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The multiscale simulation of heterogeneous materials is a popular andimportant subject in solid mechanics and materials science due to the wideapplication of composite materials. However, the classical FE2 (finiteelement2) scheme can be costly, especially when the microproblem is nonlinear.In this paper, we consider the case when the microproblem is the phase fieldformulation for fracture. We adopt the locally linear embedding (LLE) manifoldlearning approach, a method for non-linear dimension reduction, to extract themanifold that contains a collection of phase-field-represented initialmicrocrack patterns in the representative volume element (RVE). Then the outputdata corresponding to any other microcrack pattern, e.g., the evolved phasefield at a fixed load, can be accurately reconstructed using the learnedmanifold with minimum computation. The method has two features: a minimumnumber of parameters for the scheme, and an input-specific error bar. Thelatter feature enables an adaptive strategy for any new input on whether to usethe proposed, less expensive reconstruction, or to use an accurate but costlyhigh-fidelity computation instead.

A Manifold Learning Approach to Accelerate Phase Field Fracture Simulations in the Representative Volume Element