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首页> 外文期刊>Discrete and continuous dynamical systems▼hSeries S >A SURVEY ON SECOND-ORDER FREE BOUNDARY VALUE PROBLEMS MODELLING MEMS WITH GENERAL PERMITTIVITY PROFILE
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A SURVEY ON SECOND-ORDER FREE BOUNDARY VALUE PROBLEMS MODELLING MEMS WITH GENERAL PERMITTIVITY PROFILE

机译：具有通用介电常数的MEMS建模的二阶自由边值问题研究

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摘要

In this survey we review some recent results on microelectrome-chanical 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 semi-or 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.
机译：在这项调查中，我们回顾了有关具有一般介电常数分布的微机电系统的一些最新结果。根据特定的应用，通过考虑各种物理建模方面来推导不同的微分方程系统。在任何情况下，弹性膜位移的半或准线性双曲或抛物线演化问题都与椭圆运动边界问题有关，椭圆运动边界问题决定了弹性膜和刚性接地板所占据区域的静电势。在所有模型中，特别令人感兴趣的是不同类别的介电常数分布的影响。随后的分析研究仅限于耗散占主导地位的膜位移。对于由此产生的抛物线演化问题，分别研究了局部适定性，整体存在性，有限时间奇异性的发生以及对所谓小纵横比模型的解的收敛性。此外，要解决的一个问题是要注意的，直到考虑到非恒定的介电常数分布时为止-膜的偏转方向，或者用数学的话说，就是解决演化问题的标志。该调查是通过介绍一些数值结果而完成的，这些数值结果通过揭示广泛使用的小长宽比模型和耦合问题的解的实质性差异，特别证明了考虑耦合问题的合理性。

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