Details
| Original language | English |
|---|---|
| Pages (from-to) | 745-771 |
| Number of pages | 27 |
| Journal | Discrete and Continuous Dynamical Systems - Series S |
| Volume | 10 |
| Issue number | 4 |
| Publication status | Published - Jul 2017 |
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 semior 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 inuence 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 deection 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.
Keywords
- Finite-time singularity, Free boundary value problem, General permittivity profile, MEMS
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Mathematics(all)
- Applied Mathematics
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In: Discrete and Continuous Dynamical Systems - Series S, Vol. 10, No. 4, 07.2017, p. 745-771.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A survey on second-order free boundary value problems modelling MEMs with general permittivity profile
AU - Escher, Joachim
AU - Lienstromberg, Christina
PY - 2017/7
Y1 - 2017/7
N2 - 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 semior 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 inuence 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 deection 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.
AB - 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 semior 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 inuence 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 deection 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.
KW - Finite-time singularity
KW - Free boundary value problem
KW - General permittivity profile
KW - MEMS
UR - http://www.scopus.com/inward/record.url?scp=85018456400&partnerID=8YFLogxK
U2 - 10.3934/dcdss.2017038
DO - 10.3934/dcdss.2017038
M3 - Article
AN - SCOPUS:85018456400
VL - 10
SP - 745
EP - 771
JO - Discrete and Continuous Dynamical Systems - Series S
JF - Discrete and Continuous Dynamical Systems - Series S
SN - 1937-1632
IS - 4
ER -