Perturbation stochastic finite element-based homogenization of polycrystalline materials (original) (raw)
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Micromechanically based stochastic finite elements: length scales and anisotropy
Probabilistic Engineering Mechanics, 1996
The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the infinite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elements' stiffness matrices from the microstructure over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.
Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM
International Journal for Numerical Methods in Engineering, 2011
An extended finite element method (XFEM) coupled with a Monte Carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials. The methodology allows for an arbitrary number, aspect ratio, location and orientation of elliptic inclusions within a matrix, without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure. Moreover, the number of degrees of freedom in the enriched elements is dynamically reallocated for each Monte Carlo sample run based on the given volume fraction. The main advantage of the proposed XFEM-based methodology is a major reduction in the computational effort in extensive Monte Carlo simulations compared with the standard FEM approach. Monte Carlo and XFEM appear to work extremely efficiently together. The Monte Carlo approach allows for the modeling of the size, aspect ratios, orientations, and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions. Numerical results are presented and the uncertainty of the homogenized elastic properties is discussed.
Computer Methods in Applied Mechanics and Engineering, 2019
An inverse Mean-Field Homogenization (MFH) process is developed to improve the computational efficiency of non-linear stochastic multiscale analyzes by relying on a micro-mechanics model. First full-field simulations of composite Stochastic Volume Element (SVE) realizations are performed to characterize the homogenized stochastic behavior. The uncertainties observed in the non-linear homogenized response, which result from the uncertainties of their micro-structures, are then translated to an incrementalsecant MFH formulation by defining the MFH input parameters as random effective properties. These effective input parameters, which correspond to the micro-structure geometrical information and to the material phases model parameters, are identified by conducting an inverse analysis from the full-field homogenized responses. Compared to the direct finite element analyzes on SVEs, the resulting stochastic MFH process reduces not only the computational cost, but also the order of uncertain parameters in the composite micro-structures, leading to a stochastic Mean-Field Reduced Order Model (MF-ROM). A data-driven stochastic model is then built in order to generate the random effective properties under the form of a random field used
Computer Methods in Applied Mechanics and Engineering, 2016
Quantifying uncertainty in the overall elastic properties of composite materials arising from randomness in the material properties and geometry of composites at microscopic level is crucial in the stochastic analysis of composites. In this paper, a stochastic multi-scale finite element method, which couples the multi-scale computational homogenization method with the second-order perturbation technique, is proposed to calculate the statistics of the overall elasticity properties of composite materials in terms of the mean value and standard deviation. The uncertainties associated with the material properties of the constituents are considered. Performance of the proposed method is evaluated by comparing mean values and coefficients of variation for components of the effective elastic tensor against corresponding values calculated using Monte Carlo simulation for three numerical examples. Results demonstrate that the proposed method has sufficient accuracy to capture the variability in effective elastic properties of the composite induced by randomness in the constituent material properties. c
Stochastic finite element analysis of composite structures based on material microstructure
Composite Structures, 2015
The linking of microstructure uncertainty with the random variation of material properties at the macroscale is particularly needed in the framework of the stochastic finite element method (SFEM) where arbitrary assumptions are usually made regarding the probability distribution and correlation structure of the macroscopic mechanical properties. This linking can be accomplished in an efficient manner by exploiting the excellent synergy of the extended finite element method (XFEM) and Monte Carlo simulation (MCS) for the computation of the effective properties of random two-phase composites. The homogenization is based on Hill's energy condition and involves the generation of a large number of random realizations of the microstructure geometry based on a given volume fraction of the inclusions and other parameters (shape, number, spatial distribution and orientation). In this paper, the mean value, coefficient of variation and probability distribution of the effective elastic modulus and Poisson ratio are computed taking into account the material microstructure. The effective properties are used in the framework of SFEM to obtain the response of a composite structure and it is shown that the response variability can be significantly affected by the random microstructure.
Micromechanically based stochastic finite elements
Finite Elements in Analysis and Design, 1993
A stochastic finite element method for analysis of effects of spatial variability of material properties is developed with the help of a micromechanics approach. The method is illustrated by evaluating the first and second moments of the global response of a membrane with microstructure of a spatially random inclusionmatrix composite under a deterministic uniformly distributed load. It is shown that two mesoscale random continuum fields have to be introduced to bound the material properties and, in turn, the global response from above and from below. The intrinsic scale dependence of these two random fields is dictated by the choice of the finite element mesh.
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Computer Methods in Applied Mechanics and Engineering, 2011
A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic-plastic materials.
Composite Structures, 2014
The main purpose of this work is computational simulation of the expectations, standard deviations, skewness and kurtosis of the homogenized tensor for some composites with metallic components. The Representative Volume Element (RVE) of this composite contains a single cylindrical fiber and their components are treated as statistically homogeneous and isotropic media uniquely defined by the Gaussian elastic modulus. Probabilistic approach is based on the generalized stochastic perturbation technique allowing for large random dispersions of the input random variables and is implemented using the polynomial response functions recovered using the Least Squares Method. Homogenization technique employed is dual and consists of (1) stress version of the effective modules method and (2) its displacements counterpart based on the deformation energies of the real and homogenized composites. The cell problem is solved for the first case by the plane strain homogenization-oriented code MCCEFF and, in the 3D case, using the system ABAQUS Ò (8-node linear brick finite elements C3D8), where the uniform deformations are imposed on specific outer surfaces of the composite cell; probabilistic part is carried out in the symbolic computations package MAPLE Ò. We compare probabilistic coefficients of the effective elasticity tensor computed in this way with the corresponding coefficients for their upper and lower bounds and this is done for the composite with small and larger contrast between Young moduli of the fiber and the matrix. The main conclusion coming from the performed numerical analysis is a very good agreement of the probabilistic moments resulting from 2 and 3D computer models; this conclusion is totally independent from the contrast between elastic moduli of both composite components.
2012
This work presents a methodology for the construction of an uncertain nonlinear computational model adapted to the static analysis of a complex mechanical system. The deterministic nonlinear computational model is constructed with the finite element method using a total Lagrangian formulation. The finite element nonlinear response is then considered as a reference deterministic solution from which a reduced-order basis is constructed using the POD (Proper Orthogonal Decomposition) methodology. The mean reduced nonlinear computational model is thus obtained by projecting the reference deterministic solution on this basis. The explicit construction of the mean reduced nonlinear computational model is proposed for any type of structure modeled with three-dimensional solid finite elements. A procedure for the robust identification of the uncertain nonlinear computational model with respect to experimental responses is then given. Finally, the methodology is applied to a structure for which simulated experiments are given.