Formulation and Analysis of Blended Atomistic to Higher-Order Continuum Coupling Methods for Crystalline Defects

Concurrent multiscale methods play an important role in modeling and simulating materials with defects, aiming to achieve the balance between accuracy and efficiency. Atomistic-to-continuum (a/c) coupling methods, a typical class of concurrent multiscale methods, link atomic-scale simulations with continuum mechanics. Existing a/c methods adopt the classic second-order Cauchy-Born approximation as the continuum mechanics model. In this work, we employ a higher-order Cauchy-Born model to study the potential accuracy improvement of the coupling scheme. In particular, we develop an energy-based blended atomistic to higher-order continuum method and present a rigorous a priori error analysis. We show that the overall accuracy of the energy-based blended method is not actually improved due the coupling interface error which is of lower order and may not be improved. On the contrast, higher order accuracy is achieved by the force-based blended atomistic to higher-order continuum method. Our theoretical results are demonstrated by a detailed numerical study.