# Simulation of an Adaptive Fluid-Membrane Piezoelectric Lens

^{*}

## Abstract

**:**

^{®}. Our model shows the explicit coupling of the piezoelectric physics with the fluid dynamics physics to simulate the interaction between the piezoelectric and the fluid forces that contribute to the deformation of a flexible membrane in the adaptive lens. Furthermore, the simulation model is extended to describe the membrane deformation by additional fluid forces from the fluid thermal expansion. Subsequently, the simulation model is used to study the refractive power of the adaptive lens as a function of internal fluid pressure and analyze the effect of the fluid thermal expansion on the refractive power. Finally, the simulation results of the refractive power are compared to the experimental results at different actuation levels and temperatures validating the coupled COMSOL model very well. This is explicitly proven by explaining an observed positive drift of the refractive power at higher temperatures.

## 1. Introduction

^{®}(5.3a, COMSOL Inc, Burlington, MA, USA) to define the refractive power linearly as a function of both the fluid pressure and temperature.

^{®}is based on the finite-element method (FEM), which solves engineering problems such as structural mechanics, fluid dynamics, heat transfer by a numerical approach. In FEM, the complex geometry is divided into simpler domains. These domains are defined with the elementary partial differential equations based on the physics. Then the elementary equations are combined to form a system of global equations, which represent the complex geometry [8]. The system of global equations can be solved using FEM-based simulation software such as ANSYS, ABAQUS, ATILA, and COMSOL [9]. To simulate complex geometry with multiple physics domains, COMSOL Multiphysics

^{®}offers a methodological environment to access elementary equations and then couple them with a wide range of available physics modules [10]. Using COMSOL

^{®}, authors in [11,12,13] simulated adaptive lenses using only the piezoelectric physics module, authors in [14,15] simulated micro-pumps using the fluid-structure interaction physics module and authors in [16,17] simulated thermal actuators using the heat transfer physics module. However, the articles [11,12,13,14,15,16,17] did not simulate any kind of solid deformation produced by the fluid forces from both the piezoelectric actuation and the thermal expansion. Furthermore, COMSOL

^{®}does not provide a direct feature to couple the piezoelectric with the fluid-structure and heat transfer physics modules. Hence, in this paper, we present the explicit coupling of multiple physics modules to simulate the membrane deformation due to the fluid forces from both the piezoelectric actuation and the thermal expansion.

## 2. The Fluid-Membrane Piezoelectric Lens

## 3. Multiphysics Simulation

^{®}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.

#### 3.1. Piezoelectric Devices

#### 3.2. Fluid-Structure Interaction

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

#### 3.4. Moving Mesh

^{®}, the coupling of the piezoelectric effect and the fluid-structure interaction is not possible through a direct Multiphysics feature. Hence, the moving mesh physics module is used to couple the piezoelectric forces with the fluid forces and apply the resultant on the flexible membrane. The explicit coupling of piezoelectric and laminar flow physics is performed in a way such that the solid domain velocities Equations (8) and (9) generated by the deformation of the piezoelectric actuator are applied as the mesh velocities on the walls of the fluid chamber as shown in Figure 4c. The geometric domains with the free deformation mesh and with the fixed mesh are as shown in Figure 4a,b, respectively.

#### 3.5. Boundary Condition

#### 3.6. Mesh

## 4. Results

## 5. Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PDMS | Polydimethylsiloxane |

FSI | Fluid-structure interaction |

FEM | Finite-element method |

CNC | Computer numerical control |

PCB | Printed circuit board |

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**Figure 1.**(

**a**) 2D cut section of the adaptive lens to show the fluid chamber, flexible membrane, and the integrated actuator. (

**b**) The adaptive fluid-membrane piezoelectric lens.

**Figure 2.**The 2D cross-section of the adaptive lens showing the piezoelectric forces and fluidic forces, which form either (

**a**) plano-convex lens or (

**b**) plano-concave lens depending on the applied voltage direction.

**Figure 4.**The domains specified in the Moving Mesh module to be (

**a**) free mesh and (

**b**) fixed mesh. (

**c**) The solid domain velocity applied on the walls of the fluid chamber.

**Figure 5.**Domains with fixed boundary condition in solid mechanics physics module (

**a**), piezo 1 with default base vector system and piezo 2 with modified base vector system (

**b**), domains with temperature boundary condition in heat transfer in solids and fluids physics modules (

**c**), and domains with wall boundary condition in laminar flow physics module (

**d**).

**Figure 7.**Applied voltages on the piezoelectric actuator with (

**a**) convex lens mode and (

**b**) concave lens mode.

**Figure 8.**The 2D revolved plots showing (

**a**) aspheric convex lens and (

**b**) aspheric concave lens mode.

**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.

**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.

**Figure 12.**Experimental setup to characterize the adaptive lens at applied voltages and higher temperatures.

Component | Material | Density (kg m${}^{-3}$) | Young’s Modulus (Pa) | Thermal Expansion Coefficient (K${}^{-1}$) |
---|---|---|---|---|

Piezoelectric actuator | Piezo PZT-5H [19] | 7500 | 37 × 10^{9} | 1 × 10^{−5} |

Flexible membrane, rim | Polydimethylsiloxane [20] | 1020 | 2 × 10^{6} | 3 × 10^{−4} |

Fluid | Fomblin Y [21,22] | 1880 | - | 2.1 × 10^{−4} |

Substrate | Glass (quartz) | 2210 | 50 × 10^{9} | 4 × 10^{−5} |

Component | Thickness (mm) |
---|---|

Membrane | 0.2 |

Piezoelectric actuator | 0.2 |

Rim | 0.8 |

Substrate | 1 |

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**MDPI and ACS Style**

Bettaswamy Gowda, H.G.; Wallrabe, U.
Simulation of an Adaptive Fluid-Membrane Piezoelectric Lens. *Micromachines* **2019**, *10*, 797.
https://doi.org/10.3390/mi10120797

**AMA Style**

Bettaswamy Gowda HG, Wallrabe U.
Simulation of an Adaptive Fluid-Membrane Piezoelectric Lens. *Micromachines*. 2019; 10(12):797.
https://doi.org/10.3390/mi10120797

**Chicago/Turabian Style**

Bettaswamy Gowda, Hitesh Gowda, and Ulrike Wallrabe.
2019. "Simulation of an Adaptive Fluid-Membrane Piezoelectric Lens" *Micromachines* 10, no. 12: 797.
https://doi.org/10.3390/mi10120797