Finite element modeling microelectromechanical structures for sensing applications

A general purpose finite element method is developed for the direct numerical solution of small amplitude vibrations of crystal structures which are superposed on a static biasing state. The use of resonant structures for sensing applications is widespread in the development of microelectromechanical systems (MEMS) such as accelerometers, pressure and force sensors, rate of rotation sensors, and resonant strain gauges. Central to the design of such devices is the problem of calculating the change in resonant frequency of a vibrating structure under an imposed quasi-static mechanical loading. The direct solution to this problem, except for very simple cases, is generally intractable since the formulation usually results in a set of nonlinear partial differential equations. Perturbation analysis has been used to obtain solutions to problems of simple geometry which are valid over a small range of magnitudes of the static biasing variables. When the structural geometry becomes complicated or when the change in frequency over a large range of the biasing magnitude is needed, as it would be in modeling the performance of a sensor design over its full operational range, a powerful numerical technique is necessary. A general class of finite elements will be presented which are based on the theory of Baumhauer and Tiersten for small fields superposed on a bias. The solution of the discretized problem is handled in two parts: 1) the solution of a static problem under a loading from the variable being sensed, and 2) the construction of a vibration eigen-value problem which incorporates the static solution vector into the element stiffness matrices via finite element shape functions. Relevant examples of the modeling of various resonant microsensors mentioned above are presented using materials such as Quartz, Gallium Arsenide, and Silicon.