## 5 Numerical Examples
### Application to Lunar Lander with a Rover  
We analyzed the dynamic response of the lunar lander landing process to validate the method; the FEM model is shown in Figure 9. The main components of the lander include: (1) the center body, (2) the payloads and auxiliary equipment and (3) the landing mechanism. In this example, we only retained the main body, the solar wings and the rover. Other payloads are simplified as the distributed inertia, which adheres to the original position. The rover attaches to the main body of the lunar lander when it is transported to the moon. When arriving at the destination, it separates from the main body and starts cruising, becoming an independent system. Usually, the rover and the center body of the lander are developed by different branches separately during the prototype design process. Moreover, the development process of the rover is an iterative optimization process, and during every iteration, the modified rover should be assembled with the center body for coupling analysis. So, we used a modal substructure to model it. However, the center body and the rover usually have non-matched node distributions on the interface since their FE models are established by different design groups, as shown in Figure 10 (right). Moreover, there are some complex connecting relations inside the center body and the rover that are commonly based on nodes. Consequently, remeshing will lead to an enormous cost for rebuilding those connecting relations. Therefore, the non-matching interface method is suitable in this case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**R2**} & \multicolumn{2}{c}{**RAAE**} \\ \hline  & **NMI-IBS** & **NMI-IBS-FEM** & **NMI-IBS** & **NMI-IBS-FEM** \\ \hline Displacement & 0.99984 & 0.99971 & 0.00942 & 0.01367 \\ Velocity & 0.99963 & 0.99955 & 0.01121 & 0.01567 \\ Acceleration & 0.97822 & 0.97885 & 0.10453 & 0.10514 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Accuracy of the multi-domain assembly for the solid plate under Configuration 1. Figure 10: Finite element model of the lunar lander coupled with a rover by non-matching interfaces. In this example, the lunar lander contains two types of substructures (the center body with solar wings is an IBS substructure and the rover is an FE substructure), and the method of initial applied forces with the Newmark-\(\beta\) scheme is used to compute the approximate IRFs. There are 40 support nodes selected on the rover interface and 34 support nodes on the center body interface. The intermediate surface coincides with the rover interface, shown in Figure 11, and with the coefficient \(\alpha=1\) and \(h_{c}=0.5\) (referring to the size of the interface), the interpolation functions are generated. According to [PERSON] [15], we obtain the buffer loads acting on the center body. Take the longitudinal (X direction) buffer load generated from the Z-direction main strut, for instance [13]; its time history is shown in Figure 12.  
There are two response points (see Figure 10) selected on the lunar lander: Point 1 is on the rover and Point 2 is on the left solar wing. Using the proposed NMI-IBS, NMI-IBS-FEM and NMI-IBS-CB (all modes retained), the acceleration responses of the response points are computed. To illustrate the accuracy of these methods, an FE model with the coinciding interface is given as the reference, and the dynamic responses are obtained under the same conditions: gravity (\(g=1.63\) m/s\({}^{2}\)) and initial vertical landing velocity (\(v=4\) m/s). The results are shown in Figure 13.  
Figure 11: Distribution of interface support nodes on rover and center body. Figure 12: Longitudinal buffer load [13]. These results show that displacement and velocity have strong agreement with the reference at both of the two response points. Meanwhile, the acceleration result has a relatively high accuracy performance.