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

Wu, Ling; Lucas, Vincent; Nguyễn, Vân Dung; Golinval, Jean‐Claude; Paquay, Stéphane; Noels, Ludovic

doi:10.1016/j.cma.2016.07.042

Cited by 13 publications

(23 citation statements)

References 45 publications

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“…Details on this iterative process, with convergence improvements as compared with the literature, can be found in [37]. The properties of interest are then obtained by following the reverse path of Equations (70)-(73).…”

Section: Non-gaussian Extensionmentioning

confidence: 99%

“…Finally, results obtained for different lengths of the plate are shown in Figure 30(c), where it can be seen that the uncertainty of the resonance frequency decreases with the length of the plate. However, considering a more realistic anchor of the MEMS than a perfect clamp, as performed in [37], could change this conclusion.…”

Section: The Effect Of the Plate Dimensionsmentioning

confidence: 99%

“…In particular, by comparison with direct Monte Carlo simulations, it was shown that the generation of a spatially correlated random field allows predicting macro-scale statistical distributions independent on the SVE and macro-scale mesh sizes, as long as the distance between the macro-scale integration points remains lower than the correlation length of the meso-scale random field. This stochastic three-scale procedure was recently extended in the frame of thermo-mechanics in [37] to study the thermo-elastic damping effect in a probabilistic way.…”

Section: Introductionmentioning

confidence: 99%

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

Lucas

Golinval

Voicu

et al. 2017

Numerical Meth Engineering

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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...

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Section: Non-gaussian Extensionmentioning

confidence: 99%

Section: The Effect Of the Plate Dimensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Lucas

Golinval

Voicu

et al. 2017

Numerical Meth Engineering

Self Cite

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“…As each grain has a different size, the thermal conductivity for each grain needs to be estimated and is given in Tab. 1, see [48].…”

Section: Thermal Homogenizationmentioning

confidence: 99%

“…As the computation time increases with the RVE size, it is important to provide a critical size of the RVE allowing to capture the homogenized behavior at a reasonable computational cost [9,26,28]. However lack of representativeness of the volume element can also be exploited in order to study the material uncertainties on Statistical Volume Elements (SVEs) [38,27], allowing defining meso-scale random fields [39][40][41][42][43][44][45], or propagating material uncertainties to the structural behavior [40,[46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method

Nguyễn

Noels

2016

Comput Mech

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Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method Van-Dung Abstract This work provides a unified treatment of arbitrary kinds of microscopic boundary conditions usually considered in the multi-scale computational homogenization method for nonlinear multi-physics problems. An efficient procedure is developed to enforce the multi-point linear constraints arising from the microscopic boundary condition either by the direct constraint elimination or by the Lagrange multiplier elimination methods. The macroscopic tangent operators are computed in an efficient way from a multiple right hand sides linear system whose left hand side matrix is the stiffness matrix of the microscopic linearized system at the converged solution. The number of vectors at the right hand side is equal to the number of the macroscopic kinematic variables used to formulate the microscopic boundary condition. As the resolution of the microscopic linearized system often follows a direct factorization procedure, the computation of the macroscopic tangent operators is then performed using this factorized matrix at a reduced computational time.

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A micromechanics‐based inverse study for stochastic order reduction of elastic UD fiber reinforced composites analyses

Adam²,

Noels

2018

Numerical Meth Engineering

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This research develops a stochastic mean field homogenization process that is used as reduced order model to carry out a statistical multiscale analysis on unidirectional fiber reinforced composites. First, full-field simulations of unidirectional stochastic volume elements, whose statistical description is obtained from scanning electron microscope images, are conducted to define statistical mesoscale apparent properties. A stochastic Mori-Tanaka (M-T) mean field homogenization model is then developed through an inverse stochastic identification process performed on the apparent elastic properties obtained by full-field simulations. As a result, a random vector of the effective elastic properties of phases and microstructure information of the M-T model is inferred. In order to conduct stochastic finite element method analyses, a generator of this random vector is then constructed using the copula method, allowing predicting the statistical response of a composite ply under bending. The statistical dependence of the random vector entries is shown to be respected by the generator. Although this work is limited to the elastic response, we believe that the stochastic M-T model can be extended to nonlinear behaviors to conduct efficient stochastic multiscale simulations.

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

Cited by 13 publications

References 45 publications

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

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

Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method

A micromechanics‐based inverse study for stochastic order reduction of elastic UD fiber reinforced composites analyses

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