A free boundary value problem modelling microelectromechanical systems with general permittivity

Lienstromberg, Christina

doi:10.1016/j.nonrwa.2015.03.008

Cited by 9 publications

(7 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Results on local and global well-posedness of (33)-(36) with a constant permittivity profile p ≡ 1 have recently been established in [9,10]. Theorem 3.2 (Global Existence, [32,34]). Let q ∈ (2, ∞), ε > 0 and λ > 0.…”

Section: Simplified Modelsmentioning

confidence: 99%

See 1 more Smart Citation

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

Escher¹,

Lienstromberg²

2017

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

View full text Add to dashboard Cite

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semior quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles. The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account-the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

show abstract

Section: Simplified Modelsmentioning

confidence: 99%

“…Further investigations of qualitative properties of MEMS systems may be found in [28,29,30]. In this survey we report on recent results obtained in [7,11,12,32,33,34], treating non-constant permittivity profiles p = p(x, u(t, x)), which may depend on the spatial variable x and the membrane's displacement u(t, x).…”

mentioning

confidence: 99%

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

Escher¹,

Lienstromberg²

2017

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

View full text Add to dashboard Cite

show abstract

“…There are various variants and extensions of the governing equations for a MEMS device in order to incorporate additional physical effects. Except for [32,105,106], where the case of a non-uniform potential applied along the elastic plate is studied, and [83][84][85], where a model involving two elastic plates suspended one above each other is considered, the analytical investigations regarding extended models have been concerned so far almost exclusively with vanishing aspect ratio models like (1.9). For instance, spatial variations in the dielectric properties of the elastic plate which modify the strength of the Coulomb force may be included [127] or the device may be embedded in a circuit in an attempt to control the pull-in instability [131].…”

Section: Other Extensionsmentioning

confidence: 99%

Some singular equations modeling MEMS

Laurençot¹,

Walker²

2016

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

ABSTRACT. In the past fifteen years mathematical models for microelectromechanical systems (MEMS) have been the subject of several studies, in particular due to the interesting qualitative properties they feature. Still most research is devoted to an illustrative, but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes, the so-called vanishing (or small) aspect ratio model. The analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey.

show abstract

“…The case of a general permittivity profile without assuming a small aspect ratio is investigated in [Lie15,Lie16] for small deformations and in [EL16] for non-small defor-mations.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modelling and Analysis of Temperature Effects in MEMS

Escher

Würth

2020

Virtual Design and Validation

View full text Add to dashboard Cite

This thesis is concerned with the mathematical analysis of models for Micro-Electro-Mechanical Systems (MEMS). These models arise in the form of coupled partial differential equations with a moving boundary. Although MEMS devices are often operated in non-isothermal environments, temperature is usually neglected in the mathematical investigations. Therefore the focus of our modelling is to incorporate temperature and the related material properties. We derive two models, both of which focus on different aspects of the underlying physics. Afterwards we prove local well-posedness in time and also global well-posedness under additional assumptions on the model's parameters.Lastly, we provide some numerical results which exemplify how temperature and the model's material constants change the qualitative behaviour of the system.

show abstract

A free boundary value problem modelling microelectromechanical systems with general permittivity

Cited by 9 publications

References 23 publications

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

Some singular equations modeling MEMS

Mathematical Modelling and Analysis of Temperature Effects in MEMS

Contact Info

Product

Resources

About