Aslam Extrapolations

This test case is similar to aslam.c. However, instead of starting from the level set field, we start from a vof field, which is converted into level set, and then Aslam extrapolations are performed.

#include "grid/multigrid.h"
#include "utils.h"
#include "aslam.h"
#include "redistance.h"
#include "view.h"

We define a function that writes a picture with the map and isolines of the extended scalar fields.

void write_picture (char* name, scalar u) {
vertex scalar phi[];
foreach_vertex()
phi[] = (u[] + u[-1] + u[0,-1] + u[-1,-1])/4.;
clear();
isoline ("levelset", val = 0., lw = 2.);
isoline ("phi", n = 30);
box();
save (name);
}

void write_levelset (void) {
clear();
isoline ("levelset", val = 0., lw = 2.);
box();
save ("levelset.png");
}

#define ufunc(x,y)(x*y)
#define circle(x,y,R)(sq(R) - sq(x) - sq(y))

We declare the level set field levelset, and the field to extrapolate u.

scalar levelset[], u[], f[];

int main (void) {

We set the domain geometry and we initialize the grid.

  size (2.*pi);
origin (-pi,-pi);
init_grid (1 << 8);
double R0 = 2.;

We initialize the volume fraction field.

  fraction (f, (sq(R0) - sq(x) - sq(y)));

We reconstruct the levelset field.

  vof_to_ls (f, levelset, imax=300);
write_levelset();

We initialize the function u to be extrapolated and we call the constant_extrapolation() function. The extrapolations are perfomed using a \Delta t of 0.01 (it must be small enough to guarantee the stability of the explicit in time discretization), and using a total number of time steps equal to 300, in order to obtain a steady-state solution.

  foreach()
u[] = ufunc(x,y)*f[];
write_picture ("initial.png", u);

constant_extrapolation (u, levelset, 0.5, 300, c=f);
write_picture ("constant.png", u);
fprintf (stderr, "constant = %g\n", statsf(u).sum);

We re-initialize the function u and we apply the linear_extrapolation().

  foreach()
u[] = ufunc(x,y)*f[];
linear_extrapolation (u, levelset, 0.5, 300, c=f);
write_picture ("linear.png", u);
fprintf (stderr, "linear = %g\n", statsf(u).sum);
}

References

 [aslam2004partial] Tariq D Aslam. A partial differential equation approach to multidimensional extrapolation. Journal of Computational Physics, 193(1):349–355, 2004.