鬼力振荡对准连续体误差影响的分析
An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error
AI总结：
本文针对一维准连续体能量线性化模型，证明鬼力对位移的影响几乎抵消，并给出最优阶误差估计，表明准连续体位移以最优速率收敛到原子位移。
摘要：
原子到连续体的界面对于准连续体能量在均匀应变下表现出非零力，称为鬼力。在本文中，我们证明对于均匀应变附近的一维准连续体能量的线性化，鬼力对位移的影响几乎抵消，并且对远离界面的误差影响很小。我们给出了最优阶误差估计，表明准连续体位移在离散$\ell^\infty$范数下以最优速率$O(h)$收敛到原子位移，在$w^{1,p}$范数下对于$1 \leq p < \infty$以$O(h^{1/p})$收敛，其中$h$是原子间距。我们还证明了位移梯度误差在原子区域中距离$O(h|\log h|)$处衰减到$O(h)$，在连续体区域中距离$O(h)$处衰减到$O(h)$。E、Ming和Yang之前针对谐波原子间势给出了$w^{1,\infty}$范数收敛的反例。我们的工作给出了原子到连续体耦合误差影响随一般底层原子间势衰减的显式简化形式，并给出了上述离散$\ell^\infty$和$w^{1,p}$范数下的估计。
英文摘要：
The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the optimal rate O($h$) in the discrete $\ell^\infty$ norm and O($h^{1/p}$) in the $w^{1,p}$ norm for $1 \leq p < \infty.$ where $h$ is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O($h$) at distance O($h|\log h|$) in the atomistic region and distance O($h$) in the continuum region. E, Ming, and Yang previously gave a counterexample to convergence in the $w^{1,\infty}$ norm for a harmonic interatomic potential. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and $w^{1,p}$ norms.
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