On qualitative properties of solutions to microelectromechanical systems with general permittivity

Lienstromberg, Christina

doi:10.1007/s00605-015-0744-5

Cited by 10 publications

(8 citation statements)

References 17 publications

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“…The presentation is similar to the one in [14] for f = f (x), merely taking into account that in the present work the permittivity profile does not only depend on the spatial variable x, but also on the displacement u of the elastic membrane. The presentation is similar to the one in [14] for f = f (x), merely taking into account that in the present work the permittivity profile does not only depend on the spatial variable x, but also on the displacement u of the elastic membrane.…”

Section: Modellingmentioning

confidence: 77%

“…In [14] this topic is addressed for the case f = f (x). The situation is rather different in the case of a general permittivity profile.…”

Section: Theorem 33 (Solution To the Elliptic Problem Onmentioning

confidence: 99%

“…Always assuming the permittivity profile f to be constant, [8] deals with the above system in the stationary case, whereas in [9] the evolution problem is investigated. To the best of the author's knowledge, there is only the paper [14] which addresses the moving boundary value problem with general permittivity f = f (x). However, none of the mentioned contributions is concerned with the coupled system, including the additional feature of a general permittivity profile f = f (x, u(t, x)).…”

Section: Introductionmentioning

confidence: 99%

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A free boundary value problem modelling microelectromechanical systems with general permittivity

Lienstromberg¹

2015

Nonlinear Analysis: Real World Applications

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Section: Modellingmentioning

confidence: 77%

“…In [14] this topic is addressed for the case f = f (x). The situation is rather different in the case of a general permittivity profile.…”

Section: Theorem 33 (Solution To the Elliptic Problem Onmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A free boundary value problem modelling microelectromechanical systems with general permittivity

Lienstromberg¹

2015

Nonlinear Analysis: Real World Applications

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“…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%

“…The corresponding results are published in [33] for the case of a spatially varying permittivity p = p(x), in [11] for the case in which p depends on the membrane's displacement, and in [34] for p = p(x, u). In this survey we only state the result on the most general setting p = p(x, u) and discuss the other cases briefly in the subsequent remark.…”

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

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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.

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Mathematical Modelling and Analysis of Temperature Effects in MEMS

Escher

Würth

2020

Virtual Design and Validation

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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.

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On qualitative properties of solutions to microelectromechanical systems with general permittivity

Cited by 10 publications

References 17 publications

A free boundary value problem modelling microelectromechanical systems with general permittivity

A free boundary value problem modelling microelectromechanical systems with general permittivity

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

Mathematical Modelling and Analysis of Temperature Effects in MEMS

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