Identification of thermal conductivity by means of the gradient method and the BEM

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IDENTIFICATION OF THERMAL CONDUCTIVITY BY MEANS OF THE GRADIENT METHOD AND THE BEM

Ewa Majchrzak^{1, 2}, Mirosław Dziewoński¹, Marek Jasiński¹

1 Department for Strength of Materials and Computational Mechanics Silesian University of Technology, Gliwice, Poland

2 Institute of Mathematics and Computer Science, Czestochowa University of Technology Czestochowa, Poland, email: ewa.majchrzak@polsl.pl

Abstract. In the paper the application of the gradient method coupled with the boundary element method for numerical solution of the inverse parametric problem is presented.

On the basis of the knowledge of temperature field in the domain considered the temperature dependent thermal conductivity is identified. The non-steady state is considered and 1D problem is discussed. In the final part of the paper the results of computations are shown.

1. Direct problem

The following boundary initial problem is considered

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )

0

, ,

0 :

0 : , ,

: , , 0

0 : ,

b

T x t T x t

x L c T

t x x

T x t

x q x t T q

x T x t

x L q x t T

x

t T x t T

∂ ∂  ∂ 

< < = λ 

∂ ∂  ∂ 

= = −λ ∂ =

∂

= = −λ ∂ =

∂

= =

(1)

where c is the volumetric specific heat, λ(T) is the thermal conductivity, T, x, t denote temperature, spatial co-ordinate and time, qb is the boundary heat flux and T0 is the initial temperature.

In order to solve the problem (1) by means of the boundary element method the Kirchhoff transformation is introduced [1, 2]

( ) ( )

0

d

T

U T = λ µ µ

∫

⁽²⁾

(2)

and then the governing equations (1) take a form

( ) ( ) ( )

( ) ( )

( )

2 2

0

, ,

0 :

0 : , ,

: , , 0

0 : ,

b

U x t U x t

x L c T

t x

U x t

x q x t q

x U x t

x L q x t

x

t U x t U

∂ ∂

< < = λ

∂ ∂

= = −∂ =

∂

= = −∂ =

∂

= =

(3)

where U0=U(T0). We assume that

( )

T b1 b T2 b T3 ²

λ = + + (4)

where b1, b2, b3 are the coefficients.

If the direct problem is considered then all geometrical and thermophysical parameters appearing in the mathematical model are known.

2. Sensitivity coefficients

In this chapter the sensitivity analysis of function U(x,t) (c.f. governing equa- tions (3)) with respect to the coefficients be, e=1,2,3 (c.f. equation (4)) is discussed. Here the direct approach is applied [3, 4]. Additionally, it is assumed that the volumetric specific heat c is a constant value.

Differentiation of equations (3) with respect to be, e=1,2,3 gives

( ) ( ) ( ) ( ) ( )

( ) ( )

( )

2 2

0

, , ,

0 :

, ,

0 :

, ,

: 0

0 : ,

e e e

b

e e e

e e

U x t T U x t U x t

x L c T

t b b x x b

q x t U x t q

x b x b b

q x t U x t

x L

b x b

U x t U

t b b

∂  ∂ λ ∂ ∂ 

∂ ∂

< <  = + λ  

∂  ∂  ∂ ∂ ∂  ∂ 

∂ ∂ ∂  ∂

= = −  =

∂ ∂  ∂  ∂

∂ ∂ ∂ 

= = −  =

∂ ∂  ∂ 

∂ ∂

= =

∂ ∂

(5)

Because (c.f. equation (3))

( ) ( ) ( )

2 2

, ,

U x t c U x t

x T t

∂ ∂

∂ =λ ∂ (6)

(3)

so

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2 2

, , ,

0 :

0 : , 0

: , 0

0 : , 0

e e

e

Z x t Z x t T c U x t

x L c T

t x b T t

Z x t

x x

Z x t x L

x

t Z x t

∂ ∂ ∂ λ ∂

< < = λ +

( ) ( ) ( ) ( )

1 2 3

1 1

2 3

2

1 1 d

2 d

e

e e

e

e e

T b b

T Z T Z

b T T

b b T Z T T Z T

T T T

−

− −

∂ λ = + + =

∂ λ λ

+ + = λ +

λ λ

(12)

Finally, the equations (7) can be written in the form

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 2

1

, ,

0 :

d ,

1 , ,

d

e e

Z x t Z x t

x L c T

t x

T U x t

c Z x t T x t

T T T t

−

∂ ∂

< < = λ +

∂ ∂

 λ ∂

+ 

   

λ λ  ∂

(13)

(4)

( ) ( ) ( )

0 : , 0

: , 0

0 : , 0

e

x W x t

x L W x t

t Z x t

= =

where W x te

( )

^, = −∂Ze

( )

, , , , d

, , , , d , , , , d d

f

t x L

f f

f

t x

t x L

f f

t x

L t L

f f f f

f t

F t a F x t t J x t t

a J x t t F x t t

F x t t F x t x a Q x t F x t t x t

−

=

− −

 

ξ + ξ 

 

 

 

= ξ  +

 

 

ξ + ξ

∫

∫ ∫ ∫

(16)

where af is the mean value of thermal diffussivity for interval time t^f⁻¹,t^f , ξ is the observation point. In equation (16) ^F^*

(

^ξ^{, , ,}^{x t t}^f

)

is the fundamental solution [1, 5, 6]

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( )

( ) ⁽ ( ⁾ )

2

* 1

, , , exp

2 4

f

f f f f

F x t t x

a t t a t t

 − ξ 

 

ξ = −

−

 

π −  

(17)

( )

* , , ,^f

J ξ x t t is the heat flux resulting from the fundamental solution

( )

( ) ⁽ ( ⁾ )

2

*

, , , 3/ 2exp

4 4

f

f f f f

x x J x t t

a t t a t t

 − ξ 

− ξ  

ξ = −

−

 

 

π −   

(18)

and

( ) ( )

^,

, F x t

J x t

x

= −∂

∂ (19)

In numerical realization of the BEM the constant elements with respect to time are introduced

( ) ( )

1 , ,

, :

, ,

f

f f

f

F x t F x t

t t t

J x t J x t

−  =

 

∈   = (20)

and then the equation (16) takes a form

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1

*

0

* * 1 1

0 0

1 *

0

, , , , , d

, , , , d , , , , d

, , , , d d

f

t x L

f f f

f

t x

t x L L

f f f f f

f

t x

L t

f f

f

t

F t a J x t F x t t t

a F x t J x t t t F x t t F x t x

a Q x t F x t t t x

−

=

− −

=

−

 

ξ + ξ  =

 

 

 

ξ + ξ

 

 

 

+  ξ 

 

 

∫

∫ ∫

(21)

or

( ) ^{( )} ( ) ^{( )} ( )

( ) ( ) ^{( )} ( ) ^{( ) ( )}

, , , , 0 0,

, , , 0 0,

f f f

f f

F t g L J L t g J t

h L F L t h F t P R

ξ + ξ − ξ =

ξ − ξ + ξ + ξ (22)

where

( ) ( )

1

, * , , , d

f

t

f f

t

g x a F x t t t

−

ξ =

∫

ξ ⁽²³⁾

(6)

and

( ) ( )

1

, * , , , d

f

t

f f

t

h x a J x t t t

−

ξ =

∫

ξ ⁽²⁴⁾

while

( )

^*

(

¹

) (

¹

)

0

, , , , d

L

f f f

P ξ =

∫

F ξ x t t ⁻ F x t ⁻ x ⁽²⁵⁾ and

( ) (

¹

) ^{( )}

0

, , d

L

R ξ =

∫

Q x tf⁻ g ξ x x ⁽²⁶⁾

It should be pointed out that the integrals (23), (24) can be calculated in analytical way [1, 6].

For ξ →0⁺ and ξ →L⁻ one obtains the following system of equations

( ) ^{( )} ( ) ^{( )} ( ) ( ) ( ) ( ) ( ) ^{( ) ( )}

0, 0, , 0, 0 0,

0 , , 0 , 0 0, 0 0

, , , , 0 0,

, , , 0 0,

f f f

f f

f f f

f f

F t g L J L t g J t

h L F L t h F t P R

F L t g L L J L t g L J t

h L L F L t h L F t P L R L

+ +

− −

 + − =

 − + +



+ − =



− + +



(27)

which can be written in the matrix form

( ) ( ) ( )

( ) ^{( ) ( )} ^{( ) ( )}

11 12 11 12

21 22 21 22

0, 0, 0 0

, ,

f f

J t F t P R

G G H H

G G J L t H H F L t P L R L

     + 

  =  + 

      +

       

(28)

This system of equations allows to find the “missing” boundary vales of F or J.

The values of function F at the internal points ξ are calculated using the formula (c.f. equation (22))

( ) ^{( )} ( ) ^{( )} ( )

( ) ( ) ^{( )} ( ) ^{( ) ( )}

, , , , 0 0,

, , , 0 0,

f f f

f f

F t h L F L t h F t

g L J L t g J t P R

ξ = ξ − ξ −

ξ + ξ + ξ + ξ (29)

4. Gradient method of inverse problem solution

It is assumed that the coefficients b1, b2, b3 in equation (4) are unknown. To solve the inverse problem formulated, the additional information is necessary. Let

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(

^,

)

^, ^{1, 2,...,} ^, ^{1, 2,...,}

f f

di d i

T =T x t i= M f = F (30)

are the known (measured) temperatures at the points xi for times t^f. The following least squares criterion has been taken into account [4]

( )

²

1 1

M F

f f

i di

i f

S U U

= =

=

∑∑

− ⁽³¹⁾

where ^Uⁱ^f ⁼^{U x t}

(

ⁱ^, ^f

)

is the calculated value of function U at the point xi for time t^f, U_{d i}^f if is the value of function U corresponding to the measured value of tem- perature T_{d i}^f at the point xi for time t^f.

The necessary condition of optimum of function S leads to the following system of equations

( )

1 1

2 0, 1, 2,3

k

l l

M F f

f f i

i di

i f

l i b b

U

S U U l

b = = b =

∂ = − ∂ = =

∂

∑∑

∂ ⁽³²⁾

where b for k_l^k =0 are the arbitrary assumed values of parameters bl, while for k>0 they result from the previous iteration.

Function ^Uⁱ^f ⁼^{U x t}

(

ⁱ^, ^f

)

is expanded into the Taylor series

( )

³

(

¹

)

1 ^k

e e

k f

f f i k k

i i e e

e e b b

U U U b ⁺ b

= =

= + ∂ −

∑

∂ λ ⁽³³⁾

this means

( )

³

( )(

¹

)

1

f f k f k k

i i i e e e

e

U U Z b ⁺ b

=

= +

∑

− ⁽³⁴⁾

where

( )

k

e e

k f

f i

i e

e b b

Z U

=

=∂

∂ λ (35)

are the sensitivity coefficients.

Putting (34) into (32) one has (l=1,2,3)

( )

³

( ) (

¹

) ( )

1 1 1

0

M F f k f k k k f f k

i i e e e d i i l

i f e

U Z b ⁺ b U Z

= = =

 

+ − − =

 

 

∑∑ ∑

⁽³⁶⁾

(8)

or

( ) ( ) ( ) ( ) ( )

3

1

1 1 1 1 1

M F f k f k k k M F f f k f k

i e i l e e d i i i l

i f e i f

Z Z b ⁺ b U U Z

= = = = =

 

− =  − 

∑∑∑ ∑∑

⁽³⁷⁾

where l=1,2,3, while k=0,1,...,K is the number of iteration.

The system of equations (37) can be written in the matrix form

( )

^Z^T ^k^{Z b}^k ^k⁺¹⁼

( )

^Z^T ^k^{Z b}^k ^k⁺

( ) (

^Z^T ^k ^U^d⁻^U^k

)

⁽³⁸⁾

5. Results of computations

The plate of thickness L=0.05m has been considered. On the left surface x=0 the Neumann condition qb=9.2⋅10⁵ W/m² is assumed, on the right surface x =L the zero heat flux q(L,t)=0 has been accepted. Initial condition T0=100°C has been also given (equations (1)). Additionally, it was assumed that c=4⋅10⁶ J/(m³K) and b1=52, b2=−0.02, b3=−0.00001 (equation (4)).

Fig. 1. Heating curves

The basic problem has been solved by means of the boundary element method.

The domain has been divided into 100 internal cells and ∆t=1s. In Figure 1 the heating curves at the points 1 − x1=0, 2 − x2=0.01m and 3 − x3=0.02m are

(9)

shown. Figures 2, 3, 4 illustrate the courses of sensitivity functions at the same points.

Fig. 2. Sensitivity function Z1

Fig. 3. Sensitivity function Z₂

Next, the inverse problem has been considered. The parameters b1, b2, b3 have been identified under the assumption that b₁⁰=51, b2⁰= −0, 017, b3⁰= −0, 000018.

(10)

In the first variant of computations only one heating curve marked by 1 in Figure 1 is taken into account (curves marked by 1 in Figures 5, 6, 7), while in the second variant of computations three heating curves (Figure 1) are taken into account (curves marked by 3 in Figures 5, 6, 7).

Fig. 4. Sensitivity function Z3

Fig. 5. Identification of b1

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It is visible, that the iteration process is convergent, but even for the initial values of parameters b₁⁰, b₂⁰ and b₃⁰ close to the exact solution the number of iterations is very big. The effectiveness of the algorithm proposed considerably improves when three heating curves have been taken into account.

(12)

Summing up, the algorithm proposed constitutes the effective tool of such inverse problem solution but the number of points for which the temperature course is known has an essential effect on the number of iterations.

Acknowledgement

This work was funded by Grant No NN 507 359 233.

References

[1] Mochnacki B., Suchy J.S., Numerical methods in computations of foundry processes, PFTA, Cracow 1995.

[2] Majchrzak E., Boundary element method in heat transfer, Publ. of the Czest. Univ. of Techn., Czestochowa 2001 (in Polish).

[3] Kleiber M., Parameter sensitivity in nonlinear mechanics, J. Willey & Sons Ltd., London 1997.

[4] Kurpisz K., Nowak A.J., Inverse thermal problems, Computational Mechanics Publications, Southampton, Boston 1995, 259-298.

[5] Ingham D.B., Yuan Y., The boundary element method for solving improperly posed problems, Computational Mechanics Publications, Southampton 1994.

[6] Brebbia C.A., Dominguez J., Boundary elements, an introductory course, Computational Mechanics Publications, McGraw-Hill Book Company, London 1992.

Identification of thermal conductivity by means of the gradient method and the BEM

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