A comparison of staggered solution schemes for coupled particle–continuum systems modeled with the Arlequin method

摘要

This contribution aims at a systematic investigation of staggered solution schemes for the computation of coupled domains having different resolutions in space, a problem frequently arising in multi-scale modeling of materials. To couple a standard finite element domain with a high resolution atomistic or coarse-grained, i.e. particle-based domain, a so-called bridging domain is considered. In this handshake region a total energy, which is the sum of the weighted energies of both domains, needs to be formulated. Interactions in the particle domain are modeled by potential functions, e.g. a harmonic potential in the simplest case or the Lennard-Jones potential to consider also anharmonic interactions between the particles. The main goal is to separate the computation of finite element and particle domains as much as possible, amongst others to calculate the different domains on several CPUs. In the present work, the governing equations of the coupling method are presented. The energy functions of continuum, particle domain and bridging domain are recapitulated and the coupling constraint is set up. For the sake of simplicity, these relations are reformulated for the case of a one dimensional system. On the one hand, this system is computed monolithically without any separation of domains. On the other hand, various staggered solution schemes are derived systematically. The relevant equations of each scheme are given in detail together with the sequent iteration steps. All staggered schemes are investigated qualitatively, e.g. by their convergence behavior, as well as quantitatively by comparing the staggered solutions with the monolithic solution.