A Posteriori Error Estimation and Adaptive Algorithm for Atomistic/Continuum Coupling in Two Dimensions

Abstract

Atomistic/continuum coupling methods aim to achieve optimal balance between accuracy and efficiency. Adaptivity is the key for the efficient implementation of such methods. In this paper, we carry out a rigorous a posteriori analysis of the residual, the stability constant, and the error bound, for a consistent atomistic/continuum coupling method in 2D. We design and implement the corresponding adaptive mesh refinement algorithm, and the convergence rate with respect to degrees of freedom is optimal compare with a priori error estimates.

Adaptive QM/MM coupling for crystalline defects

“…5 where such an assumption holds. A more general numerical approach may need to combine ideas such as stress based error indicator from adaptive atomistic/continuum coupling method [31,20], and will be investigated in our future work. Remark 4.2.…”

A Posteriori Error Estimate and Adaptive Mesh Refinement Algorithm for Atomistic/Continuum Coupling with Finite Range Interactions in Two Dimensions

“…For the nearest neighbour interactions [40], we use the canonical expressions of δE (δE h ) to define σ a (σ h ). In that case, σ a (σ h ) is piecewise constant over the triangulation T a (T h ).…”

“…We will develop the adaptive mesh refinement algorithm for GRAC method with finite range interactions. In [40], we have designed the adaptive algorithm for GRAC with nearest neighbour interaction. In the nearest neighbor case, there exists a special set of reconstruction parameters [28].…”

“…In [40], we carried out a rigorous a posteriori analysis of the residual, the stability constant, and the error bound, for a consistent atomistic/continuum coupling method [28] with nearest neighbour interactions in two dimensions. Corresponding adaptive mesh refinement algorithm was designed and implemented based on the a posteriori error estimates, and the convergence rate with respect to degrees of freedom is the same as optimal a priori error estimates.…”

Analysis of the Residual-Type and the Gradient Recovery-Type a Posteriori Error Estimators for a Consistent Atomistic-to-Continuum Coupling Method in One-Dimension

“…There are several comments we need to give at this moment. First of all, the reason for which we use the hybrid error estimator instead of the gradient recovery error estimator is that the gradient recovery estimator may not correctly reflect the influence of the model error at the interface which may be a more serious problem in higher dimensions [37]. The idea behind the hybrid estimator is that we use a certain multiple of the gradient recovery estimator to approximate the residual based error estimator in the continuum region while keeping the residual based estimator on the interface whose effectiveness and efficiency have been proved.…”