# A Scalable Moving Boundary Treatment in the Lattice Boltzmann Method
## 5 Numerical Experiment
### Parallel Speedup and Efficiency  
This article tested two kinds of parallel performance: keeping the overall task volume unchanged and increasing the number of computing cores; keeping the task volume of a single node unchanged and increasing the overall task volume. Figure 9 shows two kinds of parallel tests, a schematic diagram of a single node and two cores. In the first type of parallel test, the number of usable cores was limited by the computing domain. There was a minimum computing time, which predicted the computing time of a single task under the given computing resources. In the second type of parallel test, the ratio of communication time between cores in the total task was mainly tested, and the maximum calculation scale of the task was predicted.  
In the first parallel, due to the particularity of mesh generation, only the calculation time of the flow field calculation was tested, as shown in Figure 10. Two test results are given: type 1, the number of grids was 1680 \(\times\) 400, the solid number was 3360; type 2, the number of grids was 3360 \(\times\) 800, the solid number was 6720. It can be seen from Figure 10 that as the number of cores increased, the result of the first speedup ratio was close to the ideal speedup ratio. The number of grids and solid scattered points had little effect. In the 112 cores of the seven nodes tested, when the number of cores exceeded 16 cores, as  
Figure 8: Velocity flow field at five monitoring points and flow field with static boundary: (**a**) Steps = 136,000; (**b**) Steps = 137,000; (**c**) Steps = 138,000; (**d**) Steps = 139,000; (**e**) Steps = 140,000; (**f**) Static boundary, steps = 100,000.  
the number of cores increased, the efficiency drop was small, basically staying above 80%. More grids had better efficiency. This shows that in the pipeline flow problem with moving boundaries, when the calculation scale is small, there is a good parallel effect.  
Figure 9b shows that while keeping the calculation amount of a single core unchanged, increasing the number of cores will increase the number of grids and the number of solid scattered points simultaneously. This means that when the calculation scale is increased, the resolution of the solid boundary will not be reduced, which is suitable for calculating complex boundaries. In this paper, the number of solid scattered points was allocated according to the number of cores before the mesh generation stage. During the mesh generation process, data communication between different cores was not required. It can be seen from Figure 11 that the acceleration result of the total time was close to the mesh generation. The results show that in the program, the proportion of communication was tiny, and the impact can be ignored, which is suitable for large-scale parallel calculations.  
In calculating the moving boundary problem, as the solid boundary moves, the mesh generation step needs to be repeated, which will cause a huge computational overhead. Suppose only the number of grids is allocated according to the number of cores. In that  
Figure 10: The first parallel speedup and efficiency: **(a)** Speedup; **(b)** Efficiency.  
Figure 9: Schematic diagram of two parallel performances: **(a)** The first type; **(b)** The second type.  
case, the ideal speedup ratio of the mesh generation is a linear speedup ratio, i.e., the actual speedup multiple is close to the number of cores. In this paper, the number of solid grids was allocated simultaneously, significantly reducing the time overhead of mesh generation, thereby reducing the overall calculation time. Since this method is suitable for the moving boundary problem of the pipeline model, it can be applied to the fluid-solid coupling problem of blood vessels in future research.