Mathematical Modeling of MEMS Devices

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This award supports a coherent research program of modeling, analysis and numerical simulation of microelectromechanical systems (MEMS). The mathematical investigation of such devices forces the researcher to understand the behavior and coupling of fluid, electromagnetic, thermal and mechanical forces on the micron length scale. The focus of this project is the coupling of electrostatic and mechanical forces in MEMS devices. Due to their favorable scaling with device size, electrostatic forces are often used to actuate mechanical members of such devices. The mathematical modeling of this effect requires an understanding of coupled nonlinear elliptic systems of partial differential equations. The system consists of the equations of electrostatics coupled to the Navier equations of elasticity. Generally, the equations are coupled in two ways. First, the electrostatic potential appears as a source of mechanical force in the Navier equations. Second, the boundaries of the domain for the electrostatic problem depend upon mechanical deflections and hence the solution to the Navier equations. This two way nonlinear coupling is often additionally complicated by the appearance of nonlocal terms when the device is embedded in a circuit. In this work, a sequence of mathematical models of such devices will be designed to explain phenomena associated with MEMS devices, improve existing or suggest new numerical simulation methods and ultimately aid in the design of existing and future MEMS devices. The field of microelectromechanical systems has undergone a startling revolution in recent years. It is now possible to produce functioning motors that can only be seen with the aid of a microscope, gears smaller than a grain of pollen, and needles so tiny they can deliver an injection without stimulating nerve cells. The use of existing integrated circuit technology in the design and production of MEMS devices allows these devices to be batch processed, hence made in quantity, inexpensively. This in turn is igniting a revolution in areas such as biotechnology, where devices that once could only be dreamed about have suddenly been made possible. In order to realize the full potential of MEMS, a theoretical understanding of their function is necessary. This requires the construction and analysis of mathematical models of such devices. Of particular interest are devices which utilize electrostatic forces to provide locomotion. This class of devices will be modeled and analyzed in this project. This analysis will yield insight into the operation and future design of MEMS devices. Further, by developing an understanding of electrostatically actuated MEMS on a fundamental level, the research will suggest techniques for efficient numerical simulation of more complicated structures.