Finite element analysis of electrostatic coupled systems using geometrically nonlinear mixed assumed stress finite elements

Abstract

The micro-electromechanical systems (MEMS) industry has grown incredibly fast over the past few years, due to the irresistible character and properties of MEMS. MEMS devices have been widely used in various fields such as aerospace, microelectronics, and the automobile industry. Increasing prominence is given to the development and research of MEMS; this is largely driven by the market requirements. Multi-physics coupled fields are often present in MEMS. This makes the modelling and analysis o such devices difficult and sometimes costly. The coupling between electrostatic and mechanical fields in MEMS is one of the most common and fundamental phenomena in MEMS; it is this configuration that is studied in this thesis. The following issues are addressed: 1. Due to the complexity in the structural geometry, as well as the difficulty to analyze the behaviour in the presence of coupled fields, simple analytical solutions are normally not available for MEMS. The finite element method (FEM) is therefore used to model electrostaticmechanical coupled MEMS. In this thesis, this avenue is followed. 2. In order to capture the configuration of the system accurately, with relatively little computational effort, a geometric non-linear mixed assumed stress element is developed and used in the FE analyses. It is shown that the developed geometrically non-linear mixed assumed stress element can produce an accuracy level comparable to that of the Q8 element, while the number of the degrees of freedom is that of the Q4 element. 3. Selected algorithms for solving highly non-linear coupled systems are evaluated. It is concluded that the simple, accurate and quadratic convergent Newton-Raphson algorithm remains best. To reduce the single most frustrating disadvantage of the Newton method, namely the computational cost of constructing the gradients, analytical gradients are evaluated and implemented. It is shown the CPU time is significantly reduced when the analytical gradients are used. 4. Finally, a practical engineering MEMS problem is studied. The developed geometric nonlinear mixed element is used to model the structural part of a fixed-fixed beam that experiences large axial stress due to an applied electrostatic force. The Newton method with analytical gradients is used to solve this geometrically nonlinear coupled MEMS problem.