On periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure. (English) Zbl 1442.74191
Summary: Due to high computational costs associated with stochastic computational homogenization, a highly complex random material microstructure is often replaced by simplified, parametric, ergodic, and sometimes periodic models. This replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions. In the current contribution, we perform a validation of both assumptions through various numerical examples. To this end, we compare large-scale non-simplified and non-ergodic models with simplified, ergodic, and periodic solutions. In addition, we analyze the Hill-Mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions. As an example, we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning. The effect of these novel periodic boundary conditions is also analyzed by comparing with non-ergodic simulation results.
MSC:
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N75 | Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs |
| 74A40 | Random materials and composite materials |
Keywords:
computational homogenization; random geometry; periodic boundary conditions; stochastic FEMReferences:
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