Using derivative regularization to solve inverse heat conduction problems

Figures & data

Figure 1 The heat addition used in the derivative regularization test.

Figure 2 The temperature response from the four heat pulses used in testing the derivative regularization method.

Figure 3 The condition numbers as functions of weighting factors for the ${\tilde{X}}^T\tilde{X}$ matrix.

Figure 4 The temperature response curve to pulse heating, along with the first and second derivatives.

Figure 5 The total error in estimated heat flux obtained using derivative regularization with an imposed measurement error standard deviation of 0.0001 dimensionless temperature units.

Figure 6 The total error in estimated heat flux obtained using derivative regularization with an imposed measurement error standard deviation of 0.0005 dimensionless temperature units.

Figure 7 The total error in estimated heat flux obtained using derivative regularization with an imposed measurement error standard deviation of 0.001 dimensionless temperature units.

Table 1. Estimation error for non-regularized and regularized cases.

Standard deviation of measurement errors (dimensionless temperature)	Estimation error for non-regularized cases (dimensionless heat flux)	Estimation error for regularized cases (dimensionless heat flux)
0.0001	6.65	1.31
0.0005	8.17	2.91
0.001	18.5	5.68

Table

Nomenclature
A, B	uniformly distributed random errors
C	coefficient affecting the standard deviation of temperature errors
c	specific heat (J/kg-K)
k	thermal conductivity (W/m-K)
L	material thickness (m)
m	heat flux index
n	temperature measurement index
qD	dimensional heat flux (W/m^2)
q	dimensionless Heat Flux
q	heat flux vector
t	dimensionless time
tD	dimensional time (s)
T	dimensionless temperature
TD	dimensional temperature (K)
T	temperature measurement vector
$\tilde{\mathbf{T}}$	transformed temperature measurement vector
xD	spatial dimension (m)
x	dimensionless distance
X	sensitivity matrix
$\tilde{X}$	transformed sensitivity matrix
w1	weighting factor for the first derivative of temperature
w2	weighting factor for the second derivative of temperature
$\alpha$	thermal diffusivity (m^2/sec)
${\epsilon }_q$	total error in dimensionless heat pulse estimation
${\epsilon }_T$	error applied to synthetic dimensionless temperature measurements
${\varphi }_m(t)$	temperature response as a function of time for unit heat pulse m
$\rho$	density (kg/m^3)