## 3 Results  
### Experimental Validation  
For the validation experiments, 2 mm thick 316L austenitic chromium-nickel steel sheets were used. The chemical composition of the material used was measured with spark optical emission spectroscopy and is given together with the corresponding standard DIN EN10088-3 [20] in Table 1. The steel sheet was bead-on-plate welded in complete penetration keyhole mode. The laser beam welding experiments were performed with a Trumpf 16002 laser system, using a laser power of 2.3 kW, a welding speed of 1.2 m/min, a focal diameter of 420 um, a focal position of 5 mm above the top surface of the steel sheet, and an Argon shielding gas with a flow rate of 50 L/min and a nozzle angle of 35\({}^{\circ}\). Note that all measured data utilized to validate the derived equations for the three main crystallization parameters were performed according to the ISO/TS 18166:2016 standard providing a workflow for the execution, validation, verification, and documentation of a numerical welding simulation within the field of computational welding mechanics [21]. First, the assumption of a two-dimensional crystallization problem is verified by the weld seam width in the metallographic cross-section shown in Figure 3 (left). To obtain the metallographic samples, the specimen was first polished to obtain an even surface. The corresponding microstructure was then visualized in the cross-section and the horizontal \begin{table}
\begin{tabular}{c c c c c} \hline  & **C** & **Cr** & **Ni** & **Fe** \\ \hline DIN EN 10088-3 & \(\leq\)0.03 & 16.5–18.5 & 10.0–13.0 & bal. \\ Measured & 0.03 & 16.95 & 10.57 & bal. \\ \hline \end{tabular}
\end{table} 
Table 1: Measured and standardized chemical composition of the austenitic 316L steel in wt.%section by using color etching according to Bloech and Wedl. As seen, the weld seam width is practically constant along the thickness of the steel sheet, i.e., parallel seam flanks, and differs only slightly with an averaged deviation of below 3%. For more details, see [22] as well. Hence, the solidification problem can be considered two-dimensional, and the derived equations, Equations (5)-(10), can be utilized for the analysis of the crystallization problem. Congruously, for the validation of the theoretical results, only experimental data obtained from the top surface of the sheet are needed. In Figure 3 (right), an exemplary metallographic horizontal section from the top surface of the welded sheets is shown. This was obtained by polishing and subsequent color etching according to [23]. As observed in the figure, there are several solidification lines, directly underneath the top surface, often referred to as ripples, highlighting the local weld pool boundary at the time of solidification. Using these ripples, the length and width of the rear part of the weld pool boundary can be estimated and thus reconstructed by elliptical functions or Lame curves. Note as well that the weld pool shape on the top surface could be experimentally obtained using a high-speed camera during the process, see, e.g., [13], or by theoretical correction of the weld end crater dimensions [22; 24]. For the sake of completeness, the rear part of the weld pool boundary is reconstructed using both methods. Thereby, the length and half-width of the weld pool rear part were kept constant with \(L_{\mathrm{r}}=3.33\) mm and \(b=0.87\) mm, respectively, but the values for \(m\) and \(n\) were varied to find an optimal parameter combination. The comparison of the obtained results is shown in Figure 4. It can be easily seen that the results obtained using ellipses, \(m=n=2\), lead to a distorted weld pool boundary with quite a different radius of the curvature. This deviation in the weld pool shape will directly influence the calculation of the crystal axis, and thus, as explained above, the subsequently calculated growth rate and cross-sectional area of the columnar crystal.