The two-sided method in numerical analysis of one microelectromechanical system

Abstract

In this paper, we consider the problem of numerical analysis of an electrostatic microelectromechanical system. Microelectromechanical systems are devices of microsystem technology that combine electronic and mechanical components of micron sizes. Electrostatic activation is one of the most common types of activation of microelectromechanical systems used in accelerometers, optical switches, micropumps, etc. The disadvantages of such devices consist in their pull-instability. This effect occurs when the voltage applied to the moving electrode exceeds the critical value. As a result the system loses its stationary configuration. To ensure the stable operation of the microelectromechanical system, it is proposed to control the dielectric properties of the device components. For a mathematical modeling of the process, we use a nonlinear elliptic equation with the Laplace operator and the given boundary conditions. To construct an approximate solution of the problem under consideration, we propose to use methods of nonlinear analysis in semi-ordered spaces, in particular, the results of the solvability of nonlinear operator equations with a monotone operator obtained by M.A. Krasnosel’skij. The boundary value problem that modes a microelectromechanical system is reduced to the Hammerstein integral equation using the Green's function. The paper substan-tiates the possibility of constructing two-sided approximations to a positive solution of the problem. The method is illustrated by computational experiments for the problem considered in a unit circular domain. The computational experiments are presented in a numerical and graphical format.