A stochastic-deterministic coupling method for continuum mechanics

In this paper, we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one. The coupling strategy is performed in the Arlequin framework, which is based on a volume coupling and a partition of the energy. A suitable functional space is chosen for the weak enforcement of the continuity between the two models. The choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point-wise, and enforces appropriate boundary conditions on the stochastic dimension. The proof of the existence of the solution of the mixed problem is provided. The numerical strategy is also reviewed, in particular with a view at the Monte Carlo method. Finally, examples show the interest of the method, and possible strategies for use in adaptive modeling.