## 4 Evaluation of Quality and Efficiency
### Reduction in the Number of Load Steps to Be Simulated
#### 4.6.1 Comparison of the Methods on a Notched Flat Specim  
A first comparison of the methods to describe the load-notch-strain curve is carried out on the flat specimen made from steel notched on both sides with a tensile strength of \(\mathrm{R_{m}}=800\) MPa and a stress concentration factor \(\mathrm{K_{t}}=3\) under tensile loading. Figure 19 shows the ratios of the service lives with variable amplitude loading \(\mathrm{N_{var}}\) (see Figure 12) from load-notch-strain curves with \(\mathrm{n_{H}}\) load steps and service lives from the reference load-notch-strain curve (\(\mathrm{n_{H}}=200\) load steps). As the number of load steps increases, the service life converges to the service life with 200 load steps. The number of support points is considered good if the service life deviates less than 1% from the service life from the reference load-notch-strain curve. Table 2 summarises the number of load steps from which this condition is fulfilled. The linear interpolation connects the results of the load steps with the 1 st-degree polynomials. The quality of the service life estimation with linear interpolation depends on the load level. The higher the load is, the more load steps are needed to approximate the load-notch-strain curve with linear interpolation sufficiently well. The smallest maximum deviations among the examined approaches occur in the regression with the power function. However, the power function converges more slowly than the spline interpolation with the converged solution. Depending on the geometry, load height, and material, it may be that the condition \(|\mathrm{N_{var,n}}/\mathrm{N_{var,200}}-1|~{}\leq~{}1\%\) is never fulfilled regardless of the number of load steps \(\mathrm{n_{H}}\) used. For example, the service life of a flat specimen with \(\mathrm{K_{t}}=1.5\) and \(\mathrm{R_{m}}=400\) MPa at a high load under variable amplitude loading converges with the power function to a service life that is 3% less than the service life with 200 directly calculated load steps. This is due to the fact that the regression no longer runs exactly through the FEA results with more than two support points. Thus, the true stresses and strains--i.e., the stresses and strains calculated in the FEA--are not used. The spline interpolation method converges to the exact solution with the smallest number of load steps \(\mathrm{n_{H}}\) for variable amplitude loading. The overview of the approaches on the flat specimen with \(\mathrm{K_{t}}=3\) and \(\mathrm{R_{m}}=800\) MPa at a tensile load shows that the spline interpolation is best suited to simulate the load-notch-strain curve. 6.2 Influence of Material Behaviour and Stress Concentration Factor on the Quality of the Spline Interpolation Method  
In Figure 20, the service lives under variable amplitude loading with load-notch-strain curves of a flat specimen with Kt = 1.5, 3, and 5 and Rm = 400, 800, and 1200 MPa under a tensile load are compared with the service lives from reference load-notch-strain curves determined with nH = 200 load steps. Table 3 again summarises the number of load steps nH necessary to fulfil criterion \(|\mathrm{N_{var,n}}/\mathrm{N_{var,200}}-1|\ \leq\ 1\%\). Figure 20 and Table 3 show that with a higher load, more load steps nH are necessary. For all cases investigated, a total \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Constraint**} & \multirow{2}{*}{**Load**} & \multicolumn{3}{c}{**Minimum Number of Load Steps nH**} \\  & & \begin{tabular}{c} **Linear** \\ **Interpolation** \\ \end{tabular} & \
\begin{tabular}{c} **Power** \\ **Function** \\ \end{tabular} &
\begin{tabular}{c} **Spline** \\ **Interpolation** \\ \end{tabular} \\ \hline \multirow{2}{*}{\(\left|\begin{matrix}\mathrm{N_{var,n_{H}}}\\ \mathrm{N_{var,200}}-1\end{matrix}\right|\ \leq\ 1\%\)} & Low Lmax,1 & 3 & 2 & 2 \\  & Medium Lmax,2 & 10 & 5 & 4 \\  & High Lmax,3 & 20 & 7 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of load steps nH for a flat specimen under tension loading with different approaches.