## 3 Phenomenological Model: The MINA Model
### The Use of the Welding Notebook and Grain Growth Rules
Taking into account these three main phenomena,  
Figure 3: Simplified pass description in MINA model. the algorithm calculates the resulting grain orientation, noted \(\alpha(n)\). To reproduce the slow grain turn from its first direction towards the direction of \(\alpha_{g}\), and to reproduce the selective growth, the model uses an iterative formulation with also a scale factor. Formulae are applied n times in a mesh to simulate changes in grain orientation at a smaller scale. This model is supported by image analysis. Images come from metallographic etching on welds manufactured with the same welding notebook by two different welders [60]. For a typical scale of 2 \(\times\) 2 mm\({}^{2}\), the value of \(n\) is fixed at 10. The algorithm begins by the first pass at the bottom of the weld. For additional passes, the grain orientation is calculated in meshes with \(\alpha(0)\), the grain orientation \(\alpha_{inf}\) in a previous mesh. If a previous orientation is not close to the direction \(\alpha_{g}\), the new grain orientation, \(\alpha(n)\), inclines strongly to this direction \(\alpha_{g}\). This represents the selective growth that eliminates badly oriented grains. If the angular difference is greater than 90\({}^{\circ}\), it is considered that selective growth occurs directly, as described in formulae  
\[n=1,\alpha(0)=\alpha_{inf} \tag{4}\]  
\[if(\alpha(n-1)-\alpha_{g})<\pi/2)t(n)=cos(\alpha(n-1)-\alpha_{g});elseif=t(n )=0 \tag{5}\]  
\[\alpha(n)=t(n)\alpha(n-1)+(1-t(n)\alpha_{g}) \tag{6}\]  
Figure 4 illustrates one of the multiple tests used to verify the model efficiency. An experimented welder has filled in the notebook for a thick multiple-pass weld with 31 passes. In this case, the welding position is horizontal-vertical. Two different chemical etchings reveal grain orientation (Figure 4a) and passes (Figure 4b). Simulated grain orientation is showed in Figure 4c. For image analysis, grain orientations are represented in square meshes of 2 \(\times\) 2 mm\({}^{2}\). MINA parameters are \(R_{L}\)= 0.39, \(R_{V}\) = 0.25, \(\theta_{B}\) = 36\({}^{\circ}\), \(\theta_{C}\) = 28\({}^{\circ}\) and n = 10. Angle parameters \(\theta_{B}\) and \(\theta_{C}\) are more important than those of flat welds. Part (d) in Figure 4 represents differences between measured orientations on micrographs using level lines; the darker areas correspond to higher differences. The MINA model gives orientation maps with an average error of 10 to 15\({}^{\circ}\) for classical flat welding, with a standard deviation of 10\({}^{\circ}\)[60;61]. In the case of horizontal-vertical welding, the grain map is greatly non-symmetrical, because of the remelting of passes and the manual incline of the electrode. This modelling performed with the MINA model has been a great advance in the field, as this information was absolutely not predicted before, and the purely geometrical models only provided  
Figure 4: Example of modelling results for horizontal–vertical welding. simplified descriptions of the weld. Furthermore, studies have demonstrated that if the average error reaches 20\({}^{\circ}\) or more, this indicates that there is a strong error in the welding notebook: several passes are missing, for instance, or the order of passes were not correctly reported. The MINA model has also been used in the case of a bimetallic stainless steel weld [62]. In this study, there is a difficulty, as the metallography analysis shows that the order of passes may be read in two different ways. Nevertheless, the grain orientation is correctly predicted, with a better accuracy than with an analytical model. The study confirms that the knowledge of the correct order of passes is very important for the calculation of the grain map. [PERSON] et al. use the MINA model to supply representative anisotropic weld maps in a complete UT simulation for parametric studies of defect detection [63]. They use a semi-analytical model to simulate the degradation of ultrasonic images caused by the propagation through an anisotropic austenitic material.