论文标题
有效的二维晶体缺陷的一致原子/连续耦合方法的后验误差控制
Efficient a Posteriori Error Control of a Consistent Atomistic/Continuum Coupling Method for Two Dimensional Crystalline Defects
论文作者
论文摘要
自适应原子/连续体（A/C）耦合方法是模拟具有缺陷的材料和原子系统以实现准确性和效率平衡的重要方法。在自适应算法中通常采用基于残留的后验误差估计器，以估计通过在连续区域应用原子系统的连续近似值和有限元隔离来估算所承受的应变误差。在这项工作中，我们提出了一个基于理论的近似值，用于基于残差的后验误差估计器，从而大大提高了适应性的效率。特别是，数值昂贵的建模残差仅在耦合界面周围的一个小区域中计算出来，但被该区域以外的粗糙残留物取代了理论上合理的近似值。我们基于修改后的后验误差估计量及其针对不同类型的晶体缺陷的变体提供了一系列自适应计算，其中一些在自适应A/C方法的先前相关文献中不考虑其中一些。数值结果表明，与原始的基于残余的误差估计器相比，使用修改的误差估计器具有适当选择的参数的自适应算法会导致误差的最佳收敛速率相同，但相对于自由度的数量，计算成本将计算成本降低了一个顺序。
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.
