3.3 Effective Thermal Diffusivity and Conductivity
147
the optimal result. It should be noted that the x-axis includes all middle sensors data
used in the inverse method from T1 to T4 and the x-coordinate i indicates the data
length of four middle thermocouples.
First, Fig. (3.17a), shows the sensitivities of vector P have the same order of
magnitude, although sensitivities of p 8 and p 9 are higher than others. A similar
order of magnitude means they can be retrieved at similar accuracy.
Moreover, the magnitudes of diffusivity-parameter sensitivities in Fig. (3.17a),
are significantly smaller than those of thermocouple positions in Fig. (3.17b), which
means the thermocouple positions are retrieved at a higher precision than diffusivity
parameters within a single iteration. If all diffusivity and thermocouple parameters
are searched together, the optimization method tends to give more accurate positions
with the accuracy expense of diffusivity parameters. This is not an expected method
in consideration of the goal mentioned at the beginning of Sect. (3.3.4). Therefore,
the optimization processes of different types of parameters should be implemented
separately without the mutual influence to each other in a single iteration, and the
sensitivities of searched parameters are in similar magnitude in each iteration to avoid
the accuracy expense of diffusivity parameters.
In Fig. (3.17b), the boundary-position parameters 1 and 6 affect all computational
temperature data from T2 to T5, but the middle-position parameters from 2 to 5 only
impact the computational temperature data of their respective positions. For instance,
position 2 only affects the data of T2, according to Fig. 22 and the aforementioned
inverse method. In light of the different roles of boundary positions and middle
positions, their optimization processes should also be separated.
3.3.4.2 Improved Inverse Method
In a summary of the aforementioned analyses, a new process of the inverse method
should be split into three sub-steps:
Step (1): The ten optimal diffusivity parameters should be retrieved with fixed
boundary and middle positions. For the first time, the initial value can
be chosen as the result of the single quadratic polynomial in Sect. (3.3.3),
and the nominal positions can be used for first-time initial positions.
Step (2): The four optimal middle-sensor positions, if the data from T2 to T5 are chosen as compared data in Eqs. (3.11) and (3.12), should be retrieved by the
inverse method with fixed diffusivity parameters and boundary positions.
Step (3): The two optimal boundary positions should be retrieved by the same inverse
method with fixed diffusivity parameters and middle positions. These[Step
(4):] sub-steps should be repeated to update the diffusivity parameters and
positions until all iterations in sub-steps meet the convergence criterion.
147
the optimal result. It should be noted that the x-axis includes all middle sensors data
used in the inverse method from T1 to T4 and the x-coordinate i indicates the data
length of four middle thermocouples.
First, Fig. (3.17a), shows the sensitivities of vector P have the same order of
magnitude, although sensitivities of p 8 and p 9 are higher than others. A similar
order of magnitude means they can be retrieved at similar accuracy.
Moreover, the magnitudes of diffusivity-parameter sensitivities in Fig. (3.17a),
are significantly smaller than those of thermocouple positions in Fig. (3.17b), which
means the thermocouple positions are retrieved at a higher precision than diffusivity
parameters within a single iteration. If all diffusivity and thermocouple parameters
are searched together, the optimization method tends to give more accurate positions
with the accuracy expense of diffusivity parameters. This is not an expected method
in consideration of the goal mentioned at the beginning of Sect. (3.3.4). Therefore,
the optimization processes of different types of parameters should be implemented
separately without the mutual influence to each other in a single iteration, and the
sensitivities of searched parameters are in similar magnitude in each iteration to avoid
the accuracy expense of diffusivity parameters.
In Fig. (3.17b), the boundary-position parameters 1 and 6 affect all computational
temperature data from T2 to T5, but the middle-position parameters from 2 to 5 only
impact the computational temperature data of their respective positions. For instance,
position 2 only affects the data of T2, according to Fig. 22 and the aforementioned
inverse method. In light of the different roles of boundary positions and middle
positions, their optimization processes should also be separated.
3.3.4.2 Improved Inverse Method
In a summary of the aforementioned analyses, a new process of the inverse method
should be split into three sub-steps:
Step (1): The ten optimal diffusivity parameters should be retrieved with fixed
boundary and middle positions. For the first time, the initial value can
be chosen as the result of the single quadratic polynomial in Sect. (3.3.3),
and the nominal positions can be used for first-time initial positions.
Step (2): The four optimal middle-sensor positions, if the data from T2 to T5 are chosen as compared data in Eqs. (3.11) and (3.12), should be retrieved by the
inverse method with fixed diffusivity parameters and boundary positions.
Step (3): The two optimal boundary positions should be retrieved by the same inverse
method with fixed diffusivity parameters and middle positions. These[Step
(4):] sub-steps should be repeated to update the diffusivity parameters and
positions until all iterations in sub-steps meet the convergence criterion.