They include the. Fletcher-Reeves, Polak-Ribiere, and Powell-Beale versions. Such versions are compared for test cases involving different numbers of sensors, ...
Numerical Heat Tr ansfer, Part A, 36:229 ± 249, 1999 Copyright Q 1999 Taylor & Fr ancis 1040± 7782 r 99 $12.00 H .00
C OMPAR ISON OF D IFFER ENT VER SIONS OF THE C ONJUGATE GR AD IENT METHOD OF FUNC TION ESTIMATION Marcelo J. ColacË o and Helcio R. B. Orlande Department of Mechanical Engineering, EE r COPPE, Federal Univ ersity of Rio de Janeiro, Caixa Postal 68503, RJ, 21945-970 Rio de Janeiro, Brasil The in v erse problem of estimating the spatial and transient v ariations of the heat transfer coefficient at the surface of a plate, with no information regarding its functional form, is sol v ed by applying the conjugate gradien t method with adjoin t problem. Three different v ersions of this method, correspondin g to different procedures of computing the search direction, are applied to the solution of the present in v erse problem. They include the Fletcher-Ree v es, Polak-Ribiere , and Powell-Beale versions. Such v ersions are compared for test cases in v ol ving different numbers of sensors, le vels of measurement errors, and initial guesses used for the iterati v e procedure.
INTR OD UC TION Direct heat transfer problems can be mathematically classified as well-posed. The solution of a well-posed problem is required to satisfy the conditions of existence, uniqueness, and stability with respect to the input data w 1 ] 5x . Inverse he at transfe r problems involve the estimation of at least one of the quantitie s appearing in the formulation of well-posed direct problems, such as boundary conditions. In order to overcome the lack of information due to the unknown quantities, temperature and r or heat flux me asurements need to be used for the solution of inverse he at transfe r problems. The experimental errors associated with such me asurements may result in extremely unstable solutions, so that inverse problems are classified as ill-posed. Methods of solution for inverse heat transfer problems generally involve their reformulation as minimization problems, with some stabilization technique used in order to give them a well-posed character. The conjugate gradient method is a powerful minimization technique, which can be applied to parameter and function estimations, as well as to linear and nonlinear inverse problems w 3, 4, 6 ] 19x . The conjugate gradient method with a suitable stopping criterion belongs to the class of iterative regularization techniques, where the number of iterations of the estimation procedure is determined so that stable solutions are obtained for the inverse problem w 3, 4, 6 ] 13x . The Received 3 August 1998; accepted 2 February 1999. The CPU time for this work was provided by NACAD-COPPE-UFRJ. This work was partially supported by COSIPA under Contract Number ET-120141. The support provided by CNPq is also gre atly appreciated. Address correspondence to Professor Helcio R. B. Orlande, Department of Mechanical Engineering, EE r COPPE, Federal University of Rio de Janeiro s UFRJ., Cid. Universitaria, Caixa Postal 68503, Rio de Janeiro, RJ 21945-970 , Brasil. 229
230
M. J. COLAC Ë O AND H. R. B. ORLANDE
NOMENC LATUR E A B Bi c Cp d eRM S J J9 K r S t T X, Y, Z b g ,c
u
dimensionless length of the plate dimensionless width of the plate Biot number thickness of the plate dimensionless specific heat direction of descent given by Eq. s 16. root-mean-square error defined by Eq. s 25. objective functional given by Eq. s 4. gradient of the objective functional dimensionless thermal conductivity average rate of reduction of the objective functional number of sensors time temperature dimensionless spatial coordinates se arch step size given by Eq. s 15. conjugation coefficients for the direction of descent dimensionless estimated temperatures
u `
l m
s
t t
f
f w
dimensionless temperature of the cooling fluid Lagrange multiplier dimensionless me asured temperatures standard deviation of the measurements dimensionless time dimensionless final time dimensionless initial condition dimensionless temperature at the surface Z s 0
«
Subscripts s 0
sensor number perturbed quantities reference value
Superscripts k )
number of iterations dimensional properties
method consists in choosing at each iteration a suitable direction of descent and a se arch step size along this direction, for the minimization of the objective functional. Different versions of the conjugate gradient method can be found in the literature, depending on the form used for the computation of the direction of descent w 3, 4, 6 ] 19 x . The main objective of this paper is to present a comparison of the conjugate gradient method versions due to Fletcher and Reeves w 3, 4, 6, 10 ] 17 x , Polak and Ribiere w 4, 7 ] 9, 14, 15, 18x , and Powell and Beale w 8, 14, 19x , as applied to the solution of a nonlinear function estimation problem. As a test problem, we consider here the estimation of the transient and spatial variations of the he at transfe r coefficient at the surface of a three-dimensional plate. Such a test problem was chosen for the present comparison due to the inherent difficulties associate d with the several independent variable s of the unknown function and because it involves a nonquadrati c objective functional. The accurate knowledge of the heat transfer coefficient at the surface of plates is of importance in many engineering applications, including the cooling of continuously cast slabs and of electronic chips w 11, 20 ] 22x . The inverse problem of estimating the spatial and transient variations of the heat transfer coefficient at the surface of a plate cooled by sprays was solved in Ref. w 11x , by using the FletcherReeves version of the conjugate gradient method. The method was able to recover quite accurately functions containing sharp corners and discontinuities, which are the most difficult to be estimated by inverse analysis. The effects of number of sensors and their locations on the inverse problem solution were also addressed in Ref. w 11x .
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231
A major advantag e of the use of the conjugate gradient method is that the temperature field in the plate is calculated as part of the solution of the inverse problem. Therefore, in the present paper we extend the analysis of Ref. w 11x and show that accurate estimates can be obtaine d with the conjugate gradient method for the estimation of he at transfer coefficients that are functions of the surface temperature. The basic steps of the conjugate gradient method of function estimation include the following: direct problem inverse problem sensitivity problem adjoint problem gradient equation iterative procedure stopping criterion computational algorithm Details of each of these basic steps, as applied to the estimation of the unknown he at transfer coefficient, are described below. D IR EC T PR OBLEM The physical problem considered here involve s the cooling of a plate with initial temperature distribution f s x, y, z. and with lateral boundary surfaces insulated. The temperature F s x, y, t . at the bottom surface of the plate is considered known for the problem, while he at is lost by convection to a cooling fluid at the top surface. The heat transfe r coefficient at the top surface and the cooling fluid temperature are given by h s x, y, t . and T` s x, y, t ., respectively. The direct problem is concerned with the determination of the temperature field in the plate when the heat transfer coefficient, as well as the phsycial properties, initial condition and other quantities appearing in the boundary conditions are known. The mathematical formulation of this he at conduction problem is given in dimensionless form by Cp s u .
u t
s
t
Ks u .
u
/
q
t
Ks u .
u
/
X X Y Y in 0 - X - A; 0 - Y - B; 0 - Z - 1
Ks u . Ks u .
u X u Y
q
t
Ks u .
Z t ) 0
u Z
/
s 1 a.
s0
at X s 0
and
XsA
t ) 0
s 1b.
s0
at Y s 0
and
Ys B
t ) 0
s 1c .
u s w s X , Y, t .
at Z s 0
t ) 0
s 1d .
M. J. COLAC Ë O AND H. R. B. ORLANDE
232
Ks u .
u Z
q Bi s X, Y, t . u s Bis X , Y, t . u ` s X , Y, t .
u s f s X, Y, Z .
at Z s 1
t ) 0 s 1e .
for t s 0, in 0 - X - A, 0 - Y - B, 0 - Z - 1
s 1f.
where the following dimensionless groups were defined:
u s
T y T` T0 y T`
,0
w s
,0
Xs
x c
F y T` T0 y T`
Ys
t s
,0
f s
,0
y
z
Zs
c
c
K 0U t
r
f y T` T0 y T` As
Bi s
*C0U c 2
,0
u `
s
,0
a
Bs
c
b c
T` y T`
,0
T0 y T`
,0
s 2.
hc K 0U
Here a, b, and c are the length, width, and thickness of the plate, respectively, while T0 and T` , 0 are reference values for the temperature of the plate and of the cooling fluid, respectively. In order to write the direct problem in dimensionless form, we assumed the temperature dependence of thermal conductivity and specific he at to be written as K* s T . s K 0U K s u .
s 3 a.
C pU s T . s C0U C p s u .
s 3b .
where K 0U and C0U are reference values for thermal conductivity and specific he at, respectively, while K s u . and C p s u . are dimensionless functions of u . The plate density r * was assumed constant. The superscript asterisk refers to dimensional physical properties. INVER SE PR OBLEM For the inverse problem the dimensionless he at transfe r coefficient Bi s X, Y, t . is regarded as unknown. Such a function is to be estimated by using the transient re adings of S temperature sensors located inside the plate at positions s X s , Ys , Zs ., s s 1, . . . , S, during the time interval 0 ( t ( t f . An estimate Bis X, Y, t . is obtaine d so that the following functional is minimized: J w Bi s X , Y, t . x s
1
2H
t
f
p
S
t s 0 ss 1
w u s s t ; Bi. y m
s
s t . x 2 dt
s 4.
where m s s t . are the me asured temperatures and u s s t ; Bi. are the estimated temperatures at the me asurement locations. The estimated temperatures are obtained from the solution of the direct problem, Eq. s 1., by using an estimate for Bi s X, Y, t ..
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233
In order to apply the conjugate gradient method for minimizing the functional given by Eq. s 4., we need to develop and solve two auxiliary problems, known as the sensitivity and adjoint problems, as described next. SENSITIVITY PR OBLEM The sensitivity problem is developed by assuming that the temperature u s X, Y, Z, t . is perturbed by an amount « D u s X, Y, Z, t . when the Biot number Bi s X, Y, t . is perturbed by « D Bis X, Y, t ., where « is a re al number. Because of the nonlinear characte r of the problem, a perturbation on temperature causes perturbations on the temperature dependent properties as well. Thus we can write the following perturbed quantities:
u «
su
q« D u
s 5a .
Bi « s Bi q « D Bi C p« s C p s u K« s K s u
s 5b .
q « D u . ( Cp s u . q q« D u . ( K s u . q
dC p du
dK du
« D u
s 5c .
« D u
s 5d .
The sensitivity problem is obtaine d by applying the following limiting process:
lim
L« s Bi « . y L s Bi .
«
« ª 0
s 6.
s0
where L« s Bi « . and Ls Bi. are the operator forms of the direct problem, written for the perturbed and unperturbe d quantities, respectively. The following problem results for the sensitivity function D u s X, Y, Z, t .:
Cp s u . D u
t
s
2
w Ks u . D u x
X
2
q
2
w Ks u . D u x
Y
2
in 0 - X - A; 0 - Y - B; 0 - Z - 1
w Ks u . D u x X w Ks u . D u x Y
q
2
w Ks u . D u x
Z2
s 7a .
t ) 0
s0
at X s 0 and X s A
t ) 0
s 7b .
s0
at Y s 0 and Y s B
t ) 0
s 7c .
D u s0
at Z s 0
t ) 0
s 7d .
M. J. COLAC Ë O AND H. R. B. ORLANDE
234
w Ks u . D u x
q Bi s X , Y, t . D u s D Bi s X , Y, t . s u
Z
yu . `
at Z s 1
t ) 0 s 7e .
for t s 0, in 0 - X - A; 0 - Y - B; 0 - Z - 1
D u s0
s 7f .
AD JOINT PR OBLEM An adjoint problem for a Lagrange multiplier comes into the picture because the temperature u s s t ; Bi. appearing in the functional Eq. s 4. needs to satisfy a constraint, given by the solution of the direct problem Eq. s 1.. In order to develop the adjoint problem, we multiply the differential equation, Eq. s 1 a., of the direct problem by the Lagrange multiplier l s X, Y, Z, t ., integrate over the time and space domains, and add the resulting expression to the functional given by Eq. s 4.. The following extended functional results: J w Bi x s
1
1
H H H 2H A
B
t
f
p
S
w u s s t ; Bi. y m
Xs 0 Ys 0 Zs 0 t s 0 ss 1 A
1
B
H H H H
q
t
f
Xs 0 Ys 0 Zs 0 t s 0
y
Y
u
Cp
t
t / K
y
u
Y
y
Z
s t . x 2 d s r y r s . dt dZ dYdX
t / t /
X
s
K
K
u
X
u
l s X, Y, Z, t . dt dZ dYdX s 8 .
Z
where d s . is the Dirac delta function and r s is the vector with the position of sensor s, i.e., r s s s X s , Ys , Zs .. An expression for the directional derivative of Jw Bix in the direction of the perturbation D Bi s X, Y, t . is obtaine d by applying the following limiting process: DD
Bi
J w Bi x s lim
J w Bi« x y J w Bi x
« ª 0
s 9.
«
where Jw Bi « x is the functional Eq. s 8. written for the perturbed quantitie s given by Eqs. s 5.. After performing some integrations by parts on the resulting expression for DD Bi Jw Bix and applying the initial and boundary conditions of the sensitivity problem, we let the terms containing D u s X, Y, Z, t . go to zero. The following adjoint problem is then obtained for the Lagrange multiplier l s X, Y, Z, t .: y Cp s u .
l t
y Ks u .
y Ks u .
2l Z2
2l X2
q
p
S
y Ks u .
2l Y2
w u s s t ; Bi . y m
s
s t .x d s r y rs. s 0
ss 1
in 0 - X - A; 0 - Y - B; 0 - Z - 1
0- t - t
f
s 10 a .
VERSIONS OF THE CONJUGATE GRADIENT METHOD
l X l Y
Ks u .
l Z
s0
at X s 0 and X s A
s0
at Y s 0 and Y s B
0- t - t 0- t - t
f
f
235
s 10 b . s 10 c .
l s0
at Z s 0
0- t - t
f
s 10 d .
qBi l s 0
at Z s 1
0- t - t
f
s 10 e .
for t s t f , in 0 - X - A; 0 - Y - B ; 0 - Z - 1 s 10 f .
l s0
GR AD IENT EQUATION In the process of obtaining the adjoint problem Eq. s 10., the following integral term is left: DD
Bi
J w Bix s y
t
B
H H H f
A
t s 0 Ys 0 Xs 0
wl s u `
y u . x Zs 1 D Bi s X, Y, t . dX dYdt ª 0 s 11 a .
where wl s u `
y u . x Zs 1 ’ l s X , Y, 1, t . w u ` s X , Y, t . y u s X , Y, 1, t . x
s 11b .
By assuming that Bi s X, Y, t . belongs to the Hilbert space of square integrable functions in the domain s 0, t f . = s 0, A. = s 0, B ., we can write w 4, 15x DD
Bi
J w Bi x s
t
B
H H H f
A
t s 0 Ys 0 X s 0
J9 w X , Y, t x D Bi s X , Y, t . dX dYdt
s 12 .
Therefore, by comparing Eqs. s 11 a. and s 12., we obtain the gradient equation for the functional as J9 w X , Y, t x s y w l s u `
y u . x Zs 1
s 13 .
ITER ATIVE PR OC ED UR E The iterative procedure of the conjugate gradient method w 3, 4, 6 ] 19x , as applied to the estimation of the unknown he at transfer coefficient, is given by Bi k q1 s X , Y, t . s Bi k s X , Y, t . q b
k
d k s X , Y, t .
s 14 .
where the superscript k denotes the number of iterations. The se arch step size b k is obtained by minimizing the functional Jw Bi k q1 x given by Eq. s 4. with respect to b k. By using a first-orde r Taylor series approximation for the estimated temperatures, the following expression results for the se arch
M. J. COLAC Ë O AND H. R. B. ORLANDE
236
step size:
b
k
H s
t
S
f
m
p
t s 0 ss 1
s
H
t
s t . y u s s t ; Bi k . D u s s t ; d k . dt
f
s 15 .
S
p
D u ss t ; dk .
t s 0 ss 1
2
dt
where D u s s t ; d k . is the solution of the sensitivity problem Eq. s 7. at the me asurement location r s , obtained by setting D Bi s X, Y, t . s d k s X, Y, t .. The direction of descent d k s X, Y, t . is a conjugation of the gradient direction given by Eq. s 13. with previous directions of descent. It is given in the following general form: d k s X , Y, t . s y J9 k s X , Y, t . q g
k ky 1 s
d
X , Y, t . q c
k qs
d
X, Y, t .
s 16 .
where g k and c k are conjugation coefficients. The superscript q in Eq. s 16. denotes the iteration number where a restarting strategy is applied to the iterative procedure of the conjugate gradient method. Restarting strategies were suggested for the conjugate gradient method of parameter estimation in order to improve its convergence rate w 14 x . Different versions of the conjugate gradient method can be found in the literature, depending on the form used for the computation of the direction of descent given by Eq. s 16. w 3, 4, 6 ] 19x . In the Fletcher-Reeves version the conjugation coefficients g k and c k are obtaine d from the following expressions w 3, 4, 6, 10 ] 17x : A
g
k
B
H H H s H H H
t
2
f
Xs 0 Ys 0 t s 0 t f B
A
Xs 0 Ys 0 t s 0
g c
k
0
w J9 k s X , Y, t . x dt dYdX
w J9
ky 1 s
s0
s0
2
X , Y, t . x dt dYdX
s 17a .
ks 0 s 17b .
k s 0, 1, 2, . . .
In the Polak-Ribiere version of the conjugate gradient method w 4, 7 ] 9, 14, 15, 18x the conjugation coefficients are given by
H H H s
g
A
k
B
t
f
Xs 0 Ys 0 t s 0
v w J9 k s X , Y, t . y J9
H H H A
B
t
f
Xs 0 Ys 0 t s 0
w J9
ky 1 s
ky 1 s
X , Y, t . x J9 k s X , Y, t .4 dt dYdX 2
X , Y, t . x dt dYdX s 18 a .
c
g k
0
s0
s0
ks 0 k s 0, 1, 2, . . .
s 18 b .
VERSIONS OF THE CONJUGATE GRADIENT METHOD
237
Based on a previous work by Beale w 19x , Powell w 14 x suggested the following expressions for the conjugation coefficients, which gives the so-called Powell-Beale version of the conjugate gradient method w 8, 14, 19 x : A
g
k
t
H H H s H H H B
f
X s 0 Ys 0 t s 0 t f A B
X s 0 Ys 0 t s 0
w J9 k s X , Y, t . y J9
w J9 k s X , Y, t . y J9
ky 1 s ky 1 s
X , Y, t . x J9 k s X , Y, t . dt dYdX X , Y, t . x d ky 1 s X , Y, t . dt dYdX
g A
c
k
B
H H H s H H H
t
f
X s 0 Ys 0 t s 0 t f A B
0
s0
ks0
s 19 a .
w J9
q q1 s
X , Y, t . y J9 q s X , Y, t . x J9 k s X, Y, t . dt dYdX
w J9
q q1 s
X, Y, t . y J9 q s X , Y, t . x d q s X , Y, t . dt dYdX
X s 0 Ys 0 t s 0
c
0
s0
ks0
s 19b .
In accordance with Powell w 14 x , the application of the conjugate gradient method with the conjugation coefficients given by Eqs. s 19. requires restarting when gradients at successive iterations tend to be nonorthogonal s which is a me asure of the local nonlinearity of the problem. and when the direction of descent is not sufficiently downhill. Restarting is performed by making c k s 0 in Eq. s 16.. The nonorthogonalit y of gradients at successive iterations is tested by using
ABS
tH
t
f
H H B
A
t s 0 Ys 0 Xs 0
0 0.2
t
J9
H H H B
f
ky 1 s
A
t s 0 Ys 0 Xs 0
X , Y, t . J9 k s X , Y, t . dX dYdt 2
w J9 k s X, Y, t . x dX dYdt
/ s 20 a .
where ABSs . denotes the absolute value. A nonsufficiently downhill direction of descent s i.e., the angle between the direction of descent and the negative gradient direction is too large . is identified if either of the following inequalities is satisfied: t
B
H H H f
A
t s 0 Ys 0 Xs 0
d k s X , Y, t . J9 k s X , Y, t . dX dYdt t
H H H
( y 1.2
f
B
A
t s 0 Ys 0 Xs 0
2
w J9 k s X , Y, t . x dX dYdt
s 20 b .
M. J. COLAC Ë O AND H. R. B. ORLANDE
238
or t
B
H H H f
A
t s 0 Ys 0 Xs 0
d k s X , Y, t . J9 k s X , Y, t . dX dYdt t
B
H H H
0 y 0.8
f
A
t s 0 Ys 0 Xs 0
2
w J9 k s X , Y, t . x dX dYdt
s 20 c .
We note that the coefficients 0.2, 1.2, and 0.8 appearing in Eqs. s 20. are empirical and are the same as those used by Powell w 14x . In the Powell-Beale version of the conjugate gradient method the direction of descent given by Eq. s 16. is computed in accordance with the following algorithm for k 0 1 w 14x : Step Step Step Step Step
1: Test the inequality Eq. s 20 a.. If it is true, set q s k y 1. 2: Compute g k with Eq. s 19 a.. 3: If k s q q1, set c k s 0. If k / q q 1, compute c k with Eq. s 19b .. 4: Compute the se arch direction d k s X, Y, t . with Eq. s 16 .. 5: If k / q q1, test the inequalities Eqs. s 20 b . and s 20 c .. If either is satisfied, set q s k y 1 and c k s 0. Then recompute the se arch direction with Eq. s 16 ..
The steepest descent method, with the direction of descent given by the negative gradient equation, would be recovered with g k s c k s 0 for any k in Eq. s 16.. Although simpler, the steepest descent method does not converge as fast as the conjugate gradient method w 4, 14 ] 19 x . We note that the conjugation coefficients g k given by Eqs. s 17a., s 18 a., and s 19 a. are equivalent for quadratic functionals because the gradients at different iterations are mutually orthogonal w 14, 15x . The iterative procedure of the conjugate gradient method given by Eqs. s 14. ] s 16. is applied for the estimation of the unknown he at transfer coefficient Bi s X, Y, t . until a stopping criterion based on the discrepancy principle is satisfied, as described below. STOPPING C R ITER ION We stop the iterative procedure of the conjugate gradient method when the functional given by Eq. s 4. becomes sufficiently small; that is, J w Bi k q1 s X , Y, t . x - «
s 21 .
If the me asurements are assumed to be free of experimental errors, we can specify « as a relatively small number. However, actual me asured data contain experimental errors, which will result in an unstable inverse problem solution as the estimated temperatures approac h those me asured. Such difficulty can be alleviated by utilizing the discrepancy principle w 3, 4, 6 ] 13x to stop the iterative process and to provide the conjugate gradient method with the needed regularization for a stable solution.
VERSIONS OF THE CONJUGATE GRADIENT METHOD
239
In the discrepancy principle, we assume that the inverse problem solution is sufficiently accurate when the difference between estimated and measured temperatures is of the order of the standard deviation s of the measurements. The tolerance « is then obtained from Eq. s 4. as
« s
1 2
S s 2t
f
s 22 .
C OMPUTATIONAL ALGOR ITHM The conjugate gradient method, as applied to the estimation of the unknown function Bi s X, Y, t ., can be suitably arrange d in the following computational algorithm by assuming that an estimate Bi k s X, Y, t . is available at iteration k: Step 1: Solve the direct problem given by Eqs. s 1. to obtain the estimated temperature s u s X, Y, Z, t .. Step 2: Check the stopping criterion given by Eq. s 21.. Continue if not satisfied. Step 3: Solve the adjoint problem given by Eqs. s 10. to obtain the Lagrange multiplier l s X, Y, Z, t .. Step 4: Compute the gradient of the functional J9 k s X, Y, t . from Eq. s 13.. Step 5: Compute the direction of descent d k s X, Y, t . with Eq. s 16.. In order to do so, the conjugation coefficients g k and c k of the conjugate gradient method versions of Fletcher-Reeves, Eqs. s 17a. and s 17b ., Polak-Ribere, Eqs. s 18 a. and s 18 b ., or Powell-Beale, Eqs. s 19 a. and s 19b ., can be used. Note that the Powell-Beale version requires restarting if the inequalities in Eqs. s 20 a. ] s 20 c . are satisfied. Step 6: Solve the sensitivity problem given by Eqs. s 7. to obtain D u s X, Y, Z, t . by setting D Bi s X, Y, t . s d k s X, Y, t .. Step 7: Compute the se arch step size b k from Eq. s 15.. Step 8: Compute the new estimate Bi k q1 s X, Y, t . from Eq. s 14., and return to step 1. R ESULTS AND D ISC USSION In order to compare the results obtained with the three versions of the conjugate gradient method described above, as applied to the estimation of the unknown heat transfe r coefficient Bi s X, Y, t ., we used a test problem based on practical data appearing in the cooling of continuously cast slabs by sprays w 11 x , involving the following dimensionless values: A s 1.5, B s 5, t f s 0.1, w s 0, f s 1, and u ` s 0. The direct problem was supposed linear, so that K s u . s C p s u . s 1 in Eqs. s 3 a. and s 3b .. The direct, sensitivity, and adjoint problems were solved with finite differences by using 11 = 11 = 41 grid points in the X, Y, and Z directions, respectively, and 160 time steps. This number of points and time steps were chosen by using a grid convergence analysis and by comparing the numerical solution obtained here with an analytic solution for the direct problem w 11x . The algebraic systems resulting from the discretization by finite differences were solved
240
M. J. COLAC Ë O AND H. R. B. ORLANDE
with Gauss-Seidel’s method with successive overrelaxation s SOR.. A red-black reordering w 23 x was applied in order to allow for the vectorization of the computer code. Such reordering resulted in a speed-up of 5.5 times over a nonvector version of the same code, on a Cray J-90 with four processors. We used simulated transient me asurements in the inverse analysis. These me asurements were generated from the solution of the direct problem by using an a priori assumed functional form for Bi s X, Y, t .. The measurements containing random errors were obtained by adding an error term to the errorless measurements resulting from the solution of the direct problem; that is,
m sm
exa
qv s
s 23 .
where m exa are the errorless measurements; v is a random variable with normal distribution, zero mean, and unitary standard deviation; s is the standard deviation of the me asurement errors, which is supposed constant; and m are the me asurements containing random errors. Two different levels of measurement errors considered here are s s 0 and 0.01 w 11x . In order to compare the three versions of the conjugate gradient method addressed in this paper under strict conditions, we considered the following function for generating the simulated measurements, involving exponential decays in the X and Y directions and a step variation in time: Bi s X , Y, t . s 6 e yX eyY ft s t . 2
2
s 24 a .
where ft s t . s
w
1 2
t ( s 1 r 30 . t 0 s 2 r 30 . s 1 r 30 . - t - s 2 r 30 .
s 24 b .
The exponential decays considered in Eq. s 24 a. are associated with the nonuniform wetting of the surface by the cooling sprays w 11x . By examining Eqs. s 10 f . and s 13 ., we can notice that the gradient of the functional is null at the final time t s t f . Therefore the initial guess used for Bi s X, Y, t . at t s t f is never changed by the iterative procedure of the conjugate gradient method. As a result, oscillations may appear in the solution in the neighborhood of the final time if the initial guess is too different from the exact solution. Two different approaches are considered in this paper in order to avoid such instabilities: s 1. The exact value of the function at the final time is assumed a priori known and is used as the initial guess for the iterative procedure; and s 2. a final time larger than that of interest is used in the analysis, so that the effects of the initial guess are not noticeable in the time domain that the solution is sought. Table 1 summarizes the cases examined here, for the comparison of the different versions of the conjugate gradient method. For cases 1, 2, and 3 the exact value of the function at the final time was used as the initial guess for the iterative procedure, while for cases 4 and 5 the constant value Bi 0 s X, Y, t . s 1 was used for the initial guess. The final time was taken as 0.125 for cases 4 and 5, but the solution is sought in the time interval 0 ( t ( 0.1. For all cases examined here, we
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241
considered available 161 transient measurements per sensor in the time domain 0 ( t ( 0.1 w 11 x . For cases 4 and 5, where the final time was larger than that of interest, the number of me asurements was then increased accordingly. Cases 1, 3, 4, and 5 involve the use of 16 sensors in the inverse analysis, located at Z s 0.975 and at the grid formed by the X positions 0.0, 0.45, 1.05, and 1.5, and by the Y positions 0.0, 1.5, 3.5, and 5.0. This is the minimum number of sensors that can provide accurate estimates for the unknown function w 11x . A quite large number of sensors is considered for case 2, where ideally, one sensor is located at each point of the finite difference grid used for the solution, at Z s 0.975. We note that the numerical solutions of the direct, sensitivity, and adjoint problems by finite differences require the value of Bi s X, Y, t . at all grid points at Z s 1. Hence, for the cases involving only 16 me asurements, an interpolation is needed in order to obtain values for the points where the function is not estimated, i.e., those points not coinciding with the sensor’s locations. The subroutine DBS2IN of the IMSL w 24 x was used for the interpolation, when required. Table 1 presents the results for the root-mean-square s RMS. error, eRMS , and for the average rate of reduction of the functional with respect to the number of iterations, r, obtained with the three versions of the conjugate gradient method, for each of the cases examined. The RMS error is defined as eRMS s
X IS 1
I
p
S
p
2 w Bi est s X s , Ys , t i . y Bi e xas X s , Ys , t i . x
s 25 .
is 1 ss 1
where I denotes the number of transient me asurements used per sensor in 0 ( t ( 0.1, while the subscripts est and exa refer to the estimated and exact functions, respectively. Table 1 shows that the smallest values of the RMS errors for all cases considered were obtaine d with the Powell-Beale version of the conjugate gradient method. On the other hand, the largest values of eRMS were generally obtained with the Polak-Ribiere version, except for cases 2 and 5, where the three versions have the same value for the RMS error. The values of eRMS for the Fletcher-Reeves version are identical to those obtained with the Powell-Beale version, except for case 1. We can also notice in Table 1, by comparing cases 1 and 3, as well as cases 4 and 5, the increase in the RMS error resulting from the presence of random
Table 1. Test cases with RMS errors of the estimated functions and average rates of reduction of the functional Fletcher-Reeves Case 1 2 3 4 5
S
s
16 121 16 16 16
0.00 0.00 0.01 0.00 0.01
t
f
0.1 0.1 0.1 0.125 0.125
Polak-Ribiere
Powell-Be ale
eR M S
r
e R MS
r
e R MS
r
0.4 0.1 0.5 0.3 0.5
1.5 2.5 1.2 2.6 2.1
0.8 0.1 0.9 0.5 0.5
0.7 2.5 0.6 2.3 2.0
0.2 0.1 0.5 0.3 0.5
2.4 2.5 1.2 2.8 2.1
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M. J. COLAC Ë O AND H. R. B. ORLANDE
Figure 1 a. Estimated functions for case 1.
errors in the measured data. A comparison of cases 1 and 2 reveals a decrease in the RMS error when the re adings of more sensors are used in the analysis. Such is the case because the use of more sensors allows for a better spatial resolution of the estimated function. Table 1 shows that basically no accuracy is lost by using a time interval larger than that of interest, with an initial guess different from the exact value at the final time. In fact, even some reduction on the values of the RMS error can be noticed by comparing cases 1 and 4, as well as cases 3 and 5. Figure s 1 a and 1b illustrate the results obtained for the estimated functions for cases 1 and 3, respectively, with the three versions of the conjugate gradient method, for the positions X s 0.45, Y s 0.0 and X s 0.45, Y s 1.5. Figure 1 a
Figure 1 b. Estimated functions for case 3.
VERSIONS OF THE CONJUGATE GRADIENT METHOD
243
shows the more accurate estimated functions obtained with the Powell-Beale version of the conjugate gradient method for case 1. For case 3 the results obtained with the Fletcher-Reeve and Powell-Beale versions are equivalent, as can be noticed in Figure 1b. Figure s 1 a and 1 b clearly show the very poor results obtained with the Polak-Ribiere version of the conjugate gradient method for cases 1 and 3, respectively. The qualitative comparison of the methods presented in Figures 1 a and 1 b is in accordance with the values of the RMS error shown in Table 1. Figure s 2 a ] 2 e present the reduction of the objective functional, Eq. s 4., with the number of iterations for cases 1 ] 5, respectively, obtained with the three versions of the conjugate gradient method considered in this paper. The tolerance prescribed for the stopping criterion for the cases involving errorless measurements s s s 0.0. was 10 y6 . For those cases involving measurements with random errors s s s 0.01. the tolerance was obtaine d from the discrepancy principle, Eq. s 22 ., as 8 = 10 y5 for case 3 and as 1 = 10 y4 for case 5. However, we note that the stopping criterion with such tolerances was not satisfied by some methods, depending on the case under study. For case 2 the iterative procedures of the three versions were stopped when the maximum number of iterations s 50. imposed to the program was re ached, but the final values for the functionals were quite close to the prescribed tolerance. For cases 1, 3, and 4 the Fletcher-Reeves and the Polak-Ribiere versions of the conjugate gradient method exhibited an anomalous increase of the functional, when the iterative procedure was stopped. This same behavior was observed with the Polak-Ribiere version in case 5. This behavior was prejudicial to the accuracy of the estimated function when the minimum value re ached by the functional was much larger than the stopping criterion tolerance, such as in case 1, resulting in large RMS errors. On the other hand, the increase in the functional did not result in significant loss of accuracy of the solution, when its value was about to satisfy the stopping criterion tolerance, such as for the Fletcher-Reeves version in cases 3 and 4.
Figure 2 a. Reduction of the objective functional Eq. s 4. for case 1.
244
M. J. COLAC Ë O AND H. R. B. ORLANDE
Figure 2 b. Reduction of the objective functional Eq. s 4. for case 2.
The values of the average rate of reduction of the functional with respect to the number of iterations are also shown in Table 1. Such average rate was obtained from the following approximatio n of the curves presented in Figures 2 a ] 2 e, corresponding to cases 1 ] 5, respectively: J w Bi k s X , Y, t . x s Ckyr
s 26 .
where C is a constant depending on the data, k is the number of iterations, and r is the average rate of reduction of the functional. The analysis of Table 1 reveals that the Powell-Beale version of the conjugate gradient method generally has the largest average rates of reduction of the
Figure 2 c. Reduction of the objective functional Eq. s 4. for case 3.
VERSIONS OF THE CONJUGATE GRADIENT METHOD
245
Figure 2 d. Reduction of the objective functional Eq. s 4. for case 4.
functional. As for the RMS error, the Fletcher-Reeves and Powell-Beale versions are equivalent in terms of the rate of reduction of the functional for cases 2 ] 5, although the Fletcher-Reeves rate is somewhat lower than the Powell-Beale rate for case 4. Polak-Ribiere’s rates are generally the lowest in Table 1, except for case 2, where the three versions show similar values for the rates of reduction of the functional. The results presented in Table 1 clearly reveal that the average rate of reduction of the functional is affected by the presence of random errors in the me asured temperatures, by the number of sensors used in the inverse analysis, and
Figure 2 e. Reduction of the objective functional Eq. s 4. for case 5.
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M. J. COLAC Ë O AND H. R. B. ORLANDE
by the initial guess used for the iterative procedure of the conjugate gradient method. A comparison of cases 1 and 3, as well as cases 4 and 5, shows a decrease of the rate of reduction of the functional, when me asurements containing random errors are used in the analysis instead of errorless me asurements. On the other hand, the use of more sensors in the inverse analysis resulted in the increase of such rate, as can be noticed by comparing cases 1 and 2. A similar increase in the rate of reduction of the functional is observed when a time interval larger than that of interest is used, together with an initial guess different from the exact value at the final time. This is apparent from the comparison of cases 1 and 4, as well as cases 3 and 5, in Table 1. The foregoing results show that, for the cases examined in this paper, the Powell-Beale version of the conjugate gradient method is the best among those considered in the analysis. The Powell-Beale version did not present any anomalous increase of the functional, giving an indication of its superior robustness. Also, the largest average rates of reduction of the functional were obtained with such version, and generally, it resulted in the smallest values of the RMS error for the estimated functions. The version of Fletcher-Reeves shows similar results to those obtained with the Powell-Beale version, in terms of RMS error and rate of reduction of the functional. However, the Fletcher-Reeves version presented an anomalous increase of the functional for cases 1, 3, and 4, which may result in large values for the RMS error, as for case 1. Such anomalous behavior was also observed with the use of the Polak-Ribiere version, which generally resulted in the largest RMS errors in Table 1, except for cases 2 and 5. We note that the above general conclusions were also obtained by considering in the analysis other functional forms for Bi s X, Y, t ., including functions with no variation in the Y direction s exponential decay only in the X direction. and functions with a triangular variation in time. Also, the results were not changed for nonlinear cases involving temperature-dependent thermal conductivities w 25x . The heat transfer coefficient at the surface of a plate cooled by sprays is generally a function of the surface temperature w 20 ] 22 x . Although the temperature dependence of the he at transfer coefficient was not taken into account in the formulation w 9x , it can also be estimated with the present approach, since the transient temperature field in the plate is estimated as part of the solution of the inverse problem with the conjugate gradient method. We illustrate the estimation of the spatial and temperature variations of the he at transfer coefficient, by generating simulated measurements with the following functional form: Bi s X , Y, u . s 6 e ys 0 .5 X . fu s u . 2
s 27a .
where f 0 s u . s 51.338 y 893.278 u
q 8566.53 u
q154,280 u
4
y 333,217u
q264,504u
8
y 90,201.7u
5 9
2
y 46,264.9u
q 47 ,539u
6
q 13,476.3 u
3
y 445,692 u 10
7
0.103 - u - 1 s 27b .
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The function given by Eq. s 27b . was based on the experimental data of Stewart et al. w 22x . Figure s 3 a and 3b present the results obtaine d with errorless measurements s s s 0. and with me asurements containing random errors s s s 0.01., respectively, for the estimation of Bi s X, Y, u . given by Eqs. s 27a. and s 27b .. The results were obtained by using the Powell-Beale version of the conjugate gradient method with an initial guess Bi 0 s X, Y, u . s 1 and by considering a final time of 0.125; but only the results obtained for 0 ( t ( 0.1 are presented in Figures 3 a and 3b, in order to eliminate the instabilities inherent in the conjugate gradient method in the neighborhood of the final time. Figures 3 a and 3b show that quite accurate estimates can be obtained with the present solution approach, for the estimation of
Figure 3 a. Estimation of Bi s X, Y, u . with errorless me asurements s s s 0..
Figure 3b. Estimation of Bi s X, Y, u . with measurements containing random errors s s s 0.01..
248
M. J. COLAC Ë O AND H. R. B. ORLANDE
the spatial and temperature dependencies of the heat transfer coefficient, for both errorless measurements and me asurements with random errors. C ONC LUSIONS In this paper we present a comparison of the Fletcher-Reeves, Polak-Ribiere, and Powell-Beale versions of the conjugate gradient method, as applied to the estimation of the he at transfe r coefficient at the surface of a plate. The present inverse problem is solved as a function estimation approach, by assuming that no information is available regarding the functional form of the unknown. Simulated transient temperature measurements of several sensors located inside the plate were utilized for the estimation of the spatial and timewise variations of the he at transfe r coefficient. Different cases are examined here, involving different numbers of sensors, levels of me asurement error, and initial guesses for the iterative procedure . Among the three versions tested for the conjugate gradient method, the one due to Powell and Beale appears to be the best, as applied to the cases examined in this paper. That version did not present the anomalous increase of the functional observed with the use of the other versions, and its average rates of reduction of the functional were the largest. As a result, generally, the smallest values for the RMS error of the estimated functions were obtaine d with the Powell-Beale version of the conjugate gradient method. The average rate of reduction of the functional was affected by the presence of random errors in the me asured temperatures, number of sensors used in the inverse analysis, and initial guess used for the iterative procedure of the conjugate gradient method. The use of measurements containing random errors resulted in a decrease of the rate of reduction of the objective functional. On the other hand, such rate increased when the re adings of more sensors were considered for the inverse problem solution, as well as when a time interval larger than that of interest, with an initial guess different from the exact value at the final time, was used in the analysis. The present function estimation approach was also shown to be able to recover quite accurately the dependence of the he at transfer coefficient with respect to the plate surface temperature. R EFER ENC ES 1. J. Hadamard, Lectures on Cauchy’ s Problem in Linear Partial Differential Equations, Yale University Press, New Haven, CT, 1923. 2. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problem s, V. H. Winston & Sons, Washington, DC, 1977. 3. O. M. Alifanov, Solution of an Inverse Problem of Heat Conduction by Iteration Methods, J. Eng. Phys., vol. 26, no. 4, pp. 471 ] 475, 1974. 4. O. M. Alifanov, In verse Heat Transfer Problems, Springer-Verlag, New York, 1994. 5. J. V. Beck, B. Blackwell, and C. R. St. Clair, In v erse Heat Conduction: Ill-Posed Problems, Wiley Interscience, New York, 1985. 6. E. A. Artyukhin, Iterative Algorithms for Estimating Temperature-Dependent Thermophysical Characteristics, in N. Zabaras, K. Woodburry, and M. Raynaud s eds.., First Int.
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