The effect of uncertainties in MEMS structures is more significant than in macro-scaled structures because present fabrication methods for MEMS structures are prone to bring in large error-to-size ratio. These fabrication errors in MEMS are often large enough to result in malfunction or performance degradation of the produced devices especially in mechanical responses such as resonant frequencies in vibratory micro gyroscopes or actuation stroke in a micro optical switch. There are also other uncertainties such as in operating conditions and material properties, which can do similar effects. A robust optimal design that makes the system response less sensitive to the uncertainties becomes thus a very important concept to guarantee reliable performances and improve yield rate in mass production.
In this paper a simple and cost-efficient robust optimal design formulation for MEMS structures is presented and illustrated by various design examples of MEMS devices. The basic idea of the proposed formulation is to improve robustness of the objective and constraint functions by minimizing a proper index related to uncertain variables. The methodology for the proposed robust optimal design utilizes conventional deterministic optimization, design sensitivity analysis, and Gradient Index (GI) to enhance both performance and robustness of MEMS structures. GI is an index defined as a function of gradients of performance functions with respect to uncertain variables. In the robust optimal design procedure, a deterministic optimization for performance improvement of MEMS structures is followed by design sensitivity analysis with respect to uncertainties such as fabrication errors and change of operating conditions. During the process of deterministic optimization and sensitivity analysis, dominant performance and uncertain variables are identified to define GI. The GI is incorporated as a term of objective and constraint functions of the robust optimal design formulation, which can enhance both performance and robustness. While most previous approaches for robust optimal design require definite or statistical information on design variations, the proposed GI based method needs no such information and therefore is cost-efficient and easily applicable to early design stages.
Five application examples including a micro probe, an electromagnetically driven microactuator, micro gyroscopes, and a micro accelerometer are studied comparing results for various objective functions. For the resonant-type micro probe, a robust optimum is obtained to satisfy a target for the measurement sensitivity. As a more complicated example, a laterally driven electromagnetic microactuator is optimized for both minimization of driving force and maximization of actuation stroke. A snap-through buckling behavior is found to be effective to obtain small driving force and large actuation stroke. In this case, not all optimums obtained for different objective functions are good enough because of local minimum and inappropriate selection of robust optimal design formulation, but the problem can be resolved using a new robust optimization with an additional target constraint for the performance and starting with the deterministic optimum as the initial design. The important objective in the design of vibratory micro gyroscopes is to reduce the difference between natural frequencies of the vertical (detecting) and lateral (driving) modes in order to attain high mechanical detecting sensitivity. The deterministic optimization for this goal results in good performance but is very sensitive to fabrication errors. A robustness test in terms of yield using the Monte Carlo simulation shows that the robust optimum by minimizing GI has produced twice more acceptable designs than the deterministic optimum. Improvement of robustness becomes bigger as the amount of fabrication errors is assumed larger. In case of the micro accelerometer problem, the proposed robust optimal design method is proved to handle the feasibility robustness as well using the GI for constraint functions.
In conclusion, a simple and easily applicable robust optimal design method for MEMS structures is proposed and verified through various application examples. The important point is that the formulation of minimizing the GI of the performance functions requires no statistical information on the uncertainties and yet achieves robustness cost-efficiently. This methodology, although shown for MEMS structures, can be easily applied as well to conventional mechanical structures where information on uncertainties is poor but robustness is highly important.
