Namespaces
namespace	circular

namespace	guenter
	Contains the Guenter type flux functions.

namespace	mod
	A modification flux function.

namespace	polynomial
	A polynomial approximation type flux function.

namespace	solovev
	Contains the solovev state type flux function.

namespace	taylor
	Contains the Cerfon Taylor state type flux functions.

namespace	toroidal

namespace	x

Classes
struct	aCylindricalFunctor
	Represent functions written in cylindrical coordinates that are independent of the angle phi serving as both 2d and 3d functions. More...

struct	aRealGenerator2d
	The abstract generator base class. More...

struct	aRealGeneratorX2d
	The abstract generator base class. More...

struct	BFieldP
	\( B^\varphi = R_0I/R^2\) More...

struct	BFieldR
	\( B^R = R_0\psi_Z /R\) More...

struct	BFieldT
	\( B^{\theta} = B^R\partial_R\theta + B^Z\partial_Z\theta\) More...

struct	BFieldZ
	\( B^Z = -R_0\psi_R /R\) More...

struct	BHatP
	\( \hat b^\varphi = B^\varphi/\|B\|\) More...

struct	BHatPR
	\( \partial_R \hat b^\varphi\) More...

struct	BHatPZ
	\( \partial_Z \hat b^\varphi\) More...

struct	BHatR
	\( b^R = B^R/\|B\|\) More...

struct	BHatRR
	\( \partial_R b^R\) More...

struct	BHatRZ
	\( \partial_Z b^R\) More...

struct	BHatZ
	\( b^Z = B^Z/\|B\|\) More...

struct	BHatZR
	\( \partial_R b^Z\) More...

struct	BHatZZ
	\( \partial_Z b^Z\) More...

struct	Bmodule
	\( \|B\| = R_0\sqrt{I^2+(\nabla\psi)^2}/R \) More...

struct	BR
	\( \frac{\partial \|B\| }{ \partial R} \) More...

struct	BZ
	\( \frac{\partial \|B\| }{ \partial Z} \) More...

struct	Constant
	\( f(x,y) = c\) More...

struct	CurvatureKappaR
	Approximate \( \mathcal{K}^{R}_{\vec{\kappa}}=0 \). More...

struct	CurvatureKappaZ
	Approximate \( \mathcal{K}^{Z}_{\vec{\kappa}} \). More...

struct	CurvatureNablaBR
	Approximate \( \mathcal{K}^{R}_{\nabla B} \). More...

struct	CurvatureNablaBZ
	Approximate \( \mathcal{K}^{Z}_{\nabla B} \). More...

struct	CylindricalFunctorsLvl1
	This struct bundles a function and its first derivatives. More...

struct	CylindricalFunctorsLvl2
	This struct bundles a function and its first and second derivatives. More...

struct	CylindricalSymmTensorLvl1

struct	CylindricalVectorLvl0

struct	CylindricalVectorLvl1
	This struct bundles a vector field and its divergence. More...

struct	Divb
	\( \nabla \cdot \vec b \) More...

struct	DivCurvatureKappa
	Approximate \( \vec{\nabla}\cdot \mathcal{K}_{\vec{\kappa}} \). More...

struct	DivCurvatureNablaB
	Approximate \( \vec{\nabla}\cdot \mathcal{K}_{\nabla B} \). More...

struct	DivLiseikinX
	The x-component of the divergence of the Liseikin monitor metric. More...

struct	DivLiseikinY
	The y-component of the divergence of the Liseikin monitor metric. More...

struct	DivVVP
	\( \nabla\cdot\left( \frac{ \hat b }{\hat b^\varphi}\right)\) More...

struct	DS
	Class for the evaluation of parallel derivatives. More...

struct	DSPGenerator
	A transformed field grid generator. More...

struct	Fieldaligned
	Create and manage interpolation matrices from fieldline integration. More...

struct	FluxGenerator
	A symmetry flux generator. More...

struct	FluxSurfaceAverage
	Flux surface average (differential volume average) over quantity \( \langle f\rangle(\psi_0) = \frac{1}{A} \int dR dZ \delta(\psi_p(R,Z)-\psi_0) f(R,Z)H(R,Z) \). More...

struct	FluxSurfaceIntegral
	Flux surface integral of the form \( \int dR dZ f(R,Z) \delta(\psi_p(R,Z)-\psi_0) g(R,Z) \). More...

struct	FluxVolumeIntegral
	Flux volume integral of the form \( \int dR dZ f(R,Z) \Theta(\psi_p(R,Z)-\psi_0) g(R,Z) \). More...

struct	GradLnB
	\( \nabla_\parallel \ln{(B)} \) More...

struct	Hector
	The High PrEcision Conformal grid generaTOR. More...

struct	Hoo
	Inertia factor \( \mathcal I_0 = B^2/(R_0^2\|\nabla\psi_p\|^2)\). More...

struct	InvB
	\( \|B\|^{-1} = R/R_0\sqrt{I^2+(\nabla\psi)^2} \) More...

struct	LaplacePsip
	\( \Delta\psi_p = \psi_R/R + \psi_{RR}+\psi_{ZZ} \) More...

struct	Liseikin_XX
	The xx-component of the Liseikin monitor metric. More...

struct	Liseikin_XY
	The xy-component of the Liseikin monitor metric. More...

struct	Liseikin_YY
	The yy-component of the Liseikin monitor metric. More...

struct	LnB
	\( \ln{\|B\|} \) More...

struct	LogPolarGenerator
	Log Polar coordinates (conformal) More...

struct	MagneticFieldParameters
	Meta-data about the magnetic field in particular the flux function. More...

struct	NablaPsiInv
	A weight function for the Hector algorithm. More...

struct	NablaPsiInvX
	Derivative of the weight function. More...

struct	NablaPsiInvY
	Derivative of the weight function. More...

struct	Periodify
	This function extends another function beyond the grid boundaries. More...

struct	PolarGenerator
	Polar coordinates. More...

struct	RealCurvilinearGrid2d
	A two-dimensional grid based on curvilinear coordinates. More...

struct	RealCurvilinearGridX2d
	A two-dimensional grid based on curvilinear coordinates. More...

struct	RealCurvilinearMPIGrid2d
	A two-dimensional MPI grid based on curvilinear coordinates. More...

struct	RealCurvilinearProductGrid3d
	A 2x1 curvilinear product space grid. More...

struct	RealCurvilinearProductGridX3d
	A three-dimensional grid based on curvilinear coordinates. More...

struct	RealCurvilinearProductMPIGrid3d
	A 2x1 curvilinear product space MPI grid. More...

struct	RealCurvilinearRefinedGridX2d
	A two-dimensional grid based on curvilinear coordinates. More...

struct	RealCurvilinearRefinedProductGridX3d
	A three-dimensional grid based on curvilinear coordinates. More...

struct	RealCylindricalFunctor
	Inject both 2d and 3d `operator()` to a 2d functor. More...

struct	RhoP
	\( \sqrt{1. - \psi_p/ \psi_{p,O}} \) More...

struct	Ribeiro
	A two-dimensional grid based on "almost-conformal" coordinates by Ribeiro and Scott 2010. More...

struct	RibeiroFluxGenerator
	Same as the Ribeiro class but uses \( \zeta = f_0 (\psi_p - \psi_0)\) as a flux label directly. More...

struct	RibeiroX
	A two-dimensional grid based on "almost-conformal" coordinates by Ribeiro and Scott 2010. More...

struct	SafetyFactor
	Evaluation of the safety factor q based on direct integration of \( q(\psi_0) = \frac{1}{2\pi} \int d\Theta \frac{B^\varphi}{B^\Theta} \). More...

struct	SafetyFactorAverage
	Class for the evaluation of the safety factor q based on a flux-surface integral \( q(\psi_0) = \frac{1}{2\pi} \int dRdZ \frac{I(\psi_p)}{R} \delta(\psi_p - \psi_0)H(R,Z) \). More...

struct	ScalarProduct
	Return scalar product of two vector fields \( v_0w_0 + v_1w_1 + v_2w_2\). More...

struct	SeparatrixOrthogonal
	Choose points on separatrix and construct grid from there. More...

struct	SeparatrixOrthogonalAdaptor
	An Adaptor to use SeparatrixOrthogonal as `aGenerator2d` instead of `aGeneratorX2d`. More...

struct	SimpleOrthogonal
	Generate a simple orthogonal grid. More...

struct	SquareNorm
	Return norm of scalar product of two vector fields \( \sqrt{v_0w_0 + v_1w_1 + v_2w_2}\). More...

struct	TokamakMagneticField
	A tokamak field as given by R0, Psi and Ipol plus Meta-data like shape and equilibrium. More...

struct	Topology1d
	Helper class for construction. More...

struct	TrueCurvatureKappaP
	True \( \mathcal{K}^\varphi_{\vec{\kappa}} \). More...

struct	TrueCurvatureKappaR
	True \( \mathcal{K}^R_{\vec{\kappa}} \). More...

struct	TrueCurvatureKappaZ
	True \( \mathcal{K}^Z_{\vec{\kappa}} \). More...

struct	TrueCurvatureNablaBP
	True \( \mathcal{K}^{\varphi}_{\nabla B} \). More...

struct	TrueCurvatureNablaBR
	True \( \mathcal{K}^{R}_{\nabla B} \). More...

struct	TrueCurvatureNablaBZ
	True \( \mathcal{K}^{Z}_{\nabla B} \). More...

struct	TrueDivCurvatureKappa
	True \( \vec{\nabla}\cdot \mathcal{K}_{\vec{\kappa}} \). More...

struct	TrueDivCurvatureNablaB
	True \( \vec{\nabla}\cdot \mathcal{K}_{\nabla B} \). More...

struct	WallDirection
	Determine if poloidal field points towards or away from the nearest wall. More...

struct	WallFieldlineCoordinate
	Normalized coordinate relative to wall along fieldline in phi or s coordinate. More...

struct	WallFieldlineDistance
	Distance to wall along fieldline in phi or s coordinate More...

struct	ZCutter
	\( f(R,Z)= \begin{cases} 0 \text{ if } Z < Z_X \\ 1 \text{ else } \end{cases} \) More...

Typedefs
using	CurvilinearGrid2d = dg::geo::RealCurvilinearGrid2d<double>

using	CurvilinearProductGrid3d = dg::geo::RealCurvilinearProductGrid3d<double>

using	CurvilinearGridX2d = dg::geo::RealCurvilinearGridX2d<double>

using	CurvilinearProductGridX3d = dg::geo::RealCurvilinearProductGridX3d<double>

typedef ONE	FullLimiter
	Full Limiter means there is a limiter everywhere.

typedef ZERO	NoLimiter
	No Limiter.

using	CylindricalFunctor = RealCylindricalFunctor<double>
	Most of the times we use `double`.

using	aGenerator2d = dg::geo::aRealGenerator2d<double>

using	aGeneratorX2d = dg::geo::aRealGeneratorX2d<double>

using	CurvilinearMPIGrid2d = dg::geo::RealCurvilinearMPIGrid2d<double>

using	CurvilinearProductMPIGrid3d = dg::geo::RealCurvilinearProductMPIGrid3d<double>

using	CurvilinearRefinedGridX2d = dg::geo::RealCurvilinearRefinedGridX2d<double>

using	CurvilinearRefinedProductGridX3d = dg::geo::RealCurvilinearRefinedProductGridX3d<double>

Enumerations
enum	whichMatrix { einsPlus = 0 , einsPlusT , einsMinus , einsMinusT , zeroPlus , zeroMinus , zeroPlusT , zeroMinusT , zeroForw }
	Enum for the use in Fieldaligned. More...

enum class	equilibrium { equilibrium::solovev , equilibrium::taylor , equilibrium::polynomial , equilibrium::guenter , equilibrium::toroidal , equilibrium::circular }
	How flux-function is computed. Decides how to construct magnetic field. More...

enum class	modifier { modifier::none , modifier::heaviside , modifier::sol_pfr , modifier::sol_pfr_2X }
	How flux-function is modified. More...

enum class	description { description::standardO , description::standardX , description::doubleX , description::none , description::square , description::centeredX }
	How flux function looks like. Decider on whether and what flux aligned grid to construct. More...

Functions
CylindricalFunctorsLvl1	make_NablaPsiInvCollective (const CylindricalFunctorsLvl2 &psi)
	A container class that contains all NablaPsiInv functors.

CylindricalSymmTensorLvl1	make_LiseikinCollective (const CylindricalFunctorsLvl2 &psi, double k, double eps)

template<class FieldAligned , class container >
void	assign_bc_along_field_2nd (const FieldAligned &fa, const container &fm, const container &f, const container &fp, container &fmg, container &fpg, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Assign boundary conditions along magnetic field lines interpolating a 2nd order polynomial.

template<class FieldAligned , class container >
void	assign_bc_along_field_1st (const FieldAligned &fa, const container &fm, const container &fp, container &fmg, container &fpg, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Assign boundary conditions along magnetic field lines interpolating a 1st order polynomial (a line)

template<class FieldAligned , class container >
void	swap_bc_perp (const FieldAligned &fa, const container &fm, const container &fp, container &fmg, container &fpg)
	Swap the perp boundary condition.

template<class FieldAligned , class container >
void	ds_forward (const FieldAligned &fa, double alpha, const container &f, const container &fp, double beta, container &g)
	forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

template<class FieldAligned , class container >
void	ds_forward2 (const FieldAligned &fa, double alpha, const container &f, const container &fp, const container &fpp, double beta, container &g)
	2nd order forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

template<class FieldAligned , class container >
void	ds_backward (const FieldAligned &fa, double alpha, const container &fm, const container &f, double beta, container &g)
	backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

template<class FieldAligned , class container >
void	ds_backward2 (const FieldAligned &fa, double alpha, const container &fmm, const container &fm, const container &f, double beta, container &g)
	2nd order backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

template<class FieldAligned , class container >
void	ds_centered (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	centered derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

template<class FieldAligned , class container >
void	dss_centered (const FieldAligned &fa, double alpha, const container &fm, const container &f, const container &fp, double beta, container &g)
	Centered derivative \( g = \alpha (\vec v\cdot \nabla)^2 f + \beta g \).

template<class FieldAligned , class container >
void	dssd_centered (const FieldAligned &fa, double alpha, const container &fm, const container &f, const container &fp, double beta, container &g)
	Centered derivative \( g = \alpha \nabla\cdot(\vec v \vec v\cdot \nabla) f + \beta g \).

template<class FieldAligned , class container >
void	ds_divBackward (const FieldAligned &fa, double alpha, const container &fm, const container &f, double beta, container &g)
	backward derivative \( g = \alpha \nabla \cdot \vec v f + \beta g\)

template<class FieldAligned , class container >
void	ds_divForward (const FieldAligned &fa, double alpha, const container &f, const container &fp, double beta, container &g)
	forward derivative \( g = \alpha \nabla \cdot \vec v f + \beta g\)

template<class FieldAligned , class container >
void	ds_divCentered (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	centered derivative \( g = \alpha \nabla \cdot \vec v f + \beta g\)

template<class FieldAligned , class container >
void	ds_average (const FieldAligned &, double alpha, const container &fm, const container &fp, double beta, container &g)
	Compute average along a fieldline \( g = \alpha \frac{f_{k+1} + f_{k-1}}{2} + \beta g\).

template<class FieldAligned , class container >
void	ds_slope (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	Compute simple slope along a fieldline \( g = \alpha v^\varphi\frac{f_{k+1} - f_{k-1}}{2\Delta\varphi} + \beta g\).

template<class BinaryOp , class UnaryOp >
thrust::host_vector< double >	fieldaligned_evaluate (const aProductGeometry3d &grid, const CylindricalVectorLvl0 &vec, const BinaryOp &binary, const UnaryOp &unary, unsigned p0, unsigned rounds, double eps=1e-5)
	Evaluate a 2d functor and transform to all planes along the fieldlines

int	findCriticalPoint (const CylindricalFunctorsLvl2 &psi, double &RC, double &ZC)
	This function finds critical points of psi (any point with vanishing gradient, including the X-point or O-point) via Newton iteration applied to the gradient of psi.

int	findOpoint (const CylindricalFunctorsLvl2 &psi, double &RC, double &ZC)
	This function finds O-points of psi.

void	findXpoint (const CylindricalFunctorsLvl2 &psi, double &RC, double &ZC)
	This function finds X-points of psi.

template<class Geometry3d >
dg::SparseTensor< typename Geometry3d::host_vector >	createAlignmentTensor (const dg::geo::CylindricalVectorLvl0 &bhat, const Geometry3d &g)
	\( \chi^{ij} = b^ib^j\)

template<class Geometry3d >
dg::SparseTensor< typename Geometry3d::host_vector >	createProjectionTensor (const dg::geo::CylindricalVectorLvl0 &bhat, const Geometry3d &g)
	\( h^{ij} = g^{ij} - b^ib^j\)

dg::geo::TokamakMagneticField	createGuenterField (double R_0, double I_0)
	Create a Guenter Magnetic field.

TokamakMagneticField	periodify (const TokamakMagneticField &mag, double R0, double R1, double Z0, double Z1, dg::bc bcx, dg::bc bcy)
	Use dg::geo::Periodify to periodify every function in the magnetic field.

CylindricalVectorLvl1	createEPhi (int sign)
	Contravariant components of the unit vector field (0, 0, \(\pm 1/R \)) and its Divergence and derivative (0,0) in cylindrical coordinates.

CylindricalVectorLvl0	createCurvatureNablaB (const TokamakMagneticField &mag, int sign)
	Approximate curvature vector field (CurvatureNablaBR, CurvatureNablaBZ, Constant(0))

CylindricalVectorLvl0	createCurvatureKappa (const TokamakMagneticField &mag, int sign)
	Approximate curvature vector field (CurvatureKappaR, CurvatureKappaZ, Constant(0))

CylindricalVectorLvl0	createTrueCurvatureKappa (const TokamakMagneticField &mag)
	True curvature vector field (TrueCurvatureKappaR, TrueCurvatureKappaZ, TrueCurvatureKappaP)

CylindricalVectorLvl0	createTrueCurvatureNablaB (const TokamakMagneticField &mag)
	True curvature vector field (TrueCurvatureNablaBR, TrueCurvatureNablaBZ, TrueCurvatureNablaBP)

CylindricalVectorLvl0	createGradPsip (const TokamakMagneticField &mag)
	Gradient Psip vector field (PsipR, PsipZ, 0)

CylindricalVectorLvl1	createBHat (const TokamakMagneticField &mag)
	Contravariant components of the magnetic unit vector field and its Divergence and derivative in cylindrical coordinates.

TokamakMagneticField	createMagneticField (dg::file::WrappedJsonValue gs)
	Create a Magnetic field based on the given parameters.

TokamakMagneticField	createModifiedField (dg::file::WrappedJsonValue gs, dg::file::WrappedJsonValue jsmod, CylindricalFunctor &wall, CylindricalFunctor &transition)
	Modify Magnetic Field and create wall above or below certain Psi values according to given parameters.

CylindricalFunctor	createWallRegion (dg::geo::TokamakMagneticField mag, dg::file::WrappedJsonValue jsmod)
	A convenience function call for `dg::geo::createModifiedField` that ignores the transition parameter and returns the wall functor.

CylindricalFunctor	createWallRegion (dg::file::WrappedJsonValue gs, dg::file::WrappedJsonValue jsmod)
	DEPRECATED Same as `createWallRegion( createMagneticField(gs), jsmod);`

void	createSheathRegion (dg::file::WrappedJsonValue jsmod, TokamakMagneticField mag, CylindricalFunctor wall, dg::Grid2d &sheath_walls, CylindricalFunctor &sheath)
	Create the sheath region where fieldlines intersect the boundary.

template<class BinaryOp , class UnaryOp >
MPI_Vector< thrust::host_vector< double > >	fieldaligned_evaluate (const aProductMPIGeometry3d &grid, const CylindricalVectorLvl0 &vec, const BinaryOp &binary, const UnaryOp &unary, unsigned p0, unsigned rounds, double eps=1e-5)
	Evaluate a 2d functor and transform to all planes along the fieldlines (MPI Version)

dg::geo::TokamakMagneticField	createPolynomialField (dg::geo::polynomial::Parameters gp)
	Create a Polynomial Magnetic field.

dg::geo::TokamakMagneticField	createSolovevField (dg::geo::solovev::Parameters gp)
	Create a Solovev Magnetic field.

dg::geo::TokamakMagneticField	createModifiedSolovevField (dg::geo::solovev::Parameters gp, double psi0, double alpha, double sign=-1)
	DEPRECATED Create a modified Solovev Magnetic field.

dg::geo::TokamakMagneticField	createTaylorField (dg::geo::solovev::Parameters gp)
	Create a Taylor Magnetic field.

dg::geo::TokamakMagneticField	createToroidalField (double R0)
	Create a Toroidal Magnetic field.

dg::geo::TokamakMagneticField	createCircularField (double R0, double I0, double a=1, double b=1)
	\( \psi_p = 1- \left(\frac{R-R_0}{a}\right)^2 - \left( \frac{Z}{b}\right)^2,\quad I(\psi_p) = I_0 \)

CylindricalSymmTensorLvl1	make_Xbump_monitor (const CylindricalFunctorsLvl2 &psi, double &R_X, double &Z_X, double radiusX, double radiusY)
	construct a monitor metric in which the Laplacian vanishes at the X-point

CylindricalSymmTensorLvl1	make_Xconst_monitor (const CylindricalFunctorsLvl2 &psi, double &R_X, double &Z_X)
	construct a monitor metric in which the Laplacian vanishes at the X-point

Typedef Documentation

◆ aGenerator2d

using dg::geo::aGenerator2d = dg::geo::aRealGenerator2d<double>

◆ aGeneratorX2d

using dg::geo::aGeneratorX2d = dg::geo::aRealGeneratorX2d<double>

Function Documentation

◆ make_Xbump_monitor()

CylindricalSymmTensorLvl1 dg::geo::make_Xbump_monitor	(	const CylindricalFunctorsLvl2 &	psi,
		double &	R_X,
		double &	Z_X,
		double	radiusX,
		double	radiusY )

inline

construct a monitor metric in which the Laplacian vanishes at the X-point

calls the findXpoint function to find the X-point

Parameters

psi	the flux functions
R_X	start value on input, X-point on output
Z_X	start value on input, X-point on output
radiusX	size of bump in x direction
radiusY	size of bump in y-direction

Returns: a metric tensor and its derivatives

◆ make_Xconst_monitor()

CylindricalSymmTensorLvl1 dg::geo::make_Xconst_monitor	(	const CylindricalFunctorsLvl2 &	psi,
		double &	R_X,
		double &	Z_X )

inline

construct a monitor metric in which the Laplacian vanishes at the X-point

calls the findXpoint function to find the X-point

Parameters

psi	the flux functions
R_X	start value on input, X-point on output
Z_X	start value on input, X-point on output

Returns: a metric tensor and its derivatives

Namespaces

Classes

Typedefs