Estimation of Boundary Heat Flux with Conjugate …
345
efficient method than other methods for the inverse problems [14]. After knowing
the all limiting conditions, the problem mention in the above Eq. (1a) is solved for
obtaining the temperature distribution.
3 The Inverse Problem
The heat flux value q(t) on one side of the steel rod (x = 0) is considered as being
unknown while considering the inverse heat transfer problem, but all other parameters
and the temperature values from the experiment at some particular locations and time
are regarded as available.
Let the Y (x, t) ≡ Y m (t) represents temperature reading taken from the experiment,
where M is the number of locations where temperatures are measured. We know that
the measured value of temperature Y m (t) contains errors of measurement. The inverse
problem has to be solved by considering the following minimization function:
S[t] =
t f
t=0
M
m=1
[Y m (t) − T m (t)]
2 dt
(2)
Here, T m (t) is the obtained from the solution of Eq. (1a) with the help of calculated
or estimated heat flux q
(t). The symbol ‘
’ represents estimated quantity while t f
stands for the final time.
4 Conjugate Gradient Method (CGM)
The calculation of heat flux q(t) is done using the CGM [15] considering minimization
of Eq. (2)
ˆ
q
i+1
(t) = ˆ
q
i
(t) − β
i d
i
(t)
(3a)
Here, β
i represents the search step size in the direction of moving from the
current position to the next position, d
i
(t) represents the direction of descent, and
the superscript i represents iteration (i = 0, 1, 2…). The direction of descent is given
as
d
i
(t) = S
i
(t) + ϒ
i d
i−1
(t)
(3b)
Here, S
i
(t) is the gradient of S(t) at iteration i and d
i−1
(t) is the direction of
decent at previous iteration I − 1.
Different expressions are available for conjugation coefficient ϒ
i in literature [16]
345
efficient method than other methods for the inverse problems [14]. After knowing
the all limiting conditions, the problem mention in the above Eq. (1a) is solved for
obtaining the temperature distribution.
3 The Inverse Problem
The heat flux value q(t) on one side of the steel rod (x = 0) is considered as being
unknown while considering the inverse heat transfer problem, but all other parameters
and the temperature values from the experiment at some particular locations and time
are regarded as available.
Let the Y (x, t) ≡ Y m (t) represents temperature reading taken from the experiment,
where M is the number of locations where temperatures are measured. We know that
the measured value of temperature Y m (t) contains errors of measurement. The inverse
problem has to be solved by considering the following minimization function:
S[t] =
t f
t=0
M
m=1
[Y m (t) − T m (t)]
2 dt
(2)
Here, T m (t) is the obtained from the solution of Eq. (1a) with the help of calculated
or estimated heat flux q
(t). The symbol ‘
’ represents estimated quantity while t f
stands for the final time.
4 Conjugate Gradient Method (CGM)
The calculation of heat flux q(t) is done using the CGM [15] considering minimization
of Eq. (2)
ˆ
q
i+1
(t) = ˆ
q
i
(t) − β
i d
i
(t)
(3a)
Here, β
i represents the search step size in the direction of moving from the
current position to the next position, d
i
(t) represents the direction of descent, and
the superscript i represents iteration (i = 0, 1, 2…). The direction of descent is given
as
d
i
(t) = S
i
(t) + ϒ
i d
i−1
(t)
(3b)
Here, S
i
(t) is the gradient of S(t) at iteration i and d
i−1
(t) is the direction of
decent at previous iteration I − 1.
Different expressions are available for conjugation coefficient ϒ
i in literature [16]