A variational formulation of the quasicontinuum method based on energy sampling in clusters

Abstract This contribution presents a novel quasicontinuum (QC) approach aiming at a seamless transition from the atomistic to the continuum description of crystalline solids at zero temperature, which heavily draws on the framework proposed by Knap and Ortiz [2001. An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899–1923]. Opposed to Knap and Ortiz, the energy instead of forces is subject to a cluster-based sampling scheme with adaptive resolution. We show that only the present ansatz endows the QC theory with a variational structure leading to conservative forces and symmetric stiffnesses. Equally, we show the strict symmetry in atomic interactions. This approach allows for the direct application of standard minimization methods and guarantees the existence of an equilibrium state provided that the total potential exhibits a minimum. A special focus is on the numerical error in the cluster-based summation rule for energy sampling. We compare two strategies to improve the accuracy, which are also particularly useful to account for surface effects. The fully nonlocal methodology is assessed in nanoindentation into an fcc single crystal. Compared with lattice statics good agreement is achieved with respect to the force–displacement curve, the load level and locus of dislocation nucleation and the dislocation microstructure for a small fraction of the computational costs.

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