A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

A virtual element method for the von Kármán equations

In this article we propose and analyze a Virtual Element Method (VEM) to approximate the isolated solutions of the von Kármán equations, which describe the deformation of very thin elastic plates. We consider a variational formulation in terms of two variables: the transverse displacement of the plate and the Airy stress function. The VEM scheme is conforming inH2for both variables and has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed forhsmall enough and optimal error estimates are obtained. Finally, numerical experiments are reported illustrating the behavior of the virtual scheme on different families of meshes.

Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

Consider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of {\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.