Propagation of material and surface profile uncertainties on MEMS micro‐resonators using a stochastic second‐order computational multi‐scale approach

Abstract

This paper aims at accounting for the uncertainties because of material structure and surface topology of micro-beams in a stochastic multi-scale model. For micro-resonators made of anisotropic polycrystalline materials, micro-scale uncertainties exist because of the grain size, grain orientation, and the surface profile. First, micro-scale realizations of stochastic volume elements are obtained based on experimental measurements. To account for the surface roughness, the stochastic volume elements are defined as a volume element having the same thickness as the microelectromechanical system (MEMS), with a view to the use of a plate model at the structural scale. The uncertainties are then propagated up to an intermediate scale, the meso-scale, through a second-order homogenization procedure. From the meso-scale plate-resultant material property realizations, a spatially correlated random field of the in-plane, out-of-plane, and cross-resultant material tensors can be characterized. Owing to this characterized random field, realizations of MEMS-scale problems can be defined on a plate finite element model. Samples of the macro-scale quantity of interest can then be computed by relying on a Monte Carlo simulation procedure. As a case study, the resonance frequency of MEMS micro-beams is investigated for different uncertainty cases, such as grain-preferred orientations and surface roughness effects. Figure 1. Homogenization-based multi-scale method: (a) first-order homogenization for classical macroscale continuum and (b) second-order homogenization for macro-scale Kirchhoff-Love plates.fields would require the nested resolution of meso-scale problems during the structural scale analysis, leading to a prohibitive cost. Local effects can also be treated using Monte Carlo simulations:The brittle failure of MEMS made of a poly-silicon material was studied by considering several realizations of a critical zone [24] on which the relevant loading was applied. An alternative to these approaches is to introduce in the stochastic multi-scale method a meso-scale random field, obtained from a stochastic homogenization, which is in turn used as material input by the stochastic finite element method at the structural scale. In order to ensure objectivity, the size of the (structural scale) stochastic finite elements should be small enough with respect to the (spatial) correlation length of the meso-scale random field [8], the latter depending on the size of the SVEs. In order to define a meso-scale random field, statistics and homogenization were coupled to investigate the probability convergence criterion of RVE for masonry [25], to obtain the property variations because of the grain structure of poly-silicon films [26], to extract the stochastic properties of the parameters of a meso-scale porous steel alloy material model [27], to evaluate open foams meso-scale properties [28], to extract probabilistic meso-scale cohesive laws for poly-silicon [29], to extract effective properties of random two-phase composites [30], to stud...

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

“…We recall the main equations in finite element analysis, equations (21), (22) and their matrix form (23). The material property items are the elastic tensor C C C, heat conductivity tensor κ, and thermal expansion tensor α which can be represented under the form of a random field.…”

“…The characteristic of poly-silicon micro structures can be determined using the measurements reported in [23].…”

“…For the deposition thickness of 2.0 µm obtained at different deposition temperatures [23], the average grain sizes in terms of the diameterd corresponding to the SEM measurements are reported in Table 1. Generate Gaussian random vector field…”

“…In this work, it was shown that the definition of a spatially correlated meso-scale random field is the key ingredient of the stochastic multi-scale approach; in order for the stochastic finite element method to converge, the size of the elements at the structural scale should remain lower than the spatial correlation length of the meso-scale random field, which in turn depends on the SVE size. This approach was also applied in [23] to study the effect of the geometric uncertainties resulting from the surface roughness, together with material uncertainties, on the vibration of micro-beams.…”

“…Based on experimental measurements reported in [23], the micro-scale structure, such as grain size and orientation distribution, is studied in terms of the Low Pressure Chemical Vapor Deposition (LPCVD) process temperature. The micro-scale thermo-mechanical properties are then defined in order to build the SVE realizations.…”