A stochastic approximation approach to improve the convergence behavior of hierarchical atomistic-to-continuum multiscale models

The aim of this work is to provide an improved information exchange in hierarchical atomistic-to-continuum settings by applying stochastic approximation methods. For this purpose a typical model belonging to this class is chosen and enhanced. On the macroscale of this particular two-scale model, the balance equations of continuum mechanics are solved using a nonlinear finite element formulation. The microscale, on which a canonical ensemble of statistical mechanics is simulated using molecular dynamics, replaces a classic material formulation. The constitutive behavior is computed on the microscale by computing time averages. However, these time averages are thermal noise-corrupted as the microscale may practically not be tracked for a sufficiently long period of time due to limited computational resources. This noise prevents the model from a classical convergence behavior and creates a setting that shows remarkable resemblance to iteration schemes known from stochastic approximation. This resemblance justifies the use of two averaging strategies known to improve the convergence behavior in stochastic approximation schemes under certain, fairly general, conditions. To demonstrate the effectiveness of the proposed strategies, three numerical examples are studied.