radial.h
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Radial/cylindrical coordinates
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 $(r, \theta)$.
macro VARIABLES (Point point = point, int _ig = ig, int _jg = jg, int _kg = kg)
{
VARIABLES (point, _ig, _jg, _kg);
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
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);
}