Bridging domain methods for coupled atomistic-continuum models with L2 or H1 couplings
Résumé
A bridging domain method for coupled atomistic-continuum models is proposed that enables to compare various coupling terms. The approach does not require the finite element mesh to match the lattice spacing of the atomic model. It is based on an overlapping domain decomposition method that makes use of Lagrange multipliers and weight functions in the coupling zone in order to distribute the energy between the two competing models. Two couplings are investigated. The L2 coupling enforces the continuity of displacements between the two models directly. The H1 coupling involves the definition of a strain measure. For this purpose, a moving least-square interpolant of the atomic displacement is defined. The choice of the weight functions is studied. Patch tests and a graphene sheet with a crack are studied. It is shown that both continuous and discontinuous weight functions can be used with the H1 coupling whereas the L2 coupling requires continuous weight functions. For the examples developed herein, the L2 coupling produces less error in the zone of interest. The flexibility of the H1 coupling with constant weight function may be beneficial but the results may be affected depending on the topology of the bridging zone.