Some singular equations modeling MEMS

Abstract

ABSTRACT. In the past fifteen years mathematical models for microelectromechanical systems (MEMS) have been the subject of several studies, in particular due to the interesting qualitative properties they feature. Still most research is devoted to an illustrative, but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes, the so-called vanishing (or small) aspect ratio model. The analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey.

Reinforced Limit of a MEMS Model with Heterogeneous Dielectric Properties

“…Indeed, for (x, z) ∈ R δ we note that 9) and then handle the terms separately. From (2.3) and σ δ = δσ in R δ we obtain…”

“…(u) when n = 1 for a state u = v with empty coincidence set (green) and a state u = w with non-empty coincidence set (blue) the plate. When the two plates are not prevented from touching each other, a contact of the plates commonly leads to an instability of the device-also known in the literature as "pull-in instability"-which is revealed as a singularity in the corresponding mathematical equations, e.g., see [9,16] and the references therein. In contrast, when the ground plate is coated with an insulating layer preventing a direct contact of the plates, see Fig.…”

Singular behavior of an electrostatic–elastic membrane system with an external pressure

“…This equation models (cf. [2,14]) an electrostatic-elastic membrane system with an external pressure denoted by P ≥ 0. This device consists of an elastic membrane suspended over a rigid ground plate, where the normalized distance between the membrane and the ground plate is described by u in equation (S) λ,P .…”

Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model

“…Since the pioneering works [3,7,10,18], their mathematical analysis has been the subject of numerous papers. We refer to [5,14] for a more complete account and an extensive list of references. We focus here on a model describing the evolution of the vertical deformation of the elastic plate from rest and the electrostatic potential between the plates.…”