*2.3. Heat Source*

The heat source was exerted by a heat flow density imposed on the sample surface interacting with the arc. A Gaussian heat source model was chosen (shown in Figure 4), centered on the arc axis, moving in translation along the x-axis at the welding speed *v*s. It should be noted that o represents the coordinate center of the xy plane.

**Figure 4.** The Gaussian heat source model.

The heat flow density distribution in the surface is then given by [20]:

$$q(r) = \frac{2q\_{\rm m}}{\pi d} \exp\frac{-3r^2}{d^2} \tag{2}$$

where *q*<sup>m</sup> is Gaussian amplitude, *d* is width, and *r* represents the distance to the center of the heat source, defined by:

$$
\sigma = \left(X - X\_0 - \upsilon\_s t\right)^2 + \left(Y - Y\_0\right)^2 \tag{3}
$$

Here, *X*<sup>0</sup> and *Y*<sup>0</sup> are the coordinates of the initial position of the heat source and *t* denotes the time.

The total power transmitted to the sample by this distribution, obtained by integrating Equation (2) for r from 0 to infinity, must match the effective welding power. The relationship between the distribution parameters and process parameters can be given as

$$q\_{\rm m} = \eta \cdot \mathcal{U}I \tag{4}$$

where *U* is the welding voltage, *I* is the welding current, and *η* indicates the welding efficiency.

### *2.4. Boundary and Initial Conditions*

At the beginning, the initial and ambient temperatures of the FE model for all simulations were set to 25 ◦C. The thermal boundary conditions mainly include convection in air, radiation from the surface of the workpiece toward air in light of the Stefan-Boltzmann relationship, and conduction from the workpiece toward the metal support. The heat loss from surface convection and radiation can be given as [20]:

$$q\_{\rm conv} = h(T\_{\rm c} - T\_0) \tag{5}$$

$$q\_{\rm rad} = \varepsilon \sigma \left[ \left( T\_{\rm c} - T\_{\rm abs} \right)^{4} - \left( T\_{0} - T\_{\rm abs} \right)^{4} \right] \tag{6}$$

Here, *T*<sup>c</sup> is the current temperature, *T*<sup>0</sup> indicates the ambient temperature, and *T*abs is the absolute zero temperature. In addition, *ε* represents the emissivity and *σ* is the Stefan-Boltzmann constant, which has the value of 5.68 <sup>×</sup> <sup>10</sup>−<sup>8</sup> J/K4m2s. In this paper, the convection coefficient is taken as 8 and the emissivity is 0.85, and the heat transfer coefficient has the value of *<sup>h</sup>* = 10 W/m2 · K.
