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.

The aim of this work is to study the thermo-elastic quality factor (Q) of micro-resonators with a stochastic multi-scale approach. In the design of high-Q micro-resonators, thermo-elastic damping is one of the major dissipation mechanisms, which may have detrimental effects on the quality factor, and has to be predicted accurately. Since material uncertainties are inherent to and unavoidable in micro-electromechanical systems (MEMS), the effects of those variations have to be considered in the modeling in order to ensure the required MEMS performance. To this end, a coupled thermo-mechanical stochastic multi-scale approach is developed in this paper. Thermo-mechanical micro-models of polycrystalline materials are used to represent micro-structure realizations. A computational homogenization procedure is then applied on these statistical volume elements to obtain the stochastic characterizations of the elasticity tensor, thermal expansion, and conductivity tensors at the meso-scale. Spatially correlated meso-scale random fields can thus be generated to represent the stochastic behavior of the homogenized material properties. Finally, the distribution of the thermo-elastic quality factor of MEMS resonators is studied through a stochastic finite element method using as input the generated stochastic random field.

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...

Optical measurement techniques are often employed to digitally capture three dimensional shapes of components. The digital data density output from these probes range from a few discrete points to exceeding millions of points in the point cloud. The point cloud taken as a whole represents a discretized measurement of the actual 3D shape of the surface of the component inspected to the measurement resolution of the sensor. Embedded within the measurement are the various features of the part that make up its overall shape. Part designers are often interested in the feature information since those relate directly to part function and to the analytical models used to develop the part design. Furthermore, tolerances are added to these dimensional features, making their extraction a requirement for the manufacturing quality plan of the product. The task of "extracting" these design features from the point cloud is a post processing task. Due to measurement repeatability and cycle time requirements often automated feature extraction from measurement data is required. The presence of non-ideal features such as high frequency optical noise and surface roughness can significantly complicate this feature extraction process. This research describes a robust process for extracting linear and arc segments from general 2D point clouds, to a prescribed tolerance. The feature extraction process generates the topology, specifically the number of linear and arc segments, and the geometry equations of the linear and arc segments automatically from the input 2D point clouds. This general feature extraction methodology has been employed as an integral part of the automated post processing algorithms of 3D data of fine features.

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