Abstract
The evolution of engineering applications is increasingly shifting towards the embedded nature, resulting in low-cost solutions, micro/nano dimensional and actuators being exploited as fundamental components to connect the physical nature of information with the abstract one, which is represented in the logical form in a machine. In this context, the scientific community has gained interest in modeling membrane Micro-Electro-Mechanical-Systems (MEMS), leading to a wide diffusion on an industrial level owing to their ease of modeling and realization. Physically, once the external voltage is applied, an electrostatic field, orthogonal to the tangent line of the membrane, is established inside the device, producing an electrostatic pressure that acts on the membrane, deforming it. Evidently, the greater the amplitude of the electrostatic field is, the greater the curvature of the membrane. Thus, it seems natural to consider the amplitude of the electrostatic field proportional to the curvature of the membrane. Starting with this principle, the authors are actively involved in developing a second-order semi-linear elliptic model in 1D and 2D geometries, obtaining important results regarding the existence, uniqueness and stability of solutions as well as evaluating the particular operating conditions of use of membrane MEMS devices. In this context, the idea of providing a survey matures to discussing the similarities and differences between the analytical and numerical results in detail, thereby supporting the choice of certain membrane MEMS devices according to the industrial application. Finally, some original results about the stability of the membrane in 2D geometry are presented and discussed.
Highlights
Recent industrial guidelines direct researchers and designers towards the development of low-cost devices to combine physical properties with low-level machine languages
The surveys published on MEMS membrane electrostatic devices are abundant and many of them are of high quality
A new line of research has emerged on MEMS membrane devices based on the observation that on each point of the membrane E is always orthogonal to the tangent to the membrane at the point in question, so that |E| is to be considered proportional to the curvature K of the membrane
Summary
Recent industrial guidelines direct researchers and designers towards the development of low-cost devices to combine physical properties with low-level machine languages. Analytical approaches and numerical techniques work together to obtain solutions, respecting the analytical conditions that guarantee existence and uniqueness of the solution without ghost solutions [17,19,21,22,23] In this context, the experience of the authors in the field of modeling electrostatic MEMS membrane devices with strong non-linearity has grown [16,17,18,19,20,21,22,23,24]. Once the analytical studies in 1D and 2D geometries have been compared, numerical approaches, such as the shooting procedure, the Relaxation procedure, and the Keller–Box Scheme for recovering u in both geometries, studied in [17,19,21,22,23], are presented and compared
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