Estimation of the heat transfer coefficient by means of the method of fundamental solutions

Figures & data

Figure 1. Geometry of the proposed problem.

Figure 2. Collocation points (dot), source points (circle) and the unknown boundary (cross).

Table 1. Constant shape function.

	Constant
Method	RMS error	CPU time (s)	Frequencies
SVD	6.079e-03	0.2031	2
GMRES	1.814e-03	6.250e-02	2

Table 2. Parabolic shape function.

	Parabolic
Method	RMS error	CPU time (s)	Frequencies
SVD	1.6918	98.6093	12
GMRES	0.9287	11.3281	30

Table 3. Linear shape function.

	Linear
Method	RMS error	CPU time (s)	Frequencies
SVD	1.7598	484.2656	20
GMRES	0.5231	19.3750	26

Figure 3. Estimate of the h_R – non-intrusive measurements.

Figure 4. Extra collocation points (dot) in the domain at a distance dx = 0.005 m from x = 0 m.

Table 4. Root mean square error – intrusive and non-intrusive measurements.

Distance (dx = 0.005 m)	E(T)
Non-intrusive	1.814 × 10^−3
1 dx	1.788 × 10^−3
4 dx	1.694 × 10^−3
6 dx	1.525 × 10^−3
8 dx	2.342 × 10^−4

Figure 5. Estimate of the h_R – intrusive measurements at a distance 4 dx from x = 0.

Figure 6. Isotherms: Case 1 – FDM.

Figure 7. Isotherms: Case 1 – MFS.

Table 5. Root mean square error for the parabolic function.

Distance (dx = 0.005 m)	E(T)
Non-intrusive	0.9287
1 dx	0.9150
4 dx	0.3136
6 dx	0.2651
8 dx	0.2452

Figure 8. Estimate of the heat transfer coefficient for the parabolic function.

Figure 9. Temperature distribution along the boundaries for intrusive measurements at a distance 4 dx from x = 0 m.

Figure 10. Isotherms: Case 2 – FDM.

Figure 11. Isotherms: Case 2 – MFS.

Figure 12. Estimate of h_R – error level σ = 10%.

Table 6. Root mean square error – step function.

Distance (dx = 0.005 m)	E(T)
Non-intrusive	1.8607
1 dx	1.5212
4 dx	0.4238
6 dx	0.3779
8 dx	0.2296

Figure 13. Estimate of the step function.

Figure 14. Temperature distribution along the boundaries for the step function.

Figure 15. Temperature distribution along the boundaries for the step function with σ = 10%.

Figure 16. Isotherms: Case 3 – FDM.

Figure 17. Isotherms: Case 3 – MFS.

Figure 18. Estimate of h_R for noise level σ = 10%.