import copy
import cunumpy as xp
from struphy.geometry.base import Domain
[docs]
class HollowTorus(Domain):
r"""Torus with possible hole around the magnetic axis (center of the smaller circle).
.. image:: ../../pics/mappings/hollow_torus.png
Parameters
----------
a1 : float
Inner minor radius of hollow torus (default: 0.2).
a2 : float
Outer minor radius of hollow torus (default: 1.0).
R0 : float
Major radius of torus (default: 3.0).
sfl : bool
Whether to use straight field line coordinates (True) or not (False) (default: False).
pol_period: int
Which periodicity used in the mapping, i.e. :math: `\theta = 2*\pi*\eta_2 / \mathrm{pol_period}` (piece of cake) (default: 1, only for sfl=False).
tor_period : int
Toroidal periodicity built into the mapping: :math:`\phi=2\pi\,\eta_3/\mathrm{torperiod}` (default: 3 --> one third of a torus).
"""
[docs]
@classmethod
def doc_mapping(cls):
r"""
.. math::
F: \begin{bmatrix}\eta_1\\ \eta_2\\ \eta_3\end{bmatrix}\mapsto \begin{bmatrix}
\,\,x= &\lbrace\left[\,a_1 + (a_2-a_1)\,\eta_1\,\right]\cos\left[\theta(\eta_1,\eta_2)\right]+R_0\rbrace\cos(\phantom{-}2\pi\,\eta_3 / n)\,\,\\
\,\,y= &\lbrace\left[\,a_1 + (a_2-a_1)\,\eta_1\,\right]\cos\left[\theta(\eta_1,\eta_2)\right]+R_0\rbrace\sin(-2\pi\,\eta_3 / n)\,\,\\
\,\,z= &\left[\,a_1 + (a_2-a_1)\,\eta_1\,\right]\sin\left[\theta(\eta_1,\eta_2)\right]\,\,\end{bmatrix}
with the following possible poloidal angle parametrizations:
.. math::
&\theta(\eta_1,\eta_2) = \left\{\begin{aligned}
& 2\pi\,\eta_2\,, \quad &&\textnormal{if}\quad \textnormal{sfl}=\textnormal{False}\,,
&2\arctan\left[\sqrt{\frac{1 + \epsilon(\eta_1)}{1 - \epsilon(\eta_1)}}\,\tan\left(\pi\,\eta_2\right)\right]\quad &&\textnormal{if}\quad \textnormal{sfl}=\textnormal{True}\,,
&\qquad \textrm {with}\qquad \epsilon(\eta_1) = \frac{a_1 + (a_2-a_1)\,\eta_1}{R_0}\,.
\end{aligned}\right.
"""
def __init__(
self,
a1: float = 0.1,
a2: float = 1.0,
R0: float = 3.0,
sfl: bool = False,
pol_period: int = 1,
tor_period: int = 3,
):
self.kind_map = 22
# use params setter
self.params = copy.deepcopy(locals())
self.params_numpy = self.get_params_numpy()
assert a2 <= R0, f"The minor radius must be smaller or equal than the major radius! {a2 =}, {R0 =}"
if sfl:
assert pol_period == 1, (
"Piece-of-cake is only implemented for torus coordinates, not for straight field line coordinates!"
)
# periodicity in eta3-direction and pole at eta1=0
self.periodic_eta3 = True
if a1 == 0.0:
self.pole = True
else:
self.pole = False
super().__init__()
[docs]
def inverse_map(self, x, y, z, bounded=True, change_out_order=False):
"""Analytical inverse map of HollowTorus"""
mr = xp.sqrt(x**2 + y**2) - self.params["R0"]
r = xp.sqrt(mr**2 + z**2)
eta3 = xp.arctan2(-y, x) % (2 * xp.pi / self.params["tor_period"]) / (2 * xp.pi) * self.params["tor_period"]
eta1 = (r - self.params["a1"]) / (self.params["a2"] - self.params["a1"])
if self.params["sfl"]:
# invert theta = 2*arctan(sqrt((1 + eps)/(1 - eps)) * tan(pi*eta2)) with eps = r/R0
theta = xp.arctan2(z, mr) % (2 * xp.pi)
g = xp.sqrt((1 + r / self.params["R0"]) / (1 - r / self.params["R0"]))
eta2 = xp.arctan2(xp.sin(theta / 2), g * xp.cos(theta / 2)) / xp.pi % 1.0
else:
eta2 = xp.arctan2(z, mr) % (2 * xp.pi / self.params["pol_period"]) / (2 * xp.pi / self.params["pol_period"])
if bounded:
eta1[eta1 > 1] = 1.0
eta1[eta1 < 0] = 0.0
assert xp.all(xp.logical_and(eta1 >= 0, eta1 <= 1))
assert xp.all(xp.logical_and(eta2 >= 0, eta2 <= 1))
assert xp.all(xp.logical_and(eta3 >= 0, eta3 <= 1))
if change_out_order:
return xp.transpose((eta1, eta2, eta3))
else:
return eta1, eta2, eta3
