Non-conforming element for accurate modelling of MEMS

in Finite Elements in Analysis & Design (2007), 43(10), 749-756

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 ... [more ▼]

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. (C) 2007 Elsevier B.V. All rights reserved. [less ▲]

Coupled Electro-Mechanics Simulation Methodology of the Dynamic Pull-in in Micro-Systems

Conference given outside the academic context (2005)

The aim of this paper is to deal with multi-physics simulation of micro-electro-mechanical systems (MEMS) based on an advanced numerical methodology. MEMS are very small devices in which electric as well ... [more ▼]

The aim of this paper is to deal with multi-physics simulation of micro-electro-mechanical systems (MEMS) based on an advanced numerical methodology. MEMS are very small devices in which electric as well as mechanical and fluid phenomena appear and interact. Because of their microscopic scale, strong coupling effects arise between the different physical fields, and some forces, which were negligible at macroscopic scale, have to be taken into account. In order to accurately design such micro-electro-mechanical systems, it is of primary importance to be able to handle the strong coupling between the electric and the mechanical fields. In this paper, the finite element method (FEM) is used to model the electro-mechanical interactions and to perform static and transient analyses. The application example considered here is a micro-bridge consisting in a clamped-clamped beam suspended over a substrate (the lower electrode). When a voltage is applied between the beam and the substrate, electrostatic forces appear which force the beam to bend. When the applied voltage increases, the electrostatic force becomes dominant and the plates stick together. The corresponding critical voltage is called the pull-in voltage. When the dynamic behavior of the system is taken into account, it is shown that two new parameters have to be defined: the dynamic pull-in displacement and the dynamic pull-in time. [less ▲]

Electrostatic Coupling of MEMs Structures: Transient Simulations and Dynamic Pull-in

Conference given outside the academic context (2004)

In Micro-electromechanical systems (MEMS) coupling of structures through electrostatic forces is a primordial phenomenon. Simulating the dynamics of MEMS and taking into account such strong coupling ... [more ▼]

In Micro-electromechanical systems (MEMS) coupling of structures through electrostatic forces is a primordial phenomenon. Simulating the dynamics of MEMS and taking into account such strong coupling effects allows to predict dynamical performance and stability, and is therefore an essential issue in design of highly effective and reliable devices. Analysis techniques for such systems require special attention in order to provide to the designer accurate and fast tools. We propose a finite element approach (FEM) that properly handles the strong electromechanical coupling in MEMS. In the simulation example of a micro-bridge, we show that such simulation techniques can reveal complex dynamical behaviors of MEMS such as dynamic pull-in. [less ▲]

Modeling of Electro-Mechanical Coupling Problem using the Finite Element Formulation

Conference given outside the academic context (2003)

A modeling procedure is proposed to handle strong electro-mechanical coupling appearing in micro-electromechanical systems (MEMS). The finite element method is used to discretize simultaneously the ... [more ▼]

A modeling procedure is proposed to handle strong electro-mechanical coupling appearing in micro-electromechanical systems (MEMS). The finite element method is used to discretize simultaneously the electrostatic and mechanical fields. The formulation is consistently derived from variational principles based on the electromechanical free energy. In classical weakly coupled formulations staggered iteration is used between the electrostatic and the mechanical domain. Therefore, in those approaches, linear stiffness is evaluated by finite differences and equilibrium is reached typically by relaxation techniques. The strong coupling formulation presented here allows to derive exact tangent matrices of the electro-mechanical system. Thus it allows to compute non-linear equilibrium positions using Newton-Raphson type of iterations combined with adaptive meshing in case of large displacements. Furthermore, the tangent matrix obtained in the method exposed in this paper greatly simplifies the computation of vibration modes and frequencies of the cou pled system around equilibrium configurations. The non-linear variation of frequencies with respect to voltage and stiffness can be then be investigated until pull-in appears. In order to illustrate the effectiveness of the proposed formulation numerical results are shown first for the reference problem of a simple flexible capacitor, then for the model of a micro-bridge. [less ▲]