Boltzmann equation solver for the electron energy distribution function based on the two-term approximation. More...

#include <EEDFTwoTermApproximation.h>

Detailed Description

Boltzmann equation solver for the electron energy distribution function based on the two-term approximation.

This class implements a solver for the electron energy distribution function based on a steady-state solution to the Boltzmann equation using the classical two-term expansion, applicable to weakly ionized plasmas. The numerical approach and theory are primarily derived from the work of Hagelaar and Pitchford [14].

The two-term approximation assumes that the EEDF can be represented as:

\[ f(\epsilon, \mu) = f_0(\epsilon) + \mu f_1(\epsilon), \]

where \( \epsilon \) is the electron energy and \( \mu \) is the cosine of the angle between the electron velocity vector and the electric field. The Boltzmann equation is projected onto the zeroth moment over mu to obtain an equation for \( f_0(\epsilon) \) , the isotropic part of the distribution. The first-order anisotropic term \( f_1(\epsilon) \) is not solved directly, but is approximated and substituted into the drift and collision terms. This results in a second-order differential equation for \( f_0(\epsilon) \) alone.

The governing equation for \( f_0(\epsilon) \) is discretized on an energy grid using a finite difference method and solved using a tridiagonal matrix algorithm.

Since: New in Cantera 3.2.

Warning: This class is an experimental part of Cantera and may be changed without notice.

Definition at line 48 of file EEDFTwoTermApproximation.h.

Public Member Functions
	EEDFTwoTermApproximation (PlasmaPhase *s)
	Constructor combined with the initialization function.

int	calculateDistributionFunction ()

void	setLinearGrid (double &kTe_max, size_t &ncell)

void	setGridCache ()

vector< double >	getGridEdge () const

vector< double >	getEEDFEdge () const

double	getElectronMobility () const

Protected Member Functions
void	converge (Eigen::VectorXd &f0)
	Iterate f0 (EEDF) until convergence.

Eigen::VectorXd	iterate (const Eigen::VectorXd &f0, double delta)
	An iteration of solving electron energy distribution function.

double	integralPQ (double a, double b, double u0, double u1, double g, double x0)
	The integral in [a, b] of \(x u(x) \exp[g (x_0 - x)]\) assuming that u is linear with u(a) = u0 and u(b) = u1.

vector< double >	vector_g (const Eigen::VectorXd &f0)
	Vector g is used by matrix_P() and matrix_Q().

Eigen::SparseMatrix< double >	matrix_P (const vector< double > &g, size_t k)
	The matrix of scattering-out.

Eigen::SparseMatrix< double >	matrix_Q (const vector< double > &g, size_t k)
	The matrix of scattering-in.

Eigen::SparseMatrix< double >	matrix_A (const Eigen::VectorXd &f0)
	Matrix A (Ax = b) of the equation of EEDF, which is discretized by the exponential scheme of Scharfetter and Gummel,.

double	netProductionFrequency (const Eigen::VectorXd &f0)
	Reduced net production frequency.

double	electronDiffusivity (const Eigen::VectorXd &f0)
	Diffusivity.

double	electronMobility (const Eigen::VectorXd &f0)
	Mobility.

void	initSpeciesIndexCrossSections ()
	Initialize species indices associated with cross-section data.

void	updateCrossSections ()
	Update the total cross sections based on the current state.

void	updateMoleFractions ()
	Update the vector of species mole fractions.

void	calculateTotalElasticCrossSection ()
	Compute the total elastic collision cross section.

void	calculateTotalCrossSection ()
	Compute the total (elastic + inelastic) cross section.

double	norm (const Eigen::VectorXd &f, const Eigen::VectorXd &grid)
	Compute the L1 norm of a function f defined over a given energy grid.

Protected Attributes
double	m_delta0 = 1e14
	Formerly options for the EEDF solver.

size_t	m_maxn = 200
	Maximum number of iterations.

double	m_factorM = 4.0
	The factor for step size change.

size_t	m_points = 150
	The number of points in the EEDF grid.

double	m_rtol = 1e-5
	Error tolerance for convergence.

std::string	m_growth = "temporal"
	The growth model of EEDF.

double	m_moleFractionThreshold = 0.01
	The threshold for species mole fractions.

std::string	m_firstguess = "maxwell"
	The first guess for the EEDF.

double	m_init_kTe = 2.0
	The initial electron temperature [eV].

PlasmaPhase *	m_phase
	Pointer to the PlasmaPhase object used to initialize this object.

double	m_electronMobility
	Electron mobility [m²/V·s].

Eigen::VectorXd	m_gridCenter
	Grid of electron energy (cell center) [eV].

vector< double >	m_gridEdge
	Grid of electron energy (cell boundary i-1/2) [eV].

vector< vector< size_t > >	m_j
	Location of cell j for grid cache.

vector< vector< size_t > >	m_i
	Location of cell i for grid cache.

vector< vector< vector< double > > >	m_sigma
	Cross section at the boundaries of the overlap of cell i and j.

vector< vector< vector< double > > >	m_eps
	The energy boundaries of the overlap of cell i and j.

Eigen::VectorXd	m_f0
	normalized electron energy distribution function

vector< double >	m_f0_edge
	EEDF at grid edges (cell boundaries)

vector< double >	m_totalCrossSectionCenter
	Total electron cross section on the cell center of energy grid.

vector< double >	m_totalCrossSectionEdge
	Total electron cross section on the cell boundary (i-1/2) of energy grid.

vector< double >	m_sigmaElastic
	vector of total elastic cross section weighted with mass ratio

vector< size_t >	m_kTargets
	list of target species indices in global Cantera numbering (1 index per cs)

vector< size_t >	m_klocTargets
	list of target species indices in local X EEDF numbering (1 index per cs)

vector< size_t >	m_kOthers
	Indices of species which has no cross-section data.

vector< size_t >	m_k_lg_Targets
	Local to global indices.

vector< double >	m_X_targets
	Mole fraction of targets.

vector< double >	m_X_targets_prev
	Previous mole fraction of targets used to compute eedf.

vector< int >	m_inFactor
	in factor.

double	m_gamma

bool	m_has_EEDF
	flag of having an EEDF

bool	m_first_call
	First call to calculateDistributionFunction.

Constructor & Destructor Documentation

◆ EEDFTwoTermApproximation()

EEDFTwoTermApproximation ( PlasmaPhase * s )

Constructor combined with the initialization function.

This constructor initializes the EEDFTwoTermApproximation object with everything it needs to start solving EEDF.

Parameters

s	PlasmaPhase object that will be used in the solver calls.

Definition at line 22 of file EEDFTwoTermApproximation.cpp.

Member Function Documentation

◆ calculateDistributionFunction()

int calculateDistributionFunction ( )

Definition at line 46 of file EEDFTwoTermApproximation.cpp.

◆ setLinearGrid()

void setLinearGrid	(	double &	kTe_max,
		size_t &	ncell
	)

Definition at line 31 of file EEDFTwoTermApproximation.cpp.

◆ setGridCache()

void setGridCache ( )

Definition at line 519 of file EEDFTwoTermApproximation.cpp.

◆ getGridEdge()

vector< double > getGridEdge ( ) const

inline

Definition at line 75 of file EEDFTwoTermApproximation.h.

◆ getEEDFEdge()

vector< double > getEEDFEdge ( ) const

inline

Definition at line 79 of file EEDFTwoTermApproximation.h.

◆ getElectronMobility()

double getElectronMobility ( ) const

inline

Definition at line 83 of file EEDFTwoTermApproximation.h.

◆ converge()

void converge ( Eigen::VectorXd & f0 )

protected

Iterate f0 (EEDF) until convergence.

Definition at line 86 of file EEDFTwoTermApproximation.cpp.

◆ iterate()

Eigen::VectorXd iterate	(	const Eigen::VectorXd &	f0,
		double	delta
	)

protected

An iteration of solving electron energy distribution function.

Definition at line 125 of file EEDFTwoTermApproximation.cpp.

◆ integralPQ()

double integralPQ	(	double	a,
		double	b,
		double	u0,
		double	u1,
		double	g,
		double	x0
	)

protected

The integral in [a, b] of \(x u(x) \exp[g (x_0 - x)]\) assuming that u is linear with u(a) = u0 and u(b) = u1.

Definition at line 176 of file EEDFTwoTermApproximation.cpp.

◆ vector_g()

vector< double > vector_g ( const Eigen::VectorXd & f0 )

protected

Vector g is used by matrix_P() and matrix_Q().

\[ g_i = \frac{1}{\epsilon_{i+1} - \epsilon_{i-1}} \ln(\frac{F_{0, i+1}}{F_{0, i-1}}) \]

Definition at line 203 of file EEDFTwoTermApproximation.cpp.

◆ matrix_P()

SparseMat matrix_P	(	const vector< double > &	g,
		size_t	k
	)

protected

The matrix of scattering-out.

\[ P_{i,k} = \gamma \int_{\epsilon_i - 1/2}^{\epsilon_i + 1/2} \epsilon \sigma_k exp[(\epsilon_i - \epsilon)g_i] d \epsilon \]

Definition at line 228 of file EEDFTwoTermApproximation.cpp.

◆ matrix_Q()

SparseMat matrix_Q	(	const vector< double > &	g,
		size_t	k
	)

protected

The matrix of scattering-in.

\[ Q_{i,j,k} = \gamma \int_{\epsilon_1}^{\epsilon_2} \epsilon \sigma_k exp[(\epsilon_j - \epsilon)g_j] d \epsilon \]

where the interval \([\epsilon_1, \epsilon_2]\) is the overlap of cell j, and cell i shifted by the threshold energy:

\[ \epsilon_1 = \min(\max(\epsilon_{i-1/2}+u_k, \epsilon_{j-1/2}),\epsilon_{j+1/2}), \]

\[ \epsilon_2 = \min(\max(\epsilon_{i+1/2}+u_k, \epsilon_{j-1/2}),\epsilon_{j+1/2}) \]

Definition at line 247 of file EEDFTwoTermApproximation.cpp.

◆ matrix_A()

SparseMat matrix_A ( const Eigen::VectorXd & f0 )

protected

Matrix A (Ax = b) of the equation of EEDF, which is discretized by the exponential scheme of Scharfetter and Gummel,.

\[ \left[ \tilde{W} F_0 - \tilde{D} \frac{d F_0}{\epsilon} \right]_{i+1/2} = \frac{\tilde{W}_{i+1/2} F_{0,i}}{1 - \exp[-z_{i+1/2}]} + \frac{\tilde{W}_{i+1/2} F_{0,i+1}}{1 - \exp[z_{i+1/2}]} \]

where \( z_{i+1/2} = \tilde{w}_{i+1/2} / \tilde{D}_{i+1/2} \) (Peclet number).

Definition at line 267 of file EEDFTwoTermApproximation.cpp.

◆ netProductionFrequency()

double netProductionFrequency ( const Eigen::VectorXd & f0 )

protected

Reduced net production frequency.

Equation (10) of ref. [1] divided by N.

Parameters

f0 EEDF

Definition at line 362 of file EEDFTwoTermApproximation.cpp.

◆ electronDiffusivity()

double electronDiffusivity ( const Eigen::VectorXd & f0 )

protected

Diffusivity.

Definition at line 381 of file EEDFTwoTermApproximation.cpp.

◆ electronMobility()

double electronMobility ( const Eigen::VectorXd & f0 )

protected

Mobility.

Definition at line 397 of file EEDFTwoTermApproximation.cpp.

◆ initSpeciesIndexCrossSections()

void initSpeciesIndexCrossSections ( )

protected

Initialize species indices associated with cross-section data.

Definition at line 415 of file EEDFTwoTermApproximation.cpp.

◆ updateCrossSections()

void updateCrossSections ( )

protected

Update the total cross sections based on the current state.

Definition at line 461 of file EEDFTwoTermApproximation.cpp.

◆ updateMoleFractions()

void updateMoleFractions ( )

protected

Update the vector of species mole fractions.

Definition at line 469 of file EEDFTwoTermApproximation.cpp.

◆ calculateTotalElasticCrossSection()

void calculateTotalElasticCrossSection ( )

protected

Compute the total elastic collision cross section.

Definition at line 502 of file EEDFTwoTermApproximation.cpp.

◆ calculateTotalCrossSection()

void calculateTotalCrossSection ( )

protected

Compute the total (elastic + inelastic) cross section.

Definition at line 483 of file EEDFTwoTermApproximation.cpp.

◆ norm()

double norm	(	const Eigen::VectorXd &	f,
		const Eigen::VectorXd &	grid
	)

protected

Compute the L1 norm of a function f defined over a given energy grid.

Parameters

f	Vector representing the function values (EEDF)
grid	Vector representing the energy grid corresponding to f

Definition at line 590 of file EEDFTwoTermApproximation.cpp.

Member Data Documentation

◆ m_delta0

double m_delta0 = 1e14

protected

Formerly options for the EEDF solver.

The first step size

Definition at line 92 of file EEDFTwoTermApproximation.h.

◆ m_maxn

size_t m_maxn = 200

protected

Maximum number of iterations.

Definition at line 95 of file EEDFTwoTermApproximation.h.

◆ m_factorM

double m_factorM = 4.0

protected

The factor for step size change.

Definition at line 98 of file EEDFTwoTermApproximation.h.

◆ m_points

size_t m_points = 150

protected

The number of points in the EEDF grid.

Definition at line 101 of file EEDFTwoTermApproximation.h.

◆ m_rtol

double m_rtol = 1e-5

protected

Error tolerance for convergence.

Definition at line 104 of file EEDFTwoTermApproximation.h.

◆ m_growth

std::string m_growth = "temporal"

protected

The growth model of EEDF.

Definition at line 107 of file EEDFTwoTermApproximation.h.

◆ m_moleFractionThreshold

double m_moleFractionThreshold = 0.01

protected

The threshold for species mole fractions.

Definition at line 110 of file EEDFTwoTermApproximation.h.

◆ m_firstguess

std::string m_firstguess = "maxwell"

protected

The first guess for the EEDF.

Definition at line 113 of file EEDFTwoTermApproximation.h.

◆ m_init_kTe

double m_init_kTe = 2.0

protected

The initial electron temperature [eV].

Definition at line 116 of file EEDFTwoTermApproximation.h.

◆ m_phase

PlasmaPhase* m_phase

protected

Pointer to the PlasmaPhase object used to initialize this object.

This PlasmaPhase object provides species, element, and cross-section data used by the EEDF solver. It is set during construction and is not modified afterwards. All subsequent calls to compute functions must use the same PlasmaPhase context.

Definition at line 125 of file EEDFTwoTermApproximation.h.

◆ m_electronMobility

double m_electronMobility

protected

Electron mobility [m²/V·s].

Definition at line 219 of file EEDFTwoTermApproximation.h.

◆ m_gridCenter

Eigen::VectorXd m_gridCenter

protected

Grid of electron energy (cell center) [eV].

Definition at line 222 of file EEDFTwoTermApproximation.h.

◆ m_gridEdge

vector<double> m_gridEdge

protected

Grid of electron energy (cell boundary i-1/2) [eV].

Definition at line 225 of file EEDFTwoTermApproximation.h.

◆ m_j

vector<vector<size_t> > m_j

protected

Location of cell j for grid cache.

Definition at line 228 of file EEDFTwoTermApproximation.h.

◆ m_i

vector<vector<size_t> > m_i

protected

Location of cell i for grid cache.

Definition at line 231 of file EEDFTwoTermApproximation.h.

◆ m_sigma

vector<vector<vector<double> > > m_sigma

protected

Cross section at the boundaries of the overlap of cell i and j.

Definition at line 234 of file EEDFTwoTermApproximation.h.

◆ m_eps

vector<vector<vector<double> > > m_eps

protected

The energy boundaries of the overlap of cell i and j.

Definition at line 237 of file EEDFTwoTermApproximation.h.

◆ m_f0

Eigen::VectorXd m_f0

protected

normalized electron energy distribution function

Definition at line 240 of file EEDFTwoTermApproximation.h.

◆ m_f0_edge

vector<double> m_f0_edge

protected

EEDF at grid edges (cell boundaries)

Definition at line 243 of file EEDFTwoTermApproximation.h.

◆ m_totalCrossSectionCenter

vector<double> m_totalCrossSectionCenter

protected

Total electron cross section on the cell center of energy grid.

Definition at line 246 of file EEDFTwoTermApproximation.h.

◆ m_totalCrossSectionEdge

vector<double> m_totalCrossSectionEdge

protected

Total electron cross section on the cell boundary (i-1/2) of energy grid.

Definition at line 250 of file EEDFTwoTermApproximation.h.

◆ m_sigmaElastic

vector<double> m_sigmaElastic

protected

vector of total elastic cross section weighted with mass ratio

Definition at line 253 of file EEDFTwoTermApproximation.h.

◆ m_kTargets

vector<size_t> m_kTargets

protected

list of target species indices in global Cantera numbering (1 index per cs)

Definition at line 256 of file EEDFTwoTermApproximation.h.

◆ m_klocTargets

vector<size_t> m_klocTargets

protected

list of target species indices in local X EEDF numbering (1 index per cs)

Definition at line 259 of file EEDFTwoTermApproximation.h.

◆ m_kOthers

vector<size_t> m_kOthers

protected

Indices of species which has no cross-section data.

Definition at line 262 of file EEDFTwoTermApproximation.h.

◆ m_k_lg_Targets

vector<size_t> m_k_lg_Targets

protected

Local to global indices.

Definition at line 265 of file EEDFTwoTermApproximation.h.

◆ m_X_targets

vector<double> m_X_targets

protected

Mole fraction of targets.

Definition at line 268 of file EEDFTwoTermApproximation.h.

◆ m_X_targets_prev

vector<double> m_X_targets_prev

protected

Previous mole fraction of targets used to compute eedf.

Definition at line 271 of file EEDFTwoTermApproximation.h.

◆ m_inFactor

vector<int> m_inFactor

protected

in factor.

This is used for calculating the Q matrix of scattering-in processes.

Definition at line 275 of file EEDFTwoTermApproximation.h.

◆ m_gamma

double m_gamma

protected

Definition at line 277 of file EEDFTwoTermApproximation.h.

◆ m_has_EEDF

bool m_has_EEDF

protected

flag of having an EEDF

Definition at line 280 of file EEDFTwoTermApproximation.h.

◆ m_first_call

bool m_first_call

protected

First call to calculateDistributionFunction.

Definition at line 283 of file EEDFTwoTermApproximation.h.

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

Detailed Description

Public Member Functions

Protected Member Functions

Protected Attributes

Constructor & Destructor Documentation

◆ EEDFTwoTermApproximation()

Member Function Documentation

◆ calculateDistributionFunction()

◆ setLinearGrid()

◆ setGridCache()

◆ getGridEdge()

◆ getEEDFEdge()

◆ getElectronMobility()

◆ converge()

◆ iterate()

◆ integralPQ()

◆ vector_g()

◆ matrix_P()

◆ matrix_Q()

◆ matrix_A()

◆ netProductionFrequency()

◆ electronDiffusivity()

◆ electronMobility()

◆ initSpeciesIndexCrossSections()

◆ updateCrossSections()

◆ updateMoleFractions()

◆ calculateTotalElasticCrossSection()

◆ calculateTotalCrossSection()

◆ norm()

Member Data Documentation

◆ m_delta0

◆ m_maxn

◆ m_factorM

◆ m_points

◆ m_rtol

◆ m_growth

◆ m_moleFractionThreshold

◆ m_firstguess

◆ m_init_kTe

◆ m_phase

◆ m_electronMobility

◆ m_gridCenter

◆ m_gridEdge

◆ m_j

◆ m_i

◆ m_sigma

◆ m_eps

◆ m_f0

◆ m_f0_edge

◆ m_totalCrossSectionCenter

◆ m_totalCrossSectionEdge

◆ m_sigmaElastic

◆ m_kTargets

◆ m_klocTargets

◆ m_kOthers

◆ m_k_lg_Targets

◆ m_X_targets

◆ m_X_targets_prev

◆ m_inFactor

◆ m_gamma

◆ m_has_EEDF

◆ m_first_call