Stochastic modeling of micro-electromechanical systems (MEMS).

摘要

In recent years, there has been a growing interest in analyzing and quantifying the effect of underlying fluctuations or variations, while developing reliable predictive models for physical phenomenon. State-of-the-art design methodologies for Micro-ElectroMechanical Systems (MEMS) assume that the geometrical and material properties of these devices are known in a deterministic sense. However, in reality, significant uncertainties in these properties are inevitable, and must be considered during the development of computational models. This dissertation presents a stochastic modeling framework for MEMS, which allows to quantify the effect of stochastic variations in various design parameters on device performance.;The stochastic modeling framework characterizes uncertain parameters as random processes, using which the original governing equations are reformulated as stochastic differential/integral equations. This work presents novel numerical techniques to efficiently solve these stochastic governing equations, which offer faster convergence rate than the traditionally used sampling based methods, such as Monte Carlo (MC) method. Specifically, two approaches are considered---stochastic Galerkin method and stochastic collocation method. The stochastic Galerkin method is based on representing input and unknown field variables in terms of orthogonal polynomials (termed as generalized polynomial chaos (GPC)) in the random domain. Following this, the unknown coefficients of the polynomial expansion are determined using Galerkin projections. Based on this approach, in the first part, a stochastic Lagrangian framework is developed for static analysis of MEMS, which allows considering geometrical variations and is applicable for multiphysics problems.;In the second part, a stochastic collocation framework is developed, which is based on approximating the unknown stochastic solution by a polynomial interpolation function in the multi-dimensional random domain. This approach offers high resolution similar to the Galerkin method, as well as ease of implementation as the sampling based methods. The approximation is constructed based on sparse grid interpolation using Smolyak algorithm, which leads to orders of magnitude reduction in the number of support nodes as compared to usual tensor products. For collocation methods based on standard Smolyak construction, the convergence rate may significantly deteriorate in the presence of discontinuities in the random domain. To this end, a novel domain-decomposition based adaptive collocation scheme is proposed, which is suited for handling discontinuities (such as pull-in instability in MEMS) and sharp variations in the random domain. Moreover, existing approaches do not take into account the probability measures during the construction of the sparse grids, which leads to an approximation based on support nodes sampled uniformly from the random domain. This work proposes a weighted Smolyak algorithm, which allows to incorporate the information regarding arbitrary non-uniform probability measures during the construction of sparse grids. The proposed algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density, leading to significant reduction in computational effort for highly skewed or localized probability measures.;In the final part, a data-driven stochastic collocation approach is presented, which seeks to characterize uncertain input parameters based on available experimental information. This approach models the uncertain parameters as independent random variables, for which the distributions are estimated based on experimental observations, using a nonparametric diffusion mixing based estimator. The efficiency and applicability of the developed stochastic modeling framework is demonstrated by simulating several MEMS devices, such as MEMS switches, resonators, comb-drives etc.