The volume coupling method, also known as the Arlequin framework for multimodel coupling, is a Lagrange multiplier/penalty functionbased approach that enforces constraint conditions on (the solutions of) the submodels in an overlapping domain. Within the overlapping domain, it is further required that the virtual work is the weighted sum of the works done respectively by the submodels. Each work is weighted by a smooth and continuous function w that monotonically increases from zero to one, as shown in the figure above. This means that the coupled governing equations need to be modified with the respective coupling weights that sum to one at a given location (building a partition of unity). Before choosing the constrained variables, e.g., displacement/velocity in our case, one needs to bring them to the same scale. We use CG to enrich the homogenization process with a length scale independent of both particle and finite element sizes.

