# A Comparison between Two Reduction Strategies for Shrouded Bladed Disks
## 4 Numerical Results
### First Test Cases for Accuracy and Efficiency Analysis
\begin{table}
\begin{tabular}{c c c c} \hline
**Direct Method** & \multicolumn{3}{c}{**Two-Step Method**} \\ \hline  & **1 st Set of ROMs** & **2 nd Set of ROMs** & **3 rd Set of ROMs** \\ Z & (\(Z_{1};Z_{2}\)) & (\(Z_{1};Z_{2}\)) & (\(Z_{1};Z_{2}\)) \\
20 & 60; 20 & 120; 20 & 180; 20 \\
40 & 60; 40 & 120; 40 & 180; 40 \\
60 & 60; 60 & 120; 60 & 180; 60 \\ \hline \end{tabular}
\end{table} 
Table 2: List of ROMs generated and used for comparison. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline
**Full** & \multicolumn{3}{c}{**Direct Method**} & \multicolumn{3}{c}{**Two-Step Method**} \\ \hline  & & & & \(Z_{1}=60\) & & & \(Z_{1}=120\) & & \(Z_{1}=180\) & \\
**(Hz)** & \(Z=20\) & \(Z=40\) & \(Z=60\) & \(Z_{2}=20\) & \(Z_{2}=40\) & \(Z_{2}=60\) & \(Z_{2}=20\) & \(Z_{2}=40\) & \(Z_{2}=60\) & \(Z_{2}=20\) & \(Z_{2}=40\) & \(Z_{2}=60\) \\ \hline
**200.18** & 0.00\% & 0.00\% & 0.00\% & 0.00\% &blade mode can change its position in the mode sequence (e.g., the 2 nd blade bending mode can be mode #3 when \(h=1\) and it can be mode #2 when \(h=13\)). Afterwards, the two-step method was implemented with the objective of generating a ROM of the same size (or smaller) with the same accuracy as the one of the ROMs generated with the direct method. The first set of ROMs was generated by selecting 60 slave modes in the 1 st reduction step (Z\({}_{1}=60\)). Results are summarized in Figures 5 and 6. In some cases (e.g., Figure 6), 60 modes in the 1 st reduction are enough to obtain an accuracy comparable to the direct methods in all of the first 10 natural frequencies. In this case, the results obtained with the direct method (Figure 4-Low) are almost identical to each other. The reason is that, for high harmonic indices, as \(h=13\), the disk behavior resembles a rigid body and fixed (direct method) and cyclic (two-step method) boundary conditions are almost equivalent to each other. Unfortunately, in other cases (e.g., Figure 5), the ROM accuracy does not improve as the number of slave modes \(Z_{2}\) is increased from 40 to 60. We interpret this result as a lack of accuracy of the ROM  
Figure 4: Test Case A (linear elements)—accuracy of the direct method for \(h=1\) (Up) and \(h=13\) (Low). generated in the 1 st step because of the fixed-interface boundary conditions applied to the sector interface nodes \(\mathbf{q_{H}}\) and \(\mathbf{q_{L}}\). Thus, in order to increase the accuracy of the ROM generated with the two-step approach, the 2 nd and the 3 rd sets of ROMs are generated with \(Z_{1}=120\) and \(Z_{1}=180\), respectively. In Figure 7, the effect of the number of modes of the 1 st reduction (\(Z_{1}\)) is shown for the first 10 natural frequencies associated with \(h=1\). All of the curves refer to ROMs with the same size (\(Z_{2}=60\)) as the final one generated with the direct method (Figure 4). It is observed that increasing the number of modes retained in the 1 st step significantly affects the accuracy of the final ROM, specifically for modes #9 and #10. This behavior agrees with the interpretation given about the results in Figure 5 because of the (fixed) boundary conditions applied to the disk interfaces, and the reduction basis used in the 1 st step must include a  
Figure 5: Test Case A (linear elements, \(Z_{1}=60\), \(h=1\))—accuracy of the two-step method. Figure 6: Test Case A (linear elements, \(Z_{1}=60\), \(h=13\))—accuracy of the two-step method. number of modes (\(Z_{1}\)) significantly higher (up to three times) than the number of modes retained in the 2 nd step (\(Z_{2}\)), to accurately represent the dynamics of the bladed disk. While accuracy is by far the most relevant performance index for reduction techniques, it is not the only one. Since the amount of time necessary to generate ROMs is not negligible, numerical efficiency is another parameter, which can drive the reduction technique selection.