## 4 Discussion on Numerical Efficiency
In addition, Table 12 details the effect of enabling multi-processing. The used computing hardware features up to 24 cores; however, these were addressed only by the parallelized workers in the Python code functions, as indicated in Figure 4. The averaged degree of utilization invoking multi-processing, observed as a one-minute load average, was up to 13, depending on the time fraction spent in the multi-processing modules. The peak computing load \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Input Mesh** & **Method \#** & **Overall** & **Curvature** & **Cell Fractions** & **\# Vertices** & **\# Elements** \\ \hline circle\_lin\_ & 1 * & 15 s & 2 s & 5 s & 42 & 796 \\ R1 mm\_v1 & 2 & 14 s & 2 s & 4 s & 42 & 442 \\ R1 mm\_v1 & 3 & 15 s & 2 s & 5 s & 42 & 796 \\ \hline ellipsis\_ & 1 * & 65 s & 9 s & 47 s & 100 & 4770 \\ lin\_rot0 deg\_v1 & 2 & 40 s & 9 s & 23 s & 100 & 1603 \\  & 3 & 65 s & 9 s & 47 s & 100 & 4770 \\ \hline defect\_elongated\_ & 1 * & 155 s & 25 s & 120 s & 178 & 7954 \\  & 2 & 108 s & 24 s & 72 s & 178 & 3535 \\ lin\_v1 & 3 & 155 s & 25 s & 120 s & 178 & 7954 \\ \hline circle\_fine\_ & 1 * & 992 s & 118 s & 850 s & 400 & 20,093 \\ lin\_R1 mm & 2 & 469 s & 118 s & 335 s & 400 & 6533 \\  & 3 & 992 s & 118 s & 850 s & 400 & 20,093 \\ \hline defect\_elongated\_ & 1 * & 1460 s & 95 s & 1320 s & 356 & 31,656 \\ fine\_lin\_v1 & 2 & 618 s & 93 s & 504 s & 356 & 13,519 \\  & 3 & 1460 s & 95 s & 1320 s & 356 & 31,656 \\ \hline \hline \end{tabular}  
* Computation time estimated for single load vector based on superposition method.  
\end{table} 
Table 10: Measured computational effort for pre-processing of the three characteristic defect geometries using single-core processing for one load vector (\(\varphi=0^{\circ}\) ). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Input Mesh** & **Method \#** & **Overall** & **Derive Sub-Cases** & **Animation** & **Odb2 Mat** & **Local Fatigue Assessment** \\ \hline circle\_lin\_ & 1 * & 21 s & 0 s & 1 s & 4 s & 16 s \\ R1 mm\_v1 & 2 & 9 s & 0 s & 1 s & 4 s & 3 s \\  & 3 & 24 s & 3 s & 3 s & 1 s & 16 s \\ \hline ellipsis\_ & 1 * & 73 s & 0 s & 1 s & 5 s & 67 s \\ lin\_rot0 deg\_v1 & 2 & 33 s & 0 s & 1 s & 5 s & 26 s \\  & 3 & 76 s & 4 s & 3 s & 1 s & 67 s \\ \hline defect\_elongated\_ & 1 * & 148 s & 0 s & 1 s & 5 s & 142 s \\  & 2 & 95 s & 0 s & 1 s & 5 s & 88 s \\ lin\_v1 & 3 & 151 s & 5 s & 3 s & 1 s & 142 s \\ \hline circle\_fine\_ & 1 * & 536 s & 0 s & 1 s & 7 s & 528 s \\ lin\_R1 mm & 2 & 169 s & 0 s & 1 s & 7 s & 161 s \\  & 3 & 539 s & 5 s & 5 s & 1 s & 528 s \\ \hline defect\_elongated\_ & 1 * & 740 s & 0 s & 1 s & 8 s & 731 s \\ fine\_lin\_v1 & 2 & 271 s & 0 s & 1 s & 8 s & 260 s \\  & 3 & 746 s & 7 s & 7 s & 1 s & 731 s \\ \hline \hline \end{tabular}  
* Computation time estimated for single load vector based on sub-case method, except local fatigue assessment time is taken from superposition method.  
\end{table} 
Table 11: Measured computational effort for post-processing of the three characteristic defect geometries using single-core processing for one load vector (\(\varphi=0^{\circ}\)). factor reached the available number of cores in multi-processing, reflecting the computing power of the invoked Python workers. The comparison in Table 12 indicates that both of the developed methods--either the sub-case procedure (Method 2) or superposition procedure (Method 3)--are preferable over Method 1, which sets up individual models for each load vector, in terms of computational burden. This benefit of multiple workers in the Python code is clearly recognizable for Methods 2 and 3. If the planar model possesses a refined mesh with tens of thousands of elements, multi-processing can further reduce the comparative overall computing time. Finally, Figure 29a illustrates the application of the developed sub-case method to a pore network, following ASTM standard E446 class A1.