  The multiscale simulation of heterogeneous materials is a popular and
important subject in solid mechanics and materials science due to the wide
application of composite materials. However, the classical FE2 (finite
element2) scheme can be costly, especially when the microproblem is nonlinear.
In this paper, we consider the case when the microproblem is the phase field
formulation for fracture. We adopt the locally linear embedding (LLE) manifold
learning approach, a method for non-linear dimension reduction, to extract the
manifold that contains a collection of phase-field-represented initial
microcrack patterns in the representative volume element (RVE). Then the output
data corresponding to any other microcrack pattern, e.g., the evolved phase
field at a fixed load, can be accurately reconstructed using the learned
manifold with minimum computation. The method has two features: a minimum
number of parameters for the scheme, and an input-specific error bar. The
latter feature enables an adaptive strategy for any new input on whether to use
the proposed, less expensive reconstruction, or to use an accurate but costly
high-fidelity computation instead.
