sandbox/Antoonvh/coil.c

    The fractal dimension of a coil

    Similar to the analysis that was done for the Koch fractal curve, the fractal dimension of a coil is studied.

    #include "grid/octree.h"
#include "utils.h"

int main(){

    First we calculate 150000 points on a coil. The ratio between the radius and winding distance is approx. R/d=40, there are about 60 windings in total.

      int jmax=150000;
  double jm,jmd;
  double xn[jmax],yn[jmax],zn[jmax];
  jm= (double) jmax;
  for (int j=0;j<jmax;j++){
    jmd = (double) j;
    xn[j]=0.75*sin((jmd/jm)*3.14*120.);
    yn[j]=0.75*cos((jmd/jm)*3.14*120.);
    zn[j]=2.*(((jmd)/jm)-0.5);
  }
  FILE * fp1 = fopen("cyl.dat","w");
  for (int j=0;j<jmax;j++){
    fprintf(fp1,"%g %g %g\n",xn[j],yn[j],zn[j]);
  }
  fclose(fp1);
    The plot below shows the parameterized coil.
      set xr [-1.2:1.2]
  set yr [-1.2:1.2]
  set view equal xyz
  set xlabel 'z'
  set ylabel 'x'
  set zlabel 'y'
  splot 'cyl.dat' using 3:1:2 title 'Coil' with lines
    (script)

    (script)

    Looks OK, now the algorithm is applied.

      FILE * fpit = fopen("cells.dat","w");
  L0=2.5;
  X0=-L0/2;
  Y0=X0;
  Z0=X0;
  init_grid(1<<3);
  scalar c[];
  for (int m=0;m<7;m++){
    foreach(){
      c[]=0.;
    }
    for(int i=0;i<jmax;i++){
      Point point = locate(xn[i],yn[i],zn[i]);
      if (point.level>0)
	c[]+=1.0;
    }
    int gcc=0;
    int gcn=0;
    foreach(reduction(+:gcc) reduction(+:gcn)){
      gcn++;
      if(c[]>0.5)
	gcc++;
    }
    fprintf(fpit,"%d\t%d\t%d\n",m,gcc,gcn);
    fflush(fpit);

    boundary({c});
    while(adapt_wavelet({c},(double[]){0.1},(4+m),3,{c}).nf)
      boundary({c});
    int i=m;
    scalar lev[];
    foreach()
      lev[]=level;
    static FILE * fp3 =
      popen ("gfsview-batch3D coil.hop.gfv | ppm2gif > fracz.gif --delay 100","w");
    output_gfs(fp3);
    fprintf (fp3, "Save stdout { format = PPM width = 600 height = 600}\n");
    fflush(fp3);
    fprintf(ferr,"%d\t",m);
  }
}

    Results

    First a visualization of the c=1 isosurface is given:

    Visualization of the refinement iterations with the c=1 isosurface, colored with the level of refinement. The results form the last iteration are poorly displayed as it entails data at a sub-pixel resolution

    Visualization of the refinement iterations with the c=1 isosurface, colored with the level of refinement. The results form the last iteration are poorly displayed as it entails data at a sub-pixel resolution

    It appears that the isosurface starts as a cylinder before the coil structre is retrieved at a higher resolution.

    Now we study the number of grid cells that lay on the curve as a function of the refinement iteration.

    set xr [ -0.5:6.5]
set yr [100:100000]
set logscale y
set xlabel 'refinement iteration' 
set ylabel 'Grid cells on the curve'
set key top left box 3
plot "cells.dat" title "Number of cells on the curve" ,\
      (2**(1*x))*1200 lw 2 title "1D scaling",\
      (2**(2*x))*150 lw 2 title "2D scaling"
    (script)

    (script)

    For the coarser resolutions the coil appears to be two-dimensional, wheareas at higher resolutions the curve seems to be one-dimensional. This corresponds with the transition from a cylinder-like (i.e. 2D) appearance of the coil at coarser resolutions to the line (i.e. 1D) apearance a finer resolutoins.