## 5 Conclusions  
In this paper, the theoretical background of notch root approximations is reviewed as far as it is necessary for practical implementation in the FKM Guideline nonlinear [25]. The application of notch root approximations requires the solution of a nonlinear equation. This solution is usually performed using [PERSON]'s method. The implementation of this method is the subject of this paper. In this context, the aspects of stability, performance, formulation of the root finding problem and accuracy are considered in more detail. This is interesting, for example, if the notch root approximation is to be applied to each node in the scope of variant calculations or in a postprocessing of a linear-elastic FE analysis.  
The FKM Guideline nonlinear uses the extended Neuber's rule and the notch root approximation according to Seeger and Beste. Both were examined more closely in this paper with regard to the aspects mentioned above. The results obtained show that (not depending on the notch root approximation method):  
* 10 Newton steps, as suggested in [25], almost always lead to a desired accuracy but are not required in most cases.
* In the case of a script language, here represented by MATLAB, the formulation of the root finding problem in the form of a quotient between the strain calculated via the material law and the strain calculated via the notch root approximation leads to the best computational performance.
* In the case of a language compiled in machine code, here represented by Fortran, the formulation of the root finding problem in the form of a difference in the strain calculated via the material law and strain calculated via the notch root approximation leads to the best computational performance. In particular, for the notch root approximation according to [PERSON] and [PERSON], the following findings were also obtained:
* If [PERSON] and [PERSON]'s notch root approximations are used and an incorrect termination criterion is selected, then there is a risk that the automated calculation will produce complex numbers that cannot be interpreted meaningfully in terms of a service life calculation. Here, the use of the termination criteria with a desired accuracy leads to the most stable solution, in the sense that no complex number arose in the examined parameter field.
* The analytical derivative of the notch root approximation according to [PERSON] and [PERSON] cannot be trivially used. The numerical approximation via the difference quotient is much easier to handle and leads to minor losses in performance when used in MATLAB and to comparable performance when used in Fortran.  
For implementation in engineering practice, the overall recommendation for both notch approximation methods considered here is therefore to use the termination criterion where the quotient of the strain \(\varepsilon_{\text{el,pl,RO}}/\varepsilon_{\text{el,pl,NRA}}\) converges, which leads to a constant relative accuracy over all orders of magnitude, especially when using a script language. The source code implementing all considered variants in this study are provided as a reference implementation in [43].  
Conceptualization, R.B., L.M., M.W. and A.E.; methodology, R.B. and M.W.; software, R.B.; validation, L.M., M.W. and A.E.; investigation, R.B.; resources, L.M.; writing--original draft preparation, R.B.; writing--review and editing, M.W., L.M. and A.E.; visualization, R.B. and M.W.; supervision, A.E.; project administration, M.W. and A.E. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding. However, we acknowledge support from the Open Access Publishing Fund of Clausthal University of Technology.  
The source code implementing all considered variants in this study is openly available as MATLAB source in [43].  
The authors declare no conflict of interest.