| In the bridging domain method, a system is decomposed into overlapping atomic and continuum subdomains. The total energy is a weighted combination of the atomic and continuum components, and Lagrange multiplier constraints that enforce compatibility between the two subdomains. When the Lagrange multipliers are computed without approximation, linear momentum, angular momentum, and energy are conserved. When the constraint equations are approximated by diagonalization, the system dissipates energy. It is shown that this diagonalized constraint approach effectively eliminates spurious reflections at the coupling interface. When the atomic system has a composite lattice, the bridging domain method is suboptimal due to the suppression of internal modes by the compatibility constraints. To overcome this deficiency, a relaxed bridging domain method is proposed, where the atoms are divided into primary and secondary atoms. Only primary atoms are constrained in the coupling region, which allows internal mode relaxation. The increased accuracy of the relaxed bridging domain method is demonstrated in one- and two-dimensional examples.;In the second half of this dissertation, the elastic properties and fracture of graphene are studied by multiscale methods that link quantum mechanics (QM) with continuum mechanics (CM) simulations. An anisotropic strain energy density function is developed for graphene by a hierarchical QM/CM approach. The strain energy density is assumed to be hyperelastic and the parameters are determined through density functional theory (DFT) computations. The constitutive law is applied in continuum calculations of nanoindentation experiments of single layer graphene. The predicted force-de ection curves show excellent agreement with experimental results. Graphene fracture is investigated by a newly-developed concurrent, adaptive QM/CM method, where an extended finite element method is used to model the graphene sheet, and the bond breaking process is modeled with DFT. In this analysis, the DFT region adaptively follows the crack tip during fracture. Edge cracks in the armchair and zigzag directions are studied. It is found that cracks in zigzag direction grow self-similarly whereas cracks in armchair direction exhibit substantial zigzag. |
