## 4 Results and Discussion
### Three-Dimensional Examples  
We also study the 3D topology optimization problem of an axisymmetric object with minimum drag in a uniform flow, which takes longer to solve compared with the 2D problem. Figure 9 shows the modeling and definitive solution conditions of the 3D problem. The pink area is the design domain, and the gray area is the flow domain. The purple one is the structure that we solved. The boundary and initial value conditions of 3D model are similar to the setting of the 2D problem. The length, width, and height of flow domain and design domain are 5L, 2L, 2L, and 3L, 1L, 1L, respectively. The pressure on the outlet surface is zero, and the velocity of the remaining surfaces is U = 1 in the X-direction. Except where noted, the resulting 3D topology optimizations are the same modeling and the same meshing. Of course, other factors such as boundary conditions and initial conditions are the same. Therefore, the settings of each topology structure are not explained one by one. Figure 8: The mesh representation of the result in Figure 6i. (**a**,**b**) respectively show the mesh representation and local enlarged view of the structures corresponding to \(J_{1}\) and \(J_{2}\) when the Reynolds number is 500. The optimization body shapes at various Reynolds numbers are governed by objective function \(J_{1}\) shown in Figure 10. The 3D topology optimization is a standard ellipsoid with \(\mathrm{Re}=0\). When Reynolds number tends to zero, it expresses a state that the inertia force is infinitesimal and can be ignored and there is hardly any memory of the flow state of the previous moment. We express it as \(\mathrm{Re}=0\) for convenience. The structures are similar for \(\mathrm{Re}\) of 1 and 25, whose shapes are pointed cone-shaped at the head, with a gradually narrowing column from the middle section to the tail section, and a concave area at the tail. As the Reynolds number continues to increase from 50 to 500, the 3D body shapes present a change approximated with the 2D results, with the body shape growing laterally and narrowing, and the head and the tail parts becoming pointed. The overall shape appears to be a spindle-shaped column with the wide middle and narrow ends. This kind of rotating body with a nearly circular cross-section conforms to a streamlined body shape. The standard rotation body obtained is related to the implemented condition with velocity only in the X-direction. Figure 10: Three-dimensional optimal body shapes for drag reduction corresponding to \(J_{1}\) at various Reynolds numbers. Figure 9: Schematic of the 3D modelling for drag reduction problem. The convergence history on the result in Figure 10a is given in Figure 11 including the evolution of the material density. Similarly, the convergence history for the result in the Figure 10f is given in Figure 12.  
It is worth mentioning that the surface smoothness of the body shape is correlated with the mesh density. The surface of the structure generated by the finite element method presents a sawtooth or stepped shape, but with the gradual increase of the mesh density, the surface of the structure will become smooth. As the number of iterations increases, there will also be more details such as the undulations and textures presented on the structural surface. Of course, these are the characteristics of the structure itself, independent of smoothness or roughness. When the mesh density is sufficiently fine, we can clearly see the changes of the entire surface. As shown in Table 1, the mesh density in the appropriate range does not affect the drag results. At the same time, the higher the mesh density, the longer the calculation time. In order to give consideration to the accuracy of results and computational efficiency, the mesh density of the 3D model is selected as 10. The number of the mesh in the computational domain of the 3D problem is 36,000. Therefore, the results  
Figure 11: Convergence performance of topology optimization for drag reduction on the body structure in Figure 10a.