surf_henneberg

Module Contents

class surf_henneberg.SurfaceHenneberg(nfp: int = 1, alpha_fac: int = 1, mmax: int = 1, nmax: int = 0, quadpoints_phi=None, quadpoints_theta=None, dofs=None)

Bases: spectre.surfaces.surf.Surface

This class represents a toroidal surface using the parameterization in Henneberg, Helander, and Drevlak, Journal of Plasma Physics 87, 905870503 (2021). The main benefit of this representation is that there is no freedom in the poloidal angle, i.e. \(\theta\) is uniquely defined, in contrast to other parameterizations like SurfaceRZFourier. Stellarator symmetry is assumed.

In this representation by Henneberg et al, the cylindrical coordinates \((R,\phi,Z)\) are written in terms of a unique poloidal angle \(\theta\) as follows:

\[\begin{split}R(\theta,\phi) = R_0^H(\phi) + \rho(\theta,\phi) \cos(\alpha\phi) - \zeta(\theta,\phi) \sin(\alpha\phi), \\ Z(\theta,\phi) = Z_0^H(\phi) + \rho(\theta,\phi) \sin(\alpha\phi) + \zeta(\theta,\phi) \cos(\alpha\phi),\end{split}\]

where

\[\begin{split}R_0^H(\phi) &=& \sum_{n=0}^{nmax} R_{0,n}^H \cos(n_{fp} n \phi), \\ Z_0^H(\phi) &=& \sum_{n=1}^{nmax} Z_{0,n}^H \sin(n_{fp} n \phi), \\ \zeta(\theta,\phi) &=& \sum_{n=0}^{nmax} b_n \cos(n_{fp} n \phi) \sin(\theta - \alpha \phi), \\ \rho(\theta,\phi) &=& \sum_{n,m} \rho_{n,m} \cos(m \theta - n_{fp} n \phi - \alpha \phi).\end{split}\]

The continuous degrees of freedom are \(\{\rho_{m,n}, b_n, R_{0,n}^H, Z_{0,n}^H\}\). These variables correspond to the attributes rhomn, bn, R0nH, and Z0nH respectively, which are all numpy arrays. There is also a discrete degree of freedom \(\alpha\) which should be \(\pm n_{fp}/2\) where \(n_{fp}\) is the number of field periods. The attribute alpha_fac corresponds to \(2\alpha/n_{fp}\), so alpha_fac is either 1, 0, or -1. Using alpha_fac = 0 is appropriate for axisymmetry, while values of 1 or -1 are appropriate for a stellarator, depending on the handedness of the rotating elongation.

For \(R_{0,n}^H\) and \(b_n\), \(n\) is 0 or any positive integer up through nmax (inclusive). For \(Z_{0,n}^H\), \(n\) is any positive integer up through nmax. For \(\rho_{m,n}\), \(m\) is an integer from 0 through mmax (inclusive). For positive values of \(m\), \(n\) can be any integer from -nmax through nmax. For \(m=0\), \(n\) is restricted to integers from 1 through nmax. Note that we exclude the element of \(\rho_{m,n}\) with \(m=n=0\), because this degree of freedom is already represented in \(R_{0,0}^H\).

For the 2D array rhomn, functions set_rhomn() and get_rhomn() are provided for convenience so you can specify n, since the corresponding array index is shifted by nmax. There are no corresponding functions for the 1D arrays R0nH, Z0nH, and bn since these arrays all have a first index corresponding to n=0.

Args:

nfp: The number of field periods. alpha_fac: Should be +1 or -1 for a stellarator, depending on the handedness

by which the elongation rotates, or 0 for axisymmetry.

mmax: Maximum poloidal mode number included. nmax: Maximum toroidal mode number included, divided by nfp. quadpoints_phi: Set this to a list or 1D array to set the \(\phi_j\) grid points directly. quadpoints_theta: Set this to a list or 1D array to set the \(\theta_j\) grid points directly.

nfp = 1

alpha_fac = 1

mmax = 1

nmax = 1

stellsym = True

quadpoints_theta

quadpoints_phi

allocate()

Create the arrays for the continuous degrees of freedom. Also set the names of the dofs.

get_rhomn(m, n)

Return a particular \(\rho_{m,n}\) coefficient.

set_rhomn(m, n, val)

Set a particular \(\rho_{m,n}\) coefficient.

get_dofs()

Return a 1D numpy array with all the degrees of freedom.

set_dofs(dofs)

num_dofs()

Return the number of degrees of freedom.

set_dofs_impl(v)

Set the shape coefficients from a 1D list/array

fixed_range(mmax, nmax, fixed=True)

Set the fixed property for a range of m and n values.

All modes with m <= mmax and |n| <= nmax will have their fixed property set to the value of the fixed parameter. Note that mmax and nmax are included.

Both mmax and nmax must be >= 0.

For any value of mmax, the fixed properties of R0nH, Z0nH, and rhomn are set. The fixed properties of bn are set only if mmax > 0. In other words, the bn modes are treated as having m=1.

to_RZFourier(mpol_tgt=None, ntor_tgt=None)

Return a SurfaceRZFourier object with the identical shape. This routine implements eq (4.5)-(4.6) in the Henneberg paper, plus m=0 terms for R0 and Z0.

classmethod from_RZFourier(surf, alpha_fac: int, mmax=None, nmax=None, ntheta=None, nphi=None, verbose=False)

Convert a SurfaceRZFourier surface to a SurfaceHenneberg surface.

Args:

surf: The SurfaceRZFourier object to convert. mmax: Maximum poloidal mode number to include in the new surface. If None,

the value mpol from the old surface will be used.

nmax: Maximum toroidal mode number to include in the new surface. If None,

the value ntor from the old surface will be used.

ntheta: Number of grid points in the poloidal angle used for the transformation.

If None, the value 3 * ntheta will be used.

nphi: Number of grid points in the toroidal angle used for the transformation.

If None, the value 3 * nphi will be used.

classmethod surfaces_from_spectre_xmn(xmn, bnd_rc, bnd_zs, test, alpha_fac=1, nmax=6, mmax=6, ntheta=48, nphi=24, verbose=False)

classmethod surfaces_to_spectre_xmn(surfaces, test)

classmethod surfaces_to_spectre_dxmn_dxdof(surfaces, test)

gamma()

gammadash1()

gammadash2()

gamma_lin(data, quadpoints_phi, quadpoints_theta)

Evaluate the position vector on the surface in Cartesian coordinates, for a list of (phi, theta) points.

gamma_impl(data, quadpoints_phi, quadpoints_theta)

Evaluate the position vector on the surface in Cartesian coordinates, for a tensor product grid of points in theta and phi.

gammadash1_impl(data)

Evaluate the derivative of the position vector with respect to the toroidal angle phi.

gammadash2_impl(data)

Evaluate the derivative of the position vector with respect to theta.

normal()

volume()

area()

surf_henneberg.b_min(theta, rc, zs, phi0, cosaphi, sinaphi, mpol, ntor, nfp)

This function is minimized as part of finding b.

surf_henneberg.b_max(theta, rc, zs, phi0, cosaphi, sinaphi, mpol, ntor, nfp)