src/axi.h

    Axisymmetric coordinates

    For problems with a symmetry of revolution around the z-axis of a cylindrical coordinate system. The longitudinal coordinate (z-axis) is x and the radial coordinate (\rho- or r-axis) is y. Note that y (and so Y0) cannot be negative.

    We first define a macro which will be used in some geometry-specific code (e.g. curvature computation).

    #define AXI 1

    On trees we need refinement functions.

    #if TREE
static void refine_cm_axi (Point point, scalar cm)
{
#if !EMBED
  fine(cm,0,0) = fine(cm,1,0) = y - Delta/4.;
  fine(cm,0,1) = fine(cm,1,1) = y + Delta/4.;
#else // EMBED
  if (cs[] > 0. && cs[] < 1.) {
    coord n = interface_normal (point, cs);
    // Better? involve fs (troubles w prolongation)
    // coord n = facet_normal (point, cs, fs);
    foreach_child() {
      if (cs[] > 0. && cs[] < 1.) {
	coord p;
    	double alpha = plane_alpha (cs[], n);
	plane_center (n, alpha, cs[], &p);
	cm[] = (y + Delta*p.y)*cs[];
      }
      else
	cm[] = y*cs[];
    }
  }
  else
    foreach_child()
      cm[] = y*cs[];
#endif // EMBED
}

static void refine_face_x_axi (Point point, scalar fm)
{
#if !EMBED
  if (!is_refined(neighbor(-1))) {
    fine(fm,0,0) = y - Delta/4.;
    fine(fm,0,1) = y + Delta/4.;
  }
  if (!is_refined(neighbor(1)) && neighbor(1).neighbors) {
    fine(fm,2,0) = y - Delta/4.;
    fine(fm,2,1) = y + Delta/4.;
  }
  fine(fm,1,0) = y - Delta/4.;
  fine(fm,1,1) = y + Delta/4.;
#else // EMBED
  double sig = 0., ff = 0.;
  if (cs[] > 0. && cs[] < 1.) {
    coord n = facet_normal (point, cs, fs);
    sig = sign(n.y)*Delta/4.;
  }
  if (!is_refined(neighbor(-1))) {
    ff = fine(fs.x,0,0);
    fine(fm,0,0) = (y - Delta/4. - sig*(1. - ff))*ff;
    ff = fine(fs.x,0,1);
    fine(fm,0,1) = (y + Delta/4. - sig*(1. - ff))*ff;
  }
  if (!is_refined(neighbor(1)) && neighbor(1).neighbors) {
    ff = fine(fs.x,2,0);
    fine(fm,2,0) = (y - Delta/4. - sig*(1. - ff))*ff;
    ff = fine(fs.x,2,1);
    fine(fm,2,1) = (y + Delta/4. - sig*(1. - ff))*ff;
  }
  ff = fine(fs.x,1,0);
  fine(fm,1,0) = (y - Delta/4. - sig*(1. - ff))*ff;
  ff = fine(fs.x,1,1);
  fine(fm,1,1) = (y + Delta/4. - sig*(1. - ff))*ff;
#endif // EMBED
}

static void refine_face_y_axi (Point point, scalar fm)
{
#if !EMBED
  if (!is_refined(neighbor(0,-1)))
    fine(fm,0,0) = fine(fm,1,0) = max(y - Delta/2., 1e-20);
  if (!is_refined(neighbor(0,1)) && neighbor(0,1).neighbors)
    fine(fm,0,2) = fine(fm,1,2) = y + Delta/2.;
  fine(fm,0,1) = fine(fm,1,1) = y;
#else // EMBED
  if (!is_refined(neighbor(0,-1))) {
    fine(fm,0,0) = (max(y - Delta/2., 1e-20))*fine(fs.y,0,0) ;
    fine(fm,1,0) = (max(y - Delta/2., 1e-20))*fine(fs.y,1,0);
  }
  if (!is_refined(neighbor(0,1)) && neighbor(0,1).neighbors) {
    fine(fm,0,2) = (y + Delta/2.)*fine(fs.y,0,2);
    fine(fm,1,2) = (y + Delta/2.)*fine(fs.y,1,2);
  }
  fine(fm,0,1) = y*fine(fs.y,0,1);
  fine(fm,1,1) = y*fine(fs.y,1,1);
#endif // EMBED
}
#endif

    If embedded solids are presents, cm, fm and the fluxes need to be updated consistently with the axisymmetric cylindrical coordinates and the solid fractions.

    #if EMBED
double axi_factor (Point point, coord p) {
  return (y + p.y*Delta);
}

void cm_update (scalar cm, scalar cs, face vector fs)
{
  foreach() {
    if (cs[] > 0. && cs[] < 1.) {
      coord p, n = facet_normal (point, cs, fs);
      double alpha = plane_alpha (cs[], n);
      plane_center (n, alpha, cs[], &p);
      cm[] = (y + Delta*p.y)*cs[];
    }
    else
      cm[] = y*cs[];
  }
  cm[top] = dirichlet(y*cs[]);
  cm[bottom] = dirichlet(y*cs[]);
}

void fm_update (face vector fm, scalar cs, face vector fs)
{
  foreach_face(x) {
    double sig = 0.;
    if (cs[] > 0. && cs[] < 1.) {
      coord n = facet_normal (point, cs, fs);
      sig = sign(n.y)*Delta/2.;
    }
    fm.x[] = (y - sig*(1. - fs.x[]))*fs.x[];
  }
  foreach_face(y)
    fm.y[] = max(y, 1e-20)*fs.y[];
  fm.t[top] = dirichlet(y*fs.t[]);
  fm.t[bottom] = dirichlet(y*fs.t[]);
}
#endif // EMBED

event metric (i = 0) {

    By default cm is a constant scalar field. To make it variable, we need to allocate a new field. We also move it at the begining of the list of variables: this is important to ensure the metric is defined before other fields.

      if (is_constant(cm)) {
    scalar * l = list_copy (all);
    cm = new scalar;
    free (all);
    all = list_concat ({cm}, l);
    free (l);
  }

    Metric factors must be taken into account for fluxes on embedded boundaries.

    #if EMBED
  metric_embed_factor = axi_factor;
#endif

    The volume/area of a cell is proportional to r (i.e. y). We need to set boundary conditions at the top and bottom so that cm is interpolated properly when refining/coarsening the mesh.

      scalar cmv = cm;
  foreach()
    cmv[] = y;
  cm[top] = dirichlet(y);
  cm[bottom] = dirichlet(y);

    We do the same for the length scale factors. The “length” of faces on the axis of revolution is zero (y=r=0 on the axis). To avoid division by zero we set it to epsilon (note that mathematically the limit is well posed).

      if (is_constant(fm.x)) {
    scalar * l = list_copy (all);
    fm = new face vector;
    free (all);
    all = list_concat ((scalar *){fm}, l);
    free (l);
  }
  face vector fmv = fm;
  foreach_face()
    fmv.x[] = max(y, 1./HUGE);
  fm.t[top] = dirichlet(y);
  fm.t[bottom] = dirichlet(y);

    We set our refinement/prolongation functions on trees.

    #if TREE
  cm.refine = cm.prolongation = refine_cm_axi;
  fm.x.prolongation = refine_face_x_axi;
  fm.y.prolongation = refine_face_y_axi;
#endif
}

    See also

    Usage

    Examples

    Tests