pyoculus.solvers.qfm.QFM Class Reference

Inheritance diagram for pyoculus.solvers.qfm.QFM:

Public Member Functions
	__init__ (self, ToroidalBfield problem, params=dict(), integrator=None, integrator_params=dict())
	Set up the class of the fixed point finder.
	construct_qfms (self, plist, qlist, sguesslist=None, bounding_surfaces=[0.0, 1.0], rho_label="s", niter=5, verbose=True)
	Construct the QFM surfaces for given list of p and q This subroutine will iteratively construct the QFM surfaces for the given list of p and q.
	compute_tflux ()
	straighten_boundary (self, surfaces, tol=1e-9, niter=10)
	Convert a boundary surface to have straight field line For a boundary surface with rho=constant, find the function \(\lambda(\vartheta, \zeta)\) with transformation.
	action (self, int pp, int qq, sguess=0.5, root_method="hybr", tol=1e-8)
	Construct a QFM surface based on Stuart's method, for a given orbit periodicity pp and qq.
	action_gradient (self, xx, pp, qq, a, qN, Nfft)
	Computes the action gradient, being used in root finding.
	action_gradient_jacobi (self, xx, pp, qq, a, qN, Nfft)
	Computes the jacobi matrix of action gradient, being used in root finding.
Public Member Functions inherited from pyoculus.solvers.base_solver.BaseSolver
	__init__ (self, problem, params=dict(), integrator=None, integrator_params=dict())
	Sets up the solver.
	is_successful (self)
	Returns True if the computation is successfully completed.

Public Attributes
	nfft_multiplier = params["nfft_multiplier"]
	Nfp = problem.Nfp
	sym = params["stellar_sym"]
Public Attributes inherited from pyoculus.solvers.base_solver.BaseSolver
bool	successful = False
	flagging if the computation is done and successful

Protected Member Functions
	_straighten_boundary (self, rho=1, tol=1e-9, niter=10)
	Convert a boundary surface to have straight field line, internal function For a boundary surface with rho=constant, find the function \(\lambda(\vartheta, \zeta)\) with transformation.
	_unpack_dof (self, xx, qN)
	Unpack the degrees of freedom into Fourier harmonics.
	_pack_dof (self, nv, rcn, tsn, rsn, tcn)
	Unpack the degrees of freedom into Fourier harmonics.

Protected Attributes
int	_MM = params["nfft_multiplier"] * 2
	_pqNtor = params["pqNtor"]
	_pqMpol = params["pqMpol"]
	_dz = dz
	_nlist = np.arange(0, qN + 1)
	_zeta = np.arange(0, Nfft) * dz
	_nzq = self._nlist[:, nax] * self._zeta[nax, :] / qq
	_cnzq = np.cos(self._nzq)
	_snzq = np.sin(self._nzq)
Protected Attributes inherited from pyoculus.solvers.base_solver.BaseSolver
	_integrator_type = RKIntegrator
	_params = dict(params)
	_integrator = self._integrator_type(integrator_params)
	_problem = problem
	_integrator_params = dict(integrator_params)

Constructor & Destructor Documentation

__init__()

pyoculus.solvers.qfm.QFM.__init__	(		self,
		ToroidalBfield	problem,
			params = dict(),
			integrator = None,
			integrator_params = dict() )

Set up the class of the fixed point finder.

Parameters

problem	must inherit pyoculus.problems.ToroidalBfield, the problem to solve
params	dict, the parameters for the solver
integrator	the integrator to use, must inherit \pyoculus.integrators.BaseIntegrator, if set to None by default using RKIntegrator (not used here)
integrator_params	dict, the parmaters passed to the integrator (not used here)

params['pqMpol']=8 – Fourier resolution multiplier for poloidal direction params['pqNtor']=4 – Fourier resolution multiplier for toroidal direction params['nfft_multiplier']=4 – the extended (multiplier) resolution for FFT params['stellar_sym']=True – stellarator symmetry

Member Function Documentation

_pack_dof()

pyoculus.solvers.qfm.QFM._pack_dof ( self, nv, rcn, tsn, rsn, tcn )

protected

Unpack the degrees of freedom into Fourier harmonics.

Parameters

nv
rcn
tsn
rsn
tcn

_straighten_boundary()

pyoculus.solvers.qfm.QFM._straighten_boundary ( self, rho = 1, tol = 1e-9, niter = 10 )

protected

Convert a boundary surface to have straight field line, internal function For a boundary surface with rho=constant, find the function \(\lambda(\vartheta, \zeta)\) with transformation.

Parameters

rho	the boundary surface coordinate
tol	the tolerance to stop iteration
niter	the number of iterations

Returns

tsn, tcn the sine and cosine coefficient of the map \(\lambda\) in \(\theta = \vartheta + \lambda(\vartheta, \zeta)\)

_unpack_dof()

pyoculus.solvers.qfm.QFM._unpack_dof ( self, xx, qN )

protected

Unpack the degrees of freedom into Fourier harmonics.

Parameters

xx	the packed degrees of freedom
qN	Fourier resolution

Returns

nv, rcn, tsn, rsn, tcn

action()

pyoculus.solvers.qfm.QFM.action	(		self,
		int	pp,
		int	qq,
			sguess = 0.5,
			root_method = "hybr",
			tol = 1e-8 )

Construct a QFM surface based on Stuart's method, for a given orbit periodicity pp and qq.

The surface in the old coordinate \((s, \theta, \zeta)\) will be expressed in Fourier harmonics of the reverse mapping

\[ s(\vartheta, \zeta) = \sum_{m,n} s_{c, m, n} \cos(m \vartheta - n \zeta) + s_{s, m, n} \sin(m \vartheta - n \zeta), \]

\[ \theta(\vartheta, \zeta) = \vartheta + \sum_{m,n} t_{c, m, n} \cos(m \vartheta - n \zeta) + t_{s, m, n} \sin(m \vartheta - n \zeta), \]

Parameters

pp	the orbit peroidicity. The rotational transform of such a surface will be pp/qq.
qq	the orbit peroidicity. The rotational transform of such a surface will be pp/qq.
sguess	a guess of the s coordinate for the surfaces.
root_method	root finding method being used in scipy.optimize.root
tol	the tolarence of root finding

Returns

scn_surf, tsn_surf, ssn_surf, tcn_surf - the Fourier harmonics of the surface transformation.

action_gradient()

pyoculus.solvers.qfm.QFM.action_gradient	(		self,
			xx,
			pp,
			qq,
			a,
			qN,
			Nfft )

Computes the action gradient, being used in root finding.

Parameters

xx	the packed degrees of freedom. It should contain rcn, tsn, rsn, tcn, nv.
pp	the poloidal periodicity of the island, should be an integer
qq	the toroidal periodicity of the island, should be an integer
a	the target area

Returns

ff the equtions to find zeros, see below. Construct the Fourier transform of \(B^\vartheta_i / B^\zeta_i\) and \(B^\rho_i / B^\zeta_i + \bar \nu / (J B^\zeta_i)\),

\[B^\t / B^\z & = & f^c_0 + \sum_{n=1}^{qN} \left[ f^c_n \cos(n\z/q) + f^s_n \sin(n\z/q) \right], \label{eqn:f} \]

\[B^\rho / B^\z + \bar \nu / \sqrt g B^\z & = & g^c_0 + \sum_{n=1}^{qN} \left[ g^c_n \cos(n\z/q) + g^s_n \sin(n\z/q) \right], \label{eqn:g} \]

\item The Fourier harmonics of $

action_gradient_jacobi()

pyoculus.solvers.qfm.QFM.action_gradient_jacobi	(		self,
			xx,
			pp,
			qq,
			a,
			qN,
			Nfft )

Computes the jacobi matrix of action gradient, being used in root finding.

Parameters

xx	the packed degrees of freedom. It should contain rcn, tsn, rsn, tcn, nv.
pp	the poloidal periodicity of the island, should be an integer
qq	the toroidal periodicity of the island, should be an integer
a	the target area

Returns

dff the jacobian

compute_tflux()

pyoculus.solvers.qfm.QFM.compute_tflux

(

)

construct_qfms()

pyoculus.solvers.qfm.QFM.construct_qfms	(		self,
			plist,
			qlist,
			sguesslist = None,
			bounding_surfaces = [0.0, 1.0],
			rho_label = "s",
			niter = 5,
			verbose = True )

Construct the QFM surfaces for given list of p and q This subroutine will iteratively construct the QFM surfaces for the given list of p and q.

If the system has boundaries with constants s that are flux surfaces, two bounding surfaces will be added. Otherwise, the outmost two QFM surfaces will be the new bounding surfaces.

Parameters

plist	the list of p
qlist	the list of q
sguesslist	guess of the approximate s coordinate of each surface
bounding_surfaces	instruct to add the bounding surfaces for these s coordinates, maximum two
rho_label	what to use for the new radius label \(\rho\). Can be 's', 'tflux', 'sqrttflux'.
niter	the number of iterations in constructing QFM based of straight field line coordinates
verbose	if True, print the progress

Returns

an instance of pyoculus.problems.SurfacesToroidal containing the intepolation coordinates.

straighten_boundary()

pyoculus.solvers.qfm.QFM.straighten_boundary	(		self,
			surfaces,
			tol = 1e-9,
			niter = 10 )

Convert a boundary surface to have straight field line For a boundary surface with rho=constant, find the function \(\lambda(\vartheta, \zeta)\) with transformation.

\[ \theta = \vartheta + \lambda(\vartheta, \zeta), \]

such that \(B^{\vartheta} / B^{\zeta} = 1 / q\) is a constant on the surface.

The transformation gives

\[ B^{\vartheta} = \frac{B^\theta - \lambda_\zeta B^\zeta}{1 + \lambda_\vartheta}, \]

and no change to \(B^\zeta\). Therefore, we get

\[ \frac{B^\theta}{B^\zeta}(\theta = \vartheta + \lambda(\vartheta, \zeta), \zeta) = \frac{1}{q} (1 + \lambda_\vartheta) + \lambda_\zeta \]

A fixed-point iteration method will be used. For a initial guess of \(\lambda \), we use it to compute the left-hand-side. An updated \(\lambda \) is then computed by solving the right-hand-side assuming the left-hand-side is given. This new \(\lambda \) is again substituted into the left hand side to generate a new right-hand-side. The process is repeated until converged (the difference between two iterations is lower than a threshold, or a given number of iteration is reached).

Parameters

surfaces	class SurfacesToroidal, the surfaces for which the first and last surface are bounding surfaces
tol	the tolerance to stop iteration
niter	the number of iterations

Member Data Documentation

_cnzq

pyoculus.solvers.qfm.QFM._cnzq = np.cos(self._nzq)

protected

_dz

pyoculus.solvers.qfm.QFM._dz = dz

protected

_MM

int pyoculus.solvers.qfm.QFM._MM = params["nfft_multiplier"] * 2

protected

_nlist

pyoculus.solvers.qfm.QFM._nlist = np.arange(0, qN + 1)

protected

_nzq

pyoculus.solvers.qfm.QFM._nzq = self._nlist[:, nax] * self._zeta[nax, :] / qq

protected

_pqMpol

pyoculus.solvers.qfm.QFM._pqMpol = params["pqMpol"]

protected

_pqNtor

pyoculus.solvers.qfm.QFM._pqNtor = params["pqNtor"]

protected

_snzq

pyoculus.solvers.qfm.QFM._snzq = np.sin(self._nzq)

protected

_zeta

pyoculus.solvers.qfm.QFM._zeta = np.arange(0, Nfft) * dz

protected

nfft_multiplier

pyoculus.solvers.qfm.QFM.nfft_multiplier = params["nfft_multiplier"]

Nfp

pyoculus.solvers.qfm.QFM.Nfp = problem.Nfp

sym

pyoculus.solvers.qfm.QFM.sym = params["stellar_sym"]

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The documentation for this class was generated from the following file:

• pyoculus/solvers/qfm.py