# A Comparison between Two Reduction Strategies for Shrouded Bladed Disks
## 4 Numerical Results  
### First Test Cases for Accuracy and Efficiency Analysis  
The two reduction methods described in the previous section are applied to two numerical test cases to compare their capability to obtain a CB-ROM suitable for nonlinear forced response, considering both accuracy and numerical efficiency. The analyses are performed on the FE models of the fundamental sector of two bladed disks, shown in Table 1. The contact interfaces are located at the covers located at the blade tip, called shrouds. For each sector, two different element types are used, linear and quadratic, to check the effect of the dofs number on the efficiency of the reduction methods. It is worth mentioning that quadratic elements are usually preferred to linear elements when the FE model is used not only for modal analysis but also for the stress analysis, since they allow computing non-uniform stresses inside each element. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Case** & **A** & **B** \\ \hline Blades & 27 & 81 \\ \hline \begin{tabular}{c} Element Type \\ Elements \\ Nodes \\ \end{tabular} & \begin{tabular}{c} Linear \\ 16,561 \\ 6052 \\ \end{tabular} & \begin{tabular}{c} Quadratic \\ 16,561 \\ 30,920 \\ \end{tabular} & \
\begin{tabular}{c} Linear \\ 5627 \\ 2454 \\ \end{tabular} &
\begin{tabular}{c} Quadratic \\ 5627 \\ 11,672 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of the numerical test case. To assess the efficiency and the accuracy of the two reduction techniques, multiple ROMs (Table 2) of the fundamental sectors are generated. First, the direct method is applied by selecting an increasing number \(Z\) of slave modes (\(Z=20\), 40 and 60) in order to check the convergence of the natural frequencies of the first 10 modes of the fundamental sector. Then, the two-step method is applied. Three sets of ROMs are generated with an increasing number \(Z_{1}\) of fixed-interface modes during the 1 st step (\(Z_{1}=60\), 120 and 180). For each set, three subsets are generated with an increasing number \(Z_{2}\) of slave modes (\(Z_{2}=20\), 40 and 60) in the 2 nd step. Comparisons are only performed between ROMs generated by the direct method with Z slave modes and ROMs generated by the two-step method with \(Z_{2}\) slave modes in the 2 nd step. In this way, ROMs with the same size are compared. Of course, for the two-step method, the overall calculation time, necessary to perform both steps, is given for a fair comparison. The natural frequencies of the ROMs are compared to the natural frequencies computed with the full wheel model, which, in the case of industrial applications, is of course not available. An example of the results collected for each harmonic index \(h\) is shown in Table 3, where the relative error, computed with respect to the full FE model, in computing the natural frequencies of the test case A is listed for different ROMs. In the table, errors equal to 0.00% must be interpreted as \(<\)0.01%. Figure 4 shows the relationship between the number \(Z\) of slave modes and the accuracy of the ROMs generated with the direct method for two different harmonic indexes representative of a compliant (\(h=1\)) and a stiff (\(h=13\)) disk, representative of what was observed for all of the other harmonic indexes. When \(Z=20\), some of the natural frequencies have more than 1% error, which is reduced to 0.1% with \(Z=40\) and becomes negligible when \(Z=60\). It is worth noting that, for each mode, the convergence rate depends on the harmonic index h. As shown in Figure 4, when \(Z=20\), mode #7 and mode #10 are the less accurate modes when \(h\) equals 1 and 13, respectively. This apparent inconsistency in the ROM results is due to the effect of the harmonic index on the mode shapes. When \(h=1\), the disk is more compliant than when \(h=13\), but the stiffening effect due to the harmonic index affects in different ways different blade modes. As a result, the same