Optimal Control of An Oscillating Body Using the Adjoint Equation ...

Kawahara Laboratory 27 Mar. 2010
Optimal Control of An Oscillating Body
Using the Adjoint Equation
and ALE Finite Element Methods
Hisaki SAWANOBORI
Department of Civil Engineering, Chuo University,
KASUGA 1-13-27 BUNKYO-KU,TOKYO 112-8551, JAPAN .
E-mail : sawanobori-h@civil.chuo-u.ac.jp
ABSTRACT
The purpose of this study is to determine the angle of a wing which is attached to a oscillating body located in a
transient incompressible viscous flow using the Arbitrary Lagrangian Eulerian (ALE) finite element method and
the optimal control theory in which a performance function is expressed by the velocity of the body. At present,
there are some bridges with wings to prevent oscillation by the wind and these wings are called wind-resistant
wing. When the angle of the wing changes, the state of the oscillation is also changed. Therefore, the angle of
the wing is very important so as to minimize the oscillation of bridge. In this research, we solve this problem
based on the optimal control theory. In order to minimize the oscillation of body, the performance function is
introduced. The performance function is defined by the square sum of the velocity on surface of body. This
problem can be transformed into the minimization problem by the Lagrange multiplier method. The adjoint
equations can be obtained by the stationary condition of the extended performance function. We can derive
the gradient to update the angle of the wing from solving the adjoint equations and the state equations. As a
minimization technique, the weighted gradient method is applied. In this study, the angle which the oscillation
of the body become to minimize is presented by these theory. To express the motion of fluid around a body,
the Navier-Stokes equations described in the ALE form is employed as the state equation. The motion of the
body is expressed by the motion equations. As a numerical study, the optimal control of the angle of the wing
is shown at low Reynolds number 250.0. As the numerical result, the angle of the wing which the oscillation of
the body becomes to minimize is shown.
Key words : ALE Finite Element Method, Fluid-Structure Interaction, Optimal Control Theory,
Performance Function, First Order Adjoint Equation, Weighted Gradient Method
1 INTRODUCTION
In recent years, complicated and large-scale computation can be performed because of rapid development of the
finite element method and computer hardware. That is enabled to use analytical models and techniques which is
difficult to analyze before. In the field of numerical analysis, to analyze the fluid flow becomes easily because of
these reasons. It is very important to obtain the physical quantity of fluid by numerical analysis, for example, to
analyze fluid flow of around a body. When a body is put in fluid flow, the Karman vortex occur. The Karman
vortex has various influences for structures. For example, a collapse of bridge happened to the Tacoma Narrows
1

Bridge in Washington, United States of America in 1940. The collapse of bridge happened by the Karman vortex
excitation that is occurred because of side winds. Various analysis and experiments on oscillating bodies by fluid
force have been considered various ways by physical scientist since the accident.
At present, there are some bridges with wings to prevent oscillation by the wind, and these wings are called
wind-resistant wing. Angles of these wings are determined by the wind tunnel experiment. The wind tunnel
experiment can be get a lot of data on physical quantity quickly, but the experiment requires a high cost and it
takes a lot of times to prepare models. Therefore, it is necessary to determine the angle by numerical analysis.
The analysis of oscillating body by fluids force has a moving boundary problem. It is hard to analyze the
oscillating body, but the ALE method can be solve this problem. As one of the techniques to analyze such
problems, ALE method is widely applied to this field. The ALE method is the technique which is combined the
Lagrange description with the Euler description using the moving velocity of node when the motion of fluid is
described. It is possible to use independent mesh velocity from fluid partial velocity, prevent excessive distortion of
the finite element mesh and give moving mesh suitably. An oscillating body is the phenomenon that the behavior
of the fluid and the behavior of the structure in influence each other. This phenomenon is called the fluid-structure
interaction problem. It is necessary for the fluid-structure interaction problem to find the solution while satisfying
the momentum equations concerning fluid and bodies supported by the elastic springs each other.
Therefore, an optimal control of an oscillating bridge with a wing is presented in this study. The body is assumed
to be a bridge with a wing supported by the elastic spring. the Navier-Stokes equation is employed to express
the motion of fluid around the body. The angle of wing is determined based on the optimal control theory. The
performance function is given in the optimal control theory. The vertical velocity of the body is computed so as
to minimize the performance function under the constraint conditions. The performance function is defined by the
square sum of the velocity on surface of body. The oscillation of the body can be minimized by the control variable.
The extended performance function is defined by using the Lagrange multiplier method. The first order adjoint
equations can be obtained by the stationary condition of the extended performance function. We can derive the
gradient from solving the adjoint equations and the state equations. It is necessary to change coordinate system
of the gradient into the poler coordinate system. As the minimization technique, the weighted gradient method is
applied. The performance function is minimized by calculated angle in the weighted gradient method.
In the numerical study, an optimal control of an oscillating bridge with a wing is carried out. The Reynolds
numbers are assumed 250.0 in both of the study, computational domain is done two-dimensional surface. A uniform
stream is given on inflow boundary for horizontal direction. As the numerical result, the angle of the wing which
the oscillation of the body become to minimize is shown.
2 STATE EQUATION
2.1 ALE description
Indicial notation and summation convention with repeated indices is used in this paper. A motion of a body and
fluid motion around a body are expressed by the Lagrangian, Eulerian and reference description. The reference
description is independent of the first two and possible to choose arbitrary coordinate system. According to
these description, the Lagrangian, Eulerian and reference coordinate systems can be expressed X i , x i , and χ i ,
respectively. When the function f is assumed to be arbitrary physical quantity described to the Euler description,
relation between the real time derivative and the reference time derivative is written as follows:
Putting
Df
Dt = ∂ f(χ i,t)
∂t ∣ + ∂ f(χ i,t)
χi
∂χ j
· ∂χ j(X i ,t)
∂t
∣ . (1)
Xi
2

w j = ∂χ j(X i ,t)
∂t ∣ , (2)
Xi
If f is position vector for x i at the present time, eq.(1) can be written in the following form.
∂x i (X i ,t)
∂t
∣ = ∂x i(χ i ,t)
Xi
∂t ∣ + ∂x i(χ i ,t)
· ∂χ j(X i ,t)
χi
∂χ j ∂t ∣ (3)
Xi
= ∂x i(χ i ,t)
∂t ∣ +w j · ∂χ j(X i ,t)
χi
∂t ∣ . (4)
Xi
Introducing relative velocity b i of the material point for the reference coordinate system.
where
b i = u i − ũ i = w j · ∂x i(X i ,t)
∂χ j
∣ ∣∣∣Xi
, (5)
u i = ∂x i(X i ,t)
∂t ∣ , (6)
Xi
ũ i = ∂x i(χ i ,t)
∂t ∣ . (7)
χi
Substituting the velocity relation eq.(5) into the reference time derivative equation, the reference time derivative
in Euler domain is obtained.
Df
Dt = ∂ f(χ i,t)
∂ f(χ i ,t)
∂t ∣ +b j . (8)
χi
∂x j
It is the state equation of the time derivative in the ALE method.
2.2 State equation concerning fluid
Let Ω denote the computational domain with boundary Γ, and suppose that an transient incompressible viscous
flow occupies Ω. The Navier-Stokes equations are employed and can be expressed by the non-dimensional form in
ALE method as follows:
˙u i + b j u i,j + p ,i − ν(u i,j + u j,i ) ,j = 0 in Ω, (9)
u i,i = 0 in Ω, (10)
where u i , p, f i , and ν are velocity in eq.(6), pressure, external force and kinematic viscosity coefficient, respectively.
ν is the inverse of Reynolds number. b j are convective velocity using the reference coordinate system in
eq.(5). ũ j in eq.(5) is called mesh velocity.
The boundary Γ is separated the inflow, edge, outflow and body boundary, Γ U , Γ S , Γ D and Γ B , respectively.
The boundary conditions are given as follows:
u i = û i on Γ U , (11)
u 2 = 0, t 1 = 0 on Γ S , (12)
t i = ˆt i = 0 on Γ D , (13)
u 1 = 0, b 2 = 0 on Γ B , (14)
3

Figure 1: Computational domain
where
t i = {−pδ ij + ν(u i,j + u j,i )}n j (15)
where t i is traction vector and n j is unit vector of outward normal to Γ, respectively. The initial conditions for
velocity and pressure are
where û i is constant inflow velocity.
u i = û 0 i at t = 0 in Ω, (16)
p = ˆp 0 at t = 0 in Ω, (17)
2.3 State equation concerning structure
A rigid body is assumed to have three freedom degree in two-dimension. The motion equation is expressed as
follows:
mẌ + cẊ + kX = F, (18)
X = (X,Y,Θ) T , (19)
F = (F x ,F y ,M) T , (20)
where X and F are displacement and rotational angle from the barycenter of the body and the fluid forces,
respectively. m, c and k are mass, damping and elastic coefficients, respectively. M is moment force.
Figure 2: Model of a rigid body supported
4

3 DISCRETIZATION
3.1 Spatial discretization
As for the spatial discretization, the finite element method based on the bubble function element for the velocity
and the linear interpolation for pressure are applied and expressed as follows:
1. bubble function interpolation
u i = Φ 1 u i1 + Φ 2 u i2 + Φ 3 u i3 + Φ 4 ũ i4 , (21)
ũ i4 = u i4 − 1 3 (u i1 + u i2 + u i3 ), (22)
Φ 1 = L 1 , Φ 2 = L 2 , Φ 3 = L 3 , Φ 4 = 27L 1 L 2 L 3 , (23)
2. linear interpolation
p = Ψ 1 p 1 + Ψ 2 p 2 + Ψ 3 p 3 , (24)
Ψ 1 = L 1 , Ψ 2 = L 2 , Ψ 3 = L 3 , (25)
L 1 + L 2 + L 3 = 1, (26)
1
1
4
2 3
2 3
Figure 3: Bubble function element
Figure 4: Linear element
where Φ α (α = 1,2,3,4 ) is the bubble function interpolation for velocity. The bubble function interpolation is
shown in Figure 3. The bubble function of C 0 continuous can be considered and Ψ α (α = 1,2,3 ) is the linear element
for the pressure. The linear element is shown in Figure 4. The criteria for the steady problem is used, in which the
discretization form of the function interpolation is equivalent to those from the SUPG(streamline-Upwind/Petrov-
Galerkin) method. Therefore, in the bubble function interpolation for the steady problem, the stabilized parameter
τ eβ which determines the magnitude of the streamline stabilized term. The stabilized parameter τ eβ is expressed
as follows:
where Ω e is element domain and
τ eβ =
〈φ e ,1〉 2 Ω e
(ν + ν ′ )||φ e,j || 2 Ω e
A e
, (27)
∫
∫
∫
〈u,v〉 Ωe = uvdΩ,
Ω e
||u|| 2 Ω e
= uudΩ,
Ω e
A e = dΩ.
Ω e
(28)
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The integral of the bubble function is expressed as follows:
〈φ e ,1〉 Ωe = A e
6 , ||φ e,j|| 2 Ω e
= 2A e g, g =
2∑
|Ψ α,i | 2 . (29)
From the criteria for the stabilized parameter in SUPG method, an optimal parameter τ eS can be given as follows:
i=1
τ eS =
[ (2|ui
|
) 2+ ( 4ν
) ] 2 − 1
2
, (30)
h e h 2 e
where h e is an element size.
In generally, the stabilized parameter is not equal to the optimal parameter. In this study, the bubble function
which gives an optimal viscosity is assumed to satisfy the following equation.
τ eB = τ eS , (31)
From eq.(27), (30) and (31), ν ′ can be determined. The stabilized operator control term is added only at the
barycenter point to the equation of motion,
∑N e
e=1
ν ′ ||φ e,j || 2 Ω e
b e , (32)
where N e and b e are total number of element and barycenter point, respectively.
From these discretization technique, the finite element equations of the Navier-Stokes equations are expressed
as follows:
where
M αβ ˙u βi + A αβγj b γj u βi − B αiλ p λ + D αiβj u βj = f αi , (33)
∫
∫
M αβ = (Φ α Φ β )dΩ, A αβγj = (Φ α Φ β Φ γ,j )dΩ,
Ω
Ω
B λαi u αi = 0, (34)
∫
∫
∫
D αiβj = ν (Φ α,k Φ β,k )δ ij dΩ + ν (Φ α,j Φ β,i )dΩ, B αiλ = (Φ α,i Ψ λ )dΩ,
Ω
Ω
Ω
∫
∫
B λαi = (Ψ λ Φ α,i )dΩ, f αi = (Φ α t i )dΓ, b γj = u γj − ũ γj .
Ω
Γ
Thus, the finite element equation can be written in the following form.
M ˙U i + KU i − BP = f i , (35)
B t U i = 0 , (36)
where M, K and B are mass, advection-diffusion and gradient matrix, respectively. U i and P are velocity and
6

pressure in total number of node, respectively. Variables in eqs.(35) and (36) are separated to component on moving
boundary Γ B and others.
U i = (Ui α ,U γ i )t , W i = (Wi α ,W γ i )t , f i = (0,f γ i ), (37)
where superscript γ means quantity with respect to the rigid body. α means quantity with respect to others.
The compatibility and equilibrium conditions are expressed as follows:
˙U γ i = T t Ẍ = T t ˙v, U γ i = T t Ẋ = T t v, (38)
F + Tf γ i = 0, (39)
where
⎡
⎤
1 0 1 0 1 0
T = ⎣ 0 1 · · · 0 1 · · · 0 1 ⎦, (40)
−L y1 L x1 −L yi L xi −L yobj L xobj
} [ ]{ }
cos θs −sin θ s xi
=
. (41)
L yi sin θ s cos θ s y i
{
Lxi
The matrix T expresses the geometric relation between the barycenter on the body and each node of material
surface. θ s is angle of position.
From eqs.(37) − (39), the finite element equations for the fluid-structure interaction problems are obtained.
⎡
M αα M αγ T t ⎤⎧
⎨
⎣ TM γα TM γγ T t ⎦
⎩
˙U α i
˙v
⎫ ⎡
⎬
⎭ + ⎣
K αα K αγ T t −B α ⎤⎧
⎨
TK γα TK γγ T t −TB γ ⎦
⎩
B αt B γt T t
U α i
v
P
⎫ ⎧ ⎫
⎬ ⎨ 0 ⎬
⎭ = −(m˙v + cv + kx)
⎩
⎭ (42)
0
3.2 Temporal discretization
As the temporal discretization of motion of a body, the Newmark β method is applied. The Newmark β method
is a temporal discretization technique. It is used in the field of dynamics widely. As the temporal discretization
of fluid around a body, the Predictor-Corrector method is applied, which calculates the solution of the next time
iteratively using predicted and corrected solution in each time step. This method can be solved non-linear equations
exactly. The calculative algorithm is consisted of three steps; the predictor, incremental calculation of acceleration
and pressure, and the corrector. Using these discretization techniques, variables in eq.(42) are expressed in the
first step as follows:
The first-step (the predictor): l = 0
[ fluid ] ˙U α (0)
i n+1 = 0, (43)
U α (0)
i n+1 = Ui α n + ∆t(1 − γ) ˙U i α n, (44)
P (0)
n+1 = P n , (45)
7

[ body ] ˙v (0)
n+1 = 0, (46)
v (0)
n+1 = v n + ∆t(1 − γ) ˙v n , (47)
x (0)
n+1 = x n + ∆t v n + ∆t 2 ( 1 2 − β) ˙v n, (48)
where l is the number of iteration to the next time.
The second step is consisted of four steps; calculating residues of momentum equation, to obtain the increment of
temporary accelerations, to obtain the increment of pressure, and calculating corrected increment of accelerations.
The second-step (incremental calculation of acceleration and pressure)
(a) Calculating residues of the momentum equation
R (l)
i n+1
i = −[M αα M αγ ] (l) { ˙U α (l)
T t ˙v (l)
n+1
r (l) = −m ∗(l) ˙v (l)
n+1 − c∗(l) v (l)
n+1 − kx(l) n+1
} { α (0) ] U
− [K αα K αγ ] (l) i n+1
T t v (l) + B α(l) P (l)
n+1 , (49)
n+1
−T (l) (M γα (l) ˙U α (l)
i n+1 + Kγα (l) U α (l)
i n+1 − Bγ(l) P (l)
n+1 ). (50)
where
m ∗ = m + TM γγ T t , c ∗ = c + TK γγ T t . (51)
where M,K,B, T,m ∗ and c ∗ are calculated again whenever the displacement x (l)
n+1 is updated. In the next step (b),
the increment of accelerations which is not satisfied the incompressible condition are obtained from the residue of
the momentum equation.
(b) To obtain the increment of temporary accelerations
¯M (l) ∆
˙U
α∗(l)
i = R (l)
i , (52)
¯m ∗(l) ∆ ˙v ∗(l) = r (l) , (53)
where ¯M is lumped mass matrix of M αα , and ¯m ∗ in eq.(55) is the matrix which is obtained from result applied
the Newmark β method.
¯m ∗ = m ∗ + ∆tγc ∗ + ∆ 2 βk. (54)
In the next step (c), the increment of pressure is obtained by solving simultaneous linear equation as follows:
where
(c) To obtain the increment of pressure
{ α(l)
α∗(l) }
U
L (l) ∆P (l) = −[B αt (l) B γt (l) i n+1 + γ∆t∆ ˙U i
]
T t(l) (v (l)
n+1 + , (55)
γ∆t∆˙v∗(l) )
L (l) = γ∆t(B αt ¯M −1 B α + B γt T t ¯m ∗ −1 TB γ ) (l) . (56)
In the next step (d), corrected increment of accelerations is calculated using increment of pressure.
8

(d) Calculating corrected increment of accelerations
The third step (the corrector):
∆
α(l) α∗(l)
˙U i = ∆ ˙U i + ¯M −1(l) B α(l) ∆P (l) , (57)
∆˙v (l) = ∆˙v ∗(l) + ¯m ∗ −1(l) T (l) B γ(l) ∆P (l) . (58)
[ fluid ] ˙U α (l+1)
i n+1 =
α (l) α(l) ˙U i n+1 + ∆ ˙U i , (59)
U α (l+1)
i n+1 = U α (l)
α(l)
i n+1 + γ∆t∆ ˙U i , (60)
P (l+1)
n+1 = P (l)
n+1 + ∆P (l) . (61)
[ body ] ˙v (l+1)
n+1 = ˙v (l)
n+1 + ∆˙v(l) , (62)
v (l+1)
n+1 = v (l)
n+1 + γ∆t∆˙v(l) , (63)
x (l+1)
n+1 = x (l)
n+1 + β∆t2 ∆˙v (l) . (64)
The calculation algorithm from the second step to the third step is iterated until the solution is converged.
4 FORMULATION
4.1 Performance function
The purpose of this study is to minimize the oscillation of the body. In order to minimize the oscillation of
body, the performance function J is introduced. J is defined by the square sum of the velocity for the body and
expressed as follows:
J = 1 2
∫ tf
t 0
∫Ω
(v k Q kl v l ) dΩdt, (65)
where v k and Q kl are the vertical velocity of the body and the weighting diagonal matrix, respectively. On the
optimal control theory, the optimal condition can be derived if the performance function is minimized. In case of
this study, the oscillation of the body can be minimized when the performance function is minimized.
4.2 Extended performance function
The performance function should be minimized satisfying the constraint conditions which are the state equations
(9) and (10). The Lagrange multiplier method is suitable for minimization problems with the constraint conditions.
The Lagrange multipliers for the state equations (9) and (10) are defined as the adjoint parameter u ∗ i and the adjoint
parameter p ∗ , respectively. The performance function is extended by the adding inner products between the adjoint
parameters and the state equations (9) and (10). The extended performance function J ∗ is expressed as follows:
J ∗ = 1 2
−
+
∫ tf
t 0
∫Ω
∫ tf
t 0
∫Ω
∫ tf
t 0
∫Ω
(v k Q kl v l ) dΩdt
u ∗ i { ˙u i + b j u i,j + p ,i − ν(u i,j + u j,i ) ,j }dΩdt
p ∗ u i,i dΩdt (66)
9

4.3 First order adjoint equation
The minimization problem with constraint conditions results in satisfying the stationary conditions of the
extended performance function. eq.(66) can be derived from the first variation of the extended performance
function.
δJ ∗ =
−
+
−
+
−
+
−
∫ tf
t 0
∫Ω
∫ tf
t 0
∫Ω
∫ tf
t 0
∫Ω
∫ tf
∫
t 0
∫Ω
Ω
∫ tf
t 0
∫ tf
t 0
δu ∗ i { ˙u i + b j u i,j + p ,i − ν(u i,j + u j,i ) ,j }dΩdt
δp ∗ u i,i dΩdt
δu i {− ˙u ∗ i − (b j u ∗ i ) ,j + u ∗ ju i,j + p ∗ ,i − ν(u ∗ i,j + u ∗ j,i) ,j − Q kl v l }dΩdt
δpu ∗ i,i dΩdt
u ∗ i (t f )δu i (t f )dΩ
∫
u ∗ i δt i dΓdt +
Γ U
∫
s 1 δu 1 dΓdt −
Γ S
∫ tf
t 0
∫ tf
t 0
∫
u ∗ 2δt 2 dΓdt +
Γ S
∫
s i δu i dΓdt −
Γ D
∫ tf
t 0
∫ tf
t 0
∫
u ∗ i δt i dΓdt +
Γ B
∫
s i δu i dΓdt, −
Γ B
∫ tf
t 0
∫ tf
t 0
∫
u ∗ i δt i dΓdt
Γ W
∫
Γ W
s i δu i dΓdt, (67)
where Γ W and s i are the wing boundary separated from Γ B and
s i = {b j u ∗ i − p ∗ δ ij + ν(u ∗ i,j + u ∗ j,i)}n j . (68)
The stationary condition means that the first variation of the extended performance function vanishes.
δJ ∗ = 0 (69)
Considering eq.(69), the each term of eq.(67) equals zero. Therefore, the first order adjoint equations are obtained
as follows:
− ˙u ∗ i − (b j u ∗ i ) ,j + u ∗ ju i,j + p ∗ ,i − ν(u ∗ i,j + u ∗ j,i) ,j − Q kl v l = 0 in Ω, (70)
u ∗ i,i = 0 in Ω, (71)
u ∗ i (t f ) = 0 in Ω, (72)
u ∗ i = 0 on Γ U , (73)
s 1 = 0, u ∗ 2 = 0 on Γ S , (74)
s i = 0, u ∗ i = 0 on Γ B , (75)
u ∗ i = 0 on Γ W , (76)
s ∗ i = 0 on Γ D , (77)
where eq.(70) and (71) are the adjoint equations and eq.(72) is terminal condition. Then, eq.(67) can be transformed
into the following form:
∫ tf
∫
δJ ∗ = − s i δu i dΓdt. (78)
t 0 Γ W
10

The variation δu i can be expressed by the variation of the angle δX j as:
δu i = ∂u i
∂X j
δX j = u i,j δX j . (79)
It is necessary to change the coordinate system X j into the polar coordinate system to obtain the gradient for
updating the angle.
X 1 = r cos θ, X 2 = r sin θ. (80)
where r and θ are the radius vector and the angle from the rotational center of the wing, respectively. The variation
δu i can be expressed by the variation of the angle δθ as:
Introducing eq.(81) into eq.(78), it is obtained that
∂X j
δu i = u i,j δθ. (81)
∂θ
∫ tf
∫
δJ ∗ ∂X j
= − s i u i,j
t 0 Γ W
∂θ
Thus, the gradient to update the angle of the wing can be derived by eq.(82) as follow:
δθdΓdt. (82)
∂J ∗
∂θ = −s ∂X j
iu i,j
∂θ
(83)
5 MINIMIZATION
5.1 Weighted gradient method
As the minimization technique, the weighted gradient method, which seems to be scarcely dependent on the
initial value, is applied. In this method, a modified performance function K (l) is obtained by adding a penalty
term to the extended performance function. The modified performance function is expressed as follow:
K (l) = J ∗(l) + 1 2 (θ(l+1) − θ (l) )W(θ (l+1) − θ (l) ) (84)
where l and W are iteration number and weighting parameter, respectively. When the modified performance
function converges to zero, the penalty term also be zero. The minimization of the modified performance function
is equal to the minimization of the extended performance function. When the following stationary condition is
applied to the modified performance function,
δK (l) = 0, (85)
the update angle of the wing is calculated at each iteration cycle by the following equation:
∫ tf
∫
Wθ (l+1) = Wθ (l) ∂J ∗(l)
−
dΓdt. (86)
t 0 Γ W
∂θ
11

5.2 Algorithm
The following calculative algorithm is employed for the computation.
Step1. Select an initial angle of the wing θ (0) .
Step2. Solve u (0)
i ,p (0) ,v (0) ,y (0) by eqs.(9)-(18) in Ω.
Step3. Solve u ∗(l)
i ,p ∗(l) by eq.(70)-(77) in Ω.
Step4. Compute θ (l) by eq.(86)
Step5. Update mesh around the wing.
Step6. Solve u (l+1)
i ,p (l+1) ,v (l+1) ,y (l+1) by eqs.(9)-(18) in Ω.
Step7. If |J (l+1) − J (l) | < ǫ then stop, Else go to step.8
Step8. Update a weighting parameter W (l+1) ;
If J (l+1) − J (l) < 0, then set W (l+1) = 0.9W (l) and go to step.3
else W (l+1) = 2.0W (l) and go to step.4.
6 MESH CONTROL
If a body moves, mesh control is needed in the computational domain. In this study, mesh control which is based
on the divergence theory is applied. The Laplace equation is employed as governing equation to control mesh and
expressed in the following equation.
φ ,ii = 0 in Ω. (87)
As the discretization, linear interpolation is applied for potential φ. The linear element is shown in Figure 4. Thus,
the finite element equation is expressed as follows:
K αiβi φ β = 0 in Ω. (88)
As the boundary conditions, the displacement of a body is given for the body boundary, and 0 is given for other
boundaries. The finite element mesh is updated suitably by solving this equation. Figures 5 and 6 show before
moving mesh and after moving mesh.
2
2
Y
0
Y
0
-2
-2
-2 0 2
X
-2 0 2
X
Figure 5: Before moving mesh
Figure 6: After moving mesh
12

7 NUMERICAL STUDY
As a numerical study, the optimal control of an oscillating bridge with a wing is carried out. The computational
domain and boundary conditions are shown in Figure 7. The finite element mesh is shown in Figure 8. This mesh
is separated two moving section; the one is the section that the nodes are moved based on divergence theory, the
other is the section that the nodes are moved in the same as the bridge. The section that the nodes are moved in
the same as the bridge is separated between re-meshing section to change the angle of wing and not. When the
angle of wing is 0.0 ◦ , the total number of nodes and elements are 3282 and 6250. The Reynolds number is set to
250.0 in the all cases and studies. The parameters β and γ in the Newmark β method are set to 0.25 and 0.5,
respectively. ∆t is set to 0.01 in the all cases and studies. Mass, damping and elastic coefficients are 100.0, 0.0 and
400.0, respectively. As initial condition, the angle of the wing is set 10.0 ◦
Figure 7: Computational domain and boundary conditions
Figure 8: Finite element mesh
8 NUMERICAL RESULT
As numerical result, the computed variables which are obtained by the optimal control are shown. The variation
of the performance function is shown in Figure.9. The variation of the angle of the wing is shown in Figure.10. The
initial and final velocities of the bridge are shown in Figure.11. The initial and final displacements of the bridge
are shown in Figure.12. The streamlines, pressure and vorticity are shown in Figure.13-18.
13

9 CONCLUSION
In this study, an optimal control of an oscillating bridge with a wing in an incompressible flow is presented. An
oscillating bridge located in the transient incompressible viscous flow can be analyzed by the ALE finite element
method. As numerical study, an optimal control of an oscillating bridge with a wing in an incompressible flow can
be carried out. You can see that the angle which the oscillation is minimized is obtained. As the future work, we
should change the shape of the wing and study the optimal control of the oscillating body or use a mesh that wings
are set at the front and the rear and study.
Reference
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Analysis’, Journal of Applied Mechanics Vol.2,pp.211-222,1999
2. J.Donea, S.Giuliani and J.P.Halleux,“An arbitrary Lagrangian-Eulerian finite element method for transient
dynamic fluid-structure interactions”, Comp. Meth. Appl. Mech. Engrg.,33,pp.689-723,1982.
3. J.Matsumoto and M.Kawahara,“Stable Shape Identification for Fluid-Structure Interaction Problem Using
MINI Element”, Journal of Applied Mechanics Vol. 3, pp. 263-274
4. P.Ananostopoulos and P.W.Bearman,“Response characteristics of vortex-excited cylinder at low Reynolds
numbers”,J.Fluids Struct.,6,pp.39-50,1992.
5. T.J.Hughes, W.K.Liu and T.K.Zimmerman,“Lagrangian-Eulerian finite element formulation for incompressible
viscous flows”, Comp. Meth. Appl. Mech. Engrg.,29, pp.329-349,1981
6. T.Nomura,“Application of predictor-corrector method to ALE finite element analysis of flow-structure interaction
problems and associated computational techniques”, Journal of hydraulic, Journal of Hydraulic,
Coastal and environmental Engineering, No.455/I-21, pp.55-63, 1992.10
7. T.Nomura and T.J.Hughes,“An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid
and a rigid body”, Comp. Meth. Appl. Mech. Engrg.,95, pp.411-430, 1992
8. Z.Qun and T.Hisada,“Investigations of the coupling method for FSI Analysis by FEM ”,Transactions of the
Japan Society of Mechanical Engineers, A67(662) pp,1555-1562 20011025.
14

1.1
1
Performance function
0.9
0.8
0.7
0.6
0.5
0.4
0 5 10 15 20 25
Iteration
Figure 9: The variation of Performance funciton
25
20
Degree
15
10
0 5 10 15 20 25
Iteration
Figure 10: The variation of angle of wing
15

0.02
Initial angle
Final angle
0.01
Velocity
0
-0.01
-0.02
0 20 40 60
Time
Figure 11: The velocity of the body
0.015
Initial angle
Final angle
0.01
Displacement
0.005
0
-0.005
-0.01
0 20 40 60
Time
Figure 12: The displacement of the body
16

2
2
1
1
0
0
-1
-1
-2
-2
Figure 13: Velocity at θ = 10.00
Figure 16: Velocity at θ = 23.27
4
4
2
2
0
0
-2
-2
-4
-4
Figure 14: Pressure at θ = 10.00
Figure 17: Pressure at θ = 23.27
4
4
2
2
0
0
-2
-2
-4
-4
Figure 15: Vorticity at θ = 10.00
Figure 18: Vorticity at θ = 23.27
17