## 2 Methodology
### Software Structure and Modularization  
Consolidating the requirements of the N-SIF, TCD, and ESED local fatigue approaches for numerical implementation, it is evident that tasks such as planar curvature computation, evaluating the number and fractions of elements within or across the control volume, and exploiting the principal stresses along varying paths should be set up as common processes for all approaches. Further on, the overall simulation work can be divided into the pre-processing stage, solving of linear-elastic load cases, and post-processing of required datasets. Figure 4 depicts the overall workflow for the processing stages. The pre-processing stage takes as input only a planar mesh depicting the imperfection (or the imperfection network) to be assessed. Of course, the quality of the input mesh must first be checked to avoid distorted mesh elements with negative Jacobian. The automated workflow for each method is called by a main Tcl wrapper script, which allows one to set up global variables before starting the automated evaluation. In pre-processing, the input variables are the value of the critical radius \(R_{c}\), which is related to the El Haddad length \(a_{0}\) and plane model formulation, the material properties (in terms of Young's modulus \(E\) and Poisson's ratio \(\
u\)), a flag to enable multi-processing of user-defined Python 3\({}^{\text{TM}}\) routines, and a flag to distinguish between plane strain and plane stress conditions. In addition, in the case of the sub-case approach (see Method 2 in Figure 2b), the value of the load angle increment \(\Delta\varphi\) must also be defined. The execution of the linear-elastic analysis may be triggered within the pre-processing script; however, as multiple jobs can be put into the queuing system of the high-performance cluster available at Montanuniversitat Leoben, solving the linear-elastic problem was treated as a stand-alone step using commercial finite element software. After conducting the linear analysis, the obtained output databases must be evaluated regarding local fatigue approaches; however, animations of the stress or elastic strain energy density distributions should first be recorded. The latter are not necessary for local fatigue assessment but are helpful for visualizing the change in the highly stressed volume with varying load vectors of irregularly shaped planar imperfections. Global input parameters for post-processing include the El Haddad length parameter \(a_{0}\), which is important for TCD and ESED evaluation. Next, the maximum evaluation distance of the TCD method \(L_{m,max}\) for finite life assessment has to be set. In addition, in the case of generating the derived results by superposition (see Method 3 in Figure 2c), the chosen incremental step angle \(\Delta\varphi\) must be defined. It should be noted that both methods can use different load step angle increments; however, for comparison of the numerical efficiency between the sub-case and superposition methods, the angle step was kept equal for both methods and each study case. The conducted local fatigue approaches are finally stored in a common database, which can be used for probabilistic fatigue evaluation by the user. In [66], it was suggested to use an exceedance limit of 97.5% for the maximum fatigue factor. However, continuous probability methods can also be applied to characterize the course of the local fatigue factor along the circumference of the defect; for example, by evaluating the data as a two-parametric beta distribution [69; 70]. Figure 4: Structured workflow for local fatigue assessment of planar imperfections. As linear-elastic studies can be carried out by numerous commercial solvers [42], the pre- and post-processing should be as versatile as possible. Hence, as a general framework for finite element-based software development, Altair(r) HyperWorks(r)[71] was chosen, while Abaqus(r) was selected to solve the linear-elastic study.