This implements the radial coordinate mapping illustrated below.

It works in 1D, 2D and 3D. The 3D version corresponds to cylindrical coordinates since the z-coordinate is unchanged.

The only parameter is d\theta, the total angle of the sector.

double dtheta = pi/3.;

For convenience we add definitions for the radial and angular coordinates (r, \theta).

map {
double r = x, theta = y*dtheta/L0;
NOT_UNUSED(r); NOT_UNUSED(theta);
}

event metric (i = 0) {

We initialise the scale factors, taking care to first allocate the fields if they are still constant.

  if (is_constant(cm)) {
scalar * l = list_copy (all);
cm = new scalar;
free (all);
all = list_concat ({cm}, l);
free (l);
}
if (is_constant(fm.x)) {
scalar * l = list_copy (all);
fm = new face vector;
free (all);
all = list_concat ((scalar *){fm}, l);
free (l);
}

The area (in 2D) of a mapped element is the area of an annulus defined by the two radii r-\Delta/2 and r+\Delta/2, divided by the total number of sectors N=2\pi L0/(d\theta\Delta), this gives \displaystyle \frac{\pi [(r + \Delta / 2)^2 - (r - \Delta / 2)^2]}{2 \pi L 0 / (d \theta \Delta)} = \frac{2 \pi r \Delta}{2 \pi L 0 / (d \theta \Delta)} = \frac{rd \theta}{L 0} \Delta^2 By definition, the (area) metric factor cm is the mapped area divided by the unmapped area \Delta^2.

  scalar cmv = cm;
foreach()
cmv[] = r*dtheta/L0;

It is important to set proper boundary conditions, in particular when refining the grid.

  cm[left] = dirichlet (r*dtheta/L0);
cm[right] = dirichlet (r*dtheta/L0);

The (length) metric factor fm is the ratio of the mapped length of a face to its unmapped length \Delta. In the present case, it is unity for all dimensions except for the x coordinates for which it is the ratio of the arclength by the unmapped length \Delta. We also set a small minimal value to avoid division by zero, in the case of a vanishing inner radius.

  face vector fmv = fm;
foreach_face()
fmv.x[] = 1.;
foreach_face(x)
fmv.x[] = max(r*dtheta/L0, 1e-20);

boundary ({cm, fm});
}