**3. Multiphysics Simulation**

To reduce the complexity of a 3*D* geometry and at the same time to decrease the computation time, a 2*D*-axisymmetric space dimension is chosen to model the adaptive lens (Figure 3) with the radial axis 'r' and the deformation axis 'z'. The adaptive lens components include a piezoelectric actuator, fluid, membrane, rim, and substrate. The materials for the components are chosen from the COMSOL Multiphysics-R inbuilt material library [18]. The material parameters are changed to the equivalent parameters of the materials, which are used in the manufacture of the adaptive lens. The adaptive lens components, along with the modified material parameters used in the simulation, are mentioned in Table 1 and the adaptive lens components thicknesses are mentioned in Table 2. The following section describes the physics modules used in the simulation.



**Figure 3.** The 2D axisymmetric simulation model of the adaptive lens.

**Table 2.** The thickness of the adaptive lens components.


## *3.1. Piezoelectric Devices*

The adaptive lens uses the inverse piezoelectric property of the actuator to vary the refractive power. To model the inverse piezoelectric effect, the piezoelectric devices module is used. The module couples the solid mechanics Equation (1) and the electrostatics Equation (2) physics to combine the electrical behavior and the mechanical behavior of the piezoelectric ceramics.

$$
\rho \frac{\partial^2 x}{\partial^2 t} = \nabla \cdot s + F\_v \tag{1}
$$

where *ρ* is the density, *x* is the displacement, *s* is the stress, and *Fv* is the volume force.

$$E = -\nabla \cdot V \tag{2}$$

where *E* is the electric field, and *V* is the electric potential.

The combined behavior is modelled through the coupled Equations (3) and (4) in strain-charge form.

$$S = S\_E T + d^I E \tag{3}$$

$$D = dT + \epsilon\_o \epsilon\_{rT} E \tag{4}$$

where the solid mechanics parameters are strain *S* and stress *T*, the electrostatic parameters are electric field *E* and electric displacement field *D*, and the piezoelectric material parameters are compliance coefficient *SE*, piezoelectric coefficient *dT*, and permittivity [23]. In the simulation model, the piezoelectric coefficients and compliance coefficients are obtained from the piezo PZT-5H inbuilt material library [18].

## *3.2. Fluid-Structure Interaction*

The piezoelectric actuator in the adaptive lens deforms the fluid chamber and varies the internal fluid pressure. The varied internal fluid pressure results in fluid forces, which act on the flexible membrane. The fluid forces contribute to the deformation of the flexible membrane to an aspherical surface. The fluid-structure interaction (FSI) physics module models the fluid forces acting on the membrane by coupling the solid mechanics Equation (1) and the laminar flow Equation (5) physics modules.

$$
\rho \frac{\partial \mu}{\partial t} + \rho (\mu \cdot \nabla) \mu = \mu \nabla^2 \mu + F + \rho \text{g} \tag{5}
$$

The Navier–Stokes Equation (5) models the motion of incompressible fluids, where *ρ* is the fluid density, *u* is the fluid velocity, *F* is the external force and *g* is the gravity. The FSI Multiphysics module couples the fluid inertial forces in Equation (5) with the external forces in Equation (1) [24].

#### *3.3. Heat Transfer in Solids and Fluids*

Apart from the piezoelectric actuation that contributes to the deformation of the membrane, the thermal expansion of the fluid at higher temperatures will as well cause the membrane deformation. Hence to model the fluid thermal expansion, heat transfer in solids (Equation (6)) and heat transfer in fluids (Equation (7)) physics modules are used.

$$Q = \alpha T \cdot \frac{\text{dS}}{\text{dt}} \tag{6}$$

$$Q = aT(\frac{\partial \rho}{\partial t} + \mu \cdot \nabla p) \tag{7}$$

where *Q* is the heat source, *T* is the temperature, *S* is the solid stress tensor, *α* is the coefficient of thermal expansion, *p* is the fluid pressure, and *u* is the fluid velocity.

Equations (6) and (7) define the heat source *Q* that contributes to set the complete adaptive lens domain to the required temperature *T*. Equation (7) models the thermal expansion that contributes to the fluid pressure *p*, which acts on the flexible membrane [25].
