Figure 1.
$ \lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 0.01,\; M = 29 $
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2019, 26: 54-71.
doi: 10.3934/era.2019.26.005
Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems
| 1. | Lebanese University, Khawarizmi Laboratory for Mathematics and Applications, Hadath, Mont Liban, Beirut, Lebanon |
| 2. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France |
| 3. | Lebanese International University, Department of Mathematics and Physics, Lebanon |
| 4. | Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen, Fujian, China |
This paper is dedicated to study the fully discretized semi implicit and implicit schemes of a 2D parabolic semi linear problem modeling MEMS devices. Starting with the analysis of the semi-implicit scheme, we proved the existence of the discrete solution which converges under certain conditions on the voltage $ \lambda $. On the other hand, we consider a fully implicit scheme, we proved the existence of the discrete solution, which also converges to the stationary solution under certain conditions on the voltage $ \lambda $ and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.
Keywords:
Micro-Electromechanical Systems (MEMS),
pull-in voltage,
2D semi implicit scheme,
2D implicit scheme,
numerical simulations.
Mathematics Subject Classification:
Primary: 65N06, 65N12, 35K55.
Citation:
Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements,
2019, 26: 54-71.
doi: 10.3934/era.2019.26.005
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References:
| [1] |
E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1–64.
doi: 10.1017/s0962492900002336. |
| [2] |
N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models, European Journal of Applied Mathematics, 22 (2011), 455–470.
doi: 10.1017/S0956792511000180. |
| [3] |
L. Cherfils, A. Miranville, S. Peng and C. Xu, Analysis of discretized parabolic problems modelling Electrostatic Micro-Electromechanical systems, AIMS Journals, 2018. |
| [4] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modelling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/cln/020. |
| [5] |
L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. |
| [6] |
G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touch down model of electrostatic MEMS, SIAM Jpournal on Applied Mathematics, 67 (2006/07), 434–446.
doi: 10.1137/060648866. |
| [7] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM Journal on Applied Mathematics, 38 (2006/07), 1423–1449.
doi: 10.1137/050647803. |
| [8] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅱ: Dynamic case, NoDEA Nonlinear Differential Equations and Applications, 15 (2008), 115–145.
doi: 10.1007/s00030-007-6004-1. |
| [9] |
Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅲ: Refined touchdown behavior, J. Differential Equations, 244 (2008), 2277–2309.
doi: 10.1016/j.jde.2008.02.005. |
| [10] |
Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM Journal on Applied Mathematics, 66 (2005), 309–338.
doi: 10.1137/040613391. |
| [11] |
S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, Inc., Hoboken, NJ, 2011.
doi: 10.1002/9781118111130. |
| [12] |
I. Stakgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. |
| [13] |
J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM Journal on Applied Mathematics, 62 (2002), 888–908.
doi: 10.1137/S0036139900381079. |
| [14] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. |
| [15] |
J. A. Pelesko, D. H. Bernstein and J. McCuan, Symmetry and Symmetry Breaking in Electrostatic MEMS, Proceedings of MSM, (2003), 304–307. |
| [16] |
J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, Journal of Engineering Mathematics, 41 (2001), 345–366.
doi: 10.1023/A:1012292311304. |
| [17] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. |
| [18] |
Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chinese Annals of Mathematics, 39 (2018), 129–144.
doi: 10.1007/s11401-018-1056-6. |
show all references
References:
| [1] |
E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1–64.
doi: 10.1017/s0962492900002336. |
| [2] |
N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models, European Journal of Applied Mathematics, 22 (2011), 455–470.
doi: 10.1017/S0956792511000180. |
| [3] |
L. Cherfils, A. Miranville, S. Peng and C. Xu, Analysis of discretized parabolic problems modelling Electrostatic Micro-Electromechanical systems, AIMS Journals, 2018. |
| [4] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modelling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/cln/020. |
| [5] |
L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. |
| [6] |
G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touch down model of electrostatic MEMS, SIAM Jpournal on Applied Mathematics, 67 (2006/07), 434–446.
doi: 10.1137/060648866. |
| [7] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM Journal on Applied Mathematics, 38 (2006/07), 1423–1449.
doi: 10.1137/050647803. |
| [8] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅱ: Dynamic case, NoDEA Nonlinear Differential Equations and Applications, 15 (2008), 115–145.
doi: 10.1007/s00030-007-6004-1. |
| [9] |
Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅲ: Refined touchdown behavior, J. Differential Equations, 244 (2008), 2277–2309.
doi: 10.1016/j.jde.2008.02.005. |
| [10] |
Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM Journal on Applied Mathematics, 66 (2005), 309–338.
doi: 10.1137/040613391. |
| [11] |
S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, Inc., Hoboken, NJ, 2011.
doi: 10.1002/9781118111130. |
| [12] |
I. Stakgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. |
| [13] |
J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM Journal on Applied Mathematics, 62 (2002), 888–908.
doi: 10.1137/S0036139900381079. |
| [14] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. |
| [15] |
J. A. Pelesko, D. H. Bernstein and J. McCuan, Symmetry and Symmetry Breaking in Electrostatic MEMS, Proceedings of MSM, (2003), 304–307. |
| [16] |
J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, Journal of Engineering Mathematics, 41 (2001), 345–366.
doi: 10.1023/A:1012292311304. |
| [17] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. |
| [18] |
Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chinese Annals of Mathematics, 39 (2018), 129–144.
doi: 10.1007/s11401-018-1056-6. |
Figure 2.
$ \lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 20,\; M = 29 $
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