American Institute of Mathematical Sciences

October 2019, 12(6): 1601-1621. doi: 10.3934/dcdss.2019109

Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems

 1 Université de La Rochelle, Laboratoire des Sciences de l'Ingénieur pour l'Environnement, UMR CNRS 7356, Avenue Michel Crépeau, F-17042 La Rochelle Cedex, France 2 Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling, and High Performance Scientific Computing, Xiamen, Fujian, China 3 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

* Corresponding author: Shuiran Peng

The work of this author is partialy supported by NSF of China (Grant numbers 11471274, 11421110001, 51661135011, and 91630204)

Received  November 2017 Revised  January 2018 Published  November 2018

Our aim in this paper is to study discretized parabolic problems modeling electrostatic micro-electromechanical systems (MEMS). In particular, we prove, both for semi-implicit and implicit semi-discrete schemes, that, under proper assumptions, the solutions are monotonically and pointwise convergent to the minimal solution to the corresponding elliptic partial differential equation. We also study the fully discretized semi-implicit scheme in one space dimension. We finally give numerical simulations which illustrate the behavior of the solutions both in one and two space dimensions.

Citation: 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
References:
 [1] E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1-64. [2] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61257-2. [3] J. Baranger and M. Duc-Jacquet, Matrices tridiagonales symétriques et matrices factorisables, Revue Française d'Informatique et de Recherche 0pérationnelle. Série Rouge, 5 (1971), 61-66. [4] A. Ben-Israel, A Newton-Raphson method for the solution of systems of equations, Journal of Mathematical Analysis and Applications, 15 (1966), 243-252. doi: 10.1016/0022-247X(66)90115-6. [5] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer Science & Business Media, 2003. [6] J. Bryzek, S. Roundy and B. Bircumshaw, et al, Marvelous MEMS, IEEE Circuits and Devices Magazine, 22 (2006), 8–28. [7] T. Cazenave, An Introduction to Semilinear Elliptic Equations, Editora do Instituto de Matem tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 2006. [8] C. Cowan, P. Esposito and N. Ghoussoub, et al, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Archive for Rational Mechanics and Analysis, 198 (2010), 763–787. doi: 10.1007/s00205-010-0367-x. [9] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, American Mathematical Society, 2010. doi: 10.1090/cln/020. [10] R. P. Feynman, There's plenty of room at the bottom, Engineering and Science, 23 (1960), 22-36. [11] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM Journal on Mathematical Analysis, 38 (2007), 1423-1449. doi: 10.1137/050647803. [12] 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. [13] G. H. Golub and C. F. Van Loan, Matrix Computations, JHU Press, 4$^{nd}$ edition, 2013. [14] S. B. Gueye, The exact formulation of the inverse of the tridiagonal matrix for solving the 1D Poisson equation with the finite difference method, Journal of Electromagnetic Analysis and Applications, 6 (2014), 303-308. [15] J. S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, Journal of Mathematical Analysis and Applications, 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9. [16] J. S. Guo, On the quenching rate estimate, Quarterly of Applied Mathematics, 49 (1991), 747-752. doi: 10.1090/qam/1134750. [17] Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅲ: Refined touchdown behavior, Journal of Differential Equations, 244 (2008), 2277-2309. doi: 10.1016/j.jde.2008.02.005. [18] Y. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, Journal of Differential Equations, 245 (2008), 809-844. doi: 10.1016/j.jde.2008.03.012. [19] 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. [20] T. Horiuchi and P. Kumlin, On the minimal solution for quasilinear degenerate elliptic equation and its blow-up, Journal of Mathematics of Kyoto University, 44 (2004), 381-439. doi: 10.1215/kjm/1250283558. [21] H. A. Levine, Advances in quenching, Nonlinear Diffusion Equations and Their Equilibrium States, Birkhäuser Boston, 7 (1992), 319-346. [22] F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816. [23] X. Luo, D. Ye and F. Zhou, Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection, Journal of Differential Equations, 251 (2011), 2082-2099. doi: 10.1016/j.jde.2011.07.011. [24] H. C. Nathanson and R. A. Wickstrom, A Resonant gate silicone surface transistor with high Q band pass properties, Applied Physics Letters, 7 (1965), 84-86. [25] H. C. Nathanson, W. E. Newell and R. A. Wickstrom, et al, The resonant gate transistor, IEEE Transactions on Electron Devices, 14 (1967), 117–133. [26] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Society for Industrial and Applied Mathematics, 2000. doi: 10.1137/1.9780898719468. [27] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. [28] J. I. Siddique, R. Deaton and E. Sabo, An experimental investigation of the theory of electrostatic deflections, Journal of Electrostatics, 69 (2011), 1-6. [29] D. Ye and F. Zhou, A generalized two dimensional Emden-Fowler equation with exponential nonlinearity, Calculus of Variations and Partial Differential Equations, 13 (2001), 141-158. doi: 10.1007/PL00009926.

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References:
 [1] E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1-64. [2] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61257-2. [3] J. Baranger and M. Duc-Jacquet, Matrices tridiagonales symétriques et matrices factorisables, Revue Française d'Informatique et de Recherche 0pérationnelle. Série Rouge, 5 (1971), 61-66. [4] A. Ben-Israel, A Newton-Raphson method for the solution of systems of equations, Journal of Mathematical Analysis and Applications, 15 (1966), 243-252. doi: 10.1016/0022-247X(66)90115-6. [5] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer Science & Business Media, 2003. [6] J. Bryzek, S. Roundy and B. Bircumshaw, et al, Marvelous MEMS, IEEE Circuits and Devices Magazine, 22 (2006), 8–28. [7] T. Cazenave, An Introduction to Semilinear Elliptic Equations, Editora do Instituto de Matem tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 2006. [8] C. Cowan, P. Esposito and N. Ghoussoub, et al, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Archive for Rational Mechanics and Analysis, 198 (2010), 763–787. doi: 10.1007/s00205-010-0367-x. [9] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, American Mathematical Society, 2010. doi: 10.1090/cln/020. [10] R. P. Feynman, There's plenty of room at the bottom, Engineering and Science, 23 (1960), 22-36. [11] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM Journal on Mathematical Analysis, 38 (2007), 1423-1449. doi: 10.1137/050647803. [12] 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. [13] G. H. Golub and C. F. Van Loan, Matrix Computations, JHU Press, 4$^{nd}$ edition, 2013. [14] S. B. Gueye, The exact formulation of the inverse of the tridiagonal matrix for solving the 1D Poisson equation with the finite difference method, Journal of Electromagnetic Analysis and Applications, 6 (2014), 303-308. [15] J. S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, Journal of Mathematical Analysis and Applications, 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9. [16] J. S. Guo, On the quenching rate estimate, Quarterly of Applied Mathematics, 49 (1991), 747-752. doi: 10.1090/qam/1134750. [17] Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅲ: Refined touchdown behavior, Journal of Differential Equations, 244 (2008), 2277-2309. doi: 10.1016/j.jde.2008.02.005. [18] Y. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, Journal of Differential Equations, 245 (2008), 809-844. doi: 10.1016/j.jde.2008.03.012. [19] 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. [20] T. Horiuchi and P. Kumlin, On the minimal solution for quasilinear degenerate elliptic equation and its blow-up, Journal of Mathematics of Kyoto University, 44 (2004), 381-439. doi: 10.1215/kjm/1250283558. [21] H. A. Levine, Advances in quenching, Nonlinear Diffusion Equations and Their Equilibrium States, Birkhäuser Boston, 7 (1992), 319-346. [22] F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816. [23] X. Luo, D. Ye and F. Zhou, Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection, Journal of Differential Equations, 251 (2011), 2082-2099. doi: 10.1016/j.jde.2011.07.011. [24] H. C. Nathanson and R. A. Wickstrom, A Resonant gate silicone surface transistor with high Q band pass properties, Applied Physics Letters, 7 (1965), 84-86. [25] H. C. Nathanson, W. E. Newell and R. A. Wickstrom, et al, The resonant gate transistor, IEEE Transactions on Electron Devices, 14 (1967), 117–133. [26] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Society for Industrial and Applied Mathematics, 2000. doi: 10.1137/1.9780898719468. [27] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. [28] J. I. Siddique, R. Deaton and E. Sabo, An experimental investigation of the theory of electrostatic deflections, Journal of Electrostatics, 69 (2011), 1-6. [29] D. Ye and F. Zhou, A generalized two dimensional Emden-Fowler equation with exponential nonlinearity, Calculus of Variations and Partial Differential Equations, 13 (2001), 141-158. doi: 10.1007/PL00009926.
An idealized MEMS capacitor
The branch of solutions $u(0)$ as a function of $\lambda$: (a) $f(x) = |2x|$; (b) $f(x)\equiv1$
The branch of solutions $u(0, 0)$ as a function of $\lambda$:(a) $f(x, y) = \sqrt{x^2+y^2}$; (b) $\lambda = 10$, $u_{\lambda}$, $u_{\lambda^*}$, and $u^+_{\lambda}$
The branch of solutions $u(0, 0)$ as a function of $\lambda$:(a) $f(x, y)\equiv1$; (b) $\lambda = 2.5$, $u_{\lambda}$, $u_{\lambda^*}$ and $u^+_{\lambda}$; (c) $\lambda = 1.6$, four corresponding solutions
1D, $\tau$ = 0.01. (a) Semi-implicit scheme; (b) Implicit scheme; (c) Convergence: $error = \|u^n-u_{\lambda}\|$
1D, $M = 199$, $f(x) = |2x|$. (a) $\tau = 0.01$, $\lambda = 4.0$, $u\_ini =$pointwise_ini; (b) $\tau = 0.01$, $\lambda = 4.0$, $u\_ini$ is nonsymmetric; (c) touchdown phenomenon: $\lambda$ = 4.45, $\tau$ = 0.001
2D, $\lambda$ = 10.0, $f(x, y) = \sqrt{x^2+y^2}$, $\tau$ = 0.01, $M = 29$. (a) The global solution when $t = 100\tau$; (b) Semi-implicit scheme; (c) Implicit scheme; (d) Convergence
2D, $\lambda$ = 10.0, $f(x, y) = \sqrt{x^2+y^2}$, $\tau$ = 0.01, $M = 29$. (a) Semi-implicit scheme; (b) Implicit scheme; (c) Convergence
2D, $f(x, y) = \sqrt{x^2+y^2}$, touchdown phenomenon. (a) Global solution when $t = 683\tau$; (b) Section $y = 0$
The main idea of the continuation method
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