Details
| Original language | English |
|---|---|
| Pages (from-to) | 5693-5719 |
| Number of pages | 27 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 367 |
| Issue number | 8 |
| Publication status | Published - 17 Oct 2014 |
Abstract
The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account which leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltage values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-state solutions if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.
Keywords
- Asymptotic stability, Curvature, Free boundary problem, MEMS, Small aspect ratio limit, Well-posedness
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Transactions of the American Mathematical Society, Vol. 367, No. 8, 17.10.2014, p. 5693-5719.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Dynamics of a free boundary problem with curvature modeling electrostatic MEMS
AU - Escher, Joachim
AU - Laurençot, Philippe
AU - Walker, Christoph
PY - 2014/10/17
Y1 - 2014/10/17
N2 - The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account which leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltage values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-state solutions if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.
AB - The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account which leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltage values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-state solutions if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.
KW - Asymptotic stability
KW - Curvature
KW - Free boundary problem
KW - MEMS
KW - Small aspect ratio limit
KW - Well-posedness
UR - http://www.scopus.com/inward/record.url?scp=84929111760&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2014-06320-4
DO - 10.1090/S0002-9947-2014-06320-4
M3 - Article
AN - SCOPUS:84929111760
VL - 367
SP - 5693
EP - 5719
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 8
ER -