## 4 Discussion on Numerical Efficiency  
The overall computation time for local fatigue assessment of a model with defects defines the measure of numerical efficiency. This includes pre-processing, solving, and post-processing time efforts for a combined TCD, N-SIF, and ESED assessment of an input mesh with one or more planar imperfections. The approaches according to Figure 2--namely, Method 1 consisting of individual models per load angle, Method 2 as the sub-case approach, and Method 3 as the superposition procedure--were directly compared. The computation time for Method 1 was assumed to be equal to the superposition method at a load step angle increment of \(\Delta\varphi=90^{\circ}\), as only one load step is evaluated per model in this case. For Method 1, the curvature computation routine is only counted once, as it is assumed that the evaluated notch geometry results are also valid for other load angles. However, the computation time of the element fractions within the control volume must be considered for each load angle. Alternatively, sub-modeling can be used within Method 1 for the control volume, but the time effort for this additional sub-modeling step ought to be on the same scale. Table 10 depicts the pre-processing time for the three basic defect geometries, with reference to Figure 10. Single-processing was used for all computations in order to highlight the influencing factors; the measured elapsed time includes only the evaluation of a single load vector. First, it is deducible that the curvature computation effort scales with the number of points in the defect loop, and the chosen evaluation method has no effect on the measured time effort to compute curvature. Second, the evaluation of cell fractions within the control volume depends not only on the number of vertices but also on the total element count within the mesh. This is also reasonable, as the whole input mesh is used as input for each control volume to obtain the appropriate cell fraction lists. In the case of manually setting up a model for each load vector, as in Method 1, the time effort relates to referencing the measured values of the superposition method. In conclusion, the overall pre-processing time effort is quite similar between the methods, but the fraction of control-volume-related calculation becomes more pronounced for larger models. The time to solve the finite element input files of the meshes in Table 6 using Abaqus(r) with two cores was around 12-15 s. Even when the model element count increased to 100,000 elements, the computation time was only about 20 s. Thus, the time effort for solving the linear-elastic problem can be set conservatively to about 20 s per load angle. The post-processing differed between the methods, as shown in Table 11. The sub-case procedure (Method 2) spends only a fraction of the total time to read the Abaqus(r) output as odb-files, and most time effort is put into the local fatigue assessment. It is evident that the local fatigue assessment depends not only on the number of defect points but also on the total element count, as the stress results are interpolated to the points of each vector-path. In the case of the superposition procedure (Method 3), the generation of derived sub-cases must be computed first. The time effort for animation output is about 1% in the case of Method 2, but slightly more for Method 3, as the previously derived sub-cases must be recorded in a per-load step manner first. To the obtained time measures for Method 1, the time fractions of Method 2 are largely applicable, as no additional sub-case generation is needed for a single load vector study. In the case of local fatigue assessment by Method 1,it is assumed that the original mesh is unchanged; thus, the time contribution regarding local fatigue assessment was taken from Method 3, having the same mesh element count. Regarding the time effort to assess the fatigue strength limit under variable load vectors, the subsequent computation used a load angle increment of \(\Delta\varphi=10^{\circ}\).