An Extended Bridging Domain Method for Continuum-Atomistic Simulations of Discontinuities

In this presentation we will discuss a continuum multiscale framework which combines the Bridging Domain Method (BDM) of Xiao and Belytschko [1] with the eXtended Finite Element Method (XFEM) of of Moës et al. [2]. The BDM is a hierarchical overlapping domain decomposition scheme. Material in the coarse-scale domain is modelled as a continuum using XFEM and in the fine-scale domain by Molecular Mechanics. Compatibility between the atomistic and continuum domains is enforced in the overlap region using Lagrange Multipliers. The XFEM-BDM framework allows the coarse-graining of both regions where the atomistic displacements are homogeneous and where they are discontinuous, such as across cracks and dislocation glide planes. The combination of XFEM with BDM significantly reduces the number of degrees of freedom in the systems for problems involving cracks and dislocations. The proposed framework will be compared to direct numerical simulations and its advantages and limitations will be discussed. Many multiscale methods suffer from one significant limitation; that is that, as the simulation progresses more and more of the simulation domain is converted from continuum to atomistics to accommodate the evolution of discontinuities. XFEM-BDM alleviates this problem by replacing large portions of the atomistic domain behind the crack tip with the XFEM crack approximation developed in [2]. In a similar manner, it is generally possible to replace atomistic degrees of freedom along the glide plane of a dislocation by the XFEM dislocation approximation developed by Gracie et al. [3,4]. The framework is schematically shown in Figure 1 for cracks, where only the domain near the crack tip is modelled by atomistics. The domain is decomposed into two overlapping subdomains ΩC and ΩA. In ΩC the material is modelled as a continuum and in ΩA it is modelled with Molecular Mechanics. The boundary of ΩC is consists of an outer part Γ which is completely outside of ΩA and an inner part ∂ΩC which is inside of ΩA. Essential and natural boundary conditions are only applied to Γ. The boundary of ΩA is denoted as ∂ΩA. The coupling/overlap domain is given by Ωλ = ΩC ⋂ ΩA. In the XFEM-BDM framework, the displacement approximation for a continuum subdomain with a crack defined by the zero of the level set function f(x) is