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 (ρ- 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)
  fine(cm,0,0) = fine(cm,1,0) = y - Δ/4.;
  fine(cm,0,1) = fine(cm,1,1) = y + Δ/4.;

static void refine_face_x_axi (Point point, scalar fm)
  if (!is_refined(neighbor(-1))) {
    fine(fm,0,0) = y - Δ/4.;
    fine(fm,0,1) = y + Δ/4.;
  if (!is_refined(neighbor(1)) && neighbor(1).neighbors) {
    fine(fm,2,0) = y - Δ/4.;
    fine(fm,2,1) = y + Δ/4.;
  fine(fm,1,0) = y - Δ/4.;
  fine(fm,1,1) = y + Δ/4.;

static void refine_face_y_axi (Point point, scalar fm)
  if (!is_refined(neighbor(0,-1)))
    fine(fm,0,0) = fine(fm,1,0) = max(y - Δ/2., 1e-20);
  if (!is_refined(neighbor(0,1)) && neighbor(0,1).neighbors)
    fine(fm,0,2) = fine(fm,1,2) = y + Δ/2.;
  fine(fm,0,1) = fine(fm,1,1) = y;

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);

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;
    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;
    fmv.x[] = max(y, 1e-20);
  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;
  boundary ({cm, fm});

See also