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Reconstructing the membrane detection of a 1D electrostatic-driven MEMS device by the shooting method: convergence analysis and ghost solutions identification

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Abstract

In this paper, in the domain of 1D-membrane micro-electro-mechanical systems in which the electrostatic field is expressed in terms of geometric curvature of the membrane, we present a numerical approach based on shooting techniques to reconstruct the membrane profile in the device in steady-state case. In particular, starting from known results in literature about existence achieved by Schauder–Tychonoff’s fixed point approach and uniqueness, and focusing on two physical–mathematical parameters appropriately indicative of the applied voltage and electromechanical properties of the membrane, respectively, we will discuss what operation parameters (applied voltage, amplitude of electrostatic field) and for which electromechanical membrane characteristic of the device is permitted or not a convergence of the method with respect to analytical results. Finally, we will discuss in detail the detected ghost solutions.

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Notes

  1. − is due to the opposite orientation of the vertical axis.

  2. When the deformation takes places, membrane profile, point-to-point slope and curvature are continuous functions.

  3. Functional dependency on x and u(x) for both K and \(\mu \) has been highlighted by a study carried out on well-known hemispherical benchmark in literature (Voltmer 2007).

  4. Plausible condition because the membrane does not allow tears and its slope varies with continuity.

  5. For \(\sigma \rightarrow 0, {\beta }\) assumes high values indicating less membrane deformation capacity.

  6. In our case, the exploited Green’s function is Bayley et al. (1968)

    \(G(x,s)={\left\{ \begin{array}{ll} \frac{(s+L_1) (L_1-x)}{2L_1} &{} -L_1 \le s \le x \\ \frac{(L_1-s)\cdot (x+L_1)}{2L_1} &{} x \le s \le L_1. \\ \end{array}\right. }\)      (9)

  7. Given that \((1-u)^2<1\) and \(E^2/V^2\) is the distance between the plates \((=1)\).

  8. And, for complementarity, the range of values that does not ensure convergence.

  9. Let us remember that the uniqueness of the solution for problem (7) is always guaranteed.

  10. In other words we are considering a particular engineering application.

  11. That is, we choose the material constituting the membrane.

  12. In the sense that convergence is not guaranteed.

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Authors and Affiliations

  1. DIIES Department, Cittadella Universitaria Via Graziella Feo di Vito, “Mediterranea” University Reggio Calabria, Reggio Calabria, Italy

    Giovanni Angiulli

  2. MIFT Department, Messina University, Via F. Stagno d’Alcontres, Messina, Italy

    Alessandra Jannelli

  3. DICEAM Department, Cittadella Universitaria Via Graziella Feo di Vito, “Mediterranea” University Reggio Calabria, Reggio Calabria, Italy

    F. Carlo Morabito & Mario Versaci

Authors
  1. Giovanni Angiulli
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  2. Alessandra Jannelli
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  3. F. Carlo Morabito
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  4. Mario Versaci
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Corresponding author

Correspondence to Mario Versaci.

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Communicated by Cristina Turner.

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Angiulli, G., Jannelli, A., Morabito, F.C. et al. Reconstructing the membrane detection of a 1D electrostatic-driven MEMS device by the shooting method: convergence analysis and ghost solutions identification. Comp. Appl. Math. 37, 4484–4498 (2018). https://doi.org/10.1007/s40314-017-0564-4

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  • DOI: https://doi.org/10.1007/s40314-017-0564-4

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