## 3 Functionality of Key Modules
### Local Fatigue Assessment
#### 3.4.3 Elastic Strain Energy Density
As the elastic strain energy density is computed based on the fractions of the element cells belonging to the evaluated defect point-related control volume \(\Omega\), Table 9 summarizes the effect of the mesh density setting for the circular hole-in-plate model using an increased incremental load step of \(\Delta\varphi=30^{\circ}\). The computation used the same settings as the control volume verification study detailed in Table 4, evaluating the elastic strain energy as an exceedance limit of 97.5% for each simulated load vector. Based on the ratio of the average mesh-seed-to-control radius, it is sufficient to use a mesh-seed of about one-third of the control radius \(R_{c}\) to assess the ESED properly. Further mesh refinement does not improve the accuracy. It should be noted that the results listed in Table 9 are based on computing the cell fraction of the initial mesh, and do not include additional partitioning or sub-modeling of the initial free-mesh pattern in the control volume. Even for the coarse mesh with 42 points on the defect loop, the deviation in computed elastic strain energy density within the control volume was not severe, as the obtained ESED value was only about 5% lower; see Figure 24. Thus, the ESED fatigue assessment can be applied with a coarser mesh pattern than when using the TCD method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Points on Circle** & **Average Mesh-Seed** & **ESED \(\Delta\overline{W}_{sharp}\)****1** & **ESED \(\Delta\overline{W}_{blunt}\)****1** \\ \hline
42 & 0.15 mm & \(1.157\times 10^{-5}\) mJ/mm\({}^{3}\) & \(0.954\times 10^{-5}\) mJ/mm\({}^{3}\) \\
78 & 0.10 mm & \(1.214\times 10^{-5}\) mJ/mm\({}^{3}\) & \(1.012\times 10^{-5}\) mJ/mm\({}^{3}\) \\
98 & 0.08 mm & \(1.215\times 10^{-5}\) mJ/mm\({}^{3}\) & \(1.014\times 10^{-5}\) mJ/mm\({}^{3}\) \\
131 & 0.06 mm & \(1.227\times 10^{-5}\) mJ/mm\({}^{3}\) & \(1.019\times 10^{-5}\) mJ/mm\({}^{3}\) \\
196 & 0.04 mm & \(1.227\times 10^{-5}\) mJ/mm\({}^{3}\) & \(1.019\times 10^{-5}\) mJ/mm\({}^{3}\) \\
392 & 0.02 mm & \(1.227\times 10^{-5}\) mJ/mm\({}^{3}\) & \(1.019\times 10^{-5}\) mJ/mm\({}^{3}\) \\ \hline \hline \end{tabular} \({}^{1}\) Standard value of control radius \(R_{c}=0.19\) mm is consequently used.  
\end{table} 
Table 9: Effect of mesh-seed on evaluated ESED for sharp and blunt notch assessment of circular hole-in-plate geometry. To evaluate the elastic strain energy density for the superposition of derived load cases, referring to Method 3 in Figure 2c, it is not valid to use the stress tensor transformation equations for the scalar ESED results. Figure 25 depicts the contour plots of the coarse-meshed model of a circular hole (\(\rho=1\) mm), with an angle step of \(\Delta\varphi=45^{\circ}\). The ESED scalar computation is applicable for load vector angles of \(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\), which are both computed at the first tension load case without shear contribution. However, for other load vectors, the superposition of the three basic load cases is invoked and, thus, the derived results are misleading. To apply the elastic strain energy density fatigue assessment to the superposition method as well, both the stress and strain tensors and, subsequently, the absolute maximum scalar value of the ESED tensor must be computed. The weighting of the stress tensor is defined in Equation (7), and the strain tensor is transformed similarly. \[\overline{\vec{\epsilon}}(\varphi)=f_{XX}(\varphi)\overline{\overline{\overline{ \epsilon}}_{XX}}(\varphi)+f_{YY}(\varphi)\overline{\overline{\overline{ \epsilon}}_{YY}}(\varphi)+f_{XY}(\varphi)\overline{\overline{\overline{\epsilon }}_{XY}}(\varphi) \tag{12}\]  
Figure 24: ESED evaluation of coarse-meshed circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’) by sub-case method and incremental load step of \(\Delta\varphi=15^{\circ}\): (**a**) Level plot; and (**b**) Plots along defect contour.