# OGSM: A Parallel Implicit Assembly Algorithm and Library for Overlapping Grids
## 5 Numerical Results
### A Multiple Body Trajectory Prediction Case Study  
In order to further verify the accuracy of the interpolation for the overlapping cells, we embedded OGSM into an in-house flow field solver named \"parallel simulator for multi-block grids in three-dimensional space\" (PMB3D) [28] in the form of a static library and performed a simulation for the unsteady field of a case containing three external objects. Figure 19 shows the geometry configuration of this case study, which stems from the \"multiple body trajectory prediction\" (MBTP) model. Interested readers can refer to the literature [29,30] for more information. Multi-block structured grids were generated around the wing and external objects. The wing grid has 5,214,792 nodes and 4,798,080 cells, and each external object has 1,722,242 nodes and 1,585,664 cells. Consequently, the total number of nodes and cells for the MBTP configuration are 10,291,518 and 9,555,072, respectively. To the best of our knowledge, there are no convincing wind tunnel experimental results for the multiple body trajectory prediction case. Consequently, we merely performed the numerical simulation with 64 processes. In each process, the coincident nodes and faces were merged, and the multi-block structured grids were transformed to an independent unstructured grid. When the overlapping assembly was accomplished by OGSM, the PMB3D performed the simulation of the Reynolds-averaged Navier-Stokes (RANS) equation. The convection and the viscosity terms were approximated with the Roe scheme and the SA turbulence model, respectively. A dual time-step approach was employed in the time direction. To suppress the unphysical results, a minmod limiter was included. The Mach number of the free stream was set to 0.95. The lower external object was dropped at first, followed by the upper-left one in 0.04 s and the upper-right after another 0.04 s. The ejection force applied to the store at the bottom was vertically downward, while the one at the shoulder was 45 degrees from the vertical and points outward. The action time was 0.045 s.  
Figure 20 shows the hole-cutting effect of each component grid obtained by OGSM for a cross-section perpendicular to the x-axis. The red part is the flow field domain, the  
Figure 19: Grid topological structure of the example of continuous delivery of external stores. yellow part is the automatic hole-cutting boundary, and the green part is the overlapping cells. It can be seen that OGSM successfully realized the cutting and connection of the flow field area for each component. Figure 21 shows the variation trend of the trajectory and the attitude of the three external objects within 0.51 s. A yaw motion from the head of the external object(s) towards the wing tip can be observed. At the initial stage, the ejection force exerted on the head and the tail of the external object(s) forms a head-up moment. This moment is then cancelled by the stabilization effect of the tail. In the numerical simulation, the ejection force is applied on the body shaft system fixed to the external object, such that no rolling moment is generated. Figure 22a-c show the temporal variation of the centroid trajectory, and Figure 22d-f show the attitude angles (rolling angle \(\Phi\), pitch angle \(\Theta\) and yaw angle \(\Psi\)) for the three objects, respectively. The centroid trajectory obtained by the simulation agrees well with that from the Refs. [29; 30], whereas the prediction of the attitude angles still yields certain non-negligible errors. Figure 21: Trajectory and posture change trend of external objects. Figure 20: Overlapping grid hole-cutting effect for multi-component configuration. In this example, the errors mainly come from the physical field interpolation of the overlapping cells. On the one hand, by tracking the execution of OGSM, some defective overlapping cells are detected during the implicit hole-cutting. Taking the initial status as an example, the total number of the defective overlapping cells in the whole domain is 61.