## 2 Material Modelling for Anisotropic and Heterogeneous Welds
### First Grain Modelling for Wave Propagation Simulation  
In the literature, authors proposed models of grain structures to describe materials for simulations. The description of grain structure is frequently simple. The proposed structure is mainly symmetrical for obvious practical reasons to reduce modelling complexity. But real structures are in general non-symmetrical. Figure 2 presents six examples of simulated grain structures, with the first interesting results for the analysis of UT results. In part (a), the grain structure proposed by [PERSON] [34] is represented and corresponding mathematical functions are given in Formula 3. Angles \(\theta\) between grain direction and axis. By are thus calculated and drawn in Figure 2 part (a) and part (b). The index \(i\) is equal to 1 or 2, allowing for a variation depending on the chamfer parameters, which could thus be non-symmetrical. Simulation of the wave propagation is obtained with the RAYTRAIN code, which calculates the wave central ray. \[\tan\theta_{i}=\frac{T_{i}(D_{i}+z\tan(\alpha_{i})}{y^{\eta}}\ for\ y>0 \tag{3}\]  
Therefore, this equation calculates the grain direction for the right part of the weld (\(i=1\)) and the left part of the weld (\(i=2\)). Several chamfer geometries can be defined, and also, the set of equations can be used to create a non-symmetrical grain structure. The parameter \(\eta\) range is comprised between 0 and 1. This parameter represents the orientation speed of the grains from the edge towards the vertical direction of the weld vertical axis. [PERSON] operates in the same way [35]. [PERSON] also uses this type of structure by defining several layers of transverse isotropic material to describe the weld, as shown in part (c) of Figure 2[36]. In his work, [PERSON] uses the ray code 3D-Ray-SAFT to obtain the direction of ultrasound propagation with a 3D simulated grain structure. The grain orientation of an X-veld is expressed with a three-coordinate N orientation vector [37]. The grain structure starts perpendicular to the weld edges and ends vertically in the centre of the weld. [PERSON] qualitatively compares the experimental CSCAN with the simulated CSCAN and observes a good agreement. [PERSON] [38] also chooses a simplified symmetric weld grain structure ((part (d)) of Figure 2) to make use of the EFIT (Elastodynamic Finite Integration Technique) code. Quantitative comparisons are made between experimental and calculated images (BSCAN). [PERSON] [39] uses this code with the Ogilvy grain structure in transverse isotropic symmetry. If the grain structure develops into a non-symmetrical structure, [PERSON] defines various homogeneous domains with different sizes (((part (e) and (f)) of Figure 2) [35]. [PERSON] [33] uses the ULTSON 2D finite element code enabling the visualisation of the whole beam and not only of some rays. He introduces a complex grain structure in 2 \(\times\) 2 mm\({}^{2}\) squares whose orientations correspond to those measured on a real multipass weld micrograph or proposes to model the structure using large areas by clustering in a bigger region several smaller regions of close orientations. The simulated BSCANs are comparable to the experimental BSCANs. The Champs-Sons semi-analytical code simulates the ultrasonic field for a homogeneous anisotropic material. Associated with the Mephisto code, which models the interaction of the beam with defects, it provides an efficient tool for modelling weld inspection [40; 41]. This code has the advantage of being able to simulate ultrasound propagation in 3D geometries. In contrast to the ray methods, by integrating the brush method, this code makes it possible to describe the variation in the wave amplitude. These ideas to describe an acoustically inhomogeneous anisotropic material continue to be tested regularly in various welding configurations in 2D situations, often using ray tracing models in 2D situations [43; 44].