Abstract
Atomic interactions in solid materials are described using network theory. The tools of network theory focus on understanding the properties of a system based upon the underlying interactions which govern their dynamics. While the full atomistic network is dense, we apply a spectral sparsification technique to construct a sparse interaction network model that reduces the computational complexity while preserving macroscopic conservation properties. This sparse network is compared to a reduced network created using a cut-off radius (threshold method) that is commonly used to speed-up computations while approximating interatomic forces. The approximations used to estimate the total forces on each atom are quantified to assess how local interatomic force errors propagate errors at the global or continuum scale by comparing spectral sparsification to thresholding. In particular, we quantify the performance of the spectral sparsification algorithm for the short-range Lennard-Jones potential and the long-range Coulomb potential. Spectral sparsification of the Lennard-Jones potential yields comparable results to thresholding while spectral sparsification yields improvements when considering a long-range Coulomb potential. The present network-theoretic formulation is implemented on two sample problems: relaxation of atoms near a surface and a tensile test of a solid with a circular hole.