Optimal shape design of three-dimensional MEMS with applications to electrostatic comb drives

A comb drive is one of the most important microactuators in Microelectromechanical (MEM) systems. In a standard comb drive, the capacitance varies linearly with displacement, resulting in an electrostatic driving force which is independent of the position of the moving ngers (relative to the xed ones) except at the ends of the range of travel. It is of interest in some applications to have force pro les such as linear, quadratic or cubic. Such shaped comb drives could be useful, for example, for electrostatic tuning or to get actuators with longer ranges of travel than those of standard comb drives. This paper presents a methodology for solving three-dimensional design (inverse) problems in MEM sys- tems. Design of variable shape comb drives (shape motors) is presented as an application of the general methodology. It addresses issues of simulation, sensitivity analysis and then design of three-dimensional comb drives. Direct simulation is carried out by the exterior, indirect boundary element method and shape sensitivities are obtained by the direct di erentiation approach. The inverse problem determines the height pro le of the moving ngers of a comb drive such that the driving force is a desired function of its travel distance. An available optimization code (`E04UCFʹ from the NAG package) is used to solve the inverse problem. Numerical results are presented for shape motors that produce linear or cubic force pro les as functions of travel of the moving ngers