## 3 Functionality of Key Modules
### Curvature Computation
For both elliptical holes (Figure 7b,c), the filter index was comparably low at the start point of the defect loop and opposite to it, indicating the semi-major axis, while increasing for the semi-minor axis at the perpendicular vertices. In the case of the elongated defect (Figure 7d), the curvature map also indicated the vertices with extreme curvature quite well. It should be noted that the depicted normalized curvature maps in Figure 7 visualize the curvature values over the refined and equispaced poly-line data. Thus, arc-length differences between two adjacent vertices are no longer effective for curvature computation (compare with Figure 7a, containing such mesh irregularities on purpose). The computed curvature values were mapped back to the original points of the defect contour and stored in the output database. In addition to the notch radius \(\rho\) of each point along the defect contour, the notch opening angle \(2\alpha\) must also be computed. This is obtained by filtering the previously computed adjacent curvature values, thus retrieving sections where the curvature value is within certain bounds. Figure 7: Curvature workflow exemplified for different defect geometries: (**a**) Circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’); (**b**) Elliptical hole-in-plate (mesh ‘ellipsis_lin_rot0 deg_v1’); (**c**) In-clined elliptical hole-in-plate (mesh ‘ellipsis_lin_rot30 deg_v1’); and (**d**) Elongated defect (mesh ‘defect_elongated_lin_v1’). The implemented filter distance and limit value depend on the spacing of the defect contour points, as well as the control radius (as a smoothing parameter). To illustrate the effect of chosen defect contour discretization distance on computed notch radius, Table 3 lists the results of a short study on the elliptical model shown in Figure 7b. The elliptical shape of the defect was obtained by linearly transforming the points of an evenly spaced circle with a matrix \(M\), given in compressed sparse column notation. The indices for columns and rows are zero-based. \[M=\csc\_matrix([2,0.5,1,1],[0,1,2,3],[0,1,2,3]) \tag{10}\]  
The circle with a radius of one millimeter was thereby linearly transformed into an ellipse. Both semi-axis lengths were evaluated as \(a=2\) mm and \(b=0.5\) mm, and the corresponding radii were \(\rho_{a}=0.125\) mm and \(\rho_{b}=8\) mm. The study detailed in Table 3 included the variation in the number of points, as well as the critical radius \(R_{c}\) as a smoothing parameter. If the critical radius is around the size of the minor semi-axis value, the integrated curvature smoothing becomes dominant and levels the computed radii. However, if the defect contour mesh is refined or the control radius is decreased, the computed curvature values are close to the elliptical semi-axis radii \(\rho_{a}\) and \(\rho_{b}\). Based on the curvature computation of the elliptical geometry, it can be derived that the average distance between two points of the defect contour should be about the size of the expected radius. The value of the critical radius \(R_{c}\) affects the applied Gaussian filter length regarding the computed curvature; however, this is integrated on purpose as an increase in the El Haddad length \(a_{0}\); respectively, the critical radius \(R_{c}\) has an increasing notch supporting effect as well. The course of the local notch radius \(\rho\) and notch opening angle \(2\alpha\) is given in Figure 8 for the previously introduced three geometries (i.e., circular, elliptical hole-in-plate, and elongated defect). In the circular case, the notch radius was evaluated correctly, showing negligible variations in the zone of irregularly spaced cell segments of the defect contour. The constant notch opening angle \(2\alpha\) for the circle is correct, as the curvature is constant along the circumference, and it can be treated as a U-notch.