Sensitivity Analysis of Geometrically Exact Beam

Theory (GEBT) Using the Adjoint Method with

Hydra

Qiqi Wang

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Wenbin Yu

Utah State University, Logan, UT 84322, USA

The adjoint method with dynamic checking point scheme is used to enable the sensi-tivity analysis capability of GEBT (Geometrically Exact Beam Theory), a general-purposecode for nonlinear analysis of composite beams. Such a capability can meet the design chal-lenges associated with future air vehicles featuring highly-flexible slender components madeof composites. The adjoint method computes the gradient of a user defined objective func-tion to arbitrarily many design and control parameters by backpropagating the sensitivityinformation through the physics of a simulation. As a first step, we implemented a simpli-fied version of GEBT using the Chebyshev spectral method in a code named Hydra, whichsolves the nonlinear integral-differential governing equation governing nonlinear bending,together with its adjoint equation. An example in optimal control is used to demonstratethe efficiency of this method in performing very high dimensional optimization problemsin nonlinear beam dynamics.

I. Introduction

A rapid yet confident assessment of novel air vehicle concepts, such as Micro Air Vehicle (MAVs) andSensorCraft, requires that designers are equipped with a versatile computational design framework whichat the core should adopt efficient high-fidelity engineering models to accurately analyze the complex phys-ical phenomena while maintaining the speed of conceptual design. There are two distinctive features ofthese new concepts: 1) the structures are highly flexible and could undergo large nonlinear deformation;2) the structures will be made of composites. Thus, it is necessary for the structural models to capturegeometric nonlinearity under quasi-static or dynamic loads and handle anisotropy and heterogeneity due tocomposites. Although the ultimate fidelity can be achieved by three-dimensional (3D) models, they are tootime-consuming and labor intensive to be used for effective design space exploration. We are challenged toconstruct more efficient models without significant loss of accuracy of the original 3D models.

Most flexible components are dimensionally reducible. The analysis can be simplified using lower-dimensional structural models .1 For example, one-dimensional (1D) beam models can represent slendercomponents with the cross-sectional dimension much smaller than the length. In the pursuit of efficient yetaccurate models suitable for nonlinear analysis of slender structures featuring arbitrary cross-sections andcomposites, the efficient high-fidelity approach pioneered by Hodges and his-coworkers2 stands out distinc-tively and is being accepted in both industry and research community. As shown in Figure 1, the efficienthigh-fidelity beam models are founded on two pillars: the concept of decomposition of the rotation tensor

Assistant Professor, Aeronautics and Astronautics, Cambridge, MA, USA, AIAA memberAssociate Professor, Mechanical Aerospace Engineering, Logan, UT, USA, AIAA member

1

(DRT)3 and the variational-asymptotic method.4 DRT is a powerful kinematic concept for systematicallycapturing all the geometrical nonlinearities. VAM is a mathematical method for asymptotical analysis ofthe governing variational statement. Starting from the 3D continuum formulation, we can first use DRT toformulate the kinematics in a geometrically exact manner. Then taking advantage of slenderness, we canuse VAM to rigorously reduce the original 3D problem into a 1D beam analysis along with a companiontwo-dimensional (2D) cross-sectional analysis.

3D continuum formulation

of slender structures

Dimensional reduction

using VAM

Cross-sectional analysis

(VABS)

Geometrically Exact

Beam Theory

Global structural

behaviorRecovery relations

3D displacement/stress/strain

field distributions in the structure

Beam Models

Decomposition of the

rotation tensor using DRT

Figure 1. Efficient High-Fidelity Beam Modeling

The 2D cross-sectional analysis provides necessary constitutive models for the 1D beam analysis topredict the global behavior. The cross-sectional analysis is implemented in the computer program VABS(variational-asymptotic beam sectional analysis) which is the only tool capable of realistic modeling ofinitially curved and twisted anisotropic beams with arbitrary sectional topology and materials. The 1Dbeam analysis is implemented in the computer program GEBT (geometrically exact beam theory) using themixed-formulation. Since the 1D formulation is geometrically exact, GEBT can systematically capture allgeometrical nonlinearities attainable by the Timoshenko beam model. GEBT is a general-purpose code,which can analyze structures modeled as an assembly of beams for various applications such as rotatingwings, joint wings, or flapping wings. Powered by VABS, GEBT can also model beams having arbitrarysection geometry/material with no additional cost to the 1D beam analysis. The current version of GEBT canpredict linear/nonlinear static/dynamic (both steady state and transient) behavior, and also linear/nonlinearfrequencies and mode shapes.

As the main advantage of using efficient high-fidelity modeling tools are in conceptual and preliminarydesign, we need to enable sensitivity analysis of GEBT, which computes the gradient of the output quantitiesof a nonlinear beam simulation with respect to input quantities representing design and configuration of thebeam structure. The output quantities can represent performance, cost or fatigue of the structure, whilethe input quantities can represent both the design of the structure and the uncertainties in detailed design,manufacturing process and working environment. These sensitivity gradients, computed using either forwardand adjoint modes, are a unique advantage of computational simulation over experimental methods.

Sensitivity analysis has been widely used. Forward sensitivity analysis solves the linearized model equa-tion; each solution reveals the sensitivity of many output variables with respect to a single input variable.5

Adjoint sensitivity analysis computes the input-output derivatives by solving the adjoint equation. Eachsolution reveals the sensitivity of one output variable with respect to arbitrarily many input variables.6 Com-

2

pared to forward sensitivity analysis, which computes each time the sensitivity of many output variableswith respect to a single input variable, the adjoint approach is computationally efficient when the numberof output variables is much smaller than the number of input variables. In design of modern compositebeam structures, the structural properties of each beam section and the spanwise geometry can be inputdesign variables. The material uncertainties of the many composite layers also increase the number of inputvariables. Therefore, the input variables are typically much more than the number of objective functionsand constraints, making the adjoint approach more suitable.

II. A simplified 2D model of GEBT

Direct implementation of the adjoint approach into GEBT is not an easy task difficult as GEBT is acomplex code covers virtually all aspects of nonlinear composite beam analysis. And also it will be difficultfor us to debug whether we our sensitivity analysis is correct or not. Instead, in this paper, we develop asimplified version of GEBT specifically restricted to transient analysis of an isotropic, homogeneous beamflexible in bending only, with geometrically nonlinear deflections in a 2D plane.

Let the density and moment density of a length L, initially straight beam to be M and I, let the bendingstiffness be K. Assume there is no axial deformation, then the kinetic energy and potential energy of thebeam system can be described as

K =1

2

L

0

M(x2 + y2) + I2ds

U =

L

0

1

2K

(

d

ds

)2

+Mgy ds

where x and y position of the beam as functions of the distance along the beam s is

x(s) =

s

0

cos ds, y(s) =

s

0

sin ds .

The s = 0 end of the beam is fixed (either hinged or clamped) at x(0) = y(0) = 0. We derive the equationof motion for the state of the beam (s) using Hamiltons principal of stationary action. The variation inthe Lagrangian L = K U due to a small perturbation in the trajectory of (s, t) is

L =

L

0

(

M(xx+ yy) + I Kd

ds

d

dsMgy

)

ds

Assuming the beam has a bending damping coefficient of D, its contribution to the virtual work can bemodeled as

WD =

L

0

Dd

ds

d

dsds

Assuming that a bending moment of T is applied at the s = 0 end of the beam, its virtual work is

WT = T (0)

Plug these terms into the Hamiltons extended principle

0 =

t2

t1

L+ W dt ,

Integrating by parts in time and removing the time integration, we obtain the following variational form ofthe governing equation for :

0 =

L

0

(

M(xx+ yy) I Kd

ds

d

dsMgy D

d

ds

d

ds

)

ds+ T (0) (1)

3

where

x(s) =

s

0

sin + cos 2 ds, y(s) =

s

0

cos sin 2 ds

and

x(s) =

s

0

sin ds, y(s) =

s

0

cos ds

The rest of the paper focuses on discretization and sensitivity analysis of this unsteady differential-integralequation.

III. Chebyshev spectral discretization of the beam

We use Chebyshev polynomials to represent the rotation of the beam as a function of the length alongthe beam.7 We denote k, k = 0, . . . ,K as the Chebyshev coefficients and i, i = 0, . . . ,K as the value of at Chebyshev points, i.e.

(s) =

K

k=1

kTk

(

2s 1

L

)

(2)

and

i = (si), si =

(

cos

(

i

K

)

+ 1

)

L

2, i = 0, . . . ,K

The relation between the Chebyshev coefficients k and the function value at Chebyshev points i can berepresented by the linear relation

[k] = [T ][i] , [i] = [T ]1[k]

where the rotation angle vectors in physical and Chebyshev space are denoted as

[i] =

0...

K

, [k] =

0...

K

.

The K K matrix [T ], its inverse [T ]1 and their transposes [T ]T and [T ]T can be efficiently evaluatedusing Fast Cosine Transform (O(K logK) operations as opposed to O(K2) using regular matrix-vectormultiplication). Integration and differentiation also correspond to multiplication by matrices. Suppose

d(s)

ds=

2

L

K

k=1

kTk

(

2s 1

L

)

,

then,[k] = [D][k] , [k] = [I][

k] .

Due to their sparsity, multiplication by both matrices can be evaluated with O(K) operations .7 In addition,for (s) represented using Chebyshev polynomials (2), integration can be calculated to floating point precisionusing the Clenshaw-Curtis quadrature rule

L

0

(s)ds =L

2

L

i=0

wi

(

2s 1

L

)

where wi are the Clenshaw-Curtis quadrature weights.With the Chebyshev polynomials as both basis functions and test functions, together with the discretized

operators associated with differentiation and integration, we can derive the Galerkin discretization of the

4

differential-integral equation that governs motion of the beam. Specifically, we derive a set of coupledordinary differential equations for the evolution of the Chebyshev coefficients k, k = 0, . . . ,K/2. This is

achieved by substituting each of the basis functions Tk

(

2s 1

L

)

, k = 0, . . . ,K/2 into the variation into

the continuous governing Equation (1), and use the discretized operators to approximate the differentiationand integration involved in the equations. Note that the higher order Chebyshev coefficients of (s) k, k =K/2, . . . ,K are set to 0, so that the discretized ordinary differential equation has a dimension of K/2 + 1;while in representing the intermediate variables, including x, y, x, y the full K + 1 coefficients are used.We also use all K + 1 Chebyshev points in representing any quantity in the spatial domain. Doing this isfor the purpose of dealiasing, and ensures stable time integration of the discrete system.

To simplify the notation, we denote discrete matrix operators [Ax] and [Ay] as

[Ax][i] = [T ]1[I][T ](sin[i] [i])

[Ay][i] = [T ]1[I][T ](cos[i] [i]) .

Note that both matrices depends on . Then,

[xi] = [T ]1[I][T ](sin[i] [i]) = [Ax][i]

[yi] = [T ]1[I][T ](cos[i] [i]) = [Ay][i]

where sin and operates component-wise on the elements in the vector. Similarly,

[xi] = [T ]1[I][T ](sin[i] [i] + cos[i] [i]

2) = [Ax][i] [Ay][i]2

[yi] = [T ]1[I][T ](cos[i] [i] + sin[i] [i]

2) = [Ay][i] + [Ax][i]2

Plugging these discretization into each term in Equation (1), we obtain the first term

L

0

M(xx+ yy) ds = [i]T(

[Ax]T[wM ][Ax] + [Ay]

T[wM ][Ay])

[i]

+ [i]T(

[Ax]T[wM ][Ay] + [Ay]

T[wM ][Ax])

[i]2

where the diagonal matrix [wM ] has diagonal elements wiM

(

2si 1

L

)

, i = 0, . . . ,K. wi are the Clenshaw-

Curtis quadrature weights. The second term in Equation (1) becomes

L

0

I ds = [i]T[wI ][i]

where the diagonal matrix [wI ] has diagonal elements wiI

(

2si 1

L

)

, i = 0, . . . ,K. The stiffness term in

Equation (1) becomes L

0

Kd

ds

d

dsds = [i]

T[D]T[wK ][D][i]

where the diagonal matrix [wK ] has diagonal elements wiK

(

2si 1

L

)

, i = 0, . . . ,K, and [D] = [T ]1[D][T ].

The gravity term in Equation (1) becomes

L

0

Mgy ds = g [i]T[Ay]

T[wM ] e

5

where e = [1, . . . , 1]T. The bending damping term in Equation (1) becomes

L

0

Dd

ds

d

dsds = [i]

T[D]T[wD][D][i]

where the diagonal matrix [wD] has diagonal elements wiD

(

2si 1

L

)

. Finally, the term associated with

a bending moment T at s = 0 in Equation (1) is already in an algebraic form, therefore does not requirediscretization.

For each l = 0, . . . K/2, an ordinary differential equation is obtained by substituting the lth Chebyshevbasis function into the variation

= Tl

(

2s 1

L

)

,

and the Chebyshev expansion into the state and its derivatives

(s) =

K/2

k=1

kTk

(

2s 1

L

)

, (s) =

K/2

k=1

kTk

(

2s 1

L

)

, (s) =

K/2

k=1

kTk

(

2s 1

L

)

These K/2+ 1 coupled ordinary differential equations governs the time evolution of the K/2+ 1 Chebyshevcoefficients k, k = 0, . . . ,K/2.

IV. Mathematical formulation of the adjoint method

GEBT solves the following system of equations,8

F(X, X, ) = 0, 0 < t < T (3)

where X is the state vector describing the global behavior of the beam, represents the vector of designparameters such as the mass and stiffness matrices. The equation that governs the change in X resultingfrom a small perturbation of can be obtained by linearizing the GEBT equation,

F

XX +

F

XX +

F

= 0, 0 < t < T (4)

Define our cost function as

J = JT (X(T ), ) +

T

0

J (X, ) dt (5)

which is a function of the state variables X in the time period [0, T ]. The change in the cost function dueto the small perturbation in and the resulting perturbation in X is

J =JTX

X(T ) +JT

+

T

0

J

XX +

J

dt (6)

Because (4) is always 0, we can pre-multiply it with an arbitrary time dependent vector Y T , integrate overtime, and add to J without changing its value.

J =JTX

X(T ) +JT

+

T

0

J

XX +

J

dt

+

T

0

Y T(

F

XX +

F

XX +

F

)

dt

(7)

6

Integrating by parts with respect to removing time derivative of the state vector X, we get

J =JTX

X(T ) +JT

+

T

0

J

XX +

J

dt

+ Y TF

XX

T

0

+

T

0

[

Y TF

X

d

dt

(

Y TF

X

)]

X dt+

T

0

Y TF

dt

(8)

Assuming that the perturbation X is entirely due to perturbation in , with zero change in the initialcondition of X, which means

X = 0 , t = 0, (9)

we then select the Lagrange multiplier Y such that J in Equation (8) depends only on and does notdepend on X. This can be achieved if Y satisfy the following adjoint terminal condition

Y TF

X+

JTX

= 0 at t = T (10)

and the adjoint equationd

dt

(

Y TF

X

)

= Y TF

X+

J

X= 0 (11)

As J is given by the designer,J

X, is also given. The GEBT code can be modified to solve for Y as the

computation ofF

Xis already available in the original code.

When Y satisfies both the adjoint terminal condition, Equation (10), and the adjoint equation, Equa-tion (11), we can conclude from Equation (8) the following

J =JT

+

T

0

J

dt+

T

0

Y TF

dt (12)

This formula reveals the total derivative, that is, the sensitivity, of the cost function J to the design param-eters ,

dJ

d=

JT

+

T

0

(

J

+ Y T

F

)

dt (13)

whereJ

is known when J is given and

F

can be easily obtained using dual-number automatic differ-

entiation technique. This total derivative is calculated by solving the adjoint equation (11) with terminalcondition (10) once. This is efficient compared to the finite difference method, which requires solutions ofEquation (3) as many times as the dimension of the parameter vector , which can easily exceed one hundred.

Because the adjoint solution Y must satisfy a terminal condition (10) at t = T , while the solution duringtime t [0, T ] are needed to compute the sensitivity derivative through (13), the adjoint equation must besolved backwards in time.

V. Numerical experiments

A. Validation of the spectral discretization method

The Chebyshev spectral method for solving the 2D geometrically nonlinear beam equation, described inSection III, and with the adjoint solver described in Section IV, are implement in using C in a code calledHydra. Crank-Nicolson with adaptive time stepping is used for time integration. Two unsteady cases areperformed to validate the Hydra solver and assess the rate of convergence both in the degree of Chebyshevpolynomial approximation and the time step size.

7

0.5 0.0 0.5 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

0.2

y

t=0 to t=0.68

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.71.0

0.5

0.0

0.5

1.0Hinged comparison

GEBT xGEBT yHydra xHydra y

Figure 2. Time history of a hinged flexible beam released from horizontal position.

0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=0 to t=0.68

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7t

1.0

0.5

0.0

0.5

1.0posi

tion

Clamped comparison

GEBT xGEBT yHydra xHydra y

Figure 3. Time history of a clamped flexible beam released from horizontal position.

Both test cases involve a uniform beam of length 1m. The beam is assumed to have very large tensionalstiffness, so that its tensile deformation is negligible. The beam is also thin so that rotary inertia is negligible.The weight of the beam is M = 0.15kg; the bending stiffness is K = 0.15Nm2. Damping and aerodynamicforces are assumed to be negligible. In both test cases, we release the beam at t = 0 from the horizontalcondition. The gravity constant is g = 9.8m/s2. The beam is hinged at the root s = 0 in the first test case,and is clamped at the root s = 0 in the second test case. Both simulations are performed with a K/2 = 8thorder Chebyshev polynomial to represent the rotation angle (s). The time step size is t = 0.001.

Figure 2 plots the position of the beam as solid lines for 0 t 0.68 at a frame rate of 25fps. The tip ofthe beam is plotted as dots at a frame rate of 100fps. This tip position as a function of time matches withthe output of a GEBT simulation performed by the second author. Similarly, Figure 3 plots the simulationresult for the clamped beam. The simulation also matches with GEBT output. Figure 4 demonstrates theexponential convergence of our spatial discretization with Chebyshev spectral method, and the second ordertemporal convergence of our Crank-Nicolson time integration scheme.

8

0 2 4 6 8 10Polynomial order

10-6

10-5

10-4

10-3

10-2

10-1L2

err

or

102 103

Number of time steps

10-5

10-4

10-3

L2 e

rror

Figure 4. Rate of convergence in the Chebyshev polynomial order of spatial discretization and the number of

time steps per unit time.

B. Adjoint based optimization

The adjoint solver is used to solve an optimization problem of controlling the bending moment on the root ofa hinged flexible beam to maximize the tip velocity towards the left at t = 2s. We consider a uniform beam oflength 1m. The beam is again assumed to have very large tensional stiffness, so that its tensile deformation isnegligible. The beam is also thin so that the rotary inertia is negligible. The weight of the beam is M = 1kg.The bending stiffness in the 4 problems we consider includes K = 0.01Nm2, K = 0.1Nm2, K = 1Nm2,K = 10Nm2. The damping coefficient is D = 0.01Nm2s. Aerodynamic forces are assumed to be negligible.All simulations and adjoint calculations are performed with a K/2 = 8th order Chebyshev polynomial torepresent the rotation angle (s). The time step size is t = 0.01.

In all optimization test cases, we start the beam at t = 0 from the vertical condition. A bending momentT is applied at the hinged root. This bending moment as a function of time T (t), 0 < t < 2 is optimized sothat the leftward velocity of the tip is maximized. The objective function we consider is

minT (t),0

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

yt=0 to t=0.7

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=0.7 to t=1.6

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=1.6 to t=2.0

0.0 0.5 1.0 1.5 2.0t

1.0

0.5

0.0

0.5

1.0

posi

tion

tip xtip y

2

1

0

1

2

counte

r cl

ock

wis

e t

orq

ue

torque

Figure 5. Optimized beam motion to maximize tip velocity at t = 2, while minimizing bending moment input

at the root. K = 10.0Nm2

Each iteration involves a line search to find the step size k to ensure that the Wolfe conditions with c1 = 0.1and c2 = 0.9 is satisfied.

Figure 5 shows the optimization result for a very stiff beam, with a bending stiffness of K = 10Nm2.The optimal bending moment smoothly alternates its direction according to the swing direction of the beam,using its natural dynamics to gain energy. The gained energy is then used to achieve maximum tip speed atthe target time t = 2. Under the applied bending moment, the beam rotates almost as a rigid rod withoutvisible deformation. However, from the optimized bending moment input, plotted as the dash-dotted lineon the lower right plot, we can see that starting from about t = 0.8, it appears to be trying to excite thefirst longitudinal mode of the beam, in order to gain additional tip velocity due to this vibration at t = 2.However, this effect is not visible in the time history of the tip location.

Figure 6 shows the optimization result for a moderately stiff beam, with a bending stiffness ofK = 1Nm2.The large scale behavior of the optimal bending moment is the same, using the natural dynamics to gainenergy, and use the obtained energy to achieve maximum tip speed at t = 2. The bending of the beamis visible under the applied bending moment. In addition to the rotation around the hinged root, the firstlongitudinal mode of the beam is clearly excited; the elastic potential energy stored in the bending mode isalso released at t = 2, together with the gravitational potential energy, to maximize the tip speed.

Figure 7 shows the optimization result for a flexible beam, with a bending stiffness of K = 0.1Nm2. Theoptimal bending moment uses a combination of gravitational potential energy and elastic potential energyto maximize the tip velocity at t = 2 to gain energy, and use the obtained energy to achieve maximum tipspeed at t = 2.

Figure 8 shows the optimization result for a very flexible beam, with a bending stiffness of K = 0.01Nm2.

10

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

yt=0 to t=0.7

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=0.7 to t=1.6

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=1.6 to t=2.0

0.0 0.5 1.0 1.5 2.0t

1.0

0.5

0.0

0.5

1.0

posi

tion

tip xtip y

3

2

1

0

1

2

3

counte

r cl

ock

wis

e t

orq

ue

torque

Figure 6. Optimized beam motion to maximize tip velocity at t = 2, while minimizing bending moment input

at the root. K = 1.0Nm2

The time history of the bending moment is similar to the last case, though smoother and has a smallermagnitude. Nonetheless, the dynamics history involves significantly more geometric nonlinearity, includingcomplex rotations near the root of the beam.

These four cases represent a class of optimization problems with very high dimensional input parameters.In our case, the bending moment as a function of time is discretized at 200 time steps. Therefore, thedimension of the optimization problem is 200. Yet using adjoint based optimization, we obtain optimalsolutions in 10 steps for all 4 cases. Between 30 and 60 forward and adjoint simulations are performedin each optimization problem. This is orders of magnitude more efficient than using non-gradient basedmethods, or using finite difference gradient to perform similar optimization problems with a 200 inputparameter space. This example demonstrates the efficiency of adjoint based method in high dimensionalnonlinear structural dynamics optimization problems

VI. Conclusions

This paper presents a collaborative effort between a computational mathematician and a structuralanalyst to initiate efficient yet accurate sensitivity analysis of adjoint formulation for GEBT, a general-purpose code for nonlinear analysis of composite beams. To achieve this goal, a benchmark code, Hydra, isdeveloped as a simplified version. Such a tool will enable us to effectively explore the design space of futureair vehicles featuring highly-flexible slender components exhibiting significant geometric nonlinearity underquasi-static or dynamic loads.

11

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

yt=0 to t=0.7

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=0.7 to t=1.6

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=1.6 to t=2.0

0.0 0.5 1.0 1.5 2.0t

1.0

0.5

0.0

0.5

1.0

posi

tion

tip xtip y

2

1

0

1

2

counte

r cl

ock

wis

e t

orq

ue

torque

Figure 7. Optimized beam motion to maximize tip velocity at t = 2, while minimizing bending moment input

at the root. K = 0.1Nm2

Appendix A: details on terms in the Adjoint equation

The adjoint requires the computation ofF

Xand

F

X, where the state variable

X =

[

X1

X2

]

=

[

]

and

F =

[

X1 X2

G

]

where G is the equation of motion, with individual terms discretized in the last section.

F =

[

F1

F2

]

=

X1 X2G

X2X2 +

G

X2X2 +

G

X1X1

12

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

yt=0 to t=0.7

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=0.7 to t=1.6

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0x

1.0

0.8

0.6

0.4

0.2

0.0

y

t=1.6 to t=2.0

0.0 0.5 1.0 1.5 2.0t

1.0

0.5

0.0

0.5

1.0

posi

tion

tip xtip y

1.5

1.0

0.5

0.0

0.5

1.0

1.5

counte

r cl

ock

wis

e t

orq

ue

torque

Figure 8. Optimized beam motion to maximize tip velocity at t = 2, while minimizing bending moment input

at the root. K = 0.01Nm2

So

F

X=

I 0

0G

X2

=

I 0

0G

,F

X=

0 IG

X1

G

X2

=

0 I

G

G

Denoting

Y =

[

Y1

Y2

]

=

[

]

The adjoint Equation (11) becomes

d

dt=

(

G

)T ,

d

dt= +

(

G

)T

where

=

(

G

)T

We need to derive the derivative of the equation of motion with respect to , and .

13

We first derive the derivative of the matrices Ax and Ay with respect to . From their definitions

[Ax][i]

[di] = [T ]

1[I][T ](cos[i] [di] [i]) = [Ay]([di] [i])

[Ay][i]

[di] = [T ]

1[I][T ](sin[i] [di] [i]) = [Ax]([di] [i])

With this, we derive the derivative of each term:

d

L

0

M(xx+ yy) ds = [i]T(

[Ax]T[wM ][Ax] + [Ay]

T[wM ][Ay])

[di]

+ [i]T(

[Ax]T[wM ][Ay] + [Ay]

T[wM ][Ax])

2[i] [di]

+(

[i] [di])T(

[Ay]T[wM ][Ax] + [Ax]

T[wM ][Ay])

[i]

+ [i]T(

[Ax]T[wM ][Ay] + [Ay]

T[wM ][Ax])

(

[i] [di])

+(

[i] [di])T(

[Ay]T[wM ][Ay] + [Ax]

T[wM ][Ax])

[i]2

+ [i]T(

[Ax]T[wM ][Ax] [Ay]

T[wM ][Ay])

(

[i]2 [di]

)

+ [i]T(

[Ax]T[dwM ][Ax] + [Ay]

T[dwM ][Ay])

[i]

+ [i]T(

[Ax]T[dwM ][Ay] + [Ay]

T[dwM ][Ax])

[i]2

d

L

0

I ds = [i]T[wI ][di] + [i]

T[dwI ][i]

d

L

0

Kd

ds

d

dsds = [i]

T[D]T[wK ][D][di] + [i]T[D]T[dwK ][D][i]

d

L

0

Mgy ds = g(

[i] [di])T

[Ax]T[wM ] e+ g [i]

T[Ax]T[dwM ] e

d

L

0

Dd

ds

d

dsds = [i]

T[D]T[wD][D][di] + [i]T[D]T[dwD][D][i]

d

L

0

Gd

dsds = [i]

T[D]T[w][dGi]

d

L

0

Fxx+ Fyy ds = [i]T[Ax]

T[w][dFx i](

[i] [di])T

[Ay]T[w][Fx i]

+ [i]T[Ay]

T[w][dFy i] +(

[i] [di])T

[Ax]T[w][Fy i]

In the forward tangent code, [i] is an arbitrary test function in the test space, and [di] is the tangentsolution, representing a linear perturbation in [i] caused by perturbation in control variables, includingdM , dI, dK, dD, dG, dFx and dFy. In the adjoint code, [di] and [di] are an arbitrary test functions inthe test space, while [i] is the adjoint solution, representing the sensitivity of an objective function withrespect to perturbations in [i].

Acknowledgments

The development of GEBT by the second author was supported by the 2008 Air Force Summer FacultyFellowship and the Chief Scientist Innovative Research Fund at WPAFB AFRL/RB. The views and conclu-sions contained herein are those of the authors and should not be interpreted as necessarily representing theofficial policies or endorsement, either expressed or implied, of the funding agency.

14

References

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Engineering, Georgia Institute of Technology, May 2002.2Hodges, D. H., Nonlinear Composite Beam Theory, AIAA, Reston, Virginia, 2006.3Danielson, D. A. and Hodges, D. H., Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor, Journal of

Applied Mechanics, Vol. 54, No. 2, 1987, pp. 258 262.4Berdichevsky, V. L., Variational-Asymptotic Method of Constructing a Theory of Shells, PMM , Vol. 43, No. 4, 1979,

pp. 664 687.5Borggaard, J. and Burns, J., A PDE Sensitivity Equation Method for Optimal Aerodynamic Design, Journal of Com-

putational Physics, Vol. 136, No. 2, 1997, pp. 366 384.6Giles, M. B. and Pierce, N. A., An Introduction to the Adjoint Approach to Design, Flow, Turbulence and Combustion,

Vol. 65, 200, pp. 393415.7Boyd, J. P., Chebyshev and Fourier Spectral Methods: Second Revised Edition, Dover Publications, New York, 2001.8Yu, W. and Blair, M., GEBT: A General-Purpose Tool for Nonlinear Analysis of Composite Beams, Proceedings of the

51st AIAA Structures, Structural Dynamics and Materials Conference, Orlando, Florida, April 12 15, 2010.9Jameson, A. and Kim, S., Reduction of the Adjoint Gradient Formula in the Continuous Limit, 41th AIAA Aerospace

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