A stochastic computational multiscale approach; Application to MEMS resonators

Abstract

The aim of this work is to develop a stochastic multiscale model for polycrystalline materials, which accounts for the uncertainties in the micro-structure. At the finest scale, we model the micro-structure using a random Voronoï tessellation, each grain being assigned a random orientation. Then, we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso-scale. A random field of the meso-scale elasticity tensor can then be generated based on the information obtained from the SVE simulations. Finally, using a stochastic finite element method, these meso-scale uncertainties are propagated to the coarser scale. As an illustration we study the resonance frequencies of MEMS micro-beams made of poly-silicon materials, and we show that the stochastic multiscale approach predicts results in agreement with a Monte Carlo analysis applied directly on the fine finite-element model, i.e. with an explicit discretization of the grains.

A computational stochastic multiscale methodology for MEMS structures involving adhesive contact

“…Therefore, using the constructed stochastic model, we generate N M C × N p realizations of the reduced dimension random vector H PC using Eq. (33), where N M C = 1000 is the number of FE simulations and N p = 200 is the number of sub-domains for the numerical beam length l = 600 µm. The corresponding realizations of the random vectors Q PC and of the random vector V PC are then successively evaluated using Eq.…”

“…, c N are identified in order to enhance the approximation in terms of distribution as stated by Eq. (33). During the identification process, the distribution of the random vector H is estimated from its m explicitly evaluated samples {η (1) , ...,η (m) } using the multivariate kernel density estimation detailed in AppendixB.…”

“…To reduce the computational cost, we consider here a stochastic model in the multiscale method. Although, such method was never developed for adhesive contact problems, we take advantage of previous works [31,32,33].…”

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

“…In [22], the authors, have applied the computational homogenization on a spatial sequence of SVEs to extract spatially correlated meso-scale statistical properties. A meso-scale random field of the homogenized material properties could thus be generated to feed the stochastic finite element method performed at the structural scale.…”

“…Besides the issue related to the computational cost, a finite element model based on the explicit micro-heterogeneities discretization leads to a noise field [3] instead of a smooth one [27], which prevents the use of a stochastic finite element method, such as the Neumann expansion [44] or the perturbation approximation [36]. Therefore, in order to solve the problem of structural stochasticity of heterogeneous materials, such as polycrystalline materials, at a reasonable computation cost, a stochastic multi-scale approach is required as discussed in [22].…”