## 3 Numerical Examples
### 2D Benchmark Example  
The first benchmark example was used to demonstrate the generation process and the superiority of the Voronoi-GSCS in the heat transfer problem. The size effect of the multiscale design method for the cellular structure was also surveyed.  
As shown in Figure 5b, the relative density distribution was obtained by the modified density method in the online optimization. Based on the relative density distribution and the relationship between the average aspect ratio and the relative density, the average aspect ratio distribution could be obtained, as shown in Figure 5c. It can be seen that the region with a high relative density has a smaller corresponding size function. Conversely, the region with a high relative density has a larger area size function. Then, the Voronoi skeleton could be generated by the mesh generator [34] and the coupling relationship [35], as shown in Figure 5d. The cellular structure skeleton corresponding to the region with a large size function was sparse, and the cellular structure skeleton corresponding to the region with a large size function was dense. The signed distance function of the Voronoi skeleton could be calculated by the fast marching method [36], as shown in Figure 5e. The Voronoi-GSCS was reconstructed by cutting the level set function, as shown in Figure 5f. It can be seen that the microstructures were connected smoothly in the whole design domain.  
For comparison, the optimized results obtained by the SIMP methods are given. The SIMP method is based on the work of [39]. The penalty coefficient was 3 and the minimum filter radius was 1.2. The relative density distribution is shown in Figure 7a, which shows the dendritic distribution. The value of relative density became larger when it was close to the left edge with \(T=0\). It can be observed in Figures 5b and 7a that the two relative densities showed similar distribution trends. The relative density distribution obtained by the SIMP method had more bifurcation, while the relative density distribution obtained by the method developed in this paper was more concentrated.  
As shown in Figure 7b, the heat compliance of the single-scale structure was 4438.30. The heat compliance of the cellular structure was 3737.67, which was 15.79% smaller than the single structure. This indicated that the designed cellular structure had great superiority in heat transfer.  
As we know, the size effect should be considered in the scale-separated-based multiscale design method for cellular structures. In the present multiscale method, the size effect was surveyed by adjusting the width of the struts, which determined the aspect ratio of the struts. As shown in Figure 8, the cellular structures obtained by using different widths are given. The widths of the struts for the three cellular structures were 0.08, 0.06, and 0.04, respectively. As can be seen, the three cellular structures were similar to each other. Additionally, the convergence history and objective function value were nearly the same for the full-scale reconstruction of the cellular structure, which was based on the same relative density distribution. The difference was due to the fact that the cellular structure became denser and the time cost became higher when the width of the struts  
Figure 7: Optimization results and convergence history: (**a**) the relative density distribution obtained by the SIMP method; (**b**) iteration history for the SIMP method and the multiscale method.  
became smaller. In addition, the gradient change in the structure was more obvious with a smaller width. In conclusion, it manifested that the size effect on the cellular structure design in the present multiscale method was not significant.