USING THE GREEN – RVACHEV’S QUASIFUNCTIONS METHOD IN THE NUMERICAL ANALYSIS OF ONE ELECTROSTATIC NANOELECTROMECHANICAL SYSTEM

Abstract

The problem of numerical analysis of one electrostatic nanoelectromechanical system is considered in the article. Nanoelectromechanical systems are miniature devices that combine electronic and mechanical components of micro and nano sizes. Electrostatic actuation of the mechanical components of such systems is one of the most common types of actuation and that used in accelerometers, switches, micro-mirrors, micro-resonators, etc. The disadvantages of such devices are related to the pull-in instability. This effect occurs when the voltage applied to the moving electrode exceeds a critical value, causing the system to lose its stationary configuration. A semi-linear elliptic equation with the Laplace operator and the first boundary condition was used for mathematical modeling of the process. To construct an approximate solution of the problem, it is suggested to use the methods of nonlinear analysis in semi-ordered spaces, in particular, the results of V. I. Opoitsev on the solvability of nonlinear operator equations with a heterotone operator. The boundary value problem modeling the nanoelectromechanical system is reduced to the integral Urysohn’s equation using the Green – Rvachev’s quasi-function method, which allows us to expand the application of the two-sided approximation method for domains of fairly arbitrary geometry. The article substantiates the possibility of constructing two-sided approximations to a positive solution of the problem, namely: a computational scheme is given, conditions for its convergence to a single positive solution of the problem under consideration are obtained, and an error estimate is obtained. The method is illustrated by computational experiments for a problem considered in a rectangular domain. The results of computational experiments are presented in the form of numerical and graphical information.