## 3 Numerical Validations of 3D Embedded Flaws Simulations
### Longitudinal Waves  
P waves are first considered. The first TOFD configuration is described in Figure 2a. The specimen includes a 5 mm (or 0.5 mm) high rectangular crack with a varying extension (perpendicular to the plane of Figure 2a). It is inspected using P45 waves in TOFD mode at 2.25 MHz. Figure 2: (**a**) Inspection of a 5 mm high (height represented by a red segment) and varied extent rectangular crack inspected using P45 waves in TOFD mode at 2.25 MHz; Echo magnitude versus the flaw extension (**b**) for the 0.5 mm high crack and (**c**) for the 5 mm high crack. The first numerical comparison of models consists in studying the effect of the flaw extension on maximal flaw echo amplitude (Figure 2b). For both the 0.5 mm and 5 mm high cracks, very important differences are observed between the FEM and PTD models in terms of maximal echo amplitudes. Such differences are more important for the smallest height. Nonetheless, for the 5 mm height differences are not negligible even for large extensions for which the echo magnitude tends to stabilise (25 mm is close to the beam width). For this 5 mm height, the echoes from the top and bottom edge are well dissociated, and the amplitude noticeably oscillates as a function of the extension for the FEM model. These oscillations may be due to interference between waves scattered from each edge and their corners, the latter being better predicted by FEM than by the PTD model. The PTD model used here relies on the incremental Huygens model to account for the finite edge extension [16; 26; 53; 54]. The second numerical comparison in the previous P45\({}^{\circ}\) TOFD configuration (Figure 2a) consists of studying the effect of the flaw height on maximal flaw echo amplitude for the 25 mm extent crack (extension large enough to include the beam width). The flaws' heights from 0.5 to 5 mm are studied, which correspond to adimensional wave number 0.6 < ka < 6. Again, significant differences are observed in Figure 3a between the FEM and PTD models in terms of maximal echo amplitudes, especially for the smallest heights. Whereas the echo amplitude is quite constant versus the height for the PTD, the FEM model predicts higher amplitudes when the scattered waves are mixed for small heights. By comparing the Ascans simulated by both the 3D FEM and PTD and generated by the 5 mm high and 25 mm extent rectangular crack (Figure 3b), the two models lead to a similar prediction from the top edge response, but PTD provides an underestimation for the bottom edge echo (which yields the maximal amplitude from the flaw). This is due to the use of P45\({}^{\circ}\) waves and the presence at the 38\({}^{\circ}\) inspection angle of a minimum predicted by GTD for the amplitude of the bottom edge echo versus the angle of observation in TOFD. As shown in [55], GTD and CIVA ATHENA (hybrid FEM model) both predict a minimum, but it is much more pronounced with GTD. Consequently, the difference of prediction of this minimum by the two models explains the gaps obtained in terms of echo magnitude even for large flaw heights in TOFD P45\({}^{\circ}\). The FEM model also predicts later contributions, notably due to the Rayleigh waves propagating on the flaw surfaces. The previous section concludes that the P45\({}^{\circ}\) waves TOFD configuration exhibits differences between PTD and FEM predictions due to the presence of a minimum in the directivity pattern for a particular observation angle. In the next P60\({}^{\circ}\) waves TOFD configuration, the PTD and FEM models are expected to lead to similar results for large flaws. When comparing the Ascans simulated both by the 3D FEM and PTD models using inspection at 60\({}^{\circ}\) still at 2.25 MHz (configuration shown in Figure 4a), the two models give rise to similar echoes for both the top and bottom edges (Figure 4b).