# 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
}