The scope of the present study is to develop a stochastic hierarchical multiscale computational framework for the analysis and simulation of heterogeneous materials. For the uncertainty quantification and the surrogate modeling of multi-scale materials in a hierarchical scheme, two distinct computational frameworks are being combined: the hierarchical multiscale (HMS) workflow, as defined in and the uncertainty quantification workflow in the UQpy software package.
The application of the present study is a two-scale two-dimensional computational materials model, consisting of an upper and lower scale. The upper scale model is defined in Abaqus using continuum finite elements, while the lower scale model lying at each integration point of the upper scale is defined using the open-source finite element code SfePy. The heterogeneity of the material, in the case of linear elastic analysis, is inserted at the lower-scale by simulating the Young’s modulus of the FE model at each integration point as a two-dimensional random field using the Karhunen-Loève expansion (KLE). In the case of elastoplastic analysis at the lower scale, the yield stress at each integration point is simulated using the KLE. Using this computational framework, we perform a large number of Monte Carlo simulations using the parallelization scheme in HMS and develop surrogate models for the lower-scale model using the manifold learning-based polynomial chaos expansion (m-PCE) to reduce the computational cost of the multiscale analysis without sacrificing the accuracy of the predictive models.
Numerical examples are presented for the explicit dynamic analysis and simulation of elastic and elastoplastic two-dimensional two-scale heterogenous materials.