A General Framework of Linear Elasticity Enhanced Multiscale Coupling Methods for Crystalline Defects
Yanbo Zhan, Yangshuai Wang, Hao Wang
TL;DR
The work develops QNLL, a linear-elasticity-enhanced quasinonlocal (a/c) coupling for crystalline defects in one dimension, uniting a nonlinear atomistic core with a linearized Cauchy-Born far field to reduce computational cost. It provides a rigorous a priori analysis showing that, with appropriate balancing of atomistic, nonlinear-continuum, linear-continuum, and coarse-grained regions, the QNLL method attains the same convergence order as the classic nonlinear QNL, while achieving substantial CPU-time savings. The authors extend the framework to coarse-grained continuum regions, deriving coarse-graining error bounds and optimizing mesh and region lengths to maintain accuracy. Numerical validations in 1D demonstrate that QNLL matches QNL in accuracy under optimal balancing and can outperform QNL in efficiency, particularly for larger defects or longer-range interactions. The paper also outlines future directions, including a posteriori error control, higher-dimensional extensions, and integration with other a/c methods to broaden applicability and robustness.
Abstract
The atomistic-to-continuum (a/c) coupling methods, also known as the quasicontinuum (QC) methods, are a important class of concurrent multisacle methods for modeling and simulating materials with defects. The a/c methods aim to balance the accuracy and efficiency by coupling a molecular mechanics model (also termed as the atomistic model) in the vicinity of localized defects with the Cauchy-Born approximation of the atomistic model in the elastic far field. However, since both the molecular mechanics model and its Cauchy-Born approximation are usually a nonlinear, it potentially leads to a high computational cost for large-scale simulations. In this work, we propose an advancement of the classic quasinonlocal (QNL) a/c coupling method by incorporating a linearized Cauchy-Born model to reduce the computational cost. We present a rigorous a priori error analysis for this QNL method with linear elasticity enhancement (QNLL method), and show both analytically and numerically that it achieves the same convergence behavior as the classic (nonlinear) QNL method by proper determination of certain parameters relating to domain decomposition and finite element discretization. More importantly, our numerical experiments demonstrate that the QNLL method perform an substantial improvement of the computational efficiency in term of CPU times.