Abstract
SummaryWe describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.
Highlights
Our approach is based on the finite element method (FEM)-Galerkin approximation of the 2D elliptic equations in a periodic setting by using fast assembling of the FEM stiffness matrix in a sparse matrix format, which is performed by agglomerating the Kronecker tensor products of simple 1D FEM discrete operators.[10]
We present the numerical scheme for discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in stochastic homogenization of multiscale random materials
The resulting large linear system of equations is solved by the preconditioned conjugate gradient (PCG) iteration with the convergence rate that is independent of the grid size and of the variation in jumping coefficients
Summary
Homogenization methods allow to derive the effective mechanical and physical properties of highly heterogeneous materials from the knowledge of the spatial distribution of their components.[1,2,3] In particular, stochastic homogenization via the representative volume element (RVE) methods provide means for calculating the effective large-scale characteristics related to structural and geometric properties of random composites, by utilizing a possibly large number of probabilistic. We investigate the RVE method that (approximately) extracts the effective (i.e., large-scale) behavior of the medium in form of the deterministic and homogeneous matrix Ahom from a given (stationary and ergodic) ensemble This method produces an approximation to Ahom by solving two-dimensional elliptic equations on a square of (lateral) size L with periodic boundary conditions and a specific right-hand side (the corrector equation), by taking the spatial average of the flux of these solutions, and by taking the empirical mean over N independent realizations of this coefficient field under the naturally periodized version of the ensemble.
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