## 4 Extension of the MINA Model to Gas Tungsten Arc Welding
As for the SMAW process, the extension of the MINA model takes into account the incline of the pass deposit, this incline varying according to the spatial position related to the chamfer. When a pass is at the boundary of the chamfer or when a pass rests laterally on a single pass, the grain orientation is greatly inclined, corresponding to a global inclination of the thermal gradient direction. An example of this phenomenon is shown in Figure 5a, with an incline \(\beta\). In the case of the last pass in a layer, this pass lies between the two passes that surround it, and the grain orientation is vertical. As a consequence, the directions of the temperature gradient are modified by a value \(\beta\) for all passes except for the last one for each layer. With Equation (7), [PERSON] proposes to calculate \(\beta\) as the result of the non-symmetrical balance of heat flows [65]. \[\tan\beta=\frac{H_{p}(1-R_{V})(1+\frac{H_{p}}{H_{p}+W_{p}})}{W_{p}+H_{p}(1-R_{V })(\frac{W_{p}}{H_{p}+W_{p}})} \tag{7}\]  
The orientation of a grain in a mesh (i,j) depends on other orientation directions in adjacent meshes and depends on the local temperature gradient. An epitaxy rule is defined using the difference between the average direction in the mesh (i,j) neighbourhood and the gradient direction at this location. If this difference is greater than 40\({}^{\circ}\), the grain direction follows the gradient direction. For meshes close to the chamfer, the grain directions are defined equal to the temperature gradient direction. The extension of the MINA model proves to be consistent. This has classically been assessed using the comparison between grain growth directions obtained with metallographic analysis and with directions calculated by the model. Two map examples from [65] are reported in Figure 6.  
The evaluation of grain directions is made with a mesh size of 0.5 \(\times\) 0.5 mm\({}^{2}\). A global average error \(\theta_{G}\) and the standard deviation \(\sigma_{G}\) are calculated. This global error is correlated to the grain orientation differences within the weld, but also along the chamfer and free surface. In the latter case, differences can be higher due to the geometric limitations of the model: the pass shape is defined as a rectangle, a straight chamfer that does not take into account the dilution at chamfer boundaries and simplification in the modelling of the  
Figure 5: Analysis of GTAW pass morphology. (**a**) passes shape. (**b**) bead parameters. (**c**) simplified geometry for modelling. incline of passes. A second average error \(\theta_{\text{W}}\) is then calculated within the welded area. The corresponding standard deviation is, in that case, \(\sigma_{\text{W}}\). For Figure 6a, the global statistics are \(\theta_{\text{G}}=20^{\circ}\) and \(\sigma_{\text{G}}=16.4^{\circ}\), and become \(\theta_{\text{G}}=18.2^{\circ}\) and \(\sigma_{\text{G}}=15.5^{\circ}\) in the welded zone. For Figure 6b, the global statistics are \(\theta_{\text{G}}=13.8^{\circ}\) and \(\sigma_{\text{G}}=15.2^{\circ}\), and in the welded zone, \(\theta_{\text{G}}=11.8^{\circ}\) and \(\sigma_{\text{G}}=12.7^{\circ}\). Modelling differences within the welded areas are small, a little higher than average results obtained for SMAW welds. Results demonstrate that the extension of the MINA model is in good agreement with the metallography analysis for GTAW welds.