Kernel function
Custom kernel weight functions can be incorporated easily into the solver. On a square grid of size (K,L), say the sampled point is (k.l). In the calculation of Hamiltonian Hkl, the sampled point is taken as (i,j)=(2,2) having 8 first order nearest neighbours ((1,1) to (3,3), exluding (2,2)), where i,j=1,2,3. Here, a Kernel function is used to introduce anisotropic net non-zero moment into the sampling scheme.
This is a bit of an overkill. But it does demonstrate what you can have. The kernel function used to demonstrate the capabilties in this paper is
Wij = Sij ((kijx)/(xijmAij )+(K_ij^y)/(y_ij^(n_ij ) ))^(Q_ij ) @ a given i & j, where, the matrices S_ij contains signs, K_ij^x ,K_ij^y and Q_ij contain constants, and x_ij^(m_ij ) and y_ij^(n_ij ) contain spatial coordinates raised by constants. IN this expression, common subscripts do not imply summation. The calculated weights are unit normalized as W ̅_ij=W_ij/(W_ij^' ), where, W_ij^' is the maximum of |W_ij |. For purposes of introducing flexibility, W_ij^' could be non-homogeneosuly rescaled by R as W ̅_ij=W_ij/(W_ij^' )R= W_ij/(W_ij^' )(L_ij+aR_((3,3))^R+bR_((3,3))^z+c), where, L_ij is a scalin g matrix having real values, a, b and c are real constants and R_((3,3))^R and R_((3,3))^z are 3X3 matrices having random real and integer numbers respectively. It should be noted the sub-scripts i and j used above are not the ones used to represent lattice sites in the Hamiltonian calculation. In MATLAB, they are input in “CODE NAME.m” as below:
Table.KF.01: Example definitions of KWF parameters
| Sij | Kijx | Kijy |
|---|---|---|
| sign=ones(3) | xconst=ones(3) | yconst=ones(3) |
| mij | nij | Qij |
|---|---|---|
| xpower=ones(3) | ypower=ones(3) | power=ones(3) |
Lij + aR(3,3)R + bR(3,3)Z+c |
|---|
| MultipleOfNorm = ones(3) + 0rand(3)-0+0randi(3) |
Together, the Kernel function looks like:
FIGURE.KF.01: Visualization of kernel function parameters over (3,3) sub-lattice for the values in Table.KF.01. [@repo/wiki images/Kernel function - 02.jpg]
The parameters shown in Table 1 for the calculation of W ̅_ij is visualized in above FIGURE.KF.01. Such visualizations are more useful for cases involving non-local Eulerian distance based Hamiltonian calculation and also when there more than 1 nearest neighbours.
FIGURE.KF.02: Visualization of kernel function parameters over (m,n) parent-lattice for the values in Table.KF.01. [@repo/wiki images/Kernel function - 01.jpg]
Grain structure is simulated on a 100*100 rectangular grid with square grid and the 500th and 1000th temporal slices are shown in Figure 6(i & k). Hamiltonian is calculated with using W ̅_ij visualized in Figure 6(a - i) considering 1st nearest neighbour interactions. Using different parameters shown in Table 1 for the calculation of W ̅_ij, different spatial partitioning can be generated with varying levels of symmetries. A few examples of the different parameters are provided in the appendix-C for a 32-state system. Users are urged to explore different generating parameters for W ̅_ij and report any interesting spatial partitioning. The overall asymmetry in W ̅_ij is represented in the visualization of (∑W) ̅_ij=(∑W ̅_ij)/max(∑W ̅_ij ) in Figure 6(e).