Singular perturbation analysis of a regularized MEMS model

Annalisa Iuorio, Nikola Popovic, Peter Szmolyan

Research output: Contribution to journalArticlepeer-review

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 λ. 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 ε. 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 principal model parameters ε and λ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.
Original languageEnglish
Pages (from-to)661-708
JournalSiam Journal on Applied Dynamical Systems
Volume18
Issue number2
Early online date11 Apr 2019
DOIs
Publication statusPublished - 30 Jun 2019

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