Numerical methods for quasicrystals
Numerical methods for quasicrystals
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准晶体的数值方法
DOI：
10.1016/j.jcp.2013.08.034
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发表时间：
2012-12
影响因子：
4.1
通讯作者：
Kai Jiang, Pingwen Zhang
中科院分区：
文献类型：
--
作者：
Kai Jiang, Pingwen Zhang
准晶是一种充满空间的结构。传统的晶体近似方法是利用周期结构来近似准晶。这种方法的误差来自两个部分：数值离散，和同时丢番图逼近的近似误差，这也决定了精确解所需的区域的大小。随着近似误差的减小，计算复杂度迅速增加，而且除非计算区域是全空间，否则近似误差总是存在的。在这项工作中，我们专注于发展的数值方法来计算准晶体的高精度。借助于高维倒易空间，发展了一种新的计算准晶的投影方法。这种方法使我们能够计算准晶体，而不是晶体近似。与晶体近似法相比，投影法不仅克服了同时丢番图近似法的局限性，而且可以方便地使用周期边界条件。同时，该方法通过在单胞内实现和使用伪谱方法，有效地降低了计算复杂度。为了说明的目的，我们与Lifshitz-Petrich模型，虽然我们目前的算法将适用于更一般的系统，包括准晶。我们发现投影方法可以精确地保持旋转对称性。更重要的是，该算法可以高精度地计算自由能密度。
Quasicrystals are one kind of space-filling structures. The traditional crystalline approximant method utilizes periodic structures to approximate quasicrystals. The errors of this approach come from two parts: the numerical discretization, and the approximate error of Simultaneous Diophantine Approximation which also determines the size of the domain necessary for accurate solution. As the approximate error decreases, the computational complexity grows rapidly, and moreover, the approximate error always exits unless the computational region is the full space. In this work we focus on the development of numerical method to compute quasicrystals with high accuracy. With the help of higher-dimensional reciprocal space, a new projection method is developed to compute quasicrystals. The approach enables us to calculate quasicrystals rather than crystalline approximants. Compared with the crystalline approximant method, the projection method overcomes the restrictions of the Simultaneous Diophantine Approximation, and can also use periodic boundary conditions conveniently. Meanwhile, the proposed method efficiently reduces the computational complexity through implementing in a unit cell and using pseudospectral method. For illustrative purpose we work with the Lifshitz–Petrich model, though our present algorithm will apply to more general systems including quasicrystals. We find that the projection method can maintain the rotational symmetry accurately. More significantly, the algorithm can calculate the free energy density to high precision.
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DOI：
10.1107/s0108767385001180
发表时间：
1985-11
期刊：
Acta Crystallographica Section A
影响因子：
--
作者：
H. Hiller
通讯作者：
H. Hiller
影响因子：
3.7
作者：
A. Quandt;M. Teter
通讯作者：
A. Quandt;M. Teter
影响因子：
0.7
作者：
G. Szekeres
通讯作者：
G. Szekeres
影响因子：
1.6
作者：
T. Dotera
通讯作者：
T. Dotera
DOI：
10.1016/j.jcp.2008.02.021
发表时间：
2008-05
期刊：
J. Comput. Phys.
影响因子：
--
作者：
Pingwen Zhang;Xinwei Zhang
通讯作者：
Pingwen Zhang;Xinwei Zhang