A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

Figures & data

Figure 1. The schematic shape of one-dimensional ablation problem with two unknowns Q_a(τ), Q_c(τ).

Table 1. Thermophysical properties of ablating material.

Parameter	Typical value	Unit
k	0.22	W  m^−1 K^−1
α	9.11 × 10^−8	m^2 s^−1
H	44.7 × 10^8	J m^−3
Tab	833	K
T0	298	K

Table 2. The initial and boundary conditions.

	Boundary conditions			Initial conditions
Parameter	qa,max	qc,max	T∞	T0	L0
Quantity	10^5W m^−^2	−2 × 10^4W m^−2	273 K	298 K	0.02 m

Table 3. Front surface of ablating materials.

Case number	Heat flux profile at X = 0	Heat flux profile at X = 1	Front position	Ablated thickness (mm)
Case 1	Triangular	Sinus	Xfp = 0.0103	Δx = 206
Case 2	Sinus	Triangular	Xfp = 0.0120	Δx = 0.240

Figure 2. Solution of inverse problem with noisy measurements for the first case under the study: (a) triangular profile for applied heat flux Q_a and (b) sinusoidal profile for convective heat flux Q_c.

Figure 3. Solution of inverse problem with noisy measurements for the second case under the study: (a) sinusoidal profile for applied heat flux Q_a and (b) triangular profile for convective heat flux Q_c.

Figure 4. The variation of convergence rate versus the required iteration number at two different time steps: (a) 70th time step and (b) 90th time step for the first case under the study.

Figure 5. The variation of convergence rate versus the required iteration number at two different time steps: (a) 70th time step and (b) 90th time step for the second case under the study.

Table 4. CPU run time for different test cases.

	σ = 0.05
	Case 1	Case 2
CGM	4169.54 s	3411.34 s
MCGM	1727.02 s	1562.30 s

Table 5. RMSD of estimated heat flux values.

	σ = 0.05
	Case 1		Case 2
	Qa	Qc	Qa	Qc
CGM	0.4363	0.1001	0.2712	0.1015
MCGM	0.3386	0.0492	0.2887	0.0682