Accuracy of quasicontinuum approximations near instabilities
Accuracy of quasicontinuum approximations near instabilities
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DOI：
10.1016/j.jmps.2010.06.011
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发表时间：
2009-05
影响因子：
5.3
通讯作者：
M. Dobson;M. Luskin;C. Ortner
中科院分区：
文献类型：
--
作者：
M. Dobson;M. Luskin;C. Ortner
当晶格构型失去稳定性时，即，当晶格能量泛函的海森特征值变为负值时，发生诸如裂纹、位错或晶界的晶格缺陷的形成和运动。当原子能近似的混合能量耦合原子和连续模型，近似的精度只能保证附近的原子能以及混合能量是稳定的变形。因此，我们建议，这是必不可少的评估的预测能力的原子到连续耦合方法附近的不稳定性，进行理论分析，至少对一些代表性的模型问题，确定是否保持稳定的混合能源的原子能量的不稳定性的发病。我们制定了一个一维模型的问题，最近和下一个最近的邻居的相互作用，并使用严格的分析，渐近方法，和数值实验，以获得这种尖锐的基本保守的准连续体（QC）近似的稳定性估计。我们的研究结果表明，一致的准非局部QC近似正确地再现了原子系统的稳定性，而不一致的基于能量的QC近似不正确地预测在一个显着减少的应用负载，我们描述的原子势的衍生物的分析标准不稳定。
The formation and motion of lattice defects such as cracks, dislocations, or grain boundaries, occurs when the lattice configuration loses stability, that is, when an eigenvalue of the Hessian of the lattice energy functional becomes negative. When the atomistic energy is approximated by a hybrid energy that couples atomistic and continuum models, the accuracy of the approximation can only be guaranteed near deformations where both the atomistic energy as well as the hybrid energy are stable. We propose, therefore, that it is essential for the evaluation of the predictive capability of atomistic-to-continuum coupling methods near instabilities that a theoretical analysis be performed, at least for some representative model problems, that determines whether the hybrid energies remain stable up to the onset of instability of the atomistic energy. We formulate a one-dimensional model problem with nearest and next-nearest neighbour interactions and use rigorous analysis, asymptotic methods, and numerical experiments to obtain such sharp stability estimates for the basic conservative quasicontinuum (QC) approximations. Our results show that the consistent quasi-nonlocal QC approximation correctly reproduces the stability of the atomistic system, whereas the inconsistent energy-based QC approximation incorrectly predicts instability at a significantly reduced applied load that we describe by an analytic criterion in terms of the derivatives of the atomistic potential.