Central summation in the quasicontinuum method

The quasicontinuum (QC) method [Tadmor, E.B., Phillips, R., Ortiz, M., 1996. Mixed atomistics and continuum models of deformation in solids. Langmuir 12, 4529–4534] is a multiscale methodology to significantly reduce the computational cost of atomistic simulations. The method ensures an accurate incorporation of small-scale atomistic effects in large-scale models. It essentially consists of an interpolation of the displacements of large numbers of atoms between representative atoms (repatoms) and an estimation of the total potential energy of the atomistic lattice by a so-called summation (or sampling) rule. In this paper a novel energy-based summation rule is presented for the QC method that allows for a seamless coupling between coarse domains and fully resolved domains. In the presented summation rule only the repatoms are used in combination with one extra sampling atom in the center of each interpolation triangle. The presented summation rule is therefore straightforward and computationally efficient. The performance of the proposed summation rule is evaluated for a number of two-dimensional and three-dimensional multiscale atomistic test problems.