Kawahara Lab. 22 March 2008

Optimal Control of Temperature in Fluid FlowUsing the Second Order Adjoint Equation

Daisuke YAMAZAKI† and Mutsuto KAWAHARA‡

Department of Civil Engineering, Chuo UniversityKasuga 1-13-27, Bunkyou-ku, Tokyo 112-8551, Japan

†E-mail : daiest0411@civil.chuo-u.ac.jp‡E-mail : kawa@civil.chuo-u.ac.jp

Abstract

The purpose of this paper is an optimal control problem of temperature using the Newton basedmethod and the finite element method. This method is based on the first and second order adjointtechnique allowing to obtained a better approximation to the Newton line search direction. Theformulation is based on the optimal control theory in which the performance function is expressedby the computed and target temperatures. The optimal value can be obtained by minimizing theperformance function. The gradient of the performance function is obtained by the first orderadjoint equation. The Hessian vector product is calculated by the second order adjoint equation.The thermal fluid flow is described by the Boussinesq approximated incompressible Navier-Stokesequation and the energy equation is analyzed. The Newton based method is employed as a mini-mization technique. The BFGS, the DFP and the Broyden family method are applied to calculatethe Hessian matrix which is used in the Newton based method. The traction given by the pertur-bation for the Lagrange multiplier is used in the BFGS method, the DFP and the Broyden familyMethod.

Keyword : Finite Element Method, Newton Based Method, First Order Adjoint, Second

Order Adjoint, Hessian V ector Product, BFGS Method, DFP Method, Broyden

Family Method.

1 Introduction

The thermal fluid flow is a phenomenon generated by non-homogeneity of temperature. The thermal fluidflow is divided into natural fluid flow and forced fluid flow. The Natural fluid flow is caused by the differenceof the specific gravity between warmed fluid and cool fluid. Forced fluid flow is compulsorily caused by fan orother items. In recent years, the optimal control of thermal fluid flow is used in various places in our life. Forinstance, the optimal control of the thermal fluid flow is applied to air-conditioning, heating, refrigerator andbath and so on. Therefore, the optimal control of thermal fluid flow is indispensable problem. Especially, theoptimal control in the natural fluid flow is one of most-attractive problem. This is because the control bringslow cost and good for the ecology. Floor heating system and temperature management of grass field in thefootball stadium are topical examples.

The purpose of this study is the control of temperature in the natural fluid flow. The Boussinesq approximatedincompressible Navier-Stokes and the energy equations are employed for the state equation. In the Boussinesqapproximation, the density change is approximated by the temperature change. In the optimal control theory,the control variable that makes the optimal state can be obtained by minimizing the performance function,which is composed of the square sum of difference between computed and target temperatures. The controlvariable can be converged to the target temperature in case that the performance function is minimized. TheNewton based method is employed for minimizing the performance function. It is necessary that the informationof the Hessian matrix is obtained in case that the Newton based method is employed. In this study the secondorder adjoint technique is applied to obtained the traction given by the perturbation for the Lagrange multiplierthat has the information of exact Hessian matrix. The BFGS, the DFP and the Broyden family method are

1

applied to obtain the Hessian matrix which is used in the Newton based method. The Broyden family methodis employed as combinations of the BFGS and the DFP method. The Crank-Nicolson method and the Galerkinmethod expanded by the mixed interpolation are applied to temporal and spatial discretization, respectively.The stabilized bubble function is applied to velocity and temperature fields. The linear interpolation is utilizedto pressure field.

The purpose of this study is a control of thermal fluid flow using the second order adjoint technique. Fordetails to verify the effect of the second order adjoint technique and to find the optimal control temperatureusing the second order adjoint technique.

2 State Equation

The thermal fluid flow is described by the Boussinesq approximated incompressible Navier-Stokes equationand the energy equation, which are employed for the state equation. In the Boussinesq approximation, thedensity change is approximated by the temperature change and the density change in the continuity equationcan be disregarded. In the incompressible flow, density of fluid is assumed to be constant. The Boussinesqapproximated Navier-Stokes equation and energy equation are written as follows:

ui + ujui,j + p,i − ν(ui,jj + uj,ij) = fiθ in Ω, (1)

ui,i = 0 in Ω, (2)

θ + uiθ,i − κθ,ii = 0 in Ω, (3)

where ν, κ and fi are written as follows:

ν = Pr, fi = PrRa, (4)

where ui, p and θ are velocity, pressure and temperature and the fi, ν and κ are gravitational acceleration,kinematic viscosity coefficient and diffusion coefficient, respectively. The Rayleigh number and the Prandtlnumber are denoted by Ra and Pr , respectively. The initial conditions are given as follows:

ui(t0) = u0i , (5)

θ(t0) = θ0, (6)

where ui0 and θ0 are the initial known value for velocity and temperature, respectively. The boundary conditions

are given as follows:

ui = ui on ΓuD, (7)

ti = (−pδij + ν(ui,j + uj,i))nj = ti on ΓuN , (8)

θ = θ on ΓθD, (9)

b = κθ,ini = b on ΓθN , (10)

θ = θcont on ΓθC , (11)

where subscripts D, N and C of Γ mean the Dirichlet boundary, the Neumann boundary and the controlboundary, respectively. The control variable on the control boundary Γθ

C is denoted by θcont.

2

3 Spatial Discretization

3.1 Bubble Function Interpolation

The mixed interpolation based on the bubble function and linear function is used for the spatial discretizationoriginally presented by Kawahara and his group. The bubble function interpolation is applied to the velocityand the temperature fields and the linear interpolation is applied to the pressure field as follows:

For bubble function interpolation:

ui = Φ1ui1 + Φ2ui2 + Φ3ui3 + Φ4ui4, (12)

ui4 = ui4 −1

3(ui1 + ui2 + ui3), (13)

θ = Φ1θ1 + Φ2θ2 + Φ3θ3 + Φ4θ4, (14)

θ4 = θ4 −1

3(θ1 + θ2 + θ3), (15)

Φ1 = L1, Φ2 = L2, Φ3 = L3, Φ4 = 27L1L2L3, (16)

and for linear interpolation:

p = Ψ1p1 + Ψ2p2 + Ψ3p3, (17)

Ψ1 = L1, Ψ2 = L2, Ψ3 = L3, (18)

3

2

1

4

Fig.1 : Bubble function element

3

2

1

Fig.2 : Linear element

where ui1 ∼ ui3 and θ1 ∼ θ3 are the velocity and the temperature at nodes 1 ∼ 3 of each finite element.Φα(α = 1 ∼ 4) is the bubble function for the velocity and the temperature, in which p1 ∼ p3 are the pressure atnodes 1 ∼ 3 of each finite element. The linear interpolation function is denoted by Ψα(α = 1 ∼ 3) for pressure.Area coordinate is expressed by Li.

3.2 Stabilized Form

In the bubble function, the numerical stabilization is not enough. Therefore, the stabilized parameter is usedfor stabilization. The stabilized parameter τeB can be written as follows:

τeB =〈φe, 1〉

2Ωe

ν‖φe,j‖2Ωe

Ae

, (19)

where Ωe is element domain and

< u, v >=

∫

Ωe

uvdΩ, ‖u‖2Ωe

=

∫

Ωe

uudΩ, Ae =

∫

Ωe

dΩ. (20)

The integral of bubble function is expressed as follows:

< φe, 1 >Ωe=

Ae

6, ‖ue,j‖

2Ωe

= 2Aeg, g =

2∑

i=1

|Ψα,i|2, (21)

where ν′

is stabilizing parameter. This value is determined to be equivalent to τeS using the stabilized finiteelement method. 3

τeS =

(

2|u′

i|

he

)2

+

(

4ν

h2e

)2

− 12

, u′

i =√

u21 + u2

2, (22)

where he is an element size. In generally, the stabilized parameter in Eq.(30) is not equal to the optimalparameter in Eq.(31). Thus, the bubble function which gives the optimal viscosity satisfies the followingequation.

〈φe, 1〉2

(ν + ν′)‖φe,j‖2

ΩeAe

= τeS . (23)

Eq.(32) adds stabilized operator control term only of the barycenter point to the equation of motion asfollows:

Ne∑

e=1

ν′

‖φe,j‖2Ωe

be, (24)

where Ne and be are the total number of element and the barycenter point. In the energy equation, ν is replacedwith κ.

4 Optimal Control Theory

4.1 Performance Function

In the optimal control theory, control variables can be obtained by minimizing performance function. Theperformance function J is composed of the square sum of difference between the computed and the targettemperatures. The control variable are computed to minimize the performance function under the condition ofthe state equation and boundary conditions. Minimizing the performance function means that the computedtemperature should be as close as possible to the objective temperature. The performance function is writtenas follows:

J =1

2

∫ tf

t0

∫

Ω

(θ − θobj)Q(θ − θobj)dΩdt, (25)

where θ, θobj and Q are the computed temperature, the target temperature at the object point and the weightingdiagonal matrix.

4.2 First Order Adjoint Equation

The Lagrange multiplier method is applied to minimize the performance function. The Lagrange multipliermethod is used for the minimization problem with constraints. The extended performance function J∗ isexpressed as follows:

J∗ = J +

∫ tf

t0

∫

Ω

u∗i (ui + ujui,j + p,i − ν(ui,jj + uj,ij) − fiθ)dΩdt

+

∫ tf

t0

∫

Ω

p∗(ui,i)dΩdt

+

∫ tf

t0

∫

Ω

θ∗(θ + uiθ,i − κθ,ii)dΩdt, (26)

where u∗i , p∗ and θ∗ denote the Lagrange multiplier for velocity, pressure and temperature, respectively. The

extended performance function J∗ is divided into the Hamiltonian H and the time differentiation term asfollows:

4

J∗ =

∫ tf

t0

∫

Ω

H + (u∗i ui + θ∗θ)dΩdt, (27)

where the Hamiltonian H is defined as follows:

H =1

2(θ − θobj)Q(θ − θobj)

+ u∗i ujui,j + p,i − ν(ui,jj + uj,ij) − fiθ + p∗ui,i + θ∗(ujθ,i − κθ,ii). (28)

The stationary condition is needed to minimize performance function. The first variation of J∗ should bezero as follows,

δJ∗ = 0. (29)

Therefore, the first order adjoint equation can be obtained as follows:

−u∗i − uju

∗i,j − p∗,i + ν(u∗

i,jj + u∗j,ij) + θ∗θ,i = 0 in Ω, (30)

u∗i,i = 0 in Ω. (31)

−θ∗ − uiθ∗,i − κθ∗,ii − fiu

∗i + Q(θ − θobj) = 0 in Ω, (32)

In addition, the terminal and boundary conditions for Lagrange multipliers are obtained as follows:

u∗i (tf ) = 0, (33)

θ∗(tf ) = 0, (34)

u∗i = 0 on Γu

D, (35)

(−p∗δij + ν(u∗i,j + u∗

j,i))nj = 0 on ΓuN , (36)

θ∗ = 0 on ΓθD, (37)

b∗ = κθ∗,ini = 0 on ΓθN , (38)

θ∗ = 0 on ΓθC . (39)

Moreover, the gradient of the extended performance function with respect to the control discharge can beobtained as follows:

∂J∗

∂θCont

=

∫

ΓθC

uiθ∗nidΓ +

∫

ΓθC

κθ∗,inidΓ on ΓθC . (40)

4.3 Second Order Adjoint Equation

The following perturbation equation for the state variables can be derived by the formulation that the stateequations are extended by the perturbation u′

i, p′ and θ′ and are approximated by the Tailor expansion.

u′i + ui,iu

′iδij + uju

′i,j + p′,i + ν(u′

i,jj + u′j,ij) = fiθ

′ in Ω, (41)

u′i,i = 0 in Ω, (42)

θ′ + u′iθ,i + uiθ

′,i − κθ′,ii = 0 in Ω, (43)

where the initial and boundary conditions for the perturbation equation are obtained as follows:5

u′i(t0) = u′

i(= 0), (44)

θ′(t0) = θ′(= 0), (45)

u′i = u′

i(= 0) on ΓuD, (46)

t′i = (−p′δij + ν(u′i,j + u′

j,i))nj = t′i(= 0) on ΓuN , (47)

θ′ = θ′(= 0) on ΓθD, (48)

b′ = κθ′,ini = b′(= 0) on ΓθN , (49)

θ′ = θ′cont on ΓθC . (50)

Similarly, the second order adjoint equation can be obtained by the formulation that the first order adjointequation is extended by the perturbations u′

i∗, p′∗ and θ′∗ and are approximated by the Tailor expansion as

follows:

−u′i∗ − uju

′i,j

∗ − p′,i∗ + ν(u′

i,jj∗ + u′

j,ij∗) + θ′∗θ,i = 0 in Ω, (51)

u′i,i

∗ = 0 in Ω, (52)

−θ′∗ − uiθ′,i∗ − κθ′,ii

∗ − fiu′i∗ + Qθ′ = 0 in Ω, (53)

where the terminal and boundary conditions for the second order adjoint equation are obtained as follows:

u′i∗(tf ) = 0, (54)

θ′∗(tf ) = 0, (55)

u′i∗ = 0 on Γu

D, (56)

(−p′∗δij + ν(u′i,j

∗ + u′j,i

∗))nj = 0 on ΓuN , (57)

θ′∗ = 0 on ΓθD, (58)

b′∗ = κθ′,i∗ni = 0 on Γθ

N , (59)

θ′∗ = 0 on ΓθC . (60)

Finally, the traction given by the perturbation for the Lagrange multiplier can be calculated as follows:

(∂2J∗

∂θ2Cont

)θ′Cont =

∫

ΓθC

uiθ′∗nidΓ +

∫

ΓθC

κθ′,i∗nidΓ on Γθ

C . (61)

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5 Minimization Technique

5.1 Derivation of Newton Equation

In case that J(θ(l)Cont),

∂J∂θCont

(θ(l)Cont) and ∂2J

∂θ2Cont

(θ(l)Cont) can be obtained. J(θCont) is approximated by Taylor

expansion as follows:

J(θCont) ≃ J(θ(l)Cont) +

∂J(θ(l)Cont)

∂θCont

T

(θCont − θ(l)Cont)

+ (θCont − θ(l)Cont)

T ∂2J(θ(l)Cont)

∂θ2Cont

(θCont − θ(l)Cont). (62)

It is assumed that the global minimum point for right hand side terms of the above equation is θ(l+1)Cont .

Therefor, the following equation is obtained by the stationary condition as follows:

∂J(θ(l)Cont)

∂θCont

+∂2J(θ

(l)Cont)

∂θ2Cont

(θ(l+1)Cont − θ

(l)Cont) = 0, (63)

where∂J(θ

(l)Cont

)

∂θContand

∂2J(θ(l)Cont

)

∂θ2Cont

are set as follows:

∂J(θ(l)Cont)

∂θCont

= g(l), (64)

∂2J(θ(l)Cont)

∂θ2Cont

= H(l). (65)

Therefore, the equation of stationary condition is expressed as follows:

g(l) + H(l)(θ(l+1)Cont − θ

(l)Cont) = 0, (66)

where g(l) and H(l) mean the steepest decent direction and Hessian matrix.The above equation is formulated as follows:

θ(l+1)Cont = θ

(l)Cont − H(l)−1g(l). (67)

Finally, the renewal equation of control value is obtained as follows:

θ(l+1)Cont = θ

(l)Cont + d(l), (68)

where d(l) means the Newton direction and this direction is calculated as follows:

−H(l)−1g(l) = d(l). (69)

The equation which is obtained the Newton direction is transformed as follows:

H(l)d(l) = −g(l), (70)

where this equation is called the Newton equation.

7

5.2 BFGS Method

The BFGS method ( Broyden-Fletcher-Goldfarb-Shanno method ) is one of the Newton type minimizationtechnique. This method which are formulated by the secant equation are enumerated as the methodology whichis used to obtain a Hessian matrix. This method is ensured that the Hessian matrix is positive definite andsymmetric. Therefore, This method is frequently used to calculate the Hessian matrix.

The BFGS method is applied to obtain the Hessian matrix. The update equation of Hessian matrix by usingthe BFGS method is written as follows:

H(l+1) = H(l) +g′kg′kT

g′kT θ′Contk−

H(l)θ′Contkθ′Cont

kT H(l)

θ′ContkT H(l)θ′Cont

k, (71)

where H l, θ′Cont and g′ are denoted as follows:

H(l) =∂2J∗

∂θ2Cont

, (72)

θ′Contk = θ

(l+1)Cont − θ

(l)Cont, (73)

g′k =∂J∗(l+1)

∂θCont

−∂J∗(l)

∂θCont

= g(l+1) − g(l), (74)

where s′ which means the traction given by the perturbation for the Lagrange multiplier can be written asfollows:

s′(l) = H(l)θ′Contk =

∂2J∗

∂θ2Cont

θ′Contk. (75)

Consequently, the update equation of Hessian matrix can be described as follows:

H(l+1) = H(l) +g′kg′kT

g′kT θ′Contk−

s′(l)s′(l)T

θ′ContkT s′(l)

. (76)

In generally, the product between the Hessian matrix H and the θ′Cont can not be obtained directly. Con-ventionally, this product is calculated by using an approximated Hessian matrix.

On the other hand, the traction given by the perturbation for the Lagrange multiplier expressed by exactHessian matrix can be obtained in this approach. Therefore, it is considered that the more exact Newtondirection can be obtained in comparison with conventional method.

5.3 DFP Method

The DFP method ( Davidon-Fletcher-Powell method ) is one of the Newton type minimization technique.This method which are formulated by the secant equation are the methodologies used to obtain a Hessianmatrix. This method is ensured that the Hessian matrix is positive definite and symmetric. Therefore, Thismethod is well known to calculate the Hessian matrix.

The DFP method is applied to obtain the Hessian matrix. The update equation of Hessian matrix by usingthe DFP method is written as follows:

H(l+1) = H(l) +(g′k − H(l)θ′Cont

kT )g′kT + g′k(g′k − H(l)θ′ContkT )T

g′kT θ′Contk

−(g′k − H(l)θ′Cont

kT )T θ′Contg′kg′kT

(g′kT θ′Contk)2

, (77)

where H l, θ′Cont, g′ and s′ are denoted as well as the BFGS method. Consequently, the update equation ofHessian matrix can be described as follows:

H(l+1) = H(l) +(g′k − s′(l)T )g′kT + g′k(g′k − s′(l)T )T

g′kT θ′Contk

−(g′k − s′(l)T )T θ′Contg

′kg′kT

(g′kT θ′Contk)2

, (78)

8

Therefore, the product between the Hessian matrix H and the θ′Cont is calculated by using an approximatedthe Hessian matrix.

The traction given by the perturbation for the Lagrange multiplier expressed by exact the Hessian matrixcan be obtained in this approach. Therefore, it is considered that the more exact Newton direction can beobtained in comparison with conventional method in similar the BFGS method.

5.4 Broyden family Method

The Broyden family method is employed as combinations of the BFGS and the DFP method. The updateequation of the Hessian matrix is described as follows:

H(l+1) = (1 − γ)H(l+1)BFGS + γH

(l+1)DFP , (79)

where γ is a scaler parameter that is used to determine a balance of each terms.In this research, the Broyden family method is applied to obtain the Hessian matrix. The update equation

of the Hessian matrix by using the Broyden family method is written as follows:

H(l+1) = (1 − γ)(H(l) +g′kg′kT

g′kT θ′Contk−

s′(l)s′(l)T

θ′ContkT s′(l)

)

+ γ(H(l) +(g′k − s′(l)T )g′kT + g′k(g′k − s′(l)T )T

g′kT θ′Contk

−(g′k − s′(l)T )T θ′Contg

′kg′kT

(g′kT θ′Contk)2

), (80)

where H l, θ′Cont, g′ and s′ are denoted in similar the BFGS and the DFP method.Similarly, the product between the Hessian matrix H and the θ′Cont is can not calculated directory. Conven-

tionally, this product is calculated by using an approximated Hessian matrix.The traction given by the perturbation for the Lagrange multiplier expressed by exact the Hessian matrix

can be obtained in this approach. Therefore, it is considered that the more exact Newton direction can beobtained in comparison with conventional method in similar the BFGS and the DFP method.

5.5 Newton Based Method

The computational algorithm of Newton based method is similar to that of the gradient methods. Thedifference between the Newton based method and the gradient method is the decent direction which is used forthe renewal equation of the control variables. In the gradient method, the steepest decent direction is directlyused for the renewal equation of control variables. On the other hand, in the Newton based method, the Newtondirection which is obtained by the steepest decent direction is used for the renewal equation of control variables.

The computational algorithm is written as follows:

1. Choose an initial control value θ(0)Cont, an initial Hessian Matrix H(0) and convergence criteria ǫ.

2. Compute state value u(l)i , p(l), θ(l) by the state equation.

3. Compute the performance function J (l).4. Compute the Lagrange Multiplier (u∗(l), p∗(l), θ∗(l)) and the steepest decent direction g(l).5. Solve the Newton Equation H(l)d(l) = −g(l) by the conjugate gradient method.

6. Generate a new control value θ(l+1)Cont by the Newton direction d(l+1).

7. Compute state value u(l+1)i , p(l+1), θ(l+1) by the state equation.

8. Compute the performance function J (l+1).9. Compute the Lagrange Multiplier (u∗(l+1), p∗(l+1), θ∗(l+1)) and the steepest decent direction g(l+1).

10. Check the convergence; if ‖ θ(l+1)Cont − θ

(l)Cont ‖< ǫ then stop, else go to step 11.

11. Compute perturbed solution u′i(l+1), p′(l+1), θ′(l+1) by the perturbation equation for state variables.

12. Compute the Lagrange Multiplier (u′∗(l+1), p′∗(l+1), θ′∗(l+1)) and the traction given by the perturbation forthe Lagrange Multiplier s′l.

13. Compute the Hessian Matrix H(l+1) by the (θ(l+1)Cont − θ

(l)Cont), (g

(l+1) − g(l)) and s′(l).14. Solve the Newton Equation H(l+1)d(l+1) = −g(l+1) by the conjugate gradient method and go to 6.

The initial Hessian matrix is generally set an unit matrix.

9

6 Numerical Study

6.1 Case1

In this study, the optimal control of thermal fluid flow by using the Newton based method is carry out.The finite element mesh and computation model are shown in Fig.3 and 4. The total number of nodes andelements are 1089 and 2048. The boundary condition is shown in Fig.4. In this case, the Rayleigh number,Prandtl number and the time increment ∆t are set to 1000.0, 0.71 and 0.001, respectively. On the left side, thespecified temperature θ is given in Fig.7. The object point is set on center of computational domain. To reducethe temperature at the object points from initial to target temperature, the optimal control is carried out. Thetarget temperature is set to 0.0. This optimal control problem is to find the control temperature on the controlboundaries so as to minimize the performance function. The Newton based method using three types of themethod to obtain the Hessian matrix is applied as the minimization technique. Three types of the method arethe BFGS method, the DFP method and the Broyden family method.

Fig.3 : Finite Element Mesh

u=0,v=0

u=0,v=0

u=0,

v=0

u=0,

v=0

Object Point

Control BoundarySpecified Temperature

x

y

1

1

Fig.4 : Computational Model

As a result, the performance function using the Newton based method is shown in Fig.8. The time history ofthe control temperature at the control point is shown in Fig.9. The time history of temperature at the objectpoint is shown in Fig.10. The temperature and velocity without control of non-dimensional time , T=1.5, areshown in Fig.11. The temperature and velocity with control non-dimensional time, T=1.5, are shown in Fig.12.The control temperature can be found so as to minimize the performance function in Fig.10 using the Newtonbased method. The temperature at the object point in Fig.10 is reduced using the control temperature in Fig.9.Therefore, the optimal control of thermal fluid flow using the Newton based method is successful.

6.2 Case2

In this study, the optimal control of thermal fluid flow by using the Newton based method is carry out witha mesh different of case1. The finite element mesh and computation model are shown in Fig.5 and 6. The totalnumber of nodes and elements are 1495 and 2868. The boundary condition is shown in Fig.6. In this case, theRayleigh number, Prandtl number and the time increment ∆t are set to 1000.0, 0.71 and 0.001, respectively.The specified temperature θ is given in Fig.13. The object point is set in Fig.6. To reduce the temperature atthe object points from initial to target temperature, the optimal control is carried out. The target temperatureis set to 0.0. This optimal control problem is to find the control temperature on the control boundaries so as tominimize the performance function. The Newton based method using three types of the method to obtain theHessian matrix is applied as the minimization technique. Three types of the method are the BFGS method,the DFP method and the Broyden family method.

10

Fig.5 : Finite Element Mesh

y

x

2

1

Control BoundarySpecified TemperatureObject Point

0.7

u=0v=0

Fig.6 : Computational Model

As a result, the performance function using the Newton based method is shown in Fig.14. The time history ofthe control temperature at the control point is shown in Fig.15. The time history of temperature at the objectpoint is shown in Fig.16. The temperature and velocity without control of non-dimensional time , T=1.5, areshown in Fig.17. The temperature and velocity with control non-dimensional time, T=1.5, are shown in Fig.18.The control temperature can be found so as to minimize the performance function in Fig.14 using the Newtonbased method. The temperature at the object point in Fig.16 is reduced using the control temperature inFig.15. Therefore, the optimal control of thermal fluid flow using the Newton based method is successful.

7 Conclusion

In these studies, the optimal control of thermal fluid flow is presented. The control temperature can be foundso as to minimize the performance function using the Newton based method. It is shown that the temperatureat the object points can be reduced using the control temperature. The temperature at the object points is wellcontrolled using the present methods. The Newton based method was compared with the weighted gradientmethod. In addition, the BFGS method was compared with the DFP method and the Broyden family method.In the case1, the computational time is obtained by the Newton based method become faster comparing withthe weighted gradient method. In the case2, the computational time is not obtained by the Newton basedmethod become faster comparing with the weighed gradient method.

References

1. M.Kawahara, K.Sasaki and Y.Sano, ”Parameter Identification and Optimal Control of Ground Temperature”,Int.J.Numer. Meth.Fl., vol.20,pp.789-801,19952. S.Suzuki, A.Anju and M.Kawahara, ”Management of Ground Temperature by Bang-Bang Control Based onFinite Element Application”, Int. J.Numer. Mesh.Eng., vol.39, pp.885-901,19963. J.Matsumoto, A.A.Khan, S.S. Wang, and M.Kawahara, ”Shallow Water Flow Analysis with Moving Bound-ary Technique Using Least-Squares Bubble Function”, Int. J. Comp. Fluid Dyn., vol.16(2), pp.129-134, 20024. J.Matsumoto, T.Umetsu and M.Kawahara, ”Stabilized Bubble Function Method for Shallow Water LongWave Equation”,Int. J. Comp. Fluid Dyn., vol.17(4), pp.319-325, 20035. T. E. Tezduyar, ”Stabilized Finite Element Formulations for Incompressible Flow Computations”, Advances

in Appl.Mech., vol.28, pp.1-42,19926. Aleksey K.Alekseev and I.Michael Navon ”Calculation of Uncertainty Propagation using Adjoint Equations”,International Journal of Computational Fluid Dynamics vol.17(4), pp.283-288, 20037. K. Sakuma and M. Kawahara, ”Predictive Control Applied to Flood Control Problem”, Int. Centre for Heat

and Mass Transfer 19978. S. Kato, S. Suda, K. Imazu, H. Nakane and M. Kawahara, ”A Control Analysis of Interaction Problem byFluid Force”, Commun. Numer. Meth. Eng. vol.17, pp.465-476, 20019. Aleksey K.Alekseev and I.Michael Navon ”On Estimation of Temperature Uncertainty Using the SecondOrder Adjoint Problem” ,International Journal of Computational Fluid Dynamics, Vol.16(2), pp.113-117, 200210. S.G. Nash; ”preconditioning of Truncated-Newton Methods ”, SIAM J. SCI. STAT. COMPUT., Vol.6,No.3, pp.599-616, July 198511. T.Kurahashi and M.Kawahara, ”Development of Second-order Adjoint Technique for Boundary ValueDetermination Problems”, International Journal for Numerical Methods in Fluids, (To be published)

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0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

Tem

pera

ture

Non-Dimensional Time

Specified Temperature

Fig.7 : Specified Temperature

0

20

40

60

80

100

120

140

0 5 10 15 20 25 30 35 40 45 50

Per

form

ance

Fun

ctio

n

Itaration Number

Weighted Gradient MethodBFGS Method

DFP MethodBroyden Family Method

Fig.8 : Performance Function

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.5 1 1.5 2 2.5 3

Tem

pera

ture

Non-Dimensional Time

Control Temperature

Fig.9 : Temperature at Control Point

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5 3

Tem

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ture

Non-Dimensional Time

With ControlWithout Control

Fig.10 : Temperature at Object Point

Fig.11 : Temperature Without Control Fig.12 : Temperature With Control

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0

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0.8

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0 0.5 1 1.5 2 2.5 3

Tem

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Non-Dimensional Time

Specified Temperature

Fig.13 : Specified Temperature

0

20

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60

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100

120

140

160

180

200

0 10 20 30 40 50 60

Per

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ance

Fun

ctio

n

Itaration Number

Weighted Gradient MethodBFGS Method

DFP MethodBroyden Family Method

Fig.14 : Performance Function

-1.5

-1

-0.5

0

0 0.5 1 1.5 2 2.5 3

Tem

pera

ture

Non-Dimensional Time

Control Temperature

Fig.15 : Temperature at Control Point

-0.1

0

0.1

0.2

0.3

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0.5

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0.7

0 0.5 1 1.5 2 2.5 3

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Non-Dimensional Time

With ControlWithout Control

Fig.16 : Temperature at Object Point

Fig.17 : Temperature Without Control

Fig.18 : Temperature With Control

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