## 2 Methodology  
As the TCD, N-SIF, and ESED approaches are already well-established in the field of notch fatigue design, the first step toward numerically efficient methods for arbitrarily shaped imperfections is to set up a basic simulation study of a V-notched specimen and validate the results using published data. A sharp notched flat plate under tension loading, with parameters of the double lateral V-shaped notch, in accordance with the peak stress study in [40; 41], was used as the basic geometry. The link to the elastic strain energy within the control volume has been detailed in [49]. Blunt V-notch studies relating to N-SIF and ESED assessment have been previously examined in [19]. In summary, the comprehensive analytical equations presented in [19; 49] were used to validate the basic V-notched simulation study depicted in Figure 1. A flat plate with a double lateral V-notch was simulated and geometrically defined according to the plate width \(W\), notch depth \(a\), opening angle \(2\alpha\), and (in the case of the blunt notch) radius \(\rho\). The depicted linear-elastic simulation features plane-strain condition and unified tension loading in the gross-section of the plate. The linear-elastic material properties were defined according to [PERSON]'s modulus \(E=206\) GPA and [PERSON]'s ratio \(\
u=0.3\). Moreover, the critical radius was set to \(R_{c}~{}=~{}0.19\) mm, implying a value of \(a_{0}=0.225\) mm, in accordance with the data published in [57]. This material dataset was used consistently in all following studies. Figure 1: Dimensioned simulation results of a double V-notched plate (width \(W=10\) mm, opening angle \(2\alpha=90^{\circ}\), notch depth \(a=1\) mm, unified tension loading): (a) Sharp notch (\(\rho=0\) mm); and (b) Blunt notch (\(\rho=1\) mm). The discretized geometry of any arbitrarily shaped planar defect must be assessed for each point along the defect circumference, which can be either treated as sharp V-notch, assuming a small notch root radius (see Equation (1)), or as blunt V-notch for larger root radii, implying a stress evaluation ahead of the notch tip (Equation (3)). The stress field in the vicinity of sharp and blunt V-notches was evaluated as a basic reference example, as detailed in Table 1. In addition to the numerical results, the published analytical formulae for N-SIF and ESED are included for comparison, based on matching stress paths in the vicinity of the notch tip. The stress results of the Point Method for the two investigated notches (Figure 1) are given in Table 2. The simulation result is compared to the analytical approach evaluating the stress \(\sigma_{\theta}\) along the notch bisector. The distance to evaluate the Point Method was measured from the notch tip and evaluated as \(a_{0}/2=0.113\) mm for the investigated plane strain load case. In summary, the constructed basic simulation tools provided N-SIF, ESED, and TCD results that matched the published analytical equations for sharp and blunt V-notches well. Next, we extended these basic simulation tools toward arbitrarily shaped defects and imperfection networks in a numerically efficient manner. First, if the stress (or strain) field along a complex-shaped defect is to be evaluated, one of the most straightforward solutions for engineers dealing with finite element codes is to set up individual models. Uniaxial tensile loading is defined as the characteristic load case, where the load vector orientation is changing, implying the study of the stress course within a load angle range of \(0^{\circ}\leq\varphi<180^{\circ}\). The principle approach of this method is sketched schematically in Figure 2a and denoted below as Method 1. If only the peak stress at the defect contour is to be evaluated, this first method is applicable. The necessary re-construction of the model can be carried out manually by the operator, or conducted in a script-based manner (e.g., using Abaqus(r), Python (2016), or other scripting codes).