# An SQP Algorithm for Structural Topology Optimization Based on Majorization-Minimization Method
## 5 Conclusions and Discussion  
This work introduced the MM algorithm to topology optimization, providing a new perspective for the application of the SQP algorithm in topology optimization. Numerical experiments on an MBB beam and cantilever beam verified the feasibility and efficiency of the proposed method. The contribution of our method can be concluded as follows:  
1. We proposed a novel way to calculate the Hessian matrix for structural topology optimization with minimum compliance problems. The Hessian matrix could be given almost simultaneously when solving the first derivative, and the calculation cost was almost negligible.
2. The QP subproblem is formed by the second-order Taylor expansion of the original problem, which is convex separable and easy to be imported into the QP solver. The convergence rate can be controlled by the curvature parameter.
3. To solve the QP subproblem, we proposed the QPOC method. The derivation of the update formula of this method is supported by rigorous mathematical theory, while the update formula of the OC method is heuristic.  
Compared to using general-purpose solvers, a specialized algorithm devised according to the structure of the problem might be a better option.  
Conceptualization, W.L. and H.M.; methodology, W.L. and H.M.; software, W.L.; validation, W.L. and H.M.; formal analysis, W.L., H.M. and Q.Z.; investigation, W.L. and H.M.; resources, Q.Z.; data curation, Q.Z.; writing--original draft preparation, W.L.; writing--review and editing, W.L.; visualization, W.L.; supervision, Q.Z.; project administration, Q.Z.; funding acquisition, Q.Z. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Meshes** & **Method** & **Figure 7** & **Radius \(r\)** & **Iterations** & **Compliance \(M_{\text{nd}}\) (\%)** & **Time Cost** & **Per\_Time** \\  & & & & & & & & **(s)** & **Cost (s)** \\ \hline \multirow{2}{*}{\(60\times 30\)} & MMA & a & 1.6 & 0.489 & 200 & 87.1 & 25.6 & 9.3 & 0.05 \\  & IPC & d & 1.6 & 0.500 & 48 & 70.7 & 7.1 & 3.7 & 0.07 \\ \hline \multirow{2}{*}{\(80\times 40\)} & MMA & b & 2.4 & 0.491 & 200 & 87.5 & 27.9 & 24.7 & 0.12 \\  & IPC & e & 2.4 & 0.500 & 47 & 71.0 & 8.0 & 6.2 & 0.13 \\ \hline \multirow{2}{*}{\(120\times 60\)} & MMA & c & 3.6 & 0.500 & 200 & 78.9 & 34.8 & 44.2 & 0.22 \\  & IPC & f & 3.6 & 0.496 & 116 & 67.9 & 9.1 & 42.3 & 0.36 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Optimization data for different methods and finite element meshes.  
Figure 8: Convergence curves for the cantilever beam using MMA and IPC methods: (**a**) the changes in the compliance; (**b**) the changes in the volume fractions.  
**Funding:** This research project was supported by the Central Universities Basic Research Business Fund, China (No. 76210-31610007), and the Program of \"One Hundred Talented Scholars\" of Sun Yat-Sen University, China (No. 190313). The authors would like to acknowledge the support of these funds.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors would like to thank [PERSON] of Queen Mary University of London, who was greatly helpful in writing the paper.  
**Conflicts of Interest:** The authors declare no conflict of interest.