Singular perturbation analysis of a regularized MEMS model

TL;DR

This paper analyzes a regularized MEMS model using singular perturbation techniques to understand steady states and bifurcations, especially as regularization and voltage parameters approach zero.

Contribution

It applies geometric singular perturbation theory to a MEMS model with regularization, providing detailed insights into steady states and bifurcations in the singular limit.

Findings

Describes steady-state solutions using blow-up techniques.

Analyzes bifurcation structure as regularization and voltage parameters tend to zero.

Provides a detailed bifurcation diagram for the regularized model.

Abstract

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter $\lambda$. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter $\varepsilon$. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters $\varepsilon$ and $\lambda$ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.