## 8 Conclusions  
In this work, the three-dimensional topological optimization based on the piezoelectric transducer model of the cantilever was investigated. Using the Optimality Criteria, the structure optimization was realized with the voltage and stiffness as the multi-objective function. The calculations show that the voltage value was increased, and the global stiffness of the structure was also increased. Topological optimization achieved the goal of reducing the objective value. Furthermore, the voltage was tested, and the results demonstrate that the voltage value was raised successfully. With 70% of the origin volume, this bi-objective optimization increases the global stiffness by 50.9% and the voltage by 30%. As the iteration process shows, the increasing connection to the base increases the structural stiffness, and the corresponding design of piezoelectric transducers supports the conclusion that adding a mass block at the end of the cantilever can increase the voltage output. The comparison proves the superiority of bi-objective topology optimization. This lays the foundation for future piezoelectric transducer structural optimization. Moreover, this is also the first time a piezoelectric topology code has been shared.  
**Supplementary Materials:** The following are available online at [[http://www.mdpi.com/2076-3417/10/19/6772/s1](http://www.mdpi.com/2076-3417/10/19/6772/s1)]([http://www.mdpi.com/2076-3417/10/19/6772/s1](http://www.mdpi.com/2076-3417/10/19/6772/s1)), CODE1 to CODE 7.  
**Author Contributions:** Data curation, Y.L. and Z.H.; Software, Y.G. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by National Natural Science Foundation of China (11602073, 11602149) and Natural Science Foundation of Heilongjiang Province of China (QC2018005), and the research is also supported by \"the Fundamental Research Funds for the Central Universities\" (Grant No.HIT.NSRIF.201821) and and the Fundamental Research Funds for the Central Universities under Grant No. DUT19 RC(3)006. And The APC was funded by Natural Science Foundation of Heilongjiang Province of China (QC2018005).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\begin{tabular}{l l} \(SE\) & Strain energy, the overall stiffness of the structure can be enhanced by minimizing \(SE\) \\ \(MSE\) & Mutual strain energy, the voltage output can be enlarged by maximizing \(MSE\) \\ \(g\) & Correlation coefficient between \(MSE\) and \(SE\) \\ \(U\) & Global displacement matrix \\ \(K\) & Global stiffness matrix \\ \(V\) & Global displacement matrix under unit force \\ \(U^{T},V^{T},u^{T},v^{T}\) & Transposition of \(U,V,u,v\) \\ \(x_{e}\) & Element volume fraction \\ \(x_{\text{min}}\) & Lower limit of x, commonly set as 0.001 \\ \(p\) & Penalty fraction \\ \(u_{e}\) & Element displacement under normal force \\ \(v_{e}\) & Element displacement under unit force \\ \(k_{uu}\) & Global stiffness matrix \\ \(k_{u\phi}\) & Global dielectric matrix \\ \(k_{\phi\phi}\) & Global piezoelectric matrix \\ \(k^{e}_{uu^{\prime}},k^{e}_{u\phi^{\prime}},k^{e}_{\phi\phi}\) & Element matrix of \(k_{uu},k_{\phi\phi^{\prime}},k_{u\phi}\) \\ \(V(x)\) & Current total volume \\ \(V_{0}\) & Origin volume \\ \(f\) & Global volume fraction \\ \(\Phi\) & Global potential matrix \\ \(Q\) & Global charge matrix \\ \(F\) & Force \\ \(n\)\(elx\), \(n\)\(elx\) & The x, y, z dimension of the cantilever \\ \(T\) & Stress \\ \(c^{E}\) & Elastic matrix under constant strain \\ \(S\) & Strain \\ \end{tabular}