# An SQP Algorithm for Structural Topology Optimization Based on Majorization-Minimization Method
## 4 Numerical Examples
### MBB Beam
#### 4.1.1 Optimal Solution and Optimized Design  
It can be seen in Figure 3 that, by selecting appropriate curvature parameters and filter radius, the QPOC method and IPC method could obtain the optimized structures similar to the OC method, which verifies the feasibility of the proposed method. The optimized structures obtained using the QPOC method and OC method had the phenomenon of fuzzy boundary, while the optimized structures (Figure 3h,i) obtained using the IPC method were clearer, and the structure contour was more distinct.  
The curvature parameter and the filter radius have a crucial influence on the structure, and it is necessary to explore the optimal combination of the two parameters. Because QPOC is a derivative algorithm of the OC method, its filter radius should be chosen consistently with the OC method, which is also verified by the fact that Figure 3b-d exhibits a similar structure with the same filter radius as the OC method. For the IPC method, reducing the filter radius can obtain a clearer structure as shown in Figure 3g,h, but too small a filter radius will lead to fine branches in the optimized structure and affect the fabrication, as shown in Figure 3i. It can be seen from Figure 3e-g that when the curvature parameter was increased, the filter radius could be appropriately reduced to obtain a clearer structure without fine branches.  
Figure 3: Optimized designs of the MBB beam with different methods and parameters.  
Figure 2: Design domain of the MBB beam.  
To further analyze the influence of the QPOC and IPC methods and parameters on the optimized structure, the compliance and discrete index \(M_{\text{nd}}\) were calculated, and the results are shown in Table 1. As can be seen from Table 1, the values of \(M_{\text{nd}}\) for the QPOC and OC methods were comparable, while the \(M_{\text{nd}}\) for the IPC method was much smaller than that of the other two methods. When \(q=2\) and \(r=2\), the \(M_{\text{nd}}\) of the IPC method was the smallest, 8.5%, approximately 72% smaller than that of the OC method, which indicates that the IPC method could significantly reduce the intermediate densities and reduce the boundary-blurring effect.  
In addition, the compliance of QPOC and OC methods was approximately equal; the compliance obtained by the IPC method was smaller than that of the previous two methods. It is worth noting that the compliance of the IPC (\(q=2,r=2.4\)) method was about 12% smaller than that of the OC method. Because the optimized structures of both had similar geometries, the reason for this phenomenon was that the \(M_{\text{nd}}\) of the IPC method was smaller, which means that the optimized structure had fewer intermediate densities, while SIMP penalized the stiffness of elements with intermediate density, resulting in a decrease in the integral structural stiffness.  
In summary, the optimized structure and compliance of the QPOC method were consistent with those of the OC method, and the selection of the filter radius should be the same as that of the OC method. The optimized structure of the IPC was clearer, with a distinct boundary and smaller compliance. The selection of filter radius was negatively correlated with curvature parameters.