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Curvature-dependent electrostatic field as a principle for modelling membrane MEMS device with fringing field

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Abstract

In a framework of 1D membrane MEMS theory, we consider the MEMS boundary semi-linear elliptic problem with fringing field

$$\begin{aligned} u''=-\frac{\lambda ^2(1+\delta |u'|^2)}{(1-u)^2}\;\;\;\text {in}\;\;\varOmega , \;\;u=0\;\;\text {on}\;\;\partial \varOmega , \end{aligned}$$

where \(\lambda ^2\) and \(\delta \) are positive parameters, \(\varOmega =[-L,L] \subset {\mathbb {R}}\), and u is the deflection of the membrane. In this model, since the electric field \( {\mathbf {E}} \) on the membrane is locally orthogonal to the straight line tangent to the membrane at the same point, \( | {\mathbf {E}} | \), proportional to \(\lambda ^2/(1-u)^2\), is considered locally proportional to the curvature of the membrane. Thus, we achieve interesting results of existence writing it into its equivalent integral formulation by means of a suitable Green function and applying on it the Schauder–Tychonoff fixed point theory. Therefore, the uniqueness of the solution is proved exploiting both Poincaré inequality and Gronwall Lemma. Then once the instability of the only obtained equilibrium position is verified, an interesting limitation for the potential energy dependent on the fringing field capacitance is obtained and studied.

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Notes

  1. A ghost solution is a numerical solution that does not satisfy any existence and/or uniqueness conditions.

  2. If \(\delta =0\) (i.e., without fringing field), (2) becomes (1).

  3. Because the membrane surface represents an interface between two media with different dielectric constant. Furthermore, the membrane, being a conductor, requires \( {\mathbf {E}} \) on it to be orthogonal to the straight tangent line to the membrane at the point considered.

  4. This situation is mathematically represented by a singularity.

  5. Corresponding to the value of u(x) generating the singularity in (13).

  6. Then, once we set \(\sup |u'(x)|=H\).

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale e dell’Informazione, University of Pavia, Via A. Ferrata 5, Pavia, Italy

    Paolo Di Barba

  2. Dipartimento di Ingegneria dell’Informazione Infrastrutture Energia Sostenibile, “Mediterranea” University, Via Graziella Feo di Vito, Reggio Calabria, Italy

    Luisa Fattorusso

  3. National Group for Mathematical Analysis, Probability, and their Applications (GNAMPA-INdAM), Lucca, Italy

    Luisa Fattorusso

  4. Dipartimento di Ingegneria Civile Energia Ambiente e Materiali, “Mediterranea” University, Via Graziella Feo di Vito, Reggio Calabria, Italy

    Mario Versaci

Authors
  1. Paolo Di Barba
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  2. Luisa Fattorusso
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  3. Mario Versaci
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Correspondence to Mario Versaci.

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Communicated by Baisheng Yan.

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Barba, P.D., Fattorusso, L. & Versaci, M. Curvature-dependent electrostatic field as a principle for modelling membrane MEMS device with fringing field. Comp. Appl. Math. 40, 87 (2021). https://doi.org/10.1007/s40314-021-01480-z

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  • DOI: https://doi.org/10.1007/s40314-021-01480-z

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