## 4 Optimal Placement of PZT Actuators
### Implementation of Genetic Algorithm for Single-Objective Optimization  
1. Case 1: Displacement load consists of Zernike polynomials  
Zernike polynomials are often used to represent surface errors in optical engineering. They are a series of orthogonal polynomials consisting of power functions and harmonic functions. Lower-order members of the Zernike polynomials represent typical optical wavefront aberrations, such as the power, astigmatism, coma, and spherical aberration [33]. They can be used as a set of orthogonal basis error modes to form the surface errors of reflectors or optic mirrors. The second-eighth normalized Zernike polynomials in a unit hexagonal area are employed to form a reflector surface error. Their expressions are given in Appendix A. The displacement load is written as Equation (44):  
\[S(\rho,\theta)=\frac{1}{80}\sum_{i=2}^{8}Z_{i}(\rho,\theta). \tag{44}\]  
The geometric properties of the CFRC reflector under investigation are shown in Table 2.  
The iteration processes for different numbers of actuators are shown in Figure 5. The abscissa represents the number of iterations, and vertical axis represents the minimum residual RMS error of each generation. The algorithm quickly converges and the minimum RMS error becomes stable within 200 generations, as shown in Figure 5. The initial RMS error of the reflector is 30.91 \(\upmu\)m. The residual RMS errors of 8, 10, 20, 30, 40 and 60 actuators are 11.86, 10.61, 7.52, 6.96, 6.68 and 6.46 \(\upmu\)m, respectively. The initial and residual displacement contours for different numbers of actuators are shown in Figure 6. The corresponding placement of the PZT actuators and their voltages are shown in Figure 7. As shown, the residual RMS error decreases as the number of actuators increases. However, the decreasing rate of the RMS error gradually \begin{table}
\begin{tabular}{c c c} \hline \hline
**Materials** & **Property** & **Value** \\ \hline  & Elastic modulus & \(E_{1}\) (GPa) & 26.53 \\  & Shear modulus & \(\tau_{\text{xx}}\) (GPa) & 2.15 \\  & Maximum voltages (V) & 150 \\  & Minimum voltages (V) & \(-\)100 \\  & Equivalent length (mm) & 80 \\  & Poisson’s ratio & 0.3 \\  & Piezoelectric constant (pm/V) & 360 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 5: Iteration curves with 10–60 actuators. decreases with the increasing of the actuators. This implies that the residual RMS error is stable when the number of actuators reaches a critical value. Figure 6: Initial and residual surface errors. (**a**) Initial, RMS error: 30.91 \(\upmu\)m. (**b**) Eight actuators, RMS error: 11.92 \(\upmu\)m. (**c**) Ten actuators, RMS error: 10.61 \(\upmu\)m. (**d**) Fifteen actuators, RMS error: 8.29 \(\upmu\)m. (**e**) Twenty actuators, RMS error: 7.52 \(\upmu\)m. (**f**) Twenty-five actuators, RMS error: 7.19 \(\upmu\)m. (**g**) Thirty-five actuators, RMS error: 6.77 \(\upmu\)m. (**h**) Forty-five actuators, RMS error: 6.56 \(\upmu\)m. (**i**) Sixty actuators, RMS error: 6.46 \(\upmu\)m.  
Figure 7: _Cont._  
Simulations of increasing numbers of actuators are studied. The relationship between the residual RMS error and the allowed number of actuators is shown in Figure 8.  
As shown, when the number of actuators is less than 30, the residual RMS error decreases rapidly as the number of actuators increases. After the number of actuators reaches 50, the residual RMS error does not decrease much even if the number of actuators is further increased. This means that the number of actuators has reached saturation for the shape-controlling task. Figure 8: Residual RMS error with the number of actuators. Figure 7: Arrangements of the PZT actuators with different maximum allowed numbers of actuators. (**a**) Eight actuators. (**b**) Ten actuators. (**c**) Fifteen actuators. (**d**) Twenty actuators. (**e**) Twenty-five actuators. (**f**) Thirty-five actuators. (**g**) Forty-five actuators. (**h**) Sixty actuators. (**i**) A total of 72 actuators.