Mathematical and numerical modeling of inverse …bulletin.incas.ro/files/danailachiravol_6_iss_4.pdf · Mathematical and numerical modeling of ... A boundary value problem for

INCAS BULLETIN, Volume 6, Issue 4/ 2014, pp. 23 – 39 ISSN 2066 – 8201

Mathematical and numerical modeling of inverse heat

conduction problem

Sterian DANAILA*,1

, Alina-Ioana CHIRA2

*Corresponding author

*,1

Faculty of Aerospace Engineering, “POLITEHNICA” University of Bucharest

Polizu no.1-6, RO-011061, Bucharest, Romania

[email protected] 2INCAS – National Institute for Aerospace Research “Elie Carafoli”

B-dul Iuliu Maniu 220, Bucharest 061126, Romania

[email protected]

DOI: 10.13111/2066-8201.2014.6.4.3

Abstract: The present paper refers to the assessment of three numerical methods for solving the

inverse heat conduction problem: the Alifanov’s iterative regularization method, the Tikhonov local

regularization method and the Tikhonov equation regularization method, respectively. For all

methods we developed numerical algorithms for reconstruction of the unsteady boundary condition

imposing some restrictions for the unsteady temperature field in the interior points. Numerical tests

allow evaluating the accuracy of the considered methods.

Key Words: direct problem, inverse problem, Alifanov’s regularization, Tikhonov regularization,

conjugate gradient, heat conduction.

1. INTRODUCTION

A boundary value problem for a partial differential equation is characterized by imposing a

valid equation within a given domain and the initial and boundary conditions on the

boundary of the domain. Initial and boundary conditions are formulated to identify a given

solution, from the multitude of possible solutions of the equation. In these circumstances,

there is the concept of “well posed problem” (Hadamard, [1]).

A well-posed problem actually provides the stability of the solution with respect to

small perturbations of the data, making the solution to be valid even in case of small

uncertainties of the problem data. Formulating a well-posed problem depends on the type of

the problem (elliptic, parabolic or hyperbolic). Consequently, the solution and the problem

are considered as belonging to a fully defined functional space. Consequently, a given

problem can be well-posed in a particular functional space, but may be ill-posed in another

functional space.

In a conditional well-posed problem, in the Tikhonov sense, we will have to determine a

solution that belongs to a certain class of solutions. Restricting the class of admissible

solutions for a given problem, it is possible to formulate a well-posed problem, even if it

appears to be an ill-posed problem in the classical, Hadamard sense.

From the physical point of view, inverse problems are characterized primarily by a lack

of information needed to solve as a direct problem. This lack of information has to be

compensated; consequently in solving inverse problems additional information must be

Sterian DANAILA, Alina-Ioana CHIRA 24

INCAS BULLETIN, Volume 6, Issue 4/ 2014

formulated so that we can determine a unique solution. Inverse problems can be classified in

respect to this additional necessary information.

In direct thermal conduction problem, the distribution of the temperature inside a body

is determined in respect to given initial and boundary conditions imposed on the body

surface. However, in practical applications the boundary conditions are unknown along the

body surface (or on some portion of the surface), but we can measure the temperatures in a

number of internal points. Inverse conduction problem consists of determining the heat flux

and temperature distribution inside a body if the history of change in body temperature

measured in one or more locations within the body is given. Note that because the inverse

problem “extrapolatesˮ the measured values within the body to its surface, even small

uncertainties of measurements can be amplified leading to important oscillating values on

the surface. Briefly, in direct problem, the causes are known and the effects are calculated,

while in the inverse problem a model to reconstruct an input from the corresponding output

should be used [2].

If the existence of a solution for an inverse heat transfer problem may be physically

argued, the uniqueness of the solution of inverse problems can be mathematically proved

only for some special cases [3, 4]. Moreover, inverse problems are extremely sensitive to the

initial distributions of the data. So sensitive are these problems, that even minute errors in the

data can wildly affect the computed solution [5, 6]. Generally, an ill-posed inverse problem

is solved as an approximate well-posed problem supposing the solution estimation in the

least squares sense.

Currently, several techniques of solving the inverse problems are proposed in literature

[1]). The present paper presents the solution of the inverse one-dimensional conduction

problem of estimating the right side unsteady boundary condition by using two techniques:

conjugate gradient method with adjoint problem for gradient function estimation and

Tikhonov regularization for hyperbolization of the heat conduction equation, respectively.

We examine the accuracy of both approaches by using transient simulated measurements of

several sensors located inside the domain. The inverse problem is solved for different

functional forms of the unknown boundary condition, including those containing sharp

corners and discontinuities, which are the most difficult to be recovered by an inverse

analysis. For both methods the mathematical and numerical formulations are presented and

finally the numerical results are comparatively discussed.

2. INVERSE PROBLEM FORMULATIONS

The physical problem considered here is the one-dimensional heat conduction in a solid with

constant proprieties. The mathematical formulation of this direct problem in dimensionless

form is given by:

2

20

T T

t x

, 0 x l , 0 f

t t . (1)

In order to formulate a well-posed problem, for the parabolic equation (1) the following

boundary and initial conditions are attached:

( , ) ( )T l t t , 0 ft t , (2)

(0, ) 0T

tx

, 0 f

t t , (3)

25 Mathematical and numerical modeling of inverse heat conduction problem


( ,0) 0T x , 0 x l . (4)

In the inverse problem the function ( )t , representing the temperature on the right

boundary have to be calculated in order to obtain the measured values in the point *x x :

* *( , ) ( )T x t T t , *0 x l . (5)

Such measurements may contain random errors, but all the other quantities appearing in

problem are considered to be known with sufficient degree of accuracy. We assume that no

information is available regarding the functional form of the unknown ( )t , except that it

belongs to the space of square integrable functions in the domain 0f

t t .

3. CONJUGATE GRADIENT METHOD

The first presented method is called Alifanov’s iterative regularization method that is an

infinite dimensional analog of the Conjugate Gradient method [1, 2]. In the implementation

of Alifanov’s method one has to solve the direct problem, the sensitivity problem, the adjoint

problem and to use an optimization algorithm. To avoid the contamination of the estimated

function by measurement errors one can use a stopping criterion for the iteration procedure

that takes into account the level of noise in experimental data.

We assume that the unknown function ( )t in equation (2), belongs to the Hilbert space

of square-integrable functions in the time domain:

0

( )d

ft

t t ,. (6)

The solution of the inverse problem is the function for which the functional

* * 2

0

( ) ( ( , ; ) ( )) d

ft

I T x t T t t (7)

is minimized under the constraint imposed by the direct problem. In iterative gradient

methods, the minimizing sequence ( ) , 1,2,k k K can be constructed by the rule:

( 1) ( ) ( ) ( )( ) ( ) ( )k k k kt t d t , (8)

where ( )kd is the descend direction and ( )k is the descend parameter.

The descend direction, ( )( )kd t depends on gradient of the functional (7), ( )( )kI :

( ) ( ) ( ) ( 1)( ) ( ) ( )k k k kd t I d t . (9)

Considering the unknown function in the space of square integrable functions, the variation

of the functional (7), ( )I , can be expressed as:

0

( ) d

ft

I I t , (10)



where ( )t is a small variation of ( )k . To evaluate the gradient

( )( )kI , the chosen

method, namely the adjoint problem, will be presented later in the paper.

The conjugation coefficient )(k is estimated using the Fletcher-Reeves method [21]:

12

( )

( ) 0

12

( 1)

0

( ) d

( ) d

k

k

k

I t

I t

, ,...2,1k , .0)0( (11)

The step size )(k is determined by minimizing the functional ( )I given by equation

(7) with respect to )(k , that is,

( ) ( )

2( 1) * ( 1) *

0

min ( ) min ( , ; ) d

f

k k

t

k kI T x t T t

( )

2* ( ) ( ) ( ) *

0

min ( , ; ) d

f

k

t

k k kT x t d T t

. (12)

The Taylor series expansion equation for * ( ) ( ) ( )( , , ( ) ( ))k k kT x t y d t , considering( ) ( )( )k kd t is:

* ( ) ( ) ( ) * ( ) ( ) ( )

( )( , ; ( ) ( )) ( , ; ( ))k k k k k k

k

TT x t t d t T x t t

. (13)

Denoting:

* ( ) ( )

( )( , ; )k k

k

TT x t

, (14)

the equation (13) becomes: * ( ) ( ) ( ) * ( ) ( ) ( )( , ; ( ) ( )) ( , ; ( )) ( )k k k k k kT x t t d t T x t t T . (15)

Replacing in (12) it results:

( ) ( )

2( 1) * ( ) ( ) ( ) ( ) *

0

min ( ) min ( , ; ) ( ) dt=

f

k k

t

k k k k kI T x t T T

( )

2 2 2* ( ) * ( ) ( ) ( ) ( ) * ( ) *

0

min ( , ; ) 2 ( ( , ; ) ) dt

f

k

t

k k k k k kT x t T T T T x t T

.

(16)

For small variations 2

( )kT is neglected and to minimize equation (16) we

differentiate it with respect to )(k and set the resulting expression equal to zero:

2

( 1) ( ) ( ) ( ) * ( ) *

( )

0

( ) 2 2 ( ( , ; ) ) dt=0

ft

k k k k k

kI T T T x t T

, (17)

resulting:

( ) ( ) * ( ) *

( ) 0

2( ) ( )

0

( )( ( , ; ) )dt

( ) dt

f

f

tk k k

k

t

k k

T d T x t T

T d

, (18)



where ( ) ( )( ) ( )k kT d T t d , is the small variation of temperature distribution T ,

produced by small variation of unknown boundary condition ( )( ) kt d . The function

( )T is estimated as the solution of the sensitivity problem. In conclusion, the conjugate

gradient method supposes the solution of two auxiliary problems: the sensitivity problem and

the adjoint problem, respectively.

3.1 Sensitivity problem

In sensitivity analysis it is assumed that when ( )t undergoes an increment ( )t , the

temperature ( , )T x t changes by an amount ( , )T x t . Therefore, we replace ( , )T x t by

( , ) ( , )T x t T x t and ( )t by ( ) ( )t t in the direct problem (1) and subtract from it

the original problem (1) in order to obtain the sensitivity problem:

2

20

T T

t x

, 0 x l , 0

ft t , (19)

with the following boundary and initial conditions:

( , ) ( )T l t t , 0f

t t , (20)

(0, ) 0T

tx

, 0

ft t , (21)

( ,0) 0T x , 0 x l . (22)

The problem (19)-(22) is well-posed if the increment ( )t is given.

3.2 Adjoint problem and gradient equation

To obtain the adjoint problem, we multiply equation (1) by Lagrange multiplier ( , )x t and

integrate the resulting expression over the spatial domain from x = 0 to x = 1, and then over

the time domain from 0t to ft t . The expression obtained in this manner is added to the

functional (7):

2

* * 2

2

0 0 0

( ) ( ( , ; ) ( )) d ( , ) d d

f ft t lT T

I T x t T t t x t x tt x

. (23)

Perturbing ( )t by ( )t and ( , )T x t by ( , )T x t results:

* 2 *

0 0

( ) ( ( , ; ) ( )) ( )d d

ftl

I T x t T t x x t x

2 2

2 2

0 0 0 0

( ) ( )( , ) d d ( , ) d d

f ft tl lT T T T

x t x t x t x tt x t x

,

(24)

where *( )x x is the Dirac function. The first term in (24) reads:

* 2 *

0 0

( ( , ; ) ( )) ( )d d

ftl

T x t T t x x t x

* 2 * 2 *

0 0

( ( , ) ) 2 ( ( , ) ) ( )d d

ftl

T x t T T T x t T T x x t x .

(25)



Replacing(25) in(24) and subtracting from (23) results:

2

* *

2

0 0 0 0

( ) ( )( ) 2 ( ( , ) ) ( )d d ( , ) d d

f ft tl lT T

I T T x t T x x t x x t x tt x

, (26)

where the high order terms were neglected. The second term in (26) can be integrated by

parts, then imposing conditions (20)-(22), the equation (26) becomes:

* *

0 0

( ) 2 ( ( , ) ) ( )d d

ftl

I T T x t T x x t x

2

2

0 0 0

d d d

f

f

tl l

t tT t x T t

t x

00 0

( )d d

f ft t

x l x l x

Tt T t

x x x

.

(27)

The following adjoint problem can be formulated:

2

* *

22( ( , ) ) ( , ) 0T x t T x x

t x

, 0 x l , 0

ft t , (28)

where *

*

*

1, for( , )

0, for

x xx x

x x

. (29)

The “initialˮ condition:

0 for f

t t , (30)

and the boundary conditions:

( , ) 0l t , (0, ) 0tx

, (31)

will complete the adjoint problem formulation. Note that the adjoint problem (28)-(31) is

well-posed, but the integration in time is inverted. Taking into account the adjoint problem,

from equation (27) the functional variation ( )I results:

0

( ) d

ft

x l

I tx

.

(32)

Considering ( )t least square integrable on the domain 0f

t t , the variation of the

functional ( )I can be expressed as:

0

( ) d

ft

I I t . (33)

Comparing (32) and (33) it results:

( )x l

Ix

. (34)

3.3 Stopping criterion

The stopping criterion for the iterative sequence (8) is based on the discrepancy principle [3]:



* * 2

0

( ) ( ( , ; ) ( )) d

ft

I T x t T t t . (35)

The tolerance is chosen so that smooth solutions are obtained with measurements

containing random errors. It is assumed that the solution is sufficiently accurate when:

* *( , ; ) ( )T x t T t , (36)

where is the standard deviation of measurement errors. From equation (36) it results:

2

ft , (37)

For cases involving errorless measurements, can be specified a priori as a sufficiently

small number.

3.4 Numerical discretization

In this paper a finite difference discretization is considered. For an uniform grid,

1, 1,2

i i xx x x i n

and

1, 1,2

n n tt t t n n

, an implicit forward-time-

centered-space discretization of the direct problem (1)-(4) reads [0]:

1 1 1 1

21 1

2

2,( ) 0

( )

n n n n n

i i i i iT T T T T

o t xt x

, 1,2

xi n , 0,1

tn n , (38)

with boundary conditions:

1 1

1 00n nT T , 1 ( )n

nxT t , 0,1

tn n , (39)

and initial condition:

0 0i

T , 1,2,x

i n . (40)

Consequently, the implicit scheme yields to the following tri-diagonal algebraic system

of linear equations:

1 1 1

1 1

n n n

i i i i i i ia T bT cT d

, 1,2

tn n , (41)

where:

2 2 2

1 2 1 1, , ,

( ) ( ) ( )

n

i

i i i i

Ta b c d

x x t x t

, 1,2, 1

xi n , (42)

and:

0 0 0 00, 1, 1, 0,a b c d

0, 1, 0, ( ).x x x xn n n n

a b c d t (43)

The tri-diagonal system is solved by Thomas algorithm [14]. Similarly, the forward-time

centered-space finite difference representation of the sensitivity problem results:

1 1 1

1 1

n n n

i i i i i i ia T b T c T d

, (44)

where , ,i i i

a b c are given by expressions (42)-(43) and:



n

i

i

Td

t

, 1,2, 1

xi n ,

00,d ( ).

xn nd t

(45)

For the adjoint problem (28)-(31), applying a backward-time centered space

discretization, one obtains the system:

1 1

n n n

i i i i i i ia b c d

, 1, 2, ,0n n n , (46)

where the matrix elements, , ,i i i

a b c have the identical expressions as for the direct and

sensitivity problems (42)-(43) and:

1

* *2( ( , ) ) ( , )n

i

id T x t T x x

t

, 1,2, 1

xi n ,

00,d 0.

xnd

(47)

The “initial” condition at f

t t is:

0, 0,1,tn

i xi n . (48)

3.5 Computational algorithm

Assuming an initial guess for the unknown function (0)( )t and taking 0k , the main steps

of the computational algorithm are:

1. Solve the direct problem (41)-(43).The result is the temperature field ( , )T x t for the

boundary condition on the right boundary ( )( )k t ;

2. Check the stopping criterion (35). If it is not verified, continue;

3. Solve the adjoint problem (46)-(48).

4. Calculate the gradient ( )( )kI using equation (34);

5. Calculate the parameter ( )k with (11) and the descent direction ( )( )kd t with (9);

6. Solve the sensitivity problem for ( )( ) kt d . The result is the field ( )( , ; )kT x t d .

7. Calculate the descent parameter )(k applying equation (18);

8. Calculate the new estimation of the unknown boundary condition with equation (8)

and return to step 1.

4. TIKHONOV REGULARIZATION METHOD

Another technique for solving the inverse problem is introduced by Tikhonov [3, 6, 8, 9, 11].

As already mentioned, in problems conditionally well-posed, according to Tikhonov, we

have to do not just with a solution, but with a solution that belongs to some narrower class of

solutions. To solve the inverse problem (1)-(5), the differential equation of the temperature

field (1) is approached by a well-posed hyperbolic boundary problem. Let's consider a

differential equation:

fTD )( (49)

where .D is a first kind differential operator. Let us suppose the case in which the right-hand

side of (49) is given accurate to some , f

, where:



f f

. (50)

Because the right side term is given with inaccuracy, we will try to formulate an well-

posed problem for another operator D witch possesses improved properties compared to

D :

fTD )( , (51)

where T is the approximate solution and the parameter α can be related with the inaccuracy

level in the right-hand side, i.e., α = α(δ).

As presented before, in variational methods, instead of solving equation (51), they

minimize the discrepancy functional:

2)()( fvDvI . (52)

For a bounded solution, Tikhonov regularization method introduces an additional

stabilizing functional 2

v in the discrepancy functional:

22)()( vfvDvI , (53)

where the regularization parameter α > 0 must be related to the right-hand side inaccuracy

level, δ. The approximate solution of the initial problem (49) is the extremal of the

functional:

22)(min)( vfvDTI

Hv

, (54)

H being the Hilbert space where the approximate solution belongs.

Instead of solving the variational problem (54) Tikhonov considers the related Euler

equation:

fDTDTD **, (55)

where *D is the adjoint operator.

As a general rule, the transfer from the ill-posed problem (49) to the well-posed problem

(55) can be made passing to a problem with a self-adjoint operator DD*. In order to

estimate the regularization parameter from the discrepancy criterion, we define the

function:

fTD )()( , (56)

and can be found as the solution of equation:

)( . (57)

This equation can be approximately solved using various computational procedures [7].

For instance, we can use the succession

kk q0

)( , 0q . (58)



To start, we made k = 0 and continue to a certain k = K at which equality (57) becomes

fulfilled to an acceptable accuracy. With so defined regularization parameter, only K + 1

calculations of the discrepancy (for the solutions of the Euler equations (55)) are needed. In

the present paper we apply two different approaches using the Tikhonov regularization: the

method with perturbed boundary conditions and the method with perturbed initial equation,

respectively [7, 10, 12].

4.1 Perturbed boundary condition with local regularization

For a global regularization, the solution is to be determined at all times simultaneously,

whereas in local regularization methods, the solution depends only on the pre-history, and

can be determined sequentially at separate times.

Local regularization methods take into account the specific feature of inverse problems

for evolutionary problems in maximal possible measure [10]. Let’s consider the direct and

inverse problems (1)-(5).

We consider inverse problem in which the boundary condition at the right boundary is

not given (the function ψ(t) in (2) is unknown). Instead, the additional condition at the left

boundary is given:

*(0, ) ( )T t T t , (59)

which is equivalent to (5) if * 0x .

Using, for each time value, the local Tikhonov regularization for determining the

boundary condition at the right boundary implies minimization of the smoothing functional:

2 2*

1 1 1( ) (0, ) ( ) ( )

n n nI T t T t t

, (60)

where )(t is the unknown function in the inverse problem.

The approximate ( , )T x t

can be represented as:

( , ) ( , ) ( , )o n

T x t T x t T x t

where ( , )o

T x t

is the solution of the homogeneous difference boundary value problem:

1 1 1 1

, , 1 , , 1

2

20,

( )

n n n n n

o i i o i o i o iT T T T T

t x

, 0 x l , 0f

t t , (61)

1

,0,n

o nxT

, 0

ft t , (62)

1

(0, ) 0n

oT

tx

, 0 ft t , (63)

with the initial condition 0

,0

o iT

, 0,1x

i n .

Note that n

iT

is the solution of the direct problem for a given boundary condition ( )nt .

For the non-homogeneous part ( , )n

T x t , considering the equation (1) results in:

1 1 1 1

, , 1 , , 1

2

20,

( )

n n n n

n i n i n i n iT T T T

t x

, 0 x l , 0 ft t , (64)

1

, 1( ),n

n nx nT t

(65)



1

(0, ) 0n

nT

tx

, 0f

t t ,

(66)

and the initial condition 0

,0

no iT

, 0,1x

i n . Because the equation (64) is linear, the

solution of the problem (64)-(66) can be written as:

1

1( ) ( ) ( ),n

n nT x q x t

(67)

where ( )q x is the solution of the problem:

2

20, 0 ,

q qx l

t x

(0) 0dq

dx , ( ) 1q l .

(68)

Replacing (60) and (67) in (60) we obtain:

2 2*

1 1 1 1( ) (0, ) (0) ( ) ( ) ( )

o n n n nI T t q t T t t

(69)

The minimum condition ( ) /I

leads to the equation:

*

1 1

1 2

( ) (0, )( ) (0)

(0)

n o n

n

T t T tt q

q

(70)

For a given value , the main steps of the computational algorithm to advance in time are:

1. Solve the direct problem (1)-(4); results n

iT

, 0,1,x

i n .

2. Solve the direct problem (61)-(63) to calculate 1

,

n

o iT

, 0,1,

xi n .

3. Solve direct problem (64)-(66) to calculate 1

,

n

n iT

, 0,1,

xi n .

4. Solve direct problem (68); results (0)q .

5. Estimate the value 1

( )nt

and return to step 1 for the next value of time.

For a given inaccuracy , the steps 1-4 have to be included in a loop for solving the

equation:

2*

1 1(0, ) ( )

n nT t T t

, (71)

following the sequence (58). Note that for the steps 1-3 we need to solve a tridiagonal system

of equations and the matrix of the system is the same as in previous application, (42).

4.2 Perturbed initial equation

In this case the initial differential equation is replaced by a perturbed differential equation in

order to formulate a well-posed problem. Because in the inverse problem the unknown is the

right boundary condition it is convenient to transform the initial parabolic equation into

hyperbolic equation having the x l a free boundary. Samarskii and Vabishchevich [7]

show that the desired equation is:

2 3

2 20

T T T

x t t x

, (72)

with the initial condition:



( ,0) 0T x

for 0t and 0,x l , (73)

and the final condition:

( , ) 0f

Tx t

t

for f

t t and 0,x l . (74)

The boundary conditions for equation (73) are:

*(0, ) ( )T t T t

, for 0x and 0,f

t t , (75)

(0, ) 0T

tx

, for 0x and 0,f

t t . (76)

Because the equation (72) is hyperbolic in respect to x , to solve this equation it is

necessary to advance the solution in this direction. Let's be i

x an arbitrary mesh point. For

hyperbolic equations, stable discretization is obtained using upwind schemes:

2

, , 1 , 2 2

2 2

1

2( )

( )

n n n n

i i i

i

T T TTo x

x x

, (77)

1

, 1 , 1

1

( )

n n n

i i

i

T TTo t

t t

, , , 2 2

1

( )2

n n n

i i

i

T TTo x

x x

, (78)

2 2 2

2 2 2

1 2

1

2

n n n

i i i

T T T

t x x t t

1 1 1 1

, , , , 2 , 2 , 2

2 2

2 21

2 ( ) ( )

n n n n n n

i i i i i iT T T T T T

x t t

.

(79)

Replacing in (72) results in:

1 1

, ,

,2 2 2 2

2 1

2 ( ) 2 ( ) ( ) 2 ( )

n n

i in

i

T TT

x t x t x x t

1

, 1 , 2 , 1 , 1 1 1

, 2 , 2 , 22 2

2 12

( ) 2 ( )

n n n n

i i i i n n n

i i i

T T T TT T T

x t x t

,

1,2, , 1t

n n , 2,3, , 1x

i n .

(80)

The above equation represents a tridiagonal algebraic system with the unknowns ,

n

iT

,

0,1,t

n n for const.i To close the system (80) the conditions (73) and (74) are

attached. If the discrete form of (73) is easy to obtain:

0

,0

iT , (81)

to implement condition (74) we write the difference form of the equation (72) in the point

tn n , taking into account the equality

1

, ,t tn n

i iT T

, for all i. The resulting equation is:



1

,

,2 2 2

1 1

2 ( ) ( ) 2 ( )

n

in

i

TT

x t x x t

1

, 1 , 2 , 1 , 1 1

, 2 , 22 2

2 1

( ) 2 ( )

n n n n

i i i i n n

i i

T T T TT T

x t x t

(82)

The tridiagonal system is:

1 1 1

, 1 , , 1

n n n

i i i i i i ia T bT cT d

, 1,2

tn n , (83)

where:

2

1

2 ( )i

ax t

,2 2

2 1

2 ( ) ( )i

bx t x

, 2

1

2 ( )i

cx t

,

1

, 1 , 2 , 1 , 1 1

, 2 , 22 2

2 1

( ) 2 ( )

n n n n

i i i i n n

i i i

T T T Td T T

x t x t

, 1,2, , 1

ti n ,

(84)

and:

2

1

2 ( )tna

x t

,

2 2

1 1

2 ( ) ( )tnb

x t x

,

1

, 1 , 2 , 1 , 1 1

, 2 , 22 2

2 1

( ) 2 ( )t

n n n n

i i i i n n

n i i

T T T Td T T

x t x t

(85)

To start, for 0i , we impose(75):

*

,0( )n

nT T t , (86)

and in order to implement the condition (76), we write a Taylor:

2

2 3

,1 ,0 2

0 0

1( ) ( )

2

nn

n n T TT T x x o x

x x

. (87)

Because

0

0

nT

x

, 2 1

0 0

2

00

( )

n n n nT T T To t

x t t

, (88)

It results:

1

2 30 0

,1 ,0

1( ) ( )

2

n n

n n T TT T x o x

t

. (89)

For this method the computation algorithm is represented only by one loop for solving

the equation (71). At each iteration the problem (81)-(89) is solved to find ,( )

x

n

n nT t

.

5. NUMERICAL RESULTS

For the numerical applications uniform grids with 0.01x and 1, 101x

l n were used.

Also, the assumed final time value was 1f

t , but the time step was calculated imposing

different values for tn . For each method we have solved first a direct problem considering



three shapes for the right boundary condition ( , ) ( )T l t t , denoted in the follows with

case-1, case-2, and case-3, respectively:

case-1: ( ) 4 (1 )f f

t tt

t t , case-2:

2 / ,0 / 2;( )

2( ) / , / 2 ,

f f

f f f f

t t t tt

t t t t t t

case-3:

0.5, 0 / 4;

( ) 1, / 4 3 / 4;

0.5, / 4 .

f

f f

f f

t t

t t t t

t t t

(90)

The solutions of the direct problems for all three case are represented in Figure 1 - Figure 3.

In order to formulate the input data for the inverse problem, *(0, )T t , the obtained solution of

the direct problem in left side, (0, )T t is random perturbed with an inaccuracy level . So,

the exact solution of the inverse problem will be represented by ( )t .

Figure 1. Solution of direct problem in case 1



Figure 4. Solution of inverse problem with conjugate

gradient method-case 1



Figure 5. Solution of inverse problem with Tikhonov

local regularization method-case 1


equation regularization method-case1

Figure 7. Solution of inverse problem with conjugate

gradient method-case 2


local regularization method-case 2


equation regularization method-case 2

Figure 10. Solution of inverse problem with

conjugate gradient method-case 3




Tikhonov local regularization method-case 3


Tikhonov equation regularization method-case 3

Figures 4, 5 and 6 plot the obtained numerical results for case 1. All methods are very

sensitive to the number of time steps. If the accuracy of numerical results is satisfactory for

the conjugate gradient method and for the Tikhonov equation (hyperbolic) regularization, for

the local regularization, strong oscillations on the numerical solution are present even for

low number of time layers. We appreciate that the Alifanov’s iterative regularization method

predicts smoothed results.

For the case 2, all methods present similar behavior. Figure 7, corresponding to

Alifanov’s iterative regularization method shows smoothed results in respect to the predicted

by Tikhonov equation regularization method (Figure 9). However, the last one succeeds to

capture more exactly the peak of the reference data. The Tikhonov local regularization is

again the worst prediction.

For the case 3, where we check the schemes for discontinuous input data, the results are

plotted in Figures 10-12. The plots show that all three methods are not able to capture

correctly the discontinuity. The sudden variations are replaced by contiguous ones. Again the

conjugate gradient method and the Tikhonov equation regularization predict better results.

6. CONCLUSIONS

In the present paper we try to asses three different approaches in solving the inverse thermal

conduction problem: the Alifanov’s iterative regularization method, the Tikhonov local

regularization method and the Tikhonov equation regularization method, respectively. For

the considered numerical applications, the first and the third method provide similar results

having a satisfactory accuracy in respect to exact, reference data. The local Tikhonov

regularization introduces strong oscillations in numerical prediction, being the most sensitive

to the number of performed time steps.

REFERENCES

[1] M. N. Ozisik, R. B. H. Orlande, Inverse Heat Transfer, Taylor and Francis, New York, 2000.

[2] O. M. Alifanov, Inverse Heat Transfer Problems, Springer-Verlag, New York, 1994.

[3] A. N. Tikhonov, Regularization of Incorrectly Posed Problems, Soviet Math. Dokl., 4(6), 1624-1627, 1963.

[4] J. V. Beck, B. Blackwell, A. Sheikh-Haji, Comparison of Some Inverse Heat Condition Methods Using

Experimental Data, Int. J. Heat Mass Transfer, 39(7): 3649-3657, 1996.



[5] J. V. Beck, B. Blackwell, C. R. St. Clair, Inverse Heat Conduction, Ill Posed Problems, A Wiley Interscience

Publication, New York, 1985.

[6] A. N. Tikhonov, V. Y. Arsenin, Solution of ill-posed problems, Winston & Sons, Washington, DC, 1977.

[7] A. A. Samarskii, P. N. Vabishchevich, Numerical methods for solving inverse problems of mathematical

physics, de Gruyter, ISBN 978-3-11-01966-5, Berlin, Germany.

[8] A. N. Tikhonov, Solution of Incorrectly Formulated Problems and the Regularization Method, Soviet

Math.Dokl., 4(4), 1035-1038, 1963.

[9] A. N. Tikhonov, Inverse Problems in Heat Conduction, J.Ehg. Phys., 29(1), 816-820, 1975.

[10] A. A. Samarskii, P. N. Vabishchevich, Computational Heat Transfer Vol.2. The Finite Difference

Methodology, Wiley, Chichester, 1995.

[11] A. N. Tikhonov, A. V. Samarskii, Equations of Mathematical Physics, Dover Publications, 1990.

[12] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, 1997.

[13] G. S. Dulikravich, T. J. Martin, Inverse Shape and Boundary Condition Problems and Optimization in Heat

Conduction, Chapter 10 in Advances in Numerical Heat Transfer,1, 381-426, Minkowycz, W. J. and

Sparrow, E. M. (eds.), Taylor and Francis, 1996.

[14] S. Dănăilă, C. Berbente, Numerical methods in fluid dynamics, Edit Academiei Romane, Bucharest, 2003.

[15] E. Buckingham, Model experiments and the forms of empirical equations,Trans.ASME, 37:263-296, 1915.

[16] V. L. Streeter and E. B. Wylie, Fluid Mechanics, McGraw-Hill Book Company, New York, 7th edition,

Chapter 4, 1979.

[17] A. D. Kraus, A. Aziz and J. R. Welty, Extended Surface Heat Transfer, John Wiley&Sons, Inc. NewYork,

2001.

[18] J. H. Lienhard IV, J. H. Lienhard V, A heat transfer text book ,Third Edition, paperback-august, 2003.

[19] O. M. Alifanov, Inverse Heat Transfer Problems, Springer-VerlagTelos, 1995.

[20] D. Peaceman, H. Rachford, The numerical solution of parabolic and elliptic equations, SIAM Journal, 3,28-

41, 1955.

[21] R. Fletcher and C. M. Reeves, Function Minimization by Conjugate Gradients, Computer J., 7, 149-154,

1964.

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