## 2 Estimation of the Welding Parameters
### Parameters of Welding Spots  
In welding simulations, the size of the welding spot has a large influence on the final results. The most widely used mathematical model was developed by [PERSON] et al. [24], where the welding heat sources are based on the Gaussian distribution of power density in space. The first concept of welding simulations was to use an elliptical heat source, but the results were inconsistent with the experimental results. The temperature gradient in front of the heat source was not as steep as expected. Therefore, a combination of two  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Filler Material** & **Current** & **Vollage (V)** & **Welding Heat Input** & **Electrode** & **Electrode** & **Run-Out** & **Volume** & **Real** \\
**Side.** & **(A)** & **Voltage (kJ/mm)** & **(mm)** & **(mm)** & **Length** & **(mm)** & **(mm3)** & **Valtage Area** \\ \hline EAS 2-IG & 120 & 14.4 & 0.2599 & 2.4 & - & 50 & - & -* \\ ESAB OK & 140 & 25.6 & 0.7168 & 4.0 & 300 & 130 & 3769.91 & 28.999 \\ 61.30 & 150 & 26.0 & 0.7800 & 4.0 & 300 & 125 & 3769.91 & 30.159 \\ ESAB OK & 150 & 26.0 & 0.7800 & 4.0 & 300 & 150 & 3769.91 & 25.133 \\ 61.30 & 180 & 27.2 & 0.9792 & 5.0 & 300 & 160 & 5890.47 & 36.816 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Estimated nugget areas based on the size of the electrode.  
ellipsoidal heat sources was proposed. The geometric parameters and the Gaussian volume distribution are shown in Figure 3.  
The power density distribution for the front quadrant can be expressed by the following Equation [24]:  
\[q(x,y,z,t)=\frac{6\sqrt{3}f_{f}Q}{ba_{f}d\pi\sqrt{\pi}}\mathrm{e}^{(-3x^{2}/b^{2 })}\mathrm{e}^{(-3y^{2}/d^{2})}\mathrm{e}^{(-3[z+v(\tau-t)]^{2}/a_{f}^{2})} \tag{4}\]  
In the case of the rear quadrant, a similar equation can be used for the calculation of the power density distribution:  
\[(x,y,z,t)=\frac{6\sqrt{3}f_{r}Q}{ba_{r}d\pi\sqrt{\pi}}\mathrm{e}^{(-3x^{2}/b^{ 2})}\mathrm{e}^{(-3y^{2}/d^{2})}\mathrm{e}^{(-3[z+v(\tau-t)]^{2}/a_{f}^{2})} \tag{5}\]  
where \(f_{f}\) is the fraction of the heat deposited in the front quadrant, \(f_{r}\) is the fraction of the heat deposited in the rear quadrant, \(Q\) energy input rate (W), \(\tau\) lag factor for defining the position of the source at time \(t=0\), \(v\) is the welding speed (m/s), \(a_{f}\) is the front length of the heat source, \(a_{r}\) is the rear length of the heat source, \(b\) is the width of the heat source, \(d\) is the depth of the heat source. The equations are written in a fixed \((x,y,z)\) coordinate system. The transformation between the fixed and moving coordinate systems is \(\zeta=z+\) vs. \((\tau-t)\). From these equations, it is apparent that to simulate the weld, the front length, rear length, width, and depth of the heat source must be determined. The depth and the width can be identified from a micrograph, while the other values can be estimated with the equations \(a_{f}=0.6b\) and \(a_{r}=2.0\ldots 2.5b\). If the first estimation is based on a construction plan, the so-called design throat (\(a\)) can be used to estimate the other geometric parameters. In this case, the depth and width of the heat source can be given by the simple equation \(d=a_{r}+2.0\ldots 5.0\) mm and \(b=a_{r}+1\ldots 2\) mm.