116 research outputs found
Numerical Methods for Quasicrystals
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.Comment: 27 pages, 8 figures, 6 table
Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems
In this article, we discuss the stability of soft quasicrystalline phases in
a coupled-mode Swift-Hohenberg model for three-component systems, where the
characteristic length scales are governed by the positive-definite gradient
terms. Classic two-mode approximation method and direct numerical minimization
are applied to the model. In the latter approach, we apply the projection
method to deal with the potentially quasiperiodic ground states. A variable
cell method of optimizing the shape and size of higher-dimensional periodic
cell is developed to minimize the free energy with respect to the order
parameters. Based on the developed numerical methods, we rediscover decagonal
and dodecagonal quasicrystalline phases, and find diverse periodic phases and
complex modulated phases. Furthermore, phase diagrams are obtained in various
phase spaces by comparing the free energies of different candidate structures.
It does show not only the important roles of system parameters, but also the
effect of optimizing computational domain. In particular, the optimization of
computational cell allows us to capture the ground states and phase behavior
with higher fidelity. We also make some discussions on our results and show the
potential of applying our numerical methods to a larger class of mean-field
free energy functionals.Comment: 26 pages, 13 figures; accepted by Communications in Computational
Physic
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