Boundary condition
= Boundary condition - 01
Figure-Theory-BC-01: Illustration of boundary condition (a) rectangular grid (b) toroidal boundary condition with torsion τ_η=0 and τ_ζ=0 (c) toroidal boundary condition with τ_η=π/2 and τ_ζ=0
The present model uses self-contained uniquely mapped toroidal boundary condition on the rectangular grid. The present version uses zero-twist toroidal boundary condition shown in Figure 3(b), a simple type of periodic boundary condition. The twisted toroid boundary condition shown in Figure 3(b) can also be implemented and is described now. The parent rectangular domain is treated as a toroid shown in Figure 3(b). Edges in the parent domain are ς and corners are μ. Opposite edges interact (ς_1 interacts with adjacent neighbours and ς_3; ς_2 interacts with adjacent neighbours and ς_4), adjacent opposite and diagonally opposite points interact (μ_1 interact with adjacent neighbours, adjacent points μ_2 and μ_4 and diagonally opposite point μ_3. Similar interactions apply to μ_2, μ_3 and μ_4) in the calculation of Hamiltonian H. This is done internally in Poly-XTAL Operations 9.04 using an appended element matrix, “ea”. A twist can be introduced into the boundary conditions if needed. If these boundary conditions only define the interaction of grid points lying on the periphery of the domain, then they form the peripheral boundary condition in Poly-XTAL operations. This could be of twisted type. This is visualized in an example in Figure 4(a) for the sampled grid point 14.
Figure-Theory-BC-02: