 + Kn

z Zn+1 = Rnz

and−∂2

xUn+1 + Knu Un+1 = Rn

u

where

Knz =

ρm

[1 + (∂xZn)2

]3/2

E (∆t)2[∂xUn + 1

2 (∂xZn)2] ,

Rnz =

[∆pn +

ρmh

(∆t)2(2Zn − Zn−1

)] [

1 + (∂xZn)2]3/2

Eh[∂xUn + 1

2 (∂xZn)2] ,

Knu =

ρm

E (∆t)2,

andRn

u =fn

Eh+

ρm

E (∆t)2(2Un − Un−1

)+ (∂xZn)

(∂2

xZn).

Here, fn = f(tn, x) and Un represents a finite element approximation of u(tn, x), tn = n∆t. Using the ap-proach established for the steady-state problem, we ensure at each time step that the transverse displacementhas converged within a tolerance of ε = 10−8 before proceeding to the next time step.

We used a structured grid with varying mesh sizes, polynomial degree of approximation, and time stepsizes in order to analyze the convergence properties of our scheme. All computations were performed witha density of ρm = 1200 kg/m3, membrane thickness of h = 4.5 × 10−6 m, elastic modulus of E = 2 N/m2,and a membrane strain value of r0 = 60 at the right-hand endpoint. We manufactured an exact solution bymodifying data from Lillberg, et al.,10 to include a periodic oscillation:

z(t, x) = [0.91− 1.85(x− 0.1) + 1.675(x− 0.1)(x− 0.2)− 2.3(x− 0.1)(x− 0.2)(x− 0.4)+4.8(x− 0.1)(x− 0.2)(x− 0.4)(x− 0.6)−10.1(x− 0.1)(x− 0.2)(x− 0.4)(x− 0.6)(x− 0.8)+18.7(x− 0.1)(x− 0.2)(x− 0.4)(x− 0.6)(x− 0.8)(x− 0.9)] x(x− 1) sin(πt).

Table 1 illustrates the consistency of our algorithm as the time step size is reduced. Here, the relative errorunder the L2-norm is given for a grid with mesh size h = 0.03125 and cubic polynomials. Initially, the errordrops with a linear rate of convergence. This is eventually polluted by the error associated with our choiceof mesh size and polynomial degree of approximation. Figure 2 shows the transverse membrane deflectionat various times during the simulation.

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Table 1. L2 relative error for decreasing values of ∆t, h = 0.03125, p = 3, tf = 0.4.

∆t ||Z(tf , ·)− z(tf , ·)||L2(0,1)

0.1 0.123139965902750.05 0.072248022052810.025 0.056309607724620.0125 0.051114247826320.00625 0.049443716929400.003125 0.04888251794100

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

x

Tra

nsve

rse

Mem

bran

e D

efle

ctio

n

t=0.0125t=0.025t=0.05t=0.1t=0.2t=0.4

Figure 2. Computed transverse membrane deflection for various values of t. Mesh size is h = 0.03125, polynomial degreeis p = 3, and time step size is ∆t = 0.003125.

IV. The Fluid-Structure Interaction Problem

Finally, we consider the fully coupled dynamic model, similar to that studied by Lillberg, et al.,10

T∂2z

∂x21

[1 +

(∂z

∂x1

)2]−3/2

+ ∆p = ρmh∂2z

∂t2, x = (x1, x2) ∈ Ωm (7)

ρmh∂2u

∂t2− ∂T

∂x= f(t, x1) , x ∈ Ωm (8)

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∂T

∂ξ= −σ12 , x ∈ Ωm (9)

∂v∂t

− µ

ρf

∇2v + v · ∇v +1ρf∇p = g , x ∈ Ωf (10)

∇ · v = 0 , x ∈ Ωf . (11)

Here v represents the velocity of the surrounding fluid in vector form, µ is the dynamic viscosity of thefluid, ρf is the density of the fluid, and g is the acceleration due to gravity. Equations (10) and (11) arethe familiar Navier-Stokes equations. The computational domain is shown in Figure 3. The panel width his exaggerated for illustration purposes. We will assume that Dirichlet boundary conditions are specified for

Figure 3. Fluid-structure interaction domain.

the external fluid boundary so that appropriate inflow and outflow conditions are prescribed. Additionally,continuity of velocity and surface stress will be enforced across the fluid-membrane interface.

IV.A. Arbitrary Lagrangian-Eulerian formulation

In order to account for the changing nature of our fluid domain ΩF (t), we wish to define a dynamic meshwhen discretizing in space. However, to avoid extreme mesh distortion near the interface, we also choose tomove the mesh independently of the fluid velocity on the interior of ΩF (t). Such a scheme, called an arbitraryLagrangian-Eulerian (ALE) formulation, is commonly applied when studying fluid-structure interaction.2,5, 6

In particular, we allow mesh nodes to move only in the x2-direction in order to facilitate computations withnonconforming discretizations.

Let x1 be a fixed value from x1 : 0 < x1 < L0 and let w be a grid velocity in the transverse directionsuch that w matches the velocity of the membrane at the fluid-membrane Γfm interface and w = 0 at∂Ωf \ Γfm. If γ(t, x1) represents the x2-coordinate of the interface, then let

w(t,x) =x2

γ(t, x1)γ(t, x1).

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The associated characteristic variable must satisfy

d

dtx

(s)2 (t, ξ) = w(t, x1, x

(s)2 (t, ξ)),

x(s)2 (s, ξ) = ξ

∀ξ ∈ (0, γ(s, x1)). Thus,

x(s)2 (t, ξ) = ξ

γ(t, x1)γ(s, x1)

.

Let u(t, x1, ξ) = v(t, x1, x(s)2 (t, ξ)). Then

∂u∂t

(t, x1, ξ) =∂v∂t

(t, x1, x(s)2 (t, ξ)) + w(t, x1, x

(s)2 (t, ξ))

∂v∂x2

(t, x1, x(s)2 (t, ξ))

so equation (10) becomes for j = 1, 2,[∂uj

∂t− ν∆vj + v1

∂vj

∂x1+ (v2 − w)

∂vj

∂x2+

∂p

∂xj

](t, x1, x

(s)2 (t, ξ)) = gj(t, x1, x

(s)2 (t, ξ)).

In the simple case where Hooke’s Law is applied, the resulting semi-discrete equations are given by

1∆t

(V n+1

j − V nj

)− µ

ρf∇2V n+1

j + V n1 ∂x1 V

n+1j + (V n

2 −Wn) ∂x2 Vn+1j + ∂xj P

n+1 = gn+1j , (12)

∇ · Vn+1 = 0, (13)−∂2

x1Zn+1 + Kn

z Zn+1 = Rnz , (14)

−∂2x1

Un+1 + Knu Un+1 = Rn

u. (15)

Here, φn+1j (x1, x

n2 ) = φn+1

j (x1, xn2 + Wn(x1, x

n2 )∆t) represents the analytic extension of a function φ that

is defined on the fluid domain Ωf (tn) into the computational fluid domain Ωnf . This computational domain

may contain a region of space in which no fluid is present at that time step. However, we can write equation(12) in the weak form

1∆t

∫

Ωnf

V n+1j φn

j dA − µ

ρf

∫

Ωnf

∇V n+1j · ∇φn

j dA +∫

Ωnf

[V n

1 ∂x1 Vn+1j + (V n

2 −Wn) ∂x2 Vn+1j

]φn

j dA

−∫

Ωnf

Pn+1∂xj φnj dA−

∫

Γnfm

λn+1j φn

j ds =∫

Ωnf

(gn+1

j +1

∆tV n

j

)φn

j dA,

where λn+1j = −Pn+1nj + µ

(∇V n+1j · n)

/ρf . The function λn+1j is a Lagrange multiplier from the dual

of the trace space defined on Γnfm. Our finite element spaces are now built using the corresponding two-

dimensional product spaces derived from the finite element basis presented in section 2.3. Such a formulationwas shown16 to be consistent and O(∆t).

In order to completely couple our ALE formulation for the Navier-Stokes equations with the nonlinearmembrane equations, the following interface conditions must also be enforced:

∫

Γnfm

(V n+1

2 −Qn+1)χn

1 ds = 0,

∫

Γnfm

(Zn+1 −∆tQn+1

)χn

2 ds =∫

Γnfm

Znχn2 ds,

∫

Γnfm

(λn+1 − F

) ·Θn ds = 0,

where F = (Fn, FT ) is the surface stress. The test functions χnj are from the same space as our Lagrange

multipliers λnj , while the mortar elements Qn+1 and the test functions Θn may belong to a different subspace

of that trace space. The stability of this approach was shown16 to be conditioned upon the assumption thatthe distance between the membrane and the external boundary of Ωf remains nontrivial at all times.

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IV.B. Concluding Remarks

In this paper, a nonlinear model has been introduced to model the flexible wing of a MAV. The model buildson earlier work that involved a one-dimensional linear model.14 The current model is two dimensional,incorporates geometric nonlinearity in a manner consistent with membrane assumptions, and allows for theuse of strain energy functions to include material nonlinearity. Both the transverse and axial deflection of amembrane element that models the MAV are coupled with the Navier-Stokes equations to build a consistentand stable fluid-structure interaction simulation.

Although we have presented an ALE framework that allows mesh movement only in the direction oftransverse displacement, a more realistic model problem for which the length of the membrane in the directionof axial displacement is not fixed will require a more complicated modification of the mesh at each time step.This next step in the development of our numerical methodology will be necessary in order to validate such asimulation against data collected through wind tunnel experiments. Moreover, the rates of convergence thatcan be expected for this application of hierarchic finite element basis functions remains an open problemthat will be explored in a forthcoming paper. Until this analysis is completed, it is extremely difficult toidentify an appropriate refinement scheme within each time step so that the error at each time step can bereduced through the use of higher order differencing schemes. All of these issues provide many challengesfor the future development of numerical methods for the simulation of flexible MAV wing movement.

Acknowledgments

This research is supported in part by the Department of Mathematics and Statistics at the Air ForceInstitute of Technology (AFIT/ENC), the Texas Higher Education Coordinating Board Advanced ResearchProgram (ARP 021244C399) and the National Science Foundation Computational Mathematics Program(DMS 0610026).

References

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3L. Ferguson, E. Aulisa, P. Seshaiyer and R. Gordnier. Computational modeling of highly flexible membrane wings inmicro air vehicles. AIAA Paper No. 2006-1661, AIAA/ASME/ASCE/AHS/ASC 47th Structures, Structural Dynamics andMaterials Conference, Newport, Rhode Island, May 1-4, 2006.

4M. Gad-el-Hak. Micro-air-vehicles: Can they be controlled better? Journal of Aircraft, 38(3):419-429, 2001.5C. Grandmont and V. Guimet and Y. Maday. Numerical analysis of some decoupling techniques for the approximation of

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