American Institute of Mathematical Sciences

Advanced
  2019, 26: 54-71. doi: 10.3934/era.2019.26.005

Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems

Hawraa Alsayed 1,2, , Hussein Fakih 1,3, , Alain Miranville 2,4, and  Ali Wehbe 1,

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

Received  May 2019 Revised  July 2019 Published  July 2019

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
References:
[1]

E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1–64. doi: 10.1017/s0962492900002336.  Google Scholar

[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.  Google Scholar

[3]

L. Cherfils, A. Miranville, S. Peng and C. Xu, Analysis of discretized parabolic problems modelling Electrostatic Micro-Electromechanical systems, AIMS Journals, 2018. Google Scholar

[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.  Google Scholar

[5]

L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, Inc., Hoboken, NJ, 2011. doi: 10.1002/9781118111130.  Google Scholar

[12]

I. Stakgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998.  Google Scholar

[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.  Google Scholar

[14]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[15]

J. A. Pelesko, D. H. Bernstein and J. McCuan, Symmetry and Symmetry Breaking in Electrostatic MEMS, Proceedings of MSM, (2003), 304–307. Google Scholar

[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.  Google Scholar

[17]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. Cherfils, A. Miranville, S. Peng and C. Xu, Analysis of discretized parabolic problems modelling Electrostatic Micro-Electromechanical systems, AIMS Journals, 2018. Google Scholar

[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.  Google Scholar

[5]

L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, Inc., Hoboken, NJ, 2011. doi: 10.1002/9781118111130.  Google Scholar

[12]

I. Stakgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998.  Google Scholar

[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.  Google Scholar

[14]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[15]

J. A. Pelesko, D. H. Bernstein and J. McCuan, Symmetry and Symmetry Breaking in Electrostatic MEMS, Proceedings of MSM, (2003), 304–307. Google Scholar

[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.  Google Scholar

[17]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

Figure 1.  $ \lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 0.01,\; M = 29 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  $ \lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 20,\; M = 29 $
Figure Options

Download full-size image

Download as PowerPoint slide

3(a) touchdown is observed at $t = 682\tau$ however in 3(b) it is observed at $t = 1.3339\tau$">Figure 3.  $\lambda = 11.5, \; f(x, y) = \sqrt{x^2 +y^2}, \; \tau = 0.001, \; M = 35$, in 3(a) touchdown is observed at $t = 682\tau$ however in 3(b) it is observed at $t = 1.3339\tau$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Joshua Hudson, Michael Jolly. Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations. Journal of Computational Dynamics, 2019, 6 (1) : 131-145. doi: 10.3934/jcd.2019006
[2]

Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685
[3]

Bernd Kawohl, Guido Sweers. On a formula for sets of constant width in 2d. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2117-2131. doi: 10.3934/cpaa.2019095
[4]

Ka Kit Tung, Wendell Welch Orlando. On the differences between 2D and QG turbulence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 145-162. doi: 10.3934/dcdsb.2003.3.145
[5]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[6]

Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525
[7]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297
[8]

Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations & Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57
[9]

Laurence Cherfils, Alain Miranville, Shuiran Peng, Chuanju Xu. Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1601-1621. doi: 10.3934/dcdss.2019109
[10]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021085
[11]

Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102
[12]

Brian Ryals, Robert J. Sacker. Global stability in the 2D Ricker equation revisited. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 585-604. doi: 10.3934/dcdsb.2017028
[13]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[14]

María J. Martín, Jukka Tuomela. 2D incompressible Euler equations: New explicit solutions. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4547-4563. doi: 10.3934/dcds.2019187
[15]

Leonardo Kosloff, Tomas Schonbek. Existence and decay of solutions of the 2D QG equation in the presence of an obstacle. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1025-1043. doi: 10.3934/dcdss.2014.7.1025
[16]

Yuri N. Fedorov, Luis C. García-Naranjo, Joris Vankerschaver. The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4017-4040. doi: 10.3934/dcds.2013.33.4017
[17]

Theodore Kolokolnikov, Juncheng Wei. Hexagonal spike clusters for some PDE's in 2D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 4057-4070. doi: 10.3934/dcdsb.2020039
[18]

Makram Hamouda, Chang-Yeol Jung, Roger Temam. Boundary layers for the 2D linearized primitive equations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 335-359. doi: 10.3934/cpaa.2009.8.335
[19]

A. Rousseau, Roger Temam, J. Tribbia. Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1257-1276. doi: 10.3934/dcds.2005.13.1257
[20]

Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413

2019 Impact Factor: 0.5

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences