Two-sided iterative method based on the use of the Green's function in problems of numerical analysis of some electromechanical systems

Abstract

Relevance. The paper considers the problem of numerical analysis of a nonlinear boundary value problem that models an electrostatic microelectromechanical system under the action of external pressure. Microelectromechanical systems combine mechanical and electrical components of micron size and are used in automotive, aviation, and medicine. Electrostatic activation of these systems is crucial to micromirrors, microresonators, accelerometers, etc. The main limitation of electrostatic microelectromechanical systems is the pull-in instability, leading to system destabilization. It is proposed to investigate the model’s parameters and obtain their estimates to eliminate these limitations.

Goal. Using the methods of the theory of nonlinear operators in semi-ordered spaces, develop a method of two-sided approximations for solving the given problem.

Research methods. A nonlinear elliptic equation with the Laplace operator and a homogeneous boundary condition of the first kind represents the mathematical model of the electrostatic microelectromechanical system. Using the method of Green's functions, this differential problem is transferred to the equivalent integral equation of Hammerstein, which is analyzed by methods of the theory of nonlinear operators in semi-ordered spaces.

The results. The nonlinear operator's properties included in the Hammerstein equation were studied, and the conditions for the existence of a unique positive solution to the considered problem and the conditions for the convergence of two-sided approximations were obtained. For the proposed method of two-sided approximations, a posteriori estimation of the error and estimation of the number of iterations necessary to achieve the specified accuracy were also obtained.

Conclusions. Computational experiments demonstrate the operation and efficiency of the developed method for a test problem in a circular area with different values of model parameters. The results of computational experiments are presented as numerical and graphical information.