A Posteriori Analysis and Adaptive Algorithms for Blended Type Atomistic-to-Continuum Coupling with Higher-Order Finite Elements

The efficient and accurate simulation of material systems with defects using atomistic- to-continuum (a/c) coupling methods is a topic of considerable interest in the field of computational materials science. To achieve the desired balance between accuracy and computational efficiency, the use of a posteriori analysis and adaptive algorithms is critical. In this work, we present a rigorous a posteriori error analysis for three typical blended a/c coupling methods: the blended energy-based quasi-continuum (BQCE) method, the blended force-based quasi-continuum (BQCF) method, and the atomistic/continuum blending with ghost force correction (BGFC) method. We employ first and second-order finite element methods (and potentially higher-order methods) to discretize the Cauchy-Born model in the continuum region. The resulting error estimator provides both an upper bound on the true approximation error and a lower bound up to a theory-based truncation indicator, ensuring its reliability and efficiency. Moreover, we propose an a posteriori analysis for the energy error. We have designed and implemented a corresponding adaptive mesh refinement algorithm for two typical examples of crystalline defects. In both numerical experiments, we observe optimal convergence rates with respect to degrees of freedom when compared to a priori error estimates

Adaptive Multiscale Coupling Methods of Molecular Mechanics based on a Unified Framework of a Posteriori Error Estimates

Multiscale coupling methods are significant methodologies for the modeling and simulation of materials with defects, intending to achieve the (quasi-)optimal balance of accuracy and efficiency. The a posteriori analysis and corresponding adaptive algorithms play a crucial role in the efficient implementation of multiscale coupling methods. This paper proposes a unified framework for residual-based a posteriori error estimates that can be applied to general consistent multiscale coupling methods. In particular, we prove that the error estimator based on the residual force can provide the upper bound of the true approximation error. As prototypical examples, we present a variety of adaptive computations based on this reliable error estimator for the blended atomistic-to-continuum (a/c) coupling methods, including the energy-based blended quasi-continuum (BQCE), the force-based blended quasi-continuum (BQCF) and the recently developed blended ghost force correction (BGFC) methods. We develop a coarse-grained technique for the efficient evaluation of the error estimator. A robust adaptive algorithm is therefore proposed and validated with different types of crystalline defects, some of which are not considered in previous related literature on the adaptive a/c coupling methods. The results demonstrate that the adaptive algorithm leads to the same optimal convergence rate of the error as the a priori error estimate, but with considerable computational efficiency. This study provides valuable insights into the design and implementation of adaptive multiscale methods, and represents a significant contribution to the literature on a/c coupling methods

Efficient a Posteriori Error Control of a Consistent Atomistic/Continuum Coupling Method for Two Dimensional Crystalline Defects

Adaptive atomistic/continuum (a/c) coupling method is an important method for the simulation of material and atomistic systems with defects to achieve the balance of accuracy and efficiency. Residual based a posteriori error estimator is often employed in the adaptive algorithm to provide an estimate of the error of the strain committed by applying the continuum approximation for the atomistic system and the finite element discretization in the continuum region. In this work, we propose a theory based approximation for the residual based a posteriori error estimator which greatly improves the efficiency of the adaptivity. In particular, the numerically expensive modeling residual is only computed exactly in a small region around the coupling interface but replaced by a theoretically justified approximation by the coarsening residual outside that region. We present a range of adaptive computations based on our modified a posteriori error estimator and its variants for different types of crystalline defects some of which are not considered in previous related literature of the adaptive a/c methods. The numerical results show that, compared with the original residual based error estimator, the adaptive algorithm using the modified error estimator with properly chosen parameters leads to the same optimal convergence rate of the error but reduces the computational cost by one order with respect to the number of degrees of freedom

Consistent Energy-based Atomistic/Continuum Coupling for Two-body Potentials in Three Dimensions

Very few works exist to date on development of a consistent energy-based coupling of atomistic and continuum models of materials in more than one dimension. The difficulty in constructing such a coupling consists in defining a coupled energy whose minimizers are free from uncontrollable errors on the atomistic/continuum interface. In this paper a consistent coupling in three dimensions is proposed. The main achievement of this work is to identify and efficiently treat a modified Cauchy-Born continuum model which can be coupled to the exact atomistic model. The convergence and stability of the method is confirmed with numerical tests.Comment: 29 pages, 1 Matlab code. Typos corrected, exposition improve

Atomistic/Continuum blending with ghost force correction

We combine the ideas of atomistic/continuum energy blending and ghost force correction to obtain an energy-based atomistic/continuum coupling scheme which has, for a range of benchmark problems, the same convergence rates as optimal force-based coupling schemes. We present the construction of this new scheme, numerical results exploring its accuracy in comparison with established schemes, and a rigorous error analysis for an instructive special case