Parametric identification of a heating mobile source in a three-dimensional geometry

Figures & data

Figure 1. Three-dimensional geometry.

Figure 2. Heating flux $\mathit{\varphi }(t)$.

Table 1. Parameters for direct problem resolution.

Square length $L$ in m	Plate thickness $e$ in m	Initial temperature ${\mathit{\theta }}_0$ in K	Final time $t_f$ in s
$5\times {10}^{-2}$	$2\times {10}^{-3}$	293	300
Thermal conductivity $\mathit{\lambda }$ in $\text{W}{\text{m}}^{\text{- 1}}{\text{K}}^{-1}$	Volumetric heat capacity $\mathit{\rho }c$ in $\text{J}{\text{m}}^{\text{- 3}}{\text{K}}^{-1}$	Natural convection coefficient $h$ in $\text{W}{\text{m}}^{\text{- 2}}{\text{K}}^{-1}$
1.2	$2\times {10}^6$	20

Table 2. Cost function values for several iterations (fixed source).

Iteration k	0	1	2	3	4	5	6
$J({\mathit{\varphi }}^k)$	827,622	10,709	646	489	360	184	149
$Err$	182,812	26,066	8530	5974	4180	2322	2440

Figure 3. Temperature evolution for the fixed heating source (direct problem).

Figure 4. Sensors’ locations (upper face) and source trajectory (lower face).

Figure 5. Temperature evolution on the upper face (mobile source – direct problem).

Figure 6. Basis functions $s_i(t)$.

Figure 7. Tracking error evolution (Case 1).

Figure 8. Identified heating flux (Case 1).

Figure 9. Identified heating flux (Case 2).

Figure 10. Residual temperature error (Case 2).

Figure 11. Tracking error evolution (Case 3).

Figure 12. Identified heating flux (Case 3).

Figure 13. Disturbed measurements at $C_1$ (Case 4).

Table 3. Cost function values at each iteration (Case 4).

Iteration k	0	1	2	3
$J({\mathit{\varphi }}^k)$	625	170	143.8	131.9
Err	$3.34{10}^{10}$	$8.97{10}^9$	8.71 × 10^8	$9.99{10}^8$

Figure 14. Heating flux identification (Case 4).

Figure 15. Residual temperature error (Case 4).

Table 4. Mobile source trajectory.

Time (s)	0	100	200	300
$X$ in m	−0.01	−0.02	0.02	0.02
$Z$ in m	0.02	−0.01	−0.01	−0.005

Table 5. Cost function values for several iterations (experimentation).

Iteration k	0	1	2	3	4	5	…	15	…	29	30	…	36
$J({\mathit{\varphi }}^k)$	8694	190	45	41	34	33	…	30	…	29.8	29.7	…	29.6

Figure 16. Sensors positions (upper face) and heating source trajectory (lower face).

Figure 17. Measured (square) and calculated temperature.

Figure 18. Identified experimental heating flux.

Figure 19. Residual temperature (experimentation).

Table 6. Residual temperatures (experimentation).

	Sensor $C_1$	Sensor $C_2$	Sensor $C_3$
Average value	0.08	−0.01	−0.01
Standard deviation	0.23	0.26	0.26