A geometric analysis of a model for micro-electro mechanical systems (MEMS)

13/12/2019, 14:00 - 15:00 in MAP/0G/018

Popovic, Nikola (University of Edinburgh)


Abstract:

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate; they have found numerous applications, including in medicine, optics, and telecommunications, and are encountered in a wide variety of technological devices nowadays.

Here, the electrostatic-elastic case is considered, whereby an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential. That case is commonly described by a parabolic partial differential equation that contains a singular nonlinear source term which can give rise to the “touchdown” phenomenon. Mathematically speaking, touchdown may imply the non-existence of steady states in the model, or blow-up of solutions in finite time. In a recently proposed, regularised model, a small “regularisation” parameter ε is introduced, whereby such singularities can be avoided by the introduction of an additional insulating layer between the membrane and the ground plate. Standard techniques from dynamical systems and geometric singular perturbation theory, in combination with the desingularisation technique known as “blow-up”, allow for a precise description of steady-state dynamics in the regularised model, as well as for a perturbative resolution of the resulting bifurcation diagram. The interplay between various model parameters is emphasised; in particular, the focus is on the singular limit as some of these parameters tend to zero.

Joint with Annalisa Iuorio and Peter Szmolyan, Vienna University of Technology, Austria.

Mathematical Sciences Research Centre