Class for constructing the linear/nonlinear problem for the solver. More...

#include <epot_solver.hpp>

Inheritance diagram for EpotSolver:

Public Member Functions
	EpotSolver (Geometry &geom)
	Constructor for solver from geom. More...

	EpotSolver (const EpotSolver &epsolver, Geometry &geom)
	Constructor for solver from geom. Parameters from epsolver are copied to new solver. More...

	EpotSolver (Geometry &geom, std::istream &s)
	Construct from file. More...

virtual	~EpotSolver ()
	Destructor. More...

void	set_parameters (const EpotSolver &epsolver)
	Copy parameters from solver epsolver. More...

void	set_forced_potential_volume (double force_pot, CallbackFunctorB_V *force_pot_func)
	Define forced potential volume. More...

void	set_forced_potential_volume (CallbackFunctorD_V *force_pot_func)
	Define forced potential volume. More...

void	set_initial_plasma (double Up, CallbackFunctorB_V *init_plasma_func)
	Define initial plasma to the problem. More...

void	set_pexp_plasma (double rhoe, double Te, double Up)
	Enable plasma model for positive ion extraction problem. More...

void	set_nsimp_initial_plasma (CallbackFunctorB_V *init_plasma_func)
	Define initial plasma boundary location to negative ion extraction problem. More...

void	set_nsimp_plasma (double rhop, double Ep, std::vector< double > rhoi, std::vector< double > Ei)
	Enable plasma model for negative ion extraction problem. More...

void	solve (MeshScalarField &epot, const ScalarField &scharge)
	Solve the problem. More...

bool	linear (void) const
	Return true if problem is linear. More...

const Geometry &	geometry (void) const
	Get pointer to geometry. More...

virtual void	debug_print (std::ostream &os) const =0
	Print debugging information to os. More...

virtual void	save (std::ostream &s) const =0
	Saves problem data to stream. More...

Protected Member Functions
void	pexp_newton (double &rhs, double &drhs, double epot) const
	Return non-linear right-hand-side and it's derivative for vacuum node in positive ion plasma. More...

void	nsimp_newton (double &rhs, double &drhs, double epot) const
	Return non-linear right-hand-side and it's derivative for vacuum node in negative ion plasma. More...

uint8_t	boundary_index (uint32_t i) const
	Return bitmask indicating to which boundaries the node belongs to. More...

uint8_t	boundary_index (uint32_t i, uint32_t j) const
	Return bitmask indicating to which boundaries the node belongs to. More...

uint8_t	boundary_index (uint32_t i, uint32_t j, uint32_t k) const
	Return bitmask indicating to which boundaries the node belongs to. More...

uint8_t	boundary_index_general (uint32_t i, uint32_t j, uint32_t k) const
	Return bitmask indicating to which boundaries the node belongs to. More...

void	preprocess (MeshScalarField &epot)
	Do preprocessing action before solving. More...

void	postprocess (void)
	Do postprocessing action after solving. More...

virtual void	reset_problem (void)=0
	Reset solver/problem settings. More...

MeshScalarField *	evaluate_scharge (const ScalarField &__scharge) const

virtual void	subsolve (MeshScalarField &epot, const MeshScalarField &scharge)=0
	Solve problem with given mesh based space charge. More...

Protected Attributes
Geometry &	_geom
	Geometry reference. More...

plasma_mode_e	_plasma
	Plasma simulation mode. More...

double	_rhoe
	Electron charge density (C/m3), < 0. More...

double	_Te
	Electron thermal energy, > 0. More...

double	_Up
	Plasma potential, > 0. More...

std::vector< double >	_rhoi
	Charge density for positive ions, first fast protons, then thermal ions. More...

std::vector< double >	_Ei
	Energy for positive ions, first fast protons, then thermal ions. More...

double	_force_pot
	Potential to be forced. More...

CallbackFunctorB_V *	_force_pot_func
	Force region potential function. More...

CallbackFunctorD_V *	_force_pot_func2
	Force region potential function. More...

CallbackFunctorB_V *	_init_plasma_func
	Initial plasma region function. More...

double	_plA
	Plasma parameter. For positive ion extraction: rho_th * h^2 / epsilon_0, for negative ion extraction: rho_f * h^2 / epsilon_0. More...

double	_plB
	Plasma parameter. For positive ion extraction: 1/Te, for negative ion extraction: E_f,i. More...

double	_plC
	Plasma parameter for positive ion extraction. Up/Te. More...

std::vector< double >	_plD
	Plasma parameter for negative ion extraction. rho_th,i * h^2 / epsilon_0. More...

std::vector< double >	_plE
	Plasma parameter for negative ion extraction. 1/Ti. More...

Detailed Description

Class for constructing the linear/nonlinear problem for the solver.

EpotProblem class constructs the Poisson (equation) problem in finite difference form from Geometry (mesh) and various parameters and it presents the problem to the Solver via matrix/vector representation. In case of linear problem (no plasma model), the Poisson equation

$\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$

is

$\frac{\partial^2 \phi}{\partial x^2} = -\frac{\rho}{\epsilon_0}$

in 1D, which is discretized into

$\phi_{i-1} + \phi_{i+1} - 2\phi_{i} = -h^2 \frac{\rho_{i}}{\epsilon_0}$

using finite differences. In 2D coordinates the discretized form is

$\phi_{i-1,j} + \phi_{i+1,j} + \phi_{i,j-1} + \phi_{i,j+1} - 4\phi_{i,j} = -h^2 \frac{\rho_{i,j}}{\epsilon_0}$

and in 3D it is

$\phi_{i-1,j,k} + \phi_{i+1,j,k} + \phi_{i,j-1,k} + \phi_{i,j+1,k} + \phi_{i,j,k-1} + \phi_{i,j,k+1} - 6\phi_{i,j} = -h^2 \frac{\rho_{i,j,k}}{\epsilon_0}.$

In cylindrical coordinates the Poisson equation is

$\frac{\partial^2 \phi}{\partial r^2} + \frac{1}{r} \frac{\partial \phi}{\partial r} + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = -\frac{\rho}{\epsilon_0},$

where $\frac{\partial^2 \phi}{\partial \theta^2} = 0$ because of cylindrical symmetry of the simulations. Therefore the discretized form becomes

$\phi_{i-1,j} + \phi_{i+1,j} + \left( 1 - \frac{h}{2r_j} \right) \phi_{i,j-1} + \left( 1 + \frac{h}{2r_j} \right) \phi_{i,j+1} - 4\phi_{i,j} = -h^2 \frac{\rho_{i,j}}{\epsilon_0},$

where $\frac{h}{2r_j} = \frac{h}{2jh} = \frac{1}{2j}$ because the radius $ r_j = hj $ . At the symmetry axis there is an exception because both $ r $ and $\frac{\partial \phi}{\partial r}$ approach zero. By using Bernoulli-L'Hopital rule we can evaluate

$\lim_{r \rightarrow 0} \frac{1}{r} \frac{\partial \phi}{\partial r} = \lim_{r \rightarrow 0} \frac{\partial^2 \phi}{\partial r^2},$

which turns the Poisson equation to

$2 \frac{\partial^2 \phi}{\partial r^2} + \frac{\partial^2 \phi}{\partial z^2} = -\frac{\rho}{\epsilon_0}$

on axis. Discretation of the equation gives us

$\phi_{i-1,j} + \phi_{i+1,j} + 2\phi_{i,j+1} + 2\phi_{i,j-1} - 6\phi_{i,j} = -h^2 \frac{\rho_{i,j}}{\epsilon_0}.$

Here $\phi_{i,j-1} = \phi_{i,j+1}$ so that the final form is

$\phi_{i-1,j} + \phi_{i+1,j} + 4\phi_{i,j+1} - 6\phi_{i,j} = -h^2 \frac{\rho_{i,j}}{\epsilon_0}.$

In addition to the Poisson equation the problem matrix and vector also contain finite difference representations of the boundary conditions. The Dirichlet boundary condition is defined by constant potential at boundary, i.e. $\phi_{i,j} = \phi_{\mathrm{const}}$ . The Neumann boundary condition can be defined as first order discretation

$\frac{\phi_{i+1}-\phi_{i}}{h} = N_{\mathrm{const}}$

or second order discretation

$\frac{-\phi_{i+2}+4\phi_{i+1}-3\phi_{i}}{2h} = N_{\mathrm{const}}$

selected by the user.

The plasma problems are described by using nonlinear models for screening charges in the plasma. For positive ion extraction for example, the screening charge is an electron population at the plasma potential $\phi_p$ with a thermal energy distribution with temperature $ T_e $ . The screening charge is therefore

$\rho_e = \rho_{e0} \exp \left( \frac{\phi-\phi_p}{kT_e/e} \right),$

where electron charge density at plasma potential $(\rho_{e0})$ is the same as the total positive beam space charge density for enabling plasma neutrality.

Constructor & Destructor Documentation

EpotSolver::EpotSolver ( Geometry & geom )

Constructor for solver from geom.

EpotSolver::EpotSolver	(	const EpotSolver &	epsolver,
		Geometry &	geom
	)

Constructor for solver from geom. Parameters from epsolver are copied to new solver.

EpotSolver::EpotSolver	(	Geometry &	geom,
		std::istream &	s
	)

Construct from file.

virtual EpotSolver::~EpotSolver ( )

inlinevirtual

Destructor.

Member Function Documentation

uint8_t EpotSolver::boundary_index ( uint32_t i ) const

protected

Return bitmask indicating to which boundaries the node belongs to.

1D version. The node i is checked. The return value has bits set accoding to, which boundaries the node belongs to. The lowest bit is xmin, second of xmax, third is ymin, fourth is ymax, fifth is zmin, sixth is zmax. If the node isn't a boundary node, a 0 is returned.

uint8_t EpotSolver::boundary_index	(	uint32_t	i,
		uint32_t	j
	)		const

protected

Return bitmask indicating to which boundaries the node belongs to.

2D version. The node (i,j) is checked. The return value has bits set accoding to, which boundaries the node belongs to. The lowest bit is xmin, second of xmax, third is ymin, fourth is ymax, fifth is zmin, sixth is zmax. If the node isn't a boundary node, a 0 is returned.

uint8_t EpotSolver::boundary_index	(	uint32_t	i,
		uint32_t	j,
		uint32_t	k
	)		const

protected

Return bitmask indicating to which boundaries the node belongs to.

3D version. The node (i,j,k) is checked. The return value has bits set accoding to, which boundaries the node belongs to. The lowest bit is xmin, second of xmax, third is ymin, fourth is ymax, fifth is zmin, sixth is zmax. If the node isn't a boundary node, a 0 is returned.

uint8_t EpotSolver::boundary_index_general	(	uint32_t	i,
		uint32_t	j,
		uint32_t	k
	)		const

protected

Return bitmask indicating to which boundaries the node belongs to.

Version applicable to all dimensions. The node (i,j,k) is checked. The return value has bits set accoding to, which boundaries the node belongs to. The lowest bit is xmin, second of xmax, third is ymin, fourth is ymax, fifth is zmin, sixth is zmax. If the node isn't a boundary node, a 0 is returned.

void EpotSolver::debug_print ( std::ostream & os ) const

pure virtual

Print debugging information to os.

Implemented in EpotMGSolver, EpotBiCGSTABSolver, EpotMatrixSolver, EpotGSSolver, EpotMGSubSolver, and EpotUMFPACKSolver.

const Geometry & EpotSolver::geometry ( void ) const

Get pointer to geometry.

bool EpotSolver::linear ( void ) const

Return true if problem is linear.

void EpotSolver::nsimp_newton	(	double &	rhs,
		double &	drhs,
		double	epot
	)		const

protected

Return non-linear right-hand-side and it's derivative for vacuum node in negative ion plasma.

Returns the non-linear part of the right-hand-side and it's derivative resulting from the compensating space charge in negative ion extraction case.

void EpotSolver::pexp_newton	(	double &	rhs,
		double &	drhs,
		double	epot
	)		const

protected

Return non-linear right-hand-side and it's derivative for vacuum node in positive ion plasma.

Returns the non-linear part of the right-hand-side and it's derivative resulting from the compensating space charge in positive ion extraction case.

void EpotSolver::postprocess ( void )

protected

Do postprocessing action after solving.

Return near solid neumann points as near solid. Restore distance data. Remove fixed node tags.

void EpotSolver::preprocess ( MeshScalarField & epot )

protected

Do preprocessing action before solving.

Does precalculation of plasma paramters. Sets near solid neumann points as neumann and saves distance to a vector. Mark fixed and initial plasma nodes and set potentials.

virtual void EpotSolver::reset_problem ( void )

protectedpure virtual

Reset solver/problem settings.

virtual void EpotSolver::save ( std::ostream & s ) const

pure virtual

Saves problem data to stream.

Implemented in EpotMGSolver, EpotBiCGSTABSolver, EpotMatrixSolver, EpotGSSolver, EpotMGSubSolver, and EpotUMFPACKSolver.

void EpotSolver::set_forced_potential_volume	(	double	force_pot,
		CallbackFunctorB_V *	force_pot_func
	)

Define forced potential volume.

Vacuum volume inside the volume defined by function force_pot_func will be forced to potential force_pot_func. This function is designed to be used with negative ion plasma extraction to stabilize plasma close non-physical boundaries.

This function may be removed in future.

void EpotSolver::set_forced_potential_volume ( CallbackFunctorD_V * force_pot_func )

Define forced potential volume.

The force_pot_func functor will be called for every vacuum grid node at preprocessing stage of solve. If it returns a finite value, the node will be forced to the potential defined by the returned value. Otherwise the node will be treated as a regular vacuum node.

void EpotSolver::set_initial_plasma	(	double	Up,
		CallbackFunctorB_V *	init_plasma_func
	)

Define initial plasma to the problem.

Initial plasma volume is defined in the area given by callback functor init_plasma_func.

void EpotSolver::set_nsimp_initial_plasma ( CallbackFunctorB_V * init_plasma_func )

Define initial plasma boundary location to negative ion extraction problem.

Initial plasma volume is defined in the area given by plasma_func.

void EpotSolver::set_nsimp_plasma	(	double	rhop,
		double	Ep,
		std::vector< double >	rhoi,
		std::vector< double >	Ei
	)

Enable plasma model for negative ion extraction problem.

The positive (analytic) space charges for the negative ion plasma extraction are set using this function. The positive ions consist of fast (directed) protons and any number of thermal positive ions trapped at the plasma boundary in the zero potential.

The parameters set are rhop, the space charge density of protons and Ep, the energy of protons at zero potential. Vectors rhoi and Ei are used to set the space charge densities and thermal energies of the trapped ions.

The boundary condition for the plasma should be BOUND_DIRICHLET with zero volts.

void EpotSolver::set_parameters ( const EpotSolver & epsolver )

Copy parameters from solver epsolver.

void EpotSolver::set_pexp_plasma	(	double	rhoe,
		double	Te,
		double	Up
	)

Enable plasma model for positive ion extraction problem.

Enable plasma model with background electron charge density of rhoe and electron temperature Te. The plasma potential is set to Up.

The boundary condition for the plasma should be BOUND_NEUMANN with gradient of zero V/m.

void EpotSolver::solve	(	MeshScalarField &	epot,
		const ScalarField &	scharge
	)

Solve the problem.

The epot field is used as an initial guess for the solver. The space charge density field scharge is added to the problem vector before solving. The solution is returned in epot.

virtual void EpotSolver::subsolve	(	MeshScalarField &	epot,
		const MeshScalarField &	scharge
	)

protectedpure virtual

Solve problem with given mesh based space charge.

Member Data Documentation

std::vector<double> EpotSolver::_Ei

protected

Energy for positive ions, first fast protons, then thermal ions.

double EpotSolver::_force_pot

protected

Potential to be forced.

CallbackFunctorB_V* EpotSolver::_force_pot_func

protected

Force region potential function.

CallbackFunctorD_V* EpotSolver::_force_pot_func2

protected

Force region potential function.

Geometry& EpotSolver::_geom

protected

Geometry reference.

CallbackFunctorB_V* EpotSolver::_init_plasma_func

protected

Initial plasma region function.

double EpotSolver::_plA

protected

Plasma parameter. For positive ion extraction: rho_th * h^2 / epsilon_0, for negative ion extraction: rho_f * h^2 / epsilon_0.

plasma_mode_e EpotSolver::_plasma

protected

Plasma simulation mode.

double EpotSolver::_plB

protected

Plasma parameter. For positive ion extraction: 1/Te, for negative ion extraction: E_f,i.

double EpotSolver::_plC

protected

Plasma parameter for positive ion extraction. Up/Te.

std::vector<double> EpotSolver::_plD

protected

Plasma parameter for negative ion extraction. rho_th,i * h^2 / epsilon_0.

std::vector<double> EpotSolver::_plE

protected

Plasma parameter for negative ion extraction. 1/Ti.

double EpotSolver::_rhoe

protected

Electron charge density (C/m3), < 0.

std::vector<double> EpotSolver::_rhoi

protected

Charge density for positive ions, first fast protons, then thermal ions.

double EpotSolver::_Te

protected

Electron thermal energy, > 0.

double EpotSolver::_Up

protected

Plasma potential, > 0.

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

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Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation