Using the two-sided approximations method for the numerical research of nanoelectromechanical systems under the action of the Casimir force
Abstract
Relevance. Developing the method of two-sided approximations for finding a positive solution to a nonlinear boundary value problem that models an electrostatic nanoelectromechanical system under external pressure has been considered. The presented mathematical model takes into account the influence of Casimir forces as an additional force of attraction between the components of nanosystems. Such systems feature the nonlinear phenomenon of pull-in instability, which occurs due to the interaction of conductive plates under a critical electric voltage. This phenomenon significantly limits the range of system’s stable states and is typical of many nanodevices, in particular, accelerometers, switches, micromirrors, microresonators, etc. It is suggested to study the model parameters and estimate their values in order to analyze the stable states of nanoelectromechanical systems.
Goal. To develop a method of two-sided approximations for solving the given problem by using the methods of the nonlinear operator theory in semi-ordered Banach spaces.
Research methods. The nonlinear elliptic equation that models the operation of the electrostatic nanoelectromechanical system using the Green’s function method is replaced by its Hammerstein integral equation equivalent. The specified integral equation is considered to be a nonlinear operator equation with a monotone operator in the space of continuous functions, semi-ordered by using a cone of non-negative functions. The conditions for the existence of a unique positive solution to the specified problem and the two-sided convergence of successive approximations to such a solution have been obtained.
The results. The developed method has been implemented and investigated by solving test problems. The results of computational experiment are shown in graphical and tabular form.
Conclusions. The performed computational experiments have confirmed the effectiveness of the developed method and can be used to solve the problems of mathematical modeling of nonlinear processes in micro- and nanoelectromechanical systems. The prospects for further research may lie in applying the method of two-sided approximations for models of nanoelectromechanical systems with repulsive Casimir forces.
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B. McLellan, L. Medina, C. Xu, Y. Yang “Critical pull-in curves of MEMS actuators in presence of Casimir force” ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 96, no. 12, pp. 1406-1422, 2016.
J. A. Pelesko, D. H. Bernstein Modeling MEMS and NEMS, Cleveland: CRC Press, 2002. 351 p.
J. Davila, I. Flores, I. Guerra “Multiplicity of solutions for a fourth order equation with power-type nonlinearity” Mathematische Annalen, vol. 348, no. 1, pp. 143-193, 2010.
P. Esposito, N. Ghoussoub and Y. Guo Mathematical analysis of partial differential equations modeling electrostatic MEMS, Providence: American Mathematical Society, 2010. 262 p.
Z. Guo, J. Wei “On a fourth order nonlinear elliptic equation with negative exponent” SIAM journal on mathematical analysis, vol. 40, no. 5, pp. 2034-2054, 2008.
D. Ye, F. Zhou “On a general family of nonautonomous elliptic and parabolic equations” Calculus of Variations and Partial Differential Equations, vol. 37, pp. 259-274, 2010.
A. H. Nayfeh, M. I. Younis, E. M. Abdel-Rahman “Reduced-order models for MEMS applications” Nonlinear dynamics, vol. 41, no. 1, pp. 211-236, 2005.
M. V. Sidorov “Green-Rvachev’s quasi-function method for constructing two-sided approximations to positive solution of nonlinear boundary value problems”, Carpathian Mathematical Publications, vol. 10, no. 2, pp. 360-375, 2018.
R. C. Batra, M. Porfiri, D. Spinello “Effects of Casimir force on pull-in instability in micro-membranes” Europhysics Letters, vol. 77, no. 2, pp. 20010, 2007.
J. A. Pelesko “Mathematical modeling of electrostatic MEMS with tailored dielectric properties”, SIAM Journal on Applied Mathematics, vol. 62, no. 3, pp. 888-908, 2002.
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, vol. 66, no. 1, pp. 309-338, 2005.
F. Lin and Y. Yang “Nonlinear non-local elliptic equation modeling electrostatic actuation”, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 463, no. 2081, pp. 1323-1337, 2007.
Gusejnov K. Foton: uchebnoe posobie / pod redakciej professora B.S. Ishhanova. M.: “KDU”, “Universitetskaja kniga”, 2020. 276 p. [in Russian].
J. R. Beckham, J. A. Pelesko “An electrostatic-elastic membrane system with an external pressure”, Mathematical and computer modelling, vol. 54, no. 11-12, pp. 2686-2708, 2011.
Y. Guo, Y. Zhang and F. Zhou “Singular behavior of an electrostatic–elastic membrane system with an external pressure”, Nonlinear Analysis, vol. 190, pp. 111611, 2020.
V. I. Opojtsev, T. A. Khurodze Nonlinear Operators in Spaces with a Cone, Tbilisi, USSR: Izdatel’stvo Tbilisskogo Universiteta, 1984. 246 p. [in Russian].
M. A. Krasnosel’skij Positive Solutions of Operator Equations. M: Fizmatgiz, 1962. 394 p. [in Russian].
McLellan B., Medina L., Xu C., Yang Y. Critical pull-in curves of MEMS actuators in presence of Casimir force. ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik. 2016. Vol. 96, № 12. P. 1406 1422.
Pelesko J. A., Bernstein D. H. Modeling MEMS and NEMS. Cleveland: CRC Press, 2002. 351 p.
Davila J., Flores I., Guerra I. Multiplicity of solutions for a fourth order equation with power-type nonlinearity. Mathematische Annalen. 2010. Vol. 348, № 1. P. 143–193.
Esposito P., Ghoussoub N., Guo Y. Mathematical analysis of partial differential equations modeling electrostatic MEMS. Providence: American Mathematical Society, 2010. 262 p.
Guo Z., Wei J. On a fourth order nonlinear elliptic equation with negative exponent. SIAM journal on mathematical analysis. 2008. Vol. 40, № 5. P. 2034–2054.
Ye D., Zhou F. On a general family of nonautonomous elliptic and parabolic equations. Calculus of Variations and Partial Differential Equations. 2010. Vol. 37. P. 259–274.
Nayfeh A. H., Younis M. I., Abdel-Rahman E. M. Reduced-order models for MEMS applications. Nonlinear dynamics. 2005. Vol. 41, № 1. P. 211–236.
Sidorov M. V. Green-Rvachev’s quasi-function method for constructing two-sided approximations to positive solution of nonlinear boundary value problems. Carpathian Mathematical Publications. 2018. Vol. 10, № 2. P. 360–375.
Batra R. C., Porfiri M., Spinello D. Effects of Casimir force on pull-in instability in micro-membranes. Europhysics Letters. 2007. Vol. 77, № 2. P. 20010.
Pelesko J. A. Mathematical modeling of electrostatic MEMS with tailored dielectric properties. SIAM Journal on Applied Mathematics. 2002. Vol. 62, № 3. P. 888–908.
Guo Y., Pan Z., Ward M. J. Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM Journal on Applied Mathematics. 2005. Vol. 66, № 1. P. 309–338.
Lin F., Yang Y. Nonlinear non-local elliptic equation modeling electrostatic actuation. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 2007. Vol. 463, № 2081. P. 1323–1337.
Гусейнов К. Фотон: учебное пособие / под редакцией профессора Б.С. Ишханова. Москва: “КДУ”, “Университетская книга”, 2020. 276 с.
Beckham J. R., Pelesko J. A. An electrostatic-elastic membrane system with an external pressure. Mathematical and computer modelling. 2011. Vol. 54, №11–12. P. 2686–2708.
Guo Y., Zhang Y., Zhou F. Singular behavior of an electrostatic–elastic membrane system with an external pressure. Nonlinear Analysis. 2020. Vol. 190. P. 111611.
Опойцев В. И., Хуродзе Т. А. Нелинейные операторы в пространствах с конусом. Тбилиси: Изд-во Тбилис. ун-та, 1984. 246 с.
Красносельский М. А. Положительные решения операторных уравнений. Москва: Физматгиз, 1962. 394 с.
