Application of multiplicative dimensional reduction method for uncertainty quantification and sensitivity analysis of MEMS electrostatic actuators Pham Luu Trung Duong a , Trong Toan Tran b , Nagarajan Raghavan a, ⁎ a Engineering Product Development (EPD) Pillar, Singapore University of Technology and Design, 487 372, Singapore b Faculty of Electronic Technology, Industrial University of Ho Chi Minh City, Vietnam abstract article info Article history: Received 19 May 2017 Received in revised form 24 June 2017 Accepted 26 July 2017 Available online xxxx The effect of electrostatic actuation allows MEMS devices to have specific physical movements. They have several advantages including low power, low cost, and small size and are used widely in variable capacitors, micro-accel- erometers, etc. In this study, we consider a MEMS actuator consisting of a moveable plate and a fixed plate in the presence of an applied electric field. The gap between the two plates can normally be changed by voltage control. It is known that as the gap reduces to two thirds of the original gap, the so-called “pull-in effect” tends to occur, causing the plates to collide (resulting in dielectric breakdown and actuator failure). It is therefore important to predict the onset of the “pull-in effect”. As it is practically impossible to obtain the model parameters precisely, this prediction should account for the presence of uncertainties. Sampling methods such as Monte Carlo and Quasi Monte Carlo are easy to use with the caveat of low accuracy and high computational cost. The other popular method is polynomial chaos. It has high accuracy and low computational cost under smoothness assumption for problems with small number of uncertain parameters. In this study, we consider a two-stage approach to quan- tify the parametric uncertainty of MEMS electrostatic actuators with a moderate number of causal stochastic fac- tors. In the first stage, a multiplicative dimensional reduction method is used to approximate the variance-based global sensitivity measures in order to simplify the model for the uncertainty quantification stage. The second stage involves the use of the generalized polynomial chaos (gPC) approach to quantify uncertainty of the simpli- fied model from the first stage. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction MEMS electrostatic actuators have a wide range of applications owing to their low power consumption, low cost, and small size. They have been frequently used in variable capacitors, micro-accelerometers etc. [1,2]. This work focuses on a double plate MEMS electrostatic actu- ator, which is known to have an unstable drive region after the gap is decreased to two thirds of the original gap. It is therefore necessary to predict when the actuator will fail to work in the unstable region. Un- certainties in performance are unavoidable in system models due to modeling errors, and environmental changes etc. Thus, a confident pre- diction of the range of lifetime can only be made when uncertainties are taken into account. One possibility to propagate stochastic variations and make predictions is to use Monte Carlo/Quasi Monte Carlo based methods (MC/QMC) [3,4]. However, MC/QMC based approaches are very computationally demanding since they require large sample sizes to get accurate results. Recently, generalized polynomial chaos (gPC) expansion has been applied successfully to uncertainty quantification, prediction, propagation and decision making for different engineering systems [5–9]. The method however, is only effective for cases with small number of uncertain factors. Our study here proposes a two stage approach for uncertainty quantification of double plate MEMS actuators with large number of uncertain quantities. At the first stage, a multiplicative dimensional re- duction method is used for sensitivity analysis (SA). Sensitivity analysis is a study of how “uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input” [10]. It can help one to get a better understanding of the model and simplify it, if needed. Perhaps, the most popular SA is based on variances of system output. Different approaches such as: MC, random balance design (RBD), Fourier amplitude sensitivity test (FAST) [10,11] can be used to calculate variance based quantitative indi- ces known as Sobol indices, which can be used to determine how a model output variable of interest is influenced by individual or subsets of uncertain parameters. This work is organized as follows. Section 2 provides an overview about variance based sensitivity analysis using multiplicative dimen- sional reduction method and explains how it can be used to simplify the model for the next step. In Section 3, the polynomial chaos approach method for UQ is described. The method is then demonstrated for UQ and SA of a double plate MEMS electrostatic actuator in Section 4. Microelectronics Reliability xxx (2017) xxx–xxx ⁎ Corresponding author. E-mail address: nagarajan@sutd.edu.sg (N. Raghavan). MR-12604; No of Pages 7 http://dx.doi.org/10.1016/j.microrel.2017.07.091 0026-2714/© 2017 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel Please cite this article as: P.L.T. Duong, et al., Application of multiplicative dimensional reduction method for uncertainty quantification and sensitivity analysis of MEMS electro..., Microelectronics Reliability (2017), http://dx.doi.org/10.1016/j.microrel.2017.07.091