## 4 Results and Discussion
### Three-Dimensional Examples
Figure 12: Convergence performance of topology optimization for drag reduction on the body structure in Figure 10f. we have obtained so far do not affect the exploration of the relevant regularity. We give an example of the mesh representation of the result of Figure 10a in Figure 13.  
In addition, we also investigated the 3D body shape governed by \(J_{2}\), whose integrand contains a convection term. Because the body shape corresponding to the larger Reynolds number has the solid material close to the design domain during the iterative evolution process, we set the area fraction to 0.98 to ensure that the complete body shape is obtained. Shown in Figure 14, the overall structure is a similar cylinder with a diameter change in the cross-section from head to tail. For this group of structures, the front section is conical, and the mid-section cylinder gradually thickens until the tail rapidly narrows, with the Reynolds numbers from 0 to 500. When Reynolds number is below 300, the front sections are similar, and up to 300 the head regions become blunt. In fact, the structure cannot be regarded as a standard rotating body because the cross-section of the structure contains ellipse and circle shapes with different Reynolds numbers, as shown in Figure 15.  
Figure 14: 3D optimal body shapes for drag reduction corresponding to \(J_{2}\) at various Reynolds numbers. Figure 10.0 The mesh representation of the result in Figure 10a. (**a–c**) show the mesh representation of the structures corresponding to \(J_{1}\) when Reynolds number is 0 from different perspectives. It is not difficult to see that there is a big difference in the structure of objective function \(J_{1}\) and \(J_{2}\) in three-dimensional problems. Compared with the body shapes in Figure 10, this group of structures seems to be covered with a thick shell in the middle of the spindle cylinder, which is particularly evident from the structure corresponding to \(\mathrm{Re}=1\) in Figure 14b. We guess that the effect of convection term may widen the middle of the structure in this problem. In order to verify this idea and make the result more obvious, we purposely remove the Z-axis component of the convection term in \(J_{2}\) and obtain the following results in Figure 10.1 It is obvious from Figure 16b,c that with Y-axis component of the convection term, the Y-direction of body is wider. Without Z-axis component of the convection term, there is no material growth in the central main part in the Z- direction. In addition, the convection effect enhances with the increase of the Reynolds number. Therefore, when the Reynolds number is less than 200, the shape changes little, while when it is greater than 300, the effect is obvious. Compared with the structures of Figure 10, the structures in Figure 14 have more obvious oscillating in the convergence history with the appearance of sharp corners. The convergence performance corresponding to Reynolds numbers 0 and 200 is shown in Figures 17 and 18, respectively. In addition, the mesh representation of the result in Figure 14a is shown in Figure 10.2 We can intuitively see the characteristics of the structure surface from the mesh representation. Like the curvature variation of the surface, it is a property of the structure itself, independent of the mesh density. However, the mesh density affects the smoothness or roughness of the structure surface using the finite method. The higher the mesh density, or the smaller the filtering radius, the smoother the jagged and stepped shapes in the structure surface. We use the fixed mesh achieved by mapping and sweeping operations. The number of the mesh in the computational domain of the 3D problem is 63,000. Next, we compare the change of drag value with Reynolds number between the two objective functions \(J_{1}\) and \(J_{2}\).