Stochastic modeling and uncertainty quantification in microelectromechanical systems

Uncertainty quantification (UQ) has become a necessary step in the design of most modern engineering systems due to the need to create robust devices that can tolerate variations in the manufacturing process or in the operating environment. These variations or uncertainties can be represented by stochastic variables which perturb the deterministic behaviour of the device about the nominal value for which it was designed. The UQ process consists of identifying the relevant uncertain parameters, assigning appropriate stochastic models to them and quantifying their effect on the final performance of the device. In this work, we restrict our focus to a broad category of devices that are collectively called microelectromechanical systems (MEMS). These devices have dimensions that are of the order of micrometers and are particularly sensitive to uncertainties that arise due to an inability to precisely control manufacturing tolerances. Deterministic modeling of such devices is itself quite difficult because of the number of coupled multiphysics interactions that need to be considered. When stochastic variations are also considered, it becomes very challenging to account for all the sources of variation accurately and to reproduce the effect of uncertainty using finite computational resources. Performing UQ for these devices is further complicated by the fact that the characterization data, which describes stochastic variation, is very often sparse in quantity. This work tackles all these challenges in order to develop a comprehensive framework for UQ. We do this by developing new ways to represent uncertain parameters and to estimate good stochastic models for these parameters using the limited amount of data available. We also improve the simulation tools at hand so as to reduce the computational effort required. Using a combination of methods like density estimation, stochastic process modeling, Bayesian inference, Monte Carlo sampling and stochastic collocation, we are able to successfully model device behavior in the presence of uncertainties and validate some of these results with experimental measurements. The overall contribution of this work is to make the process of UQ more tractable and reliable so that it becomes an integral part of every design scenario