**4. Results**

The time-dependent study was selected in the simulation as the velocities, which are used in the moving mesh module, are time-dependent values, and as the stationary study does not compute these instantaneous velocities. In the study, the time range was selected from 0 s to 1 s with a step of 0.1 s. The applied voltages were limited to 140 V in positive polarization direction, and −40 V in negative polarization direction to avoid piezo saturation and re-polarization [19]. A sinusoidal voltage within the voltage limits was defined using a piecewise function under global definitions. Figure 7 shows the line plot of voltages applied to the piezoelectric actuator.

Figure 7a shows the top piezo set to positive voltage limit of 140 V, and the bottom piezo set to negative voltage limit of −40 V, the corresponding voltage combination results in a plano-convex lens. In contrast, the voltage combination is reversed in Figure 7b, i.e., the top piezo is set to −40 V and the bottom piezo is set to 140 V, resulting in a plano-concave lens.

**Figure 7.** Applied voltages on the piezoelectric actuator with (**a**) convex lens mode and (**b**) concave lens mode.

The volume plots in Figure 8a,b generated by the 2*D* revolution around the symmetric axis show the adaptive lens in plano-convex and plano-concave lens modes. The revolved plot is used to visualize the aspherical deformation of the membrane.

**Figure 8.** The 2D revolved plots showing (**a**) aspheric convex lens and (**b**) aspheric concave lens mode.

The surface plots in Figure 9a,b show the membrane deformation and the corresponding dynamic internal fluid pressure during actuation. The arrows indicate the fluid velocity direction during actuation. For the plano-convex lens in Figure 9a, in which the piezoelectric actuator is set to a maximum voltage combination results in positive internal pressure of around 270 Pa and a peak deflection of around 300 μm at the center of the membrane. For the plano-concave lens in Figure 9b, in which the voltage combination is reversed compared to the former case, the actuator deforms outwards resulting in negative fluid pressure of around −270 Pa and a peak deflection of around −300 μm at the center of the membrane.

**Figure 9.** The surface simulation plots of the adaptive lens showing dynamic internal chamber pressure and solid deformation in (**a**) plano-convex and (**b**) plano-concave modes.

The heat transfer interface for solids and fluids was used to set the adaptive lens to a temperature ranging from 20 ◦C to 80 ◦C. The surface plot in Figure 10 shows the temperature distribution of the adaptive lens set to 80 ◦C with the actuator voltage set to 0 V. The expansion of the fluid at 80 ◦C corresponds to an increase in fluid pressure of around 85 Pa and a peak deflection of around 80 μm at the center of the membrane.

**Figure 10.** Temperature distribution of the adaptive lens at 80 ◦C.

The refractive power of the adaptive lens in the simulation (Figure 11) was calculated by double differentiation of the membrane boundary with respect to the deformation component (w) and the radial component (r) Equation (10). Figure 11a shows the refractive power defined as a function of internal fluid pressure. The simulation of the adaptive lens result in a refractive power range of −16 m<sup>−</sup><sup>1</sup> to 16 m<sup>−</sup><sup>1</sup> for an internal fluid pressure range of −270 Pa to 270 Pa.

$$Reffractive\ Power = \left(\frac{\text{d}^2 w}{\text{d}r^2}\right) \cdot (n - 1) \tag{10}$$

where 'w' is the deflection component, 'r' is the radial component and 'n' is the refractive index of the adaptive lens.

**Figure 11.** (**a**) The simulated refractive power of the adaptive lens as a function of fluid chamber pressure, and (**b**) the change in refractive power of the adaptive lens due to thermal expansion of the fluid.
