## 3 Functionality of Key Modules
### Curvature Computation
The ellipse, with 100 data points along the defect contour, exhibited values for the curvature distribution as evaluated for both semi-axes in Table 3. The opening angle of points near the semi-major axis was close to 180 degrees, whereas data points near the semi-minor axis exhibited minimal opening angles. The limitation of the notch radius to convex values was recognizable for the elongated defect, which also possessed concave sections. In these regions, the notch  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Points on Ellipse** & **Extreme Arc-Lengths** & \(R_{c}\) & **Semi-Major Axis Radius** & **Semi-Minor Axis Radius** \\ \hline
50 & 65 \(\upmu\)m/251 \(\upmu\)m & 0.190 mm \({}^{1}\) & 6.97 mm & 0.65 mm \\  & & 0.251 mm & 6.69 mm & 0.65 mm \\  & & 0.065 mm & 7.59 mm & 0.83 mm \\ \hline
100 & 32 \(\upmu\)m/126 \(\upmu\)m & 0.190 mm \({}  
opening angle was limited to a maximum value of 180 degrees, and the notch radius was set to an upper limit of four times the maximum extension of the evaluated defect contour. Although the notch radius and notch opening angle are the two key parameters for the evaluated defect contour, further geometric parameters, such as \(\lambda_{1}\), \(\lambda_{2}\), \(r_{0}\), \(\mu_{1}\), and \(\widetilde{\omega}_{1}\), involved in sharp and blunt notch N-SIF evaluation, are also computed within this task. Figure 9 depicts the course of William's eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\), as well as the distance \(r_{0}\) from the center of the control volume \(\Omega\) to the notch tip. The distance \(r_{0}\) is important for ESED-based evaluation of blunt notches, as it significantly affects the size of the control volume. Finally, the planar curvature evaluation must provide vector paths for each point of the defect contour. Although one could define the vector as the cross-product of the two adjacent cells, this simplification is not meaningful when the evaluation distance increases, as in the case of finite-life assessment using the TCD approach. The course of vector paths originating from defect points is displayed in Figure 10. Each computed vector path was affected by the local curvature distribution in the vicinity of the corresponding defect contour starting point and was linearly extrapolated for increasing evaluation distances. In summary, the planar curvature evaluation focuses on the notch radius and notch opening angle, but provides more geometry-related data, which are stored in the common output database '_<File>_DEFECT.h5'_, as shown in the workflow depicted in Figure 4. The output of the planar curvature evaluation is used in all local fatigue methods. Figure 8: Notch radius and notch opening angle for three defect geometries: (**a**) Circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’); (**b**) Elliptical hole-in-plate (mesh ‘ellipsis_lin_rot0 deg_v1’); and (**c**) Elongated defect (mesh ‘defect_elongated_lin_v1’). Figure 9: William’s eigenvalues and control volume distance for three defect geometries: (**a**) Circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’); (**b**) Elliptical hole-in-plate (mesh ‘ellipsis_lin_rot0 deg_v1’); and (**c**) Elongated defect (mesh ‘defect_elongated_lin_v1’).