Modeling of the influence of electric forces on mechanical structures

This paper is a summary concerning the modeling of the strong electro-mechanical coupling appearing in micro-electro-mechanical systems (MEMS). These systems are very small devices (typical size of a few microns), in which electric phenomena as well as mechanical and dynamical phenomena exist. 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 electro-mechanical 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 iterations combined with adaptive meshing in case of large displacements. Furthermore, the tangent matrix obtained in the method considered in this paper greatly simplifies the computation of vibration modes and frequencies of the coupled system around equilibrium configurations. The non-linear variation of frequencies with respect to voltage and stiffness can 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 cap acitor, then for the model of a micro-bridge.