## 2 Methods  
Welding is generally considered a complex mechanical, non-linear phenomenon. From the macroscopic perspective, welding problems are related to heat, heat transfer, and solid elastic-plastic mechanical issues. However, at the microscopic level, problems are also connected with the welding plate material, the filler material, and their solidification relationship. The present research focuses on the macroscopic level to analyze and discuss the welding process. Welding is a thermomechanical coupling problem. As it is very difficult to analyze a solid-bodied elastic-plastic energy problem with linear partial differential equations, it is highly unrealistic to seek analytical solutions in this way. Scientists and engineers therefore have found methods that chart a middle course between mathematical theory and applied science by developing numerical analysis that yields results close to analytical ones, including the finite difference method (FDM), the boundary element method (BEM), the boundary domain integral method (BDIM), the FEM, and the finite volume method (FVM). In GMAW, an electric arc is produced between the consumable wire electrodes and the workpieces, heating the metal of the workpieces and thereby melting and joining them. The welding pool thus formed has a heat flux moment, which transposes heat to the welding workpieces, causing counteractive movement. The entire welding process is therefore strongly influenced by the movement of heat, and simulation using numerical analysis is crucial for obtaining accurate data. [PERSON] et al. [PERSON] et al. (2016) combined the Gaussian normal heat source distribution model with the finite element method to simulate the welding process. The Gaussian normal heat source distribution model is the most extensive application of heat source analysis for flat welding (Figure 1); its equations are:  
\[Q_{s}=\eta UI \tag{1}\]  
\[q(x,y)=\frac{3Q_{s}}{\pi u_{a}^{2}}e^{(\frac{-3x^{2}}{\pi^{2}})} \tag{2}\]  
where: \(r_{a}\) is the radius of a heat spot and represents a heat strength of 95% for electronic arc welding applied to a workpiece; \(r\) is the radius of the welding heat source; \(q(x,y)\) stands for the heat density at a distance \(r\) from the heat center; \(Q_{s}\) is the amount of heat distribution during welding; and \(\eta\) is the efficiency of the welding heat. In 1983, [PERSON] and [PERSON] [PERSON] and [PERSON] (1983) were the first to study three-dimensional (3D) moving ellipsoid heat sources. This form of heat source model completely overcomes the weaknesses of the 2D Gaussian normal distribution heat source model, so even heat deep beneath the welding surface can be simulated. The result is similar to an actual electric arc welding outcome because it is not a simple surface analysis of the heat situation at the welding bead. The proposed equation is  
\[q(x,\ y,\ z)=\frac{6\sqrt{3}Q_{s}}{abc\pi\sqrt{\pi}}e^{(\frac{3x^{2}}{\pi^{2} }+\frac{3y^{2}}{\pi^{2}}+\frac{3x^{2}}{\pi^{2}})}. \tag{3}\]  
However, experimental results have shown that the temperature gradient is very steep for the first half of the welding pool but not for the second half. [PERSON] et al. [PERSON] et al. (2018) therefore made the welding heat source ellipsoid, combining two half ellipses into an ellipsoid of Gaussian heat source distribution, as shown in Figure 2. The modified equations are  
\[q_{f}(x,\ y,\ z)=\frac{6\sqrt{3}f_{f}Q_{s}}{abc\zeta_{r}\pi\sqrt{\pi}}e^{-( \frac{3x^{2}}{\pi^{2}}+\frac{3y^{2}}{\pi^{2}}+\frac{3y^{2}}{\pi^{2}})} \tag{4}\]  
\[q_{r}(x,\ y,\ z)=\frac{6\sqrt{3}f_{r}Q_{s}}{abc\zeta_{r}\pi\sqrt{\pi}}e^{-( \frac{3x^{2}}{\pi^{2}}+\frac{3y^{2}}{\pi^{2}}+\frac{3y^{2}}{\pi^{2}})} \tag{5}\]  
\[f_{f}+f_{r}=2, \tag{6}\]  
Figure 1: Gaussian heat source distribution shape (\(Q_{s}\)).