## 4 Evaluation of Performance and Accuracy of Different Implementations
### Implementation into Software Code  
To evaluate the abovementioned different variants of the notch root approximation implementation, two different programming languages are utilized:  
* The software environment MATLAB (version 2020b) was used to represent script languages and
* Fortran (Intel Ifort 2019 Update 5 using \"-O3\" optimization) was used to represent compiled languages in machine code.  
Script languages and compiled programming languages can be optimized differently for the respective calculation tasks. This in turn can lead to situations where a termination criterion for the Newton iteration, which results in high performance in one case, is not optimal in the other case. For this reason, both of the abovementioned programming languages were used for this study.  
All calculations were carried out separately for the variants of the termination criteria. For the notch root approximation of Seeger and Beste, in addition to the termination criterion, the type of derivation (analytical derivation according to Equations (24) and (25) or numerical approximation according to Equation (26)) is also varied here. Therefore, the following nine variants are investigated:  
1. Enhanced Neuber using the analytical derivative and termination after 10 fixed steps
2. Enhanced Neuber using the analytical derivative, termination if Equation (10) falls below \(\epsilon_{\mathrm{Vd}}\)\(=10^{-7}\)
3. Enhanced Neuber using the analytical derivative, termination if Equation (20) falls below \(\epsilon_{\mathrm{Vq}}\)\(=10^{-3}\)
4. Seeger/Beste using the analytical derivative, termination after 10 fixed steps
5. Seeger/Beste using the analytical derivative, termination if Equation (14) falls below \(\epsilon_{\mathrm{Vd}}\)\(=10^{-7}\)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Range** & **Number of** \\  & & & **Equidistant Steps in the Range** \\ \hline Limit load factor & \(\mathrm{K_{p}}\) & 1–8 & 10 \\ Tensile strength & \(\mathrm{R_{m}}\) & 300–1200 MPa & 100 \\ (resulting) Cyclic hardening coefficient & \(\mathrm{K^{\prime}}\) & 640–3500 MPa & (same as \(\mathrm{R_{m}}\)) \\ (resulting) Cyclic hardening exponent & \(\mathrm{n^{\prime}}\) & 0.187 & 1 \\ (resulting) Cyclic yield strength & \(\mathrm{R_{p0,2}^{\prime}}\) & 200–1100 MPa & (same as \(\mathrm{R_{m}}\)) \\ Maximum linear-elastic stress range & \(\Delta\sigma_{\mathrm{el,max}}\) & \(0.1\cdot\mathrm{R_{p0,2}^{\prime}}\)–0.8 \(\mathrm{R_{p0,2}^{\prime}}\cdot\mathrm{K_{p}}\) & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Investigated parameter space for the applicability and performance of the notch approximation with different termination criteria when using [PERSON]’s method.  
Seeger/Beste using the analytical derivative, termination if Equation (21) falls below \(\epsilon_{\text{Vq}}=10^{-3}\)
* Seeger/[PERSON] using the numerical approximation of the derivative, termination after 10 fixed steps
* Seeger/[PERSON] using the numerical approximation of the derivative, termination if Equation (14) falls below \(\epsilon_{\text{Vd}}=10^{-7}\)
* Seeger/Beste using the numerical approximation of the derivative, termination if Equation (21) falls below \(\epsilon_{\text{Vq}}=10^{-3}\)