# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 5 Numerical Examples
### Example 2  
The second example is a three-dimensional elastic body with a polygon section (H = 7 m, W = 4 m and L = 12 m). The material properties of the beam are: \(\rho\) = 1800 kg/m\({}^{3}\), \(E\) = 10 MPa and \(\
u\) = 0.3. The nodes on the base of the model are fixed. The FE model with 315 nodes and 945 DOFs in total is shown in Figure 8. The overall model is divided into two components numbered \"1\" and \"2\", and two interpolating schemes (Figure 9c) are adopted. In scheme (b), only triangular interpolation sub-regions are introduced. While in scheme (c), both quadrangle and triangular interpolation sub-regions are used. All the corner nodes of the sub-regions for scheme (b) and (c) are assigned as the IBNs, with the number of 11.  
Based on the results by the MCMS-fid method (scheme (a)), the relative errors of the first eight natural frequencies of the elastic body FE model are listed in Table 3. Figure 10 shows the MAC between the modes of the MCMS-fid method and the MCMS-rid method (schemes (b) and (c)). The FRF comparisons are conducted by applying a harmonic excitation at excitation level of 1N in the X direction in the middle of the beam, and the results for the first mode are plotted in Figure 11. All these results indicate that, the MCMS-rid method developed in this paper is still effective and suitable for the non-rectangle beam. Though the same number of the IBNs are kept in the two schemes, the numerical precision of (c) is better than that of (b), which means that the shape of the sub-region has important influence on the numerical precision. In general, quadrangle interpolation sub-regions are more suitable for interface DOFs reduction. However, triangular interpolation sub-regions are still needed especially when the coupling interface is irregular.  
Figure 8: Finite element model of elastic body.  
Figure 7: Frequency response functions of the beam by the MCMS-fid method and MCMS-rid method with different interpolation schemes for the first mode.  
Figure 11: Frequency response functions of the beam by the FE model, MCMS-fid and MCMS-rids with different interpolation schemes for the first mode.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{8}{c}{**Mode**} & \multirow{2}{*}{**DOFs**} \\ \cline{2-10}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline a & 2.96 & 3.73 & 5.13 & 5.25 & 9.19 & 9.42 & 10.28 & 10.75 & 210 \\ b & 2.94 & 3.77 & 5.33 & 5.27 & 9.66 & 9.68 & 10.76 & 11.16 & 126 \\  & (0.59) & (1.12) & (3.90) & (0.45) & (5.11) & (2.76) & (4.68) & (3.83) & \\ c & 2.96 & 3.56 & 5.35 & 5.26 & 9.44 & 9.59 & 10.67 & 11.01 & 126 \\  & (0.08) & (0.14) & (4.32) & (0.14) & (2.77) & (1.83) & (3.80) & (2.46) & \\ \hline \hline \end{tabular}
\end{table}
Table 3: The first eight natural frequencies of the beam (Hz) and relative error (%).  
Figure 10: Modal assurance criterion between the modes by the MCMS-fid method and MCMS-rid method different interpolation schemes. (**a**) MCMS-rid:b vs. MCMS-fid; (**b**) MCMS-rid:c vs. MCMS-fid.  
Figure 9: The coupling interface and interpolating schemes. (**a**) the coupling interface; (**b**) scheme with triangular interpolation sub-regions; (**c**) scheme with quadrangle and triangular interpolation sub-regions.