Stochastic Finite Element Analysis of Thermoelastic Effects in Micro-Resonators

In the design of micro-electromechanical systems (MEMS) such as micro-resonators, dissipation mechanisms may have detrimental effects on the quality factor. One of the major dissipation phenomena to consider in such systems is thermoelastic damping. Hence, the performance of such MEMS is directly related to their thermoelastic quality factor which has to be predicted accurately.

Moreover, the performance of MEMS can vary because manufacturing processes may leave substantial uncertainty in the geometry and in the material properties of the device. The reliability of MEMS devices is affected by the inability to accurately predict the stochastic behavior of the system due to the presence of these uncertainties. The aim of this paper is to provide a framework to account for uncertainties in the finite element analysis of the thermoelastic quality factor.

The present work focuses on second moment approaches, in which the first two statistical moments, i.e. the mean and the variance, are estimated. The perturbation stochastic finite element method is used in order to determine the mean and the variance of the thermoelastic quality factor of MEMS. The perturbation SFEM [1] consists in a deterministic analysis complemented by a sensitivity analysis with respect to the random parameters. This enables the development of a Taylor series expansion of the response, from which the mean and variance of the response can be derived knowing the mean and variance of the random parameters.

The perturbation SFEM is applied on the analysis of the thermoelastic quality factor of a micro-beam whose elastic modulus is considered as a random variable. Due to the nature of the thermoelastic problem, this study involves the calculation of eigenvalue sensitivities of a non-symmetric damped system [2]. The mean and variance of the quality factor are compared to the results obtained by Monte-Carlo simulations.

References:

[1] Kleiber, M., Hien, T.D., The stochastic finite element method: basic perturbation technique and computer implementation. Wiley & Sons, Chichester, 1992.
[2] Choi, K.M., Jo, H.K., Kim, W.H., Lee, I.W., Sensitivity analysis of non-conservative eigensystems, Journal of Sound and Vibration, v. 274, p. 997-1011, 2004.