A survey on second order free boundary value problems modelling MEMS with general permittivity profile

Joachim Escher; Christina Lienstromberg

doi:10.3934/dcdss.2017038

Article Contents

2017, Volume 10, Issue 4: 745-771. Doi: 10.3934/dcdss.2017038

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A survey on second order free boundary value problems modelling MEMS with general permittivity profile

Joachim Escher and
Christina Lienstromberg^,

Institute of Applied Mathematics, Leibniz Universität Hannover, Germany

^* Corresponding author: Christina Lienstromberg
^* Corresponding author: Christina Lienstromberg

Received: June 30, 2016

Revised: September 30, 2016

Published: July 2017

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Abstract

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles.

The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

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Mathematics Subject Classification: 35R35, 35B09, 35B44, 74M05.

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Figure 1. Sketch of the investigated idealised MEMS device

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Figure 2. Cross section of the investigated idealised MEMS device

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Figure 3. Membrane's deflection $u$ of the coupled system for $p(x)=x^8 + 0.1$ with $u_{\ast} \equiv 0$, $\lambda = 1$, and $\varepsilon \in \{0.4, 0.6\}$

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Figure 4. Membrane's deflection $u$ of the coupled system for $p(x)=x^8 + 0.1$ with $u_{\ast} \equiv 0$, $\lambda = 1$, and $\varepsilon \in \{0.1, 0.2\}$

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Figure 5. Membrane's deflection $u$ of the coupled system for $p(x)=x^8 + 0.1$ with $u_{\ast} \equiv 0$, $\lambda = 1$, and $\varepsilon = 0.15$

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Figure 6. Approximate solution $u$ to the small-aspect ratio model with $u_{\ast} \equiv 0$ for $p(x)=x^8+0.1$ and $\lambda = 1$

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