[American Institute of Aeronautics and Astronautics 51st AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics

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Thin, Flapping wings: Structural Models and Fluid-Structure Interactions

T. Fitzgerald1, M. Valdez1, S. Preidikman2, and B. Balachandran3 University of Maryland, College Park, MD 20742, USA

With a long-term goal of understanding the influence of wing flexibility on the aerodynamic performance of a flapping wing, the authors have considered a three-dimensional, insect wing inspired structure. A nonlinear finite element formulation is used to model this structure, which consists of veins and membranes that hold the veins together. Each vein contributes to the structural rigidity of the wing, and it is discretized with beam elements. The surface of the wing is modeled by using thin membrane elements. Numerical evaluation of the aerodynamic forces is carried out by using the unsteady vortex-lattice method (UVLM). In the computational framework, the fluid-structure systems are fully coupled by using a fixed-point iteration scheme. Investigations into the primary deformation modes of the coupled system are carried out to assess the contributions of spanwise, as well as chordwise deformations to the generated aerodynamic loads.

I. Introduction

uring flapping motions, it is well known that insect wings undergo large spanwise and chordwise elastic deformations due to aerodynamic, elastic, and inertia forces. However, an

important open question is the degree to which the dynamics of the fluid and the elastic wing are coupled. The roles of inertial, elastic, and aerodynamic forces during flapping flight of a variety of species have been the focus of many investigations (see for example, Ellington1; Ennos2; Lehman and Dickinson3; Sun and Tang4; Combes and Daniel5; Song et al.6). Combes and Daniel5 have presented evidence to suggest that the aerodynamic forces can be ignored relative to the inertial and elastic forces within the wing, so that dynamic shape changes can be determined a priori. Unfortunately, the inherent challenge of obtaining flow field measurements in the wake of a living insect makes such postulations difficult to test.

Mechanisms based on the elasticity of the lifting surfaces, which have been observed in moths and butterflies; include modifications and inversions of the camber, contraction and expansion of the wing surface, transversal bending, and spanwise torsion. Many investigations (e.g., Shy et al.7; Vanella et al.8) suggest that, for a particular flapping kinematics, the flexibility of the lifting surfaces has a positive impact on the performance of the flapping wing. Heathcote et al.9 carried out an experimental investigation on the effect of spanwise flexibility. The thrust of a heaving wing was found to be increased for moderate flexibility. Conventional models for studying flapping wing aerodynamics generally rely on a rigid wing on which flapping kinematics is imposed (e.g., Wang et al.10 and Vanella et al8). However, an insect wing is a highly intricate, anisotropic structure, with membranes and longitudinal veins, pointing to the need to consider a flexible wing.

1 Graduate Research Assistant, Department of Mechanical Engineering, AIAA Student Member. 2 Professor, Departamento de Estructuras, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina & Visiting Faculty, Department of Mechanical Engineering 3 Professor and Associate Chair, Department of Mechanical Engineering, 2133 Glenn L. Hall, AIAA Fellow.

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51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida

AIAA 2010-2962

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

3s

2s

1s

Stroke planeWing’s axis

Figure 1. Flapping wing kinematics description.

Early attempts to understand phenomena associated with flapping wings relied on simplified, but insightful approaches and tools (e.g., Smith et al.11). Many of these approaches and tools are based on theories and methods developed in the aeronautics field. These theories and methods include the blade-element theory and the actuator disk theory (momentum theory), widely used for the study of propeller performance (e.g., Ellington12, Weis–Fogh13, Jensen14, Spedding15); the thin airfoil theory (e.g., DeLaurier16); and the lifting-line and lifting-surface methods (e.g., Betteridge and Archer17). These methods can only provide gross estimates of the generated forces and the flapping efficiency in terms of power consumption. Furthermore, in general, they are based on restrictive assumptions such as large aspect-ratio wings, small amplitude oscillations, small angles of attack, and so on. In addition, none of these methods can be used to obtain information on the time evolution of the wakes; that is, the vortex structures generated due to the flapping motions.

To date, the use of three-dimensional models in studies of nonlinear fluid-structure interaction phenomena in flapping wing systems has been sparse. Smith18 employed an unsteady panel method based on constant strength sources and doublet distributions over each panel. The model was coupled with a finite element model for the structure. Recently, Pai et al.19 studied the fluid-structure interaction phenomena in a flapping wing system. The model consisted of a fully nonlinear finite element formulation based on nonlinear beam and membrane elements for the structure and a modified strip theory for the aerodynamic loads. The aerodynamic model, however, does not provide any information on the wake development.

II. Wing Kinematics and Discretized Wing Model

Before the discretization is considered, throughout this effort, the wing kinematics is described by four angles as shown in Figure 1. The stroke or flapping angle is the angle between the projection of the wing’s axis onto the stroke plane and the direction of the unit vector 2s . The elevation

angle is the angle between the wing’s axis and the stroke plane. The rotation angle is the angle between the wing’s cross section and the stroke plane. Finally, the orientation of the stroke plane is given by the angle .

Drawing inspiration from insects, a wing can be seen as a passive membrane, connected by veins. A seminal text on this subject is due to Comstock20. In Figure 2, the vein pattern and planform of a Musca Domestica, or the common house fly is depicted. The veins of the insect wing can be viewed as key structural elements. The membranes, on the other hand, only act as elements that transmit pressure forces on the surface of the wing to the corresponding veins. As a consequence, the structural modeling of the wing can be simplified by modeling each vein with finite elements.


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Figure 2. Diagram illustrating the vein structure in a Musca Domestica wing (Comstock20, 1918)

Figure 3. Schematic of bending of single FEM beam element, arbitrarily placed in a rotating plane.

A first approximation of the wing system can be obtained by using a model made up of linear beam finite elements and taking into account the additional inertia forces associated with a rotating plane. These finite elements are assumed to deform transversely from the reference configuration plane. In addition, the elements admit torsion, which is considered to be decoupled from bending. A single element, which is located arbitrarily in the x-y plane, is shown in Figure 3. The frame associated with the unit vectors 1 2 3

ˆ ˆ ˆ, ,b b b is a non-inertial frame, and the angular

velocity of this frame in the inertial frame is expressed as

1 1 2 2 2 2ˆ ˆ ˆN B b b b ω (1)

The frame of the element, ˆ ia , is fixed in the x-y plane. The local coordinate along the beam

element is . The transverse displacement is ( , )v t , and the length of the element is eh . The

point ir is the location of node i with respect to the ˆ b frame, and ( , )tr r is the location of a point along the element. In a similar manner as before, the torsion of the element can be assumed to be independent of transverse bending.


4

Figure 4. Representative mode shapes of FEM mesh of Musca Domestica: a) mode 1 and b) mode 2.

By using Hermite cubic polynomials as shape functions for bending, and linear shape functions for torsion, Lagrange’s equations can be formulated for a single element. In an undamped case, the element equations of motion are given by

1, 2,e e e

e e e e eM Q K H Q GF F G n ne (2)

where Te eM M is the mass matrix, Te eK K is the

stiffness matrix, e eH H ω is an angular velocity

dependent softening term. The centrifugal and Coriolis accelerations are accounted through e eF F ω

and ,e eG G ω ω , respectively. The projection of the

generalized external forces is represented by eGF .

The first two vibration modes of the considered wing are shown in Figure 4. This planform, which is being used as a proof of concept, has more degrees of freedom than needed for its description. An interesting note is that the first mode of the wing is a chordwise mode, not spanwise, as it would be expected. This was not intentionally set up to be the case, and it is attributed to the concentration of veins near the leading edge. This observation may be supportive of previous two-dimensional structural models that only focus only on chordwise flexibility8.

For further work, the wing geometry and mesh have been simplied. The geometric description of a simplified wing model that is used in the present study is shown in Figure 5. The wing is idealized as a set of beams accounting for most of the rigidity of the structure and menbranes acting as aerodynamic surfaces and supporting pressure loads as well. The beams are hold together by a rigid link as shown in the figure. This was chosen to be the case due to implementation needs of the nonlinear finite element model described below. A point ouside the wing, and rigidly attached to the link constitutes the joint about which all the rotational motions of the wing occur. As a complement to the structural model previously discussed, a model based on a geometrically exact nonlinear finite element formulation of beams and membranes is also constructed to employ finite strains and rotations, following the work carried out by Pai28. This type of model has been used in a similar effort for quasi-steady aerodynamic analysis of flapping flight19. For both the beam and membrane elements, the nonlinear Jaumann strains and stresses are used in a Total Lagrangian formulation. The main features of using these Jaumann measures are they are locally directed along a co-rotated reference frame so no further decomposition is needed, and the use of material constants from standard small-strain engineering experiments can be directly used28.


5

Joint Wing’s axis of rotation

tc

rc

s

d

Rigid link

Figure 5. Geometric description of the wing model used in the present study.

The nonlinear beams are based on the kinematics of line deformation. Although the formulation provides for the generality of initial curvatures to the reference line, here it is assumed that the beams are initially straight. This reduces the complexity of the formulation. There are 9 degrees of freedom at each node: the displacements , , u v w , the local derivative with

respect the arclength , , u v w , the torsion angle , and the transverse shears strains 5 6 , .

The 18 degrees of freedom for a two-node element are contained in q . In Figure 6, the frames of reference and deformation of the beam are illustrated. For a linearly elastic thin beam the Jaumann stresses J and strains B are related as [ ] J Q B , with the material constants of Young’s modulus E, and shear modulus G being used as

0 0 1 0 0 0

[ ] 0 0 [ ] 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

E z y y z

Q G S z

G y

(3)

This leads to [ ] ( )B S q , and ( )q is a nonlinear function of the degrees of freedom. The rotary inertias are assumed to be small and neglected, and the translational inertia is accounted for in the applied force which results in a constant inertia matrix. However, the stiffness matrix and deformation dependent loads remain nonlinear, and thus, the overall system needs to be linearized for computing the solution through an iterative Newton-Raphson scheme.

x

y

z

x

y

z

uw

v

( )Y t

( )X t

( )Z t

Moving Reference Frame

Initial configuration, fixed frame

Deformed System

Undeformed System

Figure 6. Diagram of deformed system in moving reference frame.


6

I

J

K

i

j

k

Moving Reference Frame

Fixed Frame

Deformed System

R

Undeformed System

Figure 7. Diagram of membrane element and moving reference frame.

The membrane elements are constructed by using similar procedures. This time, the degrees of freedom at each node are only the displacements , , u v w . Thus for a 4 node element, there are 12 degrees of freedom. In Figure 7, the associated reference frames are shown. Now the Jaumann stress J and strains B are related by [ ] J Q B . For an isotropic material,

1 61

2 622

12 1 61 2 62

1 0 (1 ) cos 1

[ ] 1 0 , (1 ) cos 11

0 0 (1 )sin (1 )sin

eE

Q B e

e e

(4)

where je are the engineering strains, and 6 j are the shear strains. Since these membrane

elements are very light, the rotary inertias are again neglected giving a similar set of linearized equations as for the beam elements. The assembly process is simplified since the membrane elements only have displacement degrees of freedom, which can be directly transformed into a global reference frame. Additional constraints would be required if the beam elements are arbitrarily joined together. However, by using a simplified mesh where each straight beam is only connected to membrane elements, and not one another, this can be addressed.

III. Aerodynamic Model

In contrast to the computationally expensive direct numerical simulation (DNS) method, vortex methods present a compromise between computational speed and fidelity. The unsteady vortex lattice method (UVLM) was previously used by Preidikman21 to study flutter on bridges and an airplane configurations. Fritz and Long22 employed the unsteady vortex lattice method to study oscillating plunging, pitching and flapping motions of a rigid finite aspect-ratio wing. A two-dimensional version has been used by Valdez et al.23 to study a hovering flapping wing. Willis et al.27 have used a source and doublet panel method to study flapping flight.

In this computational modeling, it is assumed that the flow field is inviscid everywhere except in the boundary layers and the wakes. The flapping wing is represented as a flat plate, whose aerodynamic equivalent is a sheet of vorticity. A vortex sheet represents a viscous shear layer in


7

the limit of the infinite Reynolds number. Along this sheet, a jump in the tangential velocity is allowed. In order for the pressure difference to vanish in the merging airstreams flowing off the upper and lower sides of the wing along its edges (i.e., separating from the wing), vorticity is shed from the trailing edge into the flow field where it moves with the fluid particles downstream and forms the wakes. The vorticity contained in the vortex sheets is discretized by using discrete vortex lines forming a lattice. The lattice that represents the boundary layer on the solid surfaces is called a bounded lattice, whereas the lattice that represents the wakes is called a free lattice. The vortex lines in the lattices form closed vortex rings as a consequence of the Helmholtz principle of spatial conservation or vortex lines. The arrangements of the panels (or vortex rings) that model the wing and the wake, together with a single vortex ring are schematically shown in Fig. 8. Since the continuous vortex sheet is discretized with vortex lines, the non-penetration condition can only be applied at a finite number of locations on the surface of the wing. These locations are called control points of the bound vortex lattice and these points correspond to the centroid of the vortex elements. By imposing the non-penetration condition, the circulations of the bound vortex rings are obtained. Then, the velocity field can be reconstructed from the vorticity on the wing and in the wakes by using the Biot-Savart law. The vortex lines located at the trailing edge are convected to satisfy the Kutta condition. Since the wake is force free, each vortex ring must move with the local stream velocity25.

III.A. Assumptions and Framework

The primary assumption underlying the unsteady vortex lattice method (UVLM) is that the viscous effects are solely confined to the thin boundary layers on the surfaces of immersed bodies and to their wakes. As a consequence, all the vorticity of the flow is also confined to these regions. Outside the wakes and boundary layers, the flow field is considered to be irrotational, and hence, a potential theory can be used to describe the evolution of the fluid variables. As the Reynolds number increases, the thickness of these regions decreases; and, in the limit of Reynolds number being infinity, they can be modeled as infinitesimally thin, continuous vortex sheets. Vortex sheets are singularity surfaces across which the tangential velocity is discontinuous.

Two types of vortex sheets are used in this model. These are a bound vortex sheet representing the boundary layers on the bodies, and a free-vortex sheet, representing the wakes. The motion of the bound vortex sheets is in general prescribed, or dictated by the dynamics of the fluid-wing system. Consequently, a jump in the pressure value occurs across this kind of sheet, results in an aerodynamic load. On the other hand, free-vortex sheets are assumed to be force free. These sheets move and deform with the local fluid velocity. Both types of vortex sheets are joined along the sharp edges where flow separation occurs such as the trailing edge of a wing. Surface singularity methods as the UVLM are based on the solutions of the continuity equation for incompressible, inviscid flows. This equation is the Laplace equation for the velocity potential field ( , )t R ; that is,

2 ( , ) 0t R (5)


8

1

23

4

G

Vortex Ring

Wake

Wing

Bound Vortex Ring

Free Vortex Ring

Figure 8. Representation of the bound and free vortex lattices in UVLM.

The gradient of velocity potential , t R is related to the velocity field as

, ,t t V R R (6)

Since the flow is considered to be inviscid, one can only specify the non-penetration condition on the surfaces of the bodies. This condition requires the normal component of the relative fluid-wing velocity to be zero at each point of the surface of the wing. This can be written as

ˆ ˆ 0S S V V n V n (7)

where SV is the local velocity of the boundary surface S and n is the local unit normal vector. In

general, SV and n vary in space and time. A regularity condition at infinity must also be

imposed. This second BC requires that the flow disturbance, due to the motion of the body (or bodies) through the fluid, must diminish far from the body. In other words, at infinity, the fluid velocity consists only of the unperturbed flow velocity V

, ,lim t lim t

R RV R R V (8)

The integral representation of the velocity field , tV R in terms of the vorticity

field , ,t t Ω R V R , is an extension of the well–known Biot–Savart law. For three–

dimensional flows, it takes the following form


9

0

0 003,

0

,1,

4 ffV t

tt dV

R

R R RV R R V

R R

(9)

where 0R is the position vector of point in the compact region 0 ,fV tR of the fluid domain.

The integrand in the surface integral (9) is zero wherever 0 , tΩ R vanishes. Thus, the region

where the flow is irrotational does not contribute to the velocity field , tV R . Consequently,

, tV R can be evaluated explicitly at each point; that is, independently of the evaluation of the

velocity at neighboring points. Due to this feature, which is absent in finite-difference methods, the evaluation of V can be confined to the viscous region; the vorticity distribution in the viscous region determines the flowfield in both, the viscous and inviscid regions.

III.B. Unsteady Vortex Lattice Method: Formulation

In UVLM, the continuous bound-vortex sheets are discretized into a grid of small straight vortex segments of constant circulation i t . These segments divide the surface into a number

of elements of area. The vortex lines in the lattice form closed vortex rings as a consequence of the Helmholtz principle of spatial conservation of the vortex lines. The model is completed by joining free vortex lines, representing the continuous free-vortex sheets, to the bound-vortex lattice along the edges of separation.

Experience with the vortex-lattice method suggests that the geometric shape of the elements in the lattice affects the accuracy and the rate of convergence. It has been found that rectangular elements work better than other shapes. Consequently, to the extent possible, the authors use rectangular, or nearly rectangular, elements everywhere except in those places where it is not feasible to accurately represent the geometry of the surface. Since the location of the vortex lines in the bound-vortex sheet are known, the unknowns of the problem are only the circulation carried by each of the vortex segments. However, to reduce the size of the problem, instead of thinking of vortex lines as unknowns, one can think of vortex rings of constant circulation as the unknowns. The circulation of each vortex ring is denoted by iG t . Since the vortex sheet has

been approximated by a lattice of vortex rings, the no-penetration condition can only be satisfied at a finite number of points in the surface. These points are called control points (CP) of the bound lattice and are located at the centroids of the elements. The problem then is reduced to finding the circulations iG t of each of the vortex rings such that the no-penetration condition is

satisfied at all control points. In order to formulate the no–penetration BC given by (7), it is convenient to divide the total velocity potential , t R into three parts, one due to the bound–

vortex sheet B , other due to the free–vortex sheet W and one due to the unperturbed flow

. Hence, Eq. (7) can be rewritten as

ˆ 0B W S V n (10)

Now, since the wakes are force free, the circulations of the vortex rings that discretize the free vortex sheets remain constant in time. Since these rings move with the local fluid velocity, their positions and their circulations are known from previous time steps. Therefore, W is known


10

as well as SV and n . Since the positions of the nodes (lines, rings, and control points) in the

bound lattice are known at each time, a matrix of aerodynamic influence coefficients ijA can be

constructed to simplify the solution of the problem. The aerodynamic coefficient ijA represents

the normal component of the velocity at the CP of the ith element associated with a vortex ring of unit circulation at the jth element. In terms of the coefficients ijA , the no–penetration condition

given by (10) is transformed into a linear system of equations whose unknowns are the circulations jG t ,

1

ˆ 1, 2, ,N

ij j W B iij

A G t i N

V n (11)

where N is the number of vortex rings in the surface.

At the end of each time step, to satisfy the Kutta condition, the vortex segments located at the trailing edge are shed into the flowfield and they become part of the grids that approximate the free vortex sheets of the wake. Since the vorticity in the wake now has been generated on, and shed from, the wing at an earlier time, the flowfield is history-dependent and so the current distribution of vorticity on the surface of the body depends to some extent on the previous distributions of vorticity. The vorticity distribution in and the shape of the wake are determined as a part of the solution; so, the history of the motion is stored in the wake.

The location and the distribution of vorticity in the wakes are unknown, and they are determined as a part of the solution. In the present scheme, the authors employ an explicit routine for generating the unsteady wake. The free discrete vortex points that previously emanated from the trailing and leading edges, and whose positions are given by tR are convected at the local

particle velocity, t V R , given by (9). The updated positions, t t R of the vortex points

in the inertial frame are computed according to

t t t t t t t R R R R V R (12)

The only force acting on the body is due to the pressure difference, L Up p , across the two

sides of the wing. The pressure jump across the wing’s cross-section is obtained by using Bernoulli’s equation for unsteady flows (Preidikman21; Katz and Plotkin26). At the CP of the ith element, this pressure jump, i

p , is defined as the difference between the pressure below the

bound–vortex sheet (point L) and the pressure above the vortex sheet (point U); that is,

2 21

2L U U LU L

i ii

ip p p V V

t t

+ (13)

where p is pressure value, V is the magnitude of the velocity and is the velocity potential. The expressions for each of the terms in (13) can be found in the work of Preidikman21. In this


11

manner, the distribution of aerodynamic load is obtained at the control points of the bound lattice.

IV. 3 D UVLM Results

For representative flapping wing motions the results shown in Figure 9 are presented to illustrate the features that can be captured with the computational fluid model. Results including fluid-structure interactions are to be presented in future work. The results shown in Figure 9 correspond to a flapping wing subjected to an incoming flow. Each subfigure contains two different points of view at the same instant in time. The one on the left corresponds to an isometric view whereas the one in the right is the front view, normal to the stroke plane. The axis of rotation of the wing is parallel to the leading edge and located at a distance of half the chord from the wing root. The tapper ratio of the wing is 4. Noting that all quantities are non-dimensional, the results have been generated for a wing span s of 1, distance d (Fig. 5) of 0.2, and a tip chord of 0.25. The incoming flow is aligned with the x axis and has a magnitude of 0.5. All the quantities are non-dimensional. The flapping kinematics; that is, the stroke and rotation angles time variations are considered to be harmonic. The stroke plane is normal to the incoming flow in this case. The kinematics variables are defined as

cos 2 cos 22 6

t ft ft

(14)

sin 22 4

t ft (15)

0t (16)

2

(17)

The flapping frequency f is chosen to be 0.25 Hz. As measured with respect to the mean wing tip velocity, this kinematics yields an advance ratio of

0.952

VJ

sf

(18)

where is the stroke angle amplitude, s is the wing span, and V is the forward speed. The contours shown in Figure 9 correspond to the magnitude of circulation in the wake. The direction of the circulation is given by the local normal vector to the surface of the wake. The scale of the contour is shown in the first sub-plot and this scale is used for the other plots as well. The formation of the tip vortex and starting vortex can be observed in the figures.


12

1sect 2 sect

3sect 4 sect

4 sect 5 sect

7 sect 8 sect

Figure 9. Three-dimensional wake visualization for forward flight combined with flapping motion.

V. Fluid-Structure Interactions

Two independent meshes, not necessarily coincident, are built for the aerodynamic model and for the structural model as shown in Fig. 10. The relevant information (pressure or loads and displacements and velocities) are carried by the control points of the aerodynamic mesh and by the nodal points of the structural mesh, respectively. Therefore, the pressure loads need to be transmitted from the aerodynamic elements to the structural nodes; and nodal velocities and displacements are to be transmitted from the structural mesh to the control points of the aerodynamic mesh. Since the aerodynamic forces highly depend on the shape of the wing surface, the element size of both meshes needs to be comparable. However, the sensitivity of the coupling method to the element size disparity is not known with certainty and this remains to be explored.


13

Model wing

Structural FEM mesh: membrane and beam

elements

Aerodynamic mesh:Vortex rings

Aerodynamic mesh

Membrane elements

Control points

Beam elements

Nodal points

a) b)

Figure 10. Schematic representation of the two interacting meshes for fluid interaction coupling scheme; a)

independent meshes and b) superimposed, non-coincident aerodynamic and structural FEM meshes.

Update tangentmatrices

Nonlinear Corrector Step

1[ ] t t t tj j jK R

1

j

t tjq

Yes No

ˆˆ ˆM q C q K q f Kinematics of the body

Aerodynamic Loads

Converged?

Predictor Step

UVLM Model

Increment timeYes No

t t t 1 k t t k t tq q

1j j

1k k 0k

0j

Figure 11. Process flow diagram of the fluid-structure coupling scheme.

To integrate forward in time, the Newmark-β28,29 algorithm is used to generate the information for the wing structure, and explicit Euler integration is used to update the fluid information. These two system integrators are coupled together as shown in Fig. 11, and described as follows. To begin the time-step, a linear prediction of the structural states is made from the tangent matrices of the previous time step, from which the position and velocities of the wing surface are extracted. This surface kinematics is then used as input to the fluid model, and the associated aerodynamic loading of the wing is computed. These loads are then fed into the corrector step to compute new states of the structural system. Since the finite element formulation is nonlinear, a Newton-Raphson procedure is employed to solve the system of equations. During this sub-iteration, the state-dependant matrices (such as the tangent and geometric stiffness matrices) are reconstructed and updated as the solver converges. Once a


14

corrected structural configuration is found, these states are then compared to the previous states of the structure to see if the overall fixed-point of the coupled system has been achieved.

VI. Concluding Remarks

A combined structural and aerodynamic model for studying the fluid-structure interactions associated with thin, flapping wing systems has been presented. The aerodynamic model is based on the well developed and established Unsteady Vortex Lattice Method (UVLM). The structural model is based on a nonlinear finite element formulation for beams and membranes in a total Lagrangian construction. Results obtained on fluid-structure interactions are to be presented in future work.

Acknowledgments

The authors gratefully acknowledge the partial support received for this work through ARO Grant No. W911NF0610369 and the Cooperative Agreement FA86501023012 between the Air Force Research Laboratory, WPAFB, Ohio and the University of Maryland, College Park, Maryland.

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