Electrostatic Micro-Electro-Mechanical-Systems (MEMS) Devices: A Comparison Among Numerical Techniques for Recovering the Membrane Profile

Abstract

In this work, numerical techniques based on Shooting procedure, Relaxation scheme and Collocation technique have been used for recovering the profile of the membrane of a 1D electrostatic Micro-Electro-Mechanical-Systems (MEMS) device whose analytic model considers |E| proportional to the membrane curvature. The comparison among these numerical techniques has put in evidence the pros and cons of each numerical procedure. Furthermore, useful convergence conditions which ensure the absence of ghost solutions, and a new condition of existence and uniqueness for the solution of the considered differential MEMS model, are obtained and discussed.

Curvature-Dependent Electrostatic Field as a Principle for Modelling Membrane-Based MEMS Devices. A Review

“…Often, many applications require that a membrane replace the deformable plate, reducing the inertial effects [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ]. Usually, the models use the deflection of the membrane, u , as an independent variable [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ]. If, on the one hand, the demand for MEMS devices is strong, on the other hand, it is not always possible to formulate models that are easy to implement [ 5 , 11 , 14 , 15 ].…”

“…If, on the one hand, the demand for MEMS devices is strong, on the other hand, it is not always possible to formulate models that are easy to implement [ 5 , 11 , 14 , 15 ]. Further, these formulations hardly allow resolution in closed form [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], so one must be satisfied with obtaining conditions that ensure both existence and uniqueness of the solution [ 14 , 16 , 17 , 20 ]. The alternative is the numerical approach, which, although interesting, does not prevent us from being present for ghost solutions [ 17 , 19 , 21 , 22 , 23 ].…”

“…Further, these formulations hardly allow resolution in closed form [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], so one must be satisfied with obtaining conditions that ensure both existence and uniqueness of the solution [ 14 , 16 , 17 , 20 ]. The alternative is the numerical approach, which, although interesting, does not prevent us from being present for ghost solutions [ 17 , 19 , 21 , 22 , 23 ]. Thus, analytical approaches and numerical techniques work together to obtain solutions, respecting the analytical conditions that guarantee existence and uniqueness of the solution without ghost solutions [ 17 , 19 , 21 , 22 , 23 ].…”

“…The alternative is the numerical approach, which, although interesting, does not prevent us from being present for ghost solutions [ 17 , 19 , 21 , 22 , 23 ]. Thus, analytical approaches and numerical techniques work together to obtain solutions, respecting the analytical conditions that guarantee existence and uniqueness of the solution without ghost solutions [ 17 , 19 , 21 , 22 , 23 ]. In this context, the experience of the authors in the field of modeling electrostatic MEMS membrane devices with strong non-linearity has grown [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ].…”

“…Thus, analytical approaches and numerical techniques work together to obtain solutions, respecting the analytical conditions that guarantee existence and uniqueness of the solution without ghost solutions [ 17 , 19 , 21 , 22 , 23 ]. In this context, the experience of the authors in the field of modeling electrostatic MEMS membrane devices with strong non-linearity has grown [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ]. In particular, by combining the physics of the problem with important results of differential geometry [ 25 ] analytical results have been obtained in terms of the existence and uniqueness of the solution in some cases [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ] and numerical solutions in the absence of ghost solutions in other cases [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ].…”