Mathematical Modeling of Electrostatic Microelectromechanical Systems

Electrostatic microelectromechanical systems (MEMS) integrate mechanical structures with electrical actuation at micron scales, enabling precision sensors, actuators and resonators. Mathematical models capture the coupling between an elastic or rigid microstructure and the electrostatic field within a narrow gap. The electrostatic potential satisfies an elliptic boundary value problem, while the moving structure follows either a parabolic or hyperbolic partial differential equation driven by the Maxwell stress. Nonlinearities arise from the inverse‐square nature of the electrostatic force and from geometric effects of large deflections. Key phenomena include pull-in instability, where a critical voltage causes the movable electrode to collapse onto the substrate, and finite-time touchdown (quenching) of the solution. Rigorous analysis encompasses existence, uniqueness and regularity of stationary and time-dependent solutions, asymptotic limits such as small aspect-ratio or reinforced-limit models via Γ-convergence, and the derivation of sharp pull-in voltage bounds. Numerical schemes—both semi-implicit and fully implicit discretizations—are employed to simulate dynamic response and to validate analytic predictions. Recent advances also consider stochastic forcing, heterogeneous dielectric properties and coupled multiphysics effects, broadening design principles for reliable and energy-efficient MEMS devices.

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Mathematical Modeling of Electrostatic Microelectromechanical Systems publication trend

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Technical terms

Pull-in voltage: The critical actuation voltage beyond which the movable electrode collapses onto the substrate.

Touchdown (quenching): A finite-time singularity in the solution representing contact between the microstructure and the counter-electrode.

Γ-convergence: A notion of variational convergence for energy functionals used to derive asymptotic models.

Well-posedness: The property that a mathematical model admits a unique solution that depends continuously on initial and boundary data.

Aspect ratio: The ratio of the gap height to the lateral dimensions of the MEMS device, often assumed small in reduced models.

Electrostatic potential: The scalar field satisfying an elliptic equation which determines the force distribution on the deformable structure.

References

1. Touchdown solutions in general MEMS models. Advances in Nonlinear Analysis (2023).
2. Impacts of noise on quenching of some models arising in MEMS technology. European Journal of Applied Mathematics (2022).
3. Reinforced Limit of a MEMS Model with Heterogeneous Dielectric Properties. Applied Mathematics & Optimization (2020).
4. On a quasilinear parabolic–hyperbolic system arising in MEMS modeling. Annali di Matematica Pura ed Applicata (1923 -) (2024).
5. Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems. Discrete and Continuous Dynamical Systems - S (2019).