Estimation of a source term in a two-dimensional heat transfer problem: application to an electron beam welding

Figures & data

Figure 1. Welding process.

Figure 2. Weld joint (dimensions in mm).

Table 1. Thermocouple locations.

Thermocouple i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
xi (mm)	2.38	2.38	2.38	1.95	1.95	1.95	1.58	1.58	1.58	1.25	1.25	0.98	0.75	0.58	0.45	0.23	0
zi (mm)	0	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60	64

Figure 3. Definition of the study plane.

Figure 4. CCT diagram.

Figure 5. Comparison between the experimental and numerical temperatures.

Figure 6. First form.

Figure 7. Computed temperatures.

Figure 8. Estimated source.

Figure 9. Residual between the exact and estimated sources.

Figure 10. Source definition.

Figure 11. Exact Gaussian source in z = 0.

Figure 12. Comparison of the temperatures.

Figure 13. The residual functional evolution.

Figure 14. Estimated source in z = 0.

Figure 15. Evolution of the fused zone.

Table

a	thermal diffusivity (m^2 s^−1)
Ci	specific heat of the phase i (J kg^−1 K^−1)
D^n	vector of descent direction
dT/dt	variation of the temperature with time (K s^−1)
Δ F0 = a Δ t/Δ x^2	delta Fourier number
h	penetration (m)
Hγ, Hα	enthalpies of the phases γ and α (J m^−3)
ΔHfus, ΔHvap	enthalpies of the phase change (J kg^−1)
I, If	current and focus current (A)
J(S)	residual functional
∇ J^n	the gradient of the residual functional
Lij, Lγ α	heat transformation of phase i to j (J m^−3)
n, n*	iteration number, last iteration
Pi, P, Peq	proportion of metallurgic phase (per volume fraction)
Pij	proportion of the transformed phase i to j
x	spatial coordinates (m)
S^n^+1, S(x, z, t)	source term
Tinf	external temperature (°C).
T, Ti	temperature in the sample (°C)
T0	initial temperature in the sample (°C)
U	voltage (V)
v	velocity (m s^−1)
Y, Yi, Ye	experimental temperatures (°C)
z	spatial coordinates (m).
γ^n	descent parameter
δ^2	convergence criterion
ε(T)	emissivity of the sample
ε1	arrest criterion
η	efficiency coefficient
λi	conductivity of the phase i (W m^−1 K^−1)
ρi	density of the phase i (kg m^−3)
σ^2	standard deviation of the temperature measurement errors
ω	a random number
θ(z, r, t)	temperature variation
Φ	beam diameter (m)
Ψ(z, r, t)	adjoint variable