THE GREEN-RVACHOV QUASI-FUNCTION METHOD IN THE NUMERICAL ANALYSIS OF MICROELECTROMECHANICAL SYSTEMS USING TWO-SIDED APPROXIMATIONS

Abstract

The work is devoted to developing a bilateral iterative method of numerical analysis of one electrostatic microelectromechanical system based on the use of the Green – Rvachev quasi-function. Microelectromechanical systems are miniature devices that combine electronic and mechanical components of micron size. Electrostatically activated microelectromechanical systems have certain disadvantages that limit the range of their operation. One of these is the pull-in instability of the functional components of the system, which occurs if the applied voltage difference is above a particular critical value. The mathematical model of the system under consideration in the work is a semi-linear elliptic equation with the Laplace operator and the Dirichlet condition. To construct an approximate solution of the problem, it is proposed to use the methods of nonlinear analysis in semi-ordered spaces, particularly the results of V. I. Opoitsev on the solvability of nonlinear operator equations with a heterotonic operator. The boundary value problem simulating the most straightforward microelectromechanical system under the action of external pressure is reduced to the Urison integral equation using the Green – Rvachev quasi-function method, which makes it possible to expand the application of the method of two-sided approximations for problems in areas of somewhat arbitrary geometry. The paper substantiates the possibility of constructing iterative sequences with two-sided convergence to a positive solution of the problem: the computational scheme is given, the conditions for its convergence to the desired solution are obtained, and the a posteriori error estimate is derived. The method is illustrated by a computational experiment for a problem considered in a rectangular area. The computational experiment results are presented in the form of the surface and lines of the level of the approximate solution, and the two-sided nature of the convergence of the proposed method is also graphically illustrated.