ssi3d {SPSL}

Site cluster on Square Isotropic 3D lattice with (1,d)-neighborhood

Description

ssi3d() function provides sites labeling of the isotropic cluster on 3D square lattice with Moore (1,d)-neighborhood.

Usage

ssi3d(x=33, p0=0.2, p1=p0/2, p2=p0/3, 
      set=(x^3+1)/2, all=TRUE, shape=c(1,1))

Arguments

x

a linear dimension of 3D square percolation lattice.

p0

a relative fraction (0<p0)&(p0<1) of accessible sites (occupation probability) for percolation lattice.

p1

p0 value, weighted by Minkowski distance: p1=p0/rhoMe1.

p2

p0 value, weighted by Minkowski distance: p2=p0/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 constant parameters p0, p1, and p2.

The isotropic cluster is formed from the accessible sites connected with initial sites subset set.

To form the cluster the condition acc[set+eN]<pN is iteratively tested for sites of the Moore (1,d)-neighborhood eN for the current cluster perimeter set, where eN is equal to e0, e1 or e2 vector; pN is equal to p0, p1 or p2 value.

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.

Forming cluster ends with the exhaustion of accessible sites in Moore (1,d)-neighborhood of the current cluster perimeter.

Value

acc

an accessibility matrix for 3D square percolation lattice:
if acc[e]<pN then acc[e] is accessible site;
if acc[e]==1 then acc[e] is non-accessible site;
if acc[e]==2 then acc[e] belongs to a sites cluster.

Author(s)

Pavel V. Moskalev

References

[1] Moskalev, P.V. Percolation modeling of porous structures. Moscow: URSS, 2018. 240 pp; in Russian.
[2] Moskalev, P.V. (2013) The structure of site percolation models on three-dimensional square lattices. Computer Research and Modeling, Vol.5, No.4, pp.607–622; in Russian.

See Also

fssi3d, ssi2d, ssi20, ssi30, ssa2d, ssa3d

Examples

# Example No.1. Axonometric projection of 3D cluster
require(lattice)
set.seed(20120507)
x <- y <- z <- seq(33)
cls <- which(ssi3d(p0=.285)>1, arr.ind=TRUE)
cloud(cls[,3] ~ cls[,1]*cls[,2],
xlim=range(x), ylim=range(y), zlim=range(z), 
col=rgb(1,0,0,0.4), xlab="x", ylab="y", zlab="z", main.cex=1,
main="Isotropic (1,1)-cluster")

# Example No.2. Z=17 slice of 3D cluster
set.seed(20120507)
cls <- ssi3d(p0=.285)
x <- y <- z <- seq(33)
image(x, y, cls[,,17], zlim=c(0,2), cex.main=1, 
main="Z=17 slice of an isotropic (1,1)-cluster")
abline(h=17, lty=2); abline(v=17, lty=2)