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)
      fine(cm,0,0) = fine(cm,1,0) = y - Delta/4.;
      fine(cm,0,1) = fine(cm,1,1) = y + Delta/4.;
    static void refine_face_x_axi (Point point, scalar fm)
      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.;
    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 - 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;
    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;

    See also