Moving Mesh simulation of contact sets in two dimensional models of elastic-electrostatic deflection problems

TL;DR

This paper introduces a moving mesh numerical method combined with singular perturbation analysis to study contact sets in electrostatic-elastic deflections of micro-electro-mechanical systems, focusing on parameter and geometry effects.

Contribution

It develops a novel adaptive moving mesh PDE approach and a geometric theory to predict contact sets in elastic-electrostatic deflection models.

Findings

The moving mesh method accurately captures singularities.

The geometric theory predicts contact set configurations.

Numerical and analytical results agree across test cases.

Abstract

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases.