## 3 Functionality of Key Modules
### Control Volume Assessment  
Computing the number and fraction of cells within the control volume \(\Omega\) requires, as input, the geometric parameters of every point along the defect circumference. Hence, the control volume assessment is conducted in sequence with the curvature computation block. The assessment is distinct between sharp and blunt notch evaluation. The center of the control sphere is located at the assessed point of the defect contour in the sharp notch case, but away from the notch tip for a blunt notch (see Figure 1 for reference). The radius of the control sphere is commonly defined as \(r_{sphere}=r_{0}+R_{c}\).  
To speed up the evaluation of the elastic strain energy density within the control volume for each defect point, the VTK point and cell datasets of the original mesh are used without additional re-meshing. For each cell across the control volume, its fraction is computed by sub-division of the cell to an edge length of less than two micrometers and counting the derived points within the control volume as spheres with a radius \(r_{0}+R_{c}\). Figure 11 displays the fraction of cells for the starting point of the defect contour for circular, elliptical, and elongated defect geometries.  
In sharp notch assessment, the center of the control sphere is always pinned to the evaluated point of the defect contour, implying a value of \(r_{0}=0\) mm.  
However, in the case of blunt-notch assessment, the value of \(r_{0}\) varies significantly (compare to Figure 9). If the control volume \(\Omega\) clips the cell boundary even a fraction, the cell is included in the member list. Only the cells which are within the control volume and chained to the assessed point of the defect loop are considered. For blunt notches, the center of the control volume sphere is shifted from the defect point along the vector path illustrated in Figure 10. This leads to a pronounced dependency of the control volume on the evaluated local curvature in blunt notch assessment.  
The selection of cells for each point of the defect contour is conducted using the VTK database of the planar input mesh. Although the element selection and partitioning may be also done in HyperView(r) using \(\mathrm{Tcl^{TM}}\) scripting, the VTK toolkit implementation in Python presents significant time benefits in computing the clipped cell fraction, due to its enhanced libraries and multi-processing features. Nevertheless, the selection and computing of cell fractions is computationally expensive. The obtained lists are stored in the common VTK database '_<File>.h5_', as depicted in Figure 4. These element fractions are further used in the ESED fatigue assessment.  
Figure 10: Computed vector paths for three defect geometries: (**a**) Circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’); (**b**) Elliptical hole-in-plate (mesh ‘ellipsis_lin_rotddeg_v1’); and (**c**) Elongated defect (mesh ‘defect_elongated_lin_v1’).  
The control volume \(\Omega\) of a bilateral notched plate, as shown in Figure 1, is used as a reference to check the fractional computation procedure. The corresponding defect geometry was a circular hole-in-plate, as depicted in Figure 7a. Partitioning of the geometry led to circular sectors for the sharp and blunt notch control volumes, where \(\Omega_{sharp,circle}=0.059\) mm\({}^{2}\) and \(\Omega_{blunt,circle}=0.184\) mm\({}^{2}\). Table 4 depicts the computed control volumes using the initial free-mesh pattern and fractioning of the cells, instead of partitioning. The control volume was correctly evaluated and independent of the mesh-seed-to-control-radius ratio.