Design of micromachines under uncertainty with the sample-average approximation method

Abstract

Variability in the features produced by microfabrication processes, as well as uncertainty in some material properties, may cause a significant deviation in the performance of micromachines within the same fabrication run. Based on an estimation of the expected process variations, the design of such devices can be optimised to achieve the design goals, even under this uncertainty. Learning from previous works on the design of microresonators, we formulate this design problem as a case of chance-constrained optimisation and expand it to a general case where both the dynamic sensitivity ought to be maximised and the natural frequency should be close to a given target. Constraints to ensure a safe operation under both static and dynamic conditions are included by means of penalty functions. We implement the ‘Sample-Average Approximation’ (SAA), known in the field of stochastic programming, to solve the problem with a single-objective genetic algorithm (CMA-ES), requiring only a numerical evaluation of the objective function—no computation of its gradient is required nor a specific analytic form. We apply this optimisation strategy to the design case of an ultrasonic transducer—‘lateral CMUT’—, using optical measurements of trench variability to estimate process variations in a hypothetical design. Comparison of different optimisation results reveals that the implementation of SAA enables the choice of a more conservative design that meets the targets in spite of variability in its features.

Keywords

• robust design
• parameter uncertainty
• micro electro-mechanical systems (mems)
• stochastic programming
• covariance matrix adaptation – evolution strategy (cma-es)