Non-conforming element for accurate modelling of MEMS

Abstract

In this work different modelling techniques are investigated to simulate the dynamic behaviour of slender structures on which electrostatic forces are acting. In particular, non-conforming elements are tested to model micro-mechanical devices (or MEMS) having a very large aspect ratio. These elements are constructed on linear shape functions enriched by internal second-order polynomials. As a consequence the element compatibility is not exactly satisfied, but such elements can efficiently model beam- or shell-like structures with a small number of degrees of freedom. The advantage of non-conforming elements compared to shell or beam elements is that they are volume elements and can therefore easily be combined with other volume finite elements. For micro-mechanical systems the structure must be coupled to the electrostatic domain with the so-called electro-mechanical elements that solve for the electrostatic potential and generate the electrostatic forces. This paper shows that constructing coupled electro-mechanical models for high aspect ratio systems is then greatly simplified when non-conforming finite elements are used. The theory is presented for small deformations and also for large displacements where geometric non-linearities must be accounted for. The elements proposed in this paper are based on non-conforming formulations published earlier. The efficiency of the non-conforming approach combined with specific electro-mechanical elements is highlighted in the analysis of two simple MEMS for which the pull-in voltage is computed.