5. Miscellaneous additions

\( q(\psi_p)\) and \( \nabla \psi_p ( R_X, Z_X) = 0\) More...

Classes
struct	dg::geo::FluxSurfaceIntegral< container >
	Flux surface integral of the form \( \int dR dZ f(R,Z) \delta(\psi_p(R,Z)-\psi_0) g(R,Z) \). More...
struct	dg::geo::FluxVolumeIntegral< container >
	Flux volume integral of the form \( \int dR dZ f(R,Z) \Theta(\psi_p(R,Z)-\psi_0) g(R,Z) \). More...
struct	dg::geo::FluxSurfaceAverage< container >
	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	dg::geo::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	dg::geo::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...

Functions
static int	dg::geo::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. More...
static int	dg::geo::findOpoint (const CylindricalFunctorsLvl2 &psi, double &RC, double &ZC)
	This function finds O-points of psi. More...
static void	dg::geo::findXpoint (const CylindricalFunctorsLvl2 &psi, double &RC, double &ZC)
	This function finds X-points of psi. More...

Detailed Description

\( q(\psi_p)\) and \( \nabla \psi_p ( R_X, Z_X) = 0\)

Function Documentation

findCriticalPoint()

static int dg::geo::findCriticalPoint ( const CylindricalFunctorsLvl2 &  psi, double &  RC, double &  ZC  )

inlinestatic

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.

Newton iteration applied to \( \nabla \psi (\vec x) = 0 \) reads

\[ \vec x_{i+1} = \vec x_i - H^{-1} \nabla \psi (\vec x_i)\]

where H is the Hessian matrix. The inverse of the Hessian matrix is computed analytically

Parameters

psi	\( \psi(R,Z)\)
RC	start value on input, critical point on output
ZC	start value on input, critical point on output

Returns

0 if no critical point or Hessian (determinant) is zero, 1 if local minimum, 2 if local maximum, 3 if saddle point

findOpoint()

static int dg::geo::findOpoint ( const CylindricalFunctorsLvl2 &  psi, double &  RC, double &  ZC  )

inlinestatic

This function finds O-points of psi.

Same as findCriticalPoint except that this function throws if it does not find a local minimum or a local maximum

Parameters

psi	\( \psi(R,Z)\)
RC	start value on input, O-point on output
ZC	start value on input, O-point on output

Returns

1 if local minimum, 2 if local maximum,

findXpoint()

static void dg::geo::findXpoint ( const CylindricalFunctorsLvl2 &  psi, double &  RC, double &  ZC  )

inlinestatic

This function finds X-points of psi.

Same as findCriticalPoint except that this function throws if it does not find a saddle point

Parameters

psi	\( \psi(R,Z)\)
RC	start value on input, X-point on output
ZC	start value on input, X-point on output