fssa3d {SPSL}

Frequency of Sites on a Square Anisotropic 3D lattice with (1,d)-neighborhood

Description

fssa3d() function calculates the relative frequency distribution of anisotropic clusters on 3D square lattice with Moore (1,d)-neighborhood.

Usage

fssa3d(n=1000, x=33, 
       p0=runif(6, max=0.4),
       p1=colMeans(matrix(p0[c(
          1,3, 2,3, 1,4, 2,4,
          1,5, 2,5, 1,6, 2,6,
          3,5, 4,5, 3,6, 4,6)], nrow=2))/2,
       p2=colMeans(matrix(p0[c(
          1,3,5, 2,3,5, 1,4,5, 2,4,5,
          1,3,6, 2,3,6, 1,4,6, 2,4,6)], nrow=3))/3,
       set=(x^3+1)/2, all=TRUE, shape=c(1,1))

Arguments

n

a sample size.

x

a linear dimension of 2D square percolation lattice.

p0

a vector of relative fractions (0<p0)&(p0<1) of accessible sites (occupation probability) for lattice directions: (-x,+x,-y,+y,-z,+z).

p1

averaged double combinations of p0-components weighted by Minkowski distance: p1=colMeans(matrix(p0[c(1,3,...)], nrow=2))/rhoMe1.

p2

averaged triple combinations of p0-components weighted by Minkowski distance: p2=colMeans(matrix(p0[c(1,3,5,...)], nrow=3))/rhoMe2.

set

a vector of linear indexes of a starting sites subset.

all

logical; if all=TRUE, mark all sites from a starting subset; if all=FALSE, mark only accessible sites from a starting subset.

shape

a vector with two shape parameters of beta-distributed random variables, weighting the percolation lattice sites.

Details

The percolation is simulated on 3D square lattice with uniformly weighted sites acc and the vectors p0, p1, and p2, distributed over the lattice directions, and their combinations.

The anisotropic cluster is formed from the accessible sites connected with the initial subset set, and depends on the direction in 3D square lattice.

Moore (1,d)-neighborhood on 3D square lattice consists of sites, at least one coordinate of which is different from the current site by one: e=c(e0,e1,e2), where
e0=c(-1, 1, -x, x, -x^2, x^2);
e1=colSums(matrix(e0[c(1,3, 2,3, 1,4, 2,4, 1,5, 2,5, 1,6, 2,6, 3,5, 4,5, 3,6, 4,6)], nrow=2));
e2=colMeans(matrix(p0[c(1,3,5, 2,3,5, 1,4,5, 2,4,5, 1,3,6, 2,3,6, 1,4,6, 2,4,6)], nrow=3)).

Minkowski distance between sites a and b depends on the exponent d:
rhoM <- function(a, b, d=1)
if (is.infinite(d)) return(apply(abs(b-a), 2, max))
else return(apply(abs(b-a)^d, 2, sum)^(1/d)).

Minkowski distance for sites from e1 and e2 subsets with the exponent d=1 is equal to rhoMe1=2 and rhoMe2=3.

Each element of the matrix frq is equal to the relative frequency with which the 3D square lattice site belongs to a cluster sample of size n.

Value

rfq

a 3D matrix of relative sampling frequencies for sites of the percolation lattice.

Author(s)

Pavel V. Moskalev <[e-mail]>

References

[1] Moskalev, P.V. Percolation modeling of porous structures. Moscow: URSS, 2018. 240 pp; in Russian.

See Also

ssa3d, fssa2d, fssa20, fssa30, fssi2d, fssi3d

Examples

x <- y <- seq(33)
rfq <- fssa3d(n=200, p0=.17*c(.5,3,.5,1.5,1,.5))
image(x, y, rfq[,,17], cex.main=1,
main="Frequencies in z=17 slice of anisotropic (1,1)-clusters")
contour(x, y, rfq[,,17], levels=seq(.05,.3,.05), add=TRUE)
abline(h=17, lty=2); abline(v=17, lty=2)