Compute the Allen-Cahn interface term with the weak form residual \( \left(\nabla (L \psi), 1/2 {\partial \kappa} \over {\partial \nabla \eta_{\alpha i}} \sum \{(\nabla \eta_{\beta j})^2 \} \right) \). More...

#include <ACInterface2DMultiPhase1.h>

Inheritance diagram for ACInterface2DMultiPhase1:

[legend]

Public Types
typedef DerivativeMaterialPropertyNameInterface::SymbolName	SymbolName

Public Member Functions
	ACInterface2DMultiPhase1 (const InputParameters &parameters)

virtual void	initialSetup ()

const GenericMaterialProperty< U, is_ad > &	getDefaultMaterialProperty (const std::string &name)

const GenericMaterialProperty< U, is_ad > &	getDefaultMaterialPropertyByName (const std::string &name)

void	validateDerivativeMaterialPropertyBase (const std::string &base)

const MaterialPropertyName	derivativePropertyName (const MaterialPropertyName &base, const std::vector< SymbolName > &c) const

const MaterialPropertyName	derivativePropertyNameFirst (const MaterialPropertyName &base, const SymbolName &c1) const

const MaterialPropertyName	derivativePropertyNameSecond (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2) const

const MaterialPropertyName	derivativePropertyNameThird (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2, const SymbolName &c3) const

GenericMaterialProperty< U, is_ad > &	declarePropertyDerivative (const std::string &base, const std::vector< VariableName > &c)

GenericMaterialProperty< U, is_ad > &	declarePropertyDerivative (const std::string &base, const std::vector< SymbolName > &c)

GenericMaterialProperty< U, is_ad > &	declarePropertyDerivative (const std::string &base, const SymbolName &c1, const SymbolName &c2="", const SymbolName &c3="")

GenericMaterialProperty< U, is_ad > &	declarePropertyDerivative (const std::string &base, const std::vector< VariableName > &c)

GenericMaterialProperty< U, is_ad > &	declarePropertyDerivative (const std::string &base, const std::vector< SymbolName > &c)

GenericMaterialProperty< U, is_ad > &	declarePropertyDerivative (const std::string &base, const SymbolName &c1, const SymbolName &c2="", const SymbolName &c3="")

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const std::vector< VariableName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const std::vector< SymbolName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const SymbolName &c1, const SymbolName &c2="", const SymbolName &c3="")

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const SymbolName &c1, unsigned int v2, unsigned int v3=libMesh::invalid_uint)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, unsigned int v1, unsigned int v2=libMesh::invalid_uint, unsigned int v3=libMesh::invalid_uint)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const std::vector< VariableName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const std::vector< SymbolName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const SymbolName &c1, const SymbolName &c2="", const SymbolName &c3="")

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, const SymbolName &c1, unsigned int v2, unsigned int v3=libMesh::invalid_uint)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivative (const std::string &base, unsigned int v1, unsigned int v2=libMesh::invalid_uint, unsigned int v3=libMesh::invalid_uint)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivativeByName (const MaterialPropertyName &base, const std::vector< VariableName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivativeByName (const MaterialPropertyName &base, const std::vector< SymbolName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivativeByName (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2="", const SymbolName &c3="")

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivativeByName (const MaterialPropertyName &base, const std::vector< VariableName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivativeByName (const MaterialPropertyName &base, const std::vector< SymbolName > &c)

const GenericMaterialProperty< U, is_ad > &	getMaterialPropertyDerivativeByName (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2="", const SymbolName &c3="")

void	validateCoupling (const MaterialPropertyName &base, const std::vector< VariableName > &c, bool validate_aux=true)

void	validateCoupling (const MaterialPropertyName &base, const VariableName &c1="", const VariableName &c2="", const VariableName &c3="")

void	validateCoupling (const MaterialPropertyName &base, const std::vector< VariableName > &c, bool validate_aux=true)

void	validateCoupling (const MaterialPropertyName &base, const VariableName &c1="", const VariableName &c2="", const VariableName &c3="")

void	validateNonlinearCoupling (const MaterialPropertyName &base, const VariableName &c1="", const VariableName &c2="", const VariableName &c3="")

void	validateNonlinearCoupling (const MaterialPropertyName &base, const VariableName &c1="", const VariableName &c2="", const VariableName &c3="")

const MaterialPropertyName	propertyName (const MaterialPropertyName &base, const std::vector< SymbolName > &c) const

const MaterialPropertyName	propertyName (const MaterialPropertyName &base, const std::vector< SymbolName > &c) const

const MaterialPropertyName	propertyNameFirst (const MaterialPropertyName &base, const SymbolName &c1) const

const MaterialPropertyName	propertyNameFirst (const MaterialPropertyName &base, const SymbolName &c1) const

const MaterialPropertyName	propertyNameSecond (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2) const

const MaterialPropertyName	propertyNameSecond (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2) const

const MaterialPropertyName	propertyNameThird (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2, const SymbolName &c3) const

const MaterialPropertyName	propertyNameThird (const MaterialPropertyName &base, const SymbolName &c1, const SymbolName &c2, const SymbolName &c3) const

Static Public Member Functions
static InputParameters	validParams ()

Protected Member Functions
virtual Real	computeQpResidual ()

virtual Real	computeQpJacobian ()

virtual Real	computeQpOffDiagJacobian (unsigned int jvar)

Real	sumSquareGradEta ()

RealGradient	gradL ()

RealGradient	gradKappa ()

RealGradient	nablaLPsi ()
	the \( \nabla(L\psi) \) term

RealGradient	kappaNablaLPsi ()
	the \( \kappa\nabla(L\psi) \) term

Protected Attributes
const MaterialProperty< RealGradient > &	_dkappadgrad_etaa
	Interfacial parameter.

const MaterialProperty< RealTensorValue > &	_d2kappadgrad_etaa

unsigned int	_num_etas
	Order parameters.

std::vector< const VariableValue * >	_eta

std::vector< const VariableGradient * >	_grad_eta

const MaterialProperty< Real > &	_L
	Mobility.

const MaterialProperty< Real > &	_kappa
	Interfacial parameter.

const bool	_variable_L
	flag set if L is a function of non-linear variables in args

const MaterialProperty< Real > &	_dkappadop
	kappa derivative w.r.t. order parameter

std::vector< const MaterialProperty< Real > * >	_dkappadarg
	kappa derivative w.r.t. other coupled variables

std::vector< const VariableGradient * >	_gradarg
	Gradients for all coupled variables.


const MaterialProperty< Real > &	_dLdop
	Mobility derivatives w.r.t. order parameter.

const MaterialProperty< Real > &	_d2Ldop2


std::vector< const MaterialProperty< Real > * >	_dLdarg
	Mobility derivative w.r.t. other coupled variables.

std::vector< const MaterialProperty< Real > * >	_d2Ldargdop

std::vector< std::vector< const MaterialProperty< Real > * > >	_d2Ldarg2

Private Member Functions
bool	haveMaterialProperty (const std::string &prop_name)

std::vector< VariableName >	buildVariableVector (const VariableName &c1, const VariableName &c2, const VariableName &c3)

void	validateCouplingHelper (const MaterialPropertyName &base, const std::vector< VariableName > &c, const System &system, std::vector< VariableName > &missing)

bool	isNotObjectVariable (const VariableName &name)

Private Attributes
FEProblemBase &	_dmi_fe_problem

Detailed Description

Compute the Allen-Cahn interface term with the weak form residual \( \left(\nabla (L \psi), 1/2 {\partial \kappa} \over {\partial \nabla \eta_{\alpha i}} \sum \{(\nabla \eta_{\beta j})^2 \} \right) \).

Definition at line 22 of file ACInterface2DMultiPhase1.h.

Constructor & Destructor Documentation

◆ ACInterface2DMultiPhase1()

ACInterface2DMultiPhase1::ACInterface2DMultiPhase1 ( const InputParameters & parameters )

Definition at line 33 of file ACInterface2DMultiPhase1.C.

  : ACInterface(parameters),
    _dkappadgrad_etaa(getMaterialProperty<RealGradient>("dkappadgrad_etaa_name")),
    _d2kappadgrad_etaa(getMaterialProperty<RealTensorValue>("d2kappadgrad_etaa_name")),
    _num_etas(coupledComponents("etas")),
    _eta(_num_etas),
    _grad_eta(_num_etas)
{
}

Member Function Documentation

◆ computeQpJacobian()

Real ACInterface2DMultiPhase1::computeQpJacobian ( )

protectedvirtual

Reimplemented from ACInterface.

Definition at line 63 of file ACInterface2DMultiPhase1.C.

{
  // dsum is the derivative \f$ \frac\partial{\partial \eta} \left( \nabla (L\psi) \right) \f$
  RealGradient dsum = _dLdop[_qp] * _phi[_j][_qp] * _grad_test[_i][_qp];
 
  // compute the derivative of the gradient of the mobility
  if (_variable_L)
  {
    RealGradient dgradL =
        _grad_phi[_j][_qp] * _dLdop[_qp] + _grad_u[_qp] * _phi[_j][_qp] * _d2Ldop2[_qp];
 
    for (unsigned int i = 0; i < _n_args; ++i)
      dgradL += (*_gradarg[i])[_qp] * _phi[_j][_qp] * (*_d2Ldargdop[i])[_qp];
 
    dsum += dgradL * _test[_i][_qp];
  }
  Real jac1 = 0.5 * dsum * _dkappadgrad_etaa[_qp] * sumSquareGradEta();
  Real jac2 =
      0.5 * nablaLPsi() * (_d2kappadgrad_etaa[_qp] * _grad_phi[_j][_qp]) * sumSquareGradEta();
  Real jac3 = nablaLPsi() * _dkappadgrad_etaa[_qp] * _grad_u[_qp] * _grad_phi[_j][_qp];
  return jac1 + jac2 + jac3;
}

◆ computeQpOffDiagJacobian()

Real ACInterface2DMultiPhase1::computeQpOffDiagJacobian ( unsigned int jvar )

protectedvirtual

Reimplemented from ACInterface.

Definition at line 87 of file ACInterface2DMultiPhase1.C.

{
  // get the coupled variable jvar is referring to
  const unsigned int cvar = mapJvarToCvar(jvar);
 
  // dsum is the derivative \f$ \frac\partial{\partial \eta} \left( \nabla (L\psi) \right) \f$
  RealGradient dsum = (*_dLdarg[cvar])[_qp] * _phi[_j][_qp] * _grad_test[_i][_qp];
 
  // compute the derivative of the gradient of the mobility
  if (_variable_L)
  {
    RealGradient dgradL = _grad_phi[_j][_qp] * (*_dLdarg[cvar])[_qp] +
                          _grad_u[_qp] * _phi[_j][_qp] * (*_d2Ldargdop[cvar])[_qp];
 
    for (unsigned int i = 0; i < _n_args; ++i)
      dgradL += (*_gradarg[i])[_qp] * _phi[_j][_qp] * (*_d2Ldarg2[cvar][i])[_qp];
 
    dsum += dgradL * _test[_i][_qp];
  }
 
  // the gradient of coupled variable etab
  _grad_eta[0] = &coupledGradient("etas", 0);
 
  Real jac1 = 0.5 * dsum * _dkappadgrad_etaa[_qp] * sumSquareGradEta();
  Real jac2 =
      -0.5 * nablaLPsi() * (_d2kappadgrad_etaa[_qp] * _grad_phi[_j][_qp]) * sumSquareGradEta();
  Real jac3 = nablaLPsi() * _dkappadgrad_etaa[_qp] * (*_grad_eta[0])[_qp] * _grad_phi[_j][_qp];
  return jac1 + jac2 + jac3;
}

◆ computeQpResidual()

Real ACInterface2DMultiPhase1::computeQpResidual ( )

protectedvirtual

Reimplemented from ACInterface.

Definition at line 57 of file ACInterface2DMultiPhase1.C.

{
  return 0.5 * nablaLPsi() * _dkappadgrad_etaa[_qp] * sumSquareGradEta();
}

◆ gradKappa()

RealGradient ACInterface::gradKappa ( )

protectedinherited

◆ gradL()

RealGradient ACInterface::gradL ( )

protectedinherited

Definition at line 73 of file ACInterface.C.

{
  RealGradient g = _grad_u[_qp] * _dLdop[_qp];
  for (unsigned int i = 0; i < _n_args; ++i)
    g += (*_gradarg[i])[_qp] * (*_dLdarg[i])[_qp];
  return g;
}

Referenced by ACInterface::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpOffDiagJacobian(), and ACInterface::nablaLPsi().

◆ initialSetup()

void ACInterface::initialSetup ( )

virtualinherited

Reimplemented in ACInterfaceChangedVariable.

Definition at line 66 of file ACInterface.C.

{
  validateCoupling<Real>("mob_name");
  validateCoupling<Real>("kappa_name");
}

◆ kappaNablaLPsi()

RealGradient ACInterface::kappaNablaLPsi ( )

protectedinherited

the \( \kappa\nabla(L\psi) \) term

Definition at line 94 of file ACInterface.C.

{
  return _kappa[_qp] * nablaLPsi();
}

Referenced by ACInterface::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpResidual(), and ACInterfaceCleavageFracture::computeQpResidual().

◆ nablaLPsi()

RealGradient ACInterface::nablaLPsi ( )

protectedinherited

the \( \nabla(L\psi) \) term

Definition at line 82 of file ACInterface.C.

{
  // sum is the product rule gradient \f$ \nabla (L\psi) \f$
  RealGradient sum = _L[_qp] * _grad_test[_i][_qp];
 
  if (_variable_L)
    sum += gradL() * _test[_i][_qp];
 
  return sum;
}

Referenced by computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), computeQpOffDiagJacobian(), ACInterface2DMultiPhase2::computeQpOffDiagJacobian(), computeQpResidual(), and ACInterface::kappaNablaLPsi().

◆ sumSquareGradEta()

Real ACInterface2DMultiPhase1::sumSquareGradEta ( )

protected

Definition at line 44 of file ACInterface2DMultiPhase1.C.

{
  // get the sum of square of gradients of all order parameters
  Real SumSquareGradOp = _grad_u[_qp] * _grad_u[_qp];
  for (unsigned int i = 0; i < _num_etas; ++i)
  {
    _grad_eta[i] = &coupledGradient("etas", i);
    SumSquareGradOp += (*_grad_eta[i])[_qp] * (*_grad_eta[i])[_qp];
  }
  return SumSquareGradOp;
}

Referenced by computeQpJacobian(), computeQpOffDiagJacobian(), and computeQpResidual().

◆ validParams()

InputParameters ACInterface2DMultiPhase1::validParams ( )

static

Definition at line 15 of file ACInterface2DMultiPhase1.C.

{
  InputParameters params = ACInterface::validParams();
  params.addClassDescription(
      "Gradient energy Allen-Cahn Kernel where the derivative of interface parameter kappa "
      "wrt the gradient of order parameter is considered.");
  params.addParam<MaterialPropertyName>("dkappadgrad_etaa_name",
                                        "dkappadgrad_etaa",
                                        "The derivative of the kappa with respect to grad_etaa");
  params.addParam<MaterialPropertyName>(
      "d2kappadgrad_etaa_name",
      "d2kappadgrad_etaa",
      "The second derivative of the kappa with respect to grad_etaa");
  params.addRequiredCoupledVar(
      "etas", "All other coupled order parameters eta_i of the multiphase problem");
  return params;
}

Member Data Documentation

◆ _d2kappadgrad_etaa

const MaterialProperty<RealTensorValue>& ACInterface2DMultiPhase1::_d2kappadgrad_etaa

protected

Definition at line 38 of file ACInterface2DMultiPhase1.h.

Referenced by computeQpJacobian(), and computeQpOffDiagJacobian().

◆ _d2Ldarg2

std::vector<std::vector<const MaterialProperty<Real> *> > ACInterface::_d2Ldarg2

protectedinherited

Definition at line 61 of file ACInterface.h.

Referenced by ACInterface::ACInterface(), ACInterface::computeQpOffDiagJacobian(), computeQpOffDiagJacobian(), and ACInterface2DMultiPhase2::computeQpOffDiagJacobian().

◆ _d2Ldargdop

std::vector<const MaterialProperty<Real> *> ACInterface::_d2Ldargdop

protectedinherited

Definition at line 60 of file ACInterface.h.

Referenced by ACInterface::ACInterface(), ACInterface::computeQpJacobian(), computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpOffDiagJacobian(), computeQpOffDiagJacobian(), and ACInterface2DMultiPhase2::computeQpOffDiagJacobian().

◆ _d2Ldop2

const MaterialProperty<Real>& ACInterface::_d2Ldop2

protectedinherited

Definition at line 52 of file ACInterface.h.

Referenced by ACInterface::computeQpJacobian(), computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), and ACInterfaceCleavageFracture::computeQpJacobian().

◆ _dkappadarg

std::vector<const MaterialProperty<Real> *> ACInterface::_dkappadarg

protectedinherited

kappa derivative w.r.t. other coupled variables

Definition at line 65 of file ACInterface.h.

Referenced by ACInterface::ACInterface(), and ACInterface::computeQpOffDiagJacobian().

◆ _dkappadgrad_etaa

const MaterialProperty<RealGradient>& ACInterface2DMultiPhase1::_dkappadgrad_etaa

protected

Interfacial parameter.

Definition at line 37 of file ACInterface2DMultiPhase1.h.

Referenced by computeQpJacobian(), computeQpOffDiagJacobian(), and computeQpResidual().

◆ _dkappadop

const MaterialProperty<Real>& ACInterface::_dkappadop

protectedinherited

kappa derivative w.r.t. order parameter

Definition at line 56 of file ACInterface.h.

Referenced by ACInterface::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), and ACInterface::computeQpOffDiagJacobian().

◆ _dLdarg

std::vector<const MaterialProperty<Real> *> ACInterface::_dLdarg

protectedinherited

Mobility derivative w.r.t. other coupled variables.

Definition at line 59 of file ACInterface.h.

Referenced by ACInterface::ACInterface(), ACInterface::computeQpOffDiagJacobian(), computeQpOffDiagJacobian(), ACInterface2DMultiPhase2::computeQpOffDiagJacobian(), and ACInterface::gradL().

◆ _dLdop

const MaterialProperty<Real>& ACInterface::_dLdop

protectedinherited

Mobility derivatives w.r.t. order parameter.

Definition at line 51 of file ACInterface.h.

Referenced by ACInterface::computeQpJacobian(), computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), and ACInterface::gradL().

◆ _eta

std::vector<const VariableValue *> ACInterface2DMultiPhase1::_eta

protected

Definition at line 42 of file ACInterface2DMultiPhase1.h.

◆ _grad_eta

std::vector<const VariableGradient *> ACInterface2DMultiPhase1::_grad_eta

protected

Definition at line 43 of file ACInterface2DMultiPhase1.h.

Referenced by computeQpOffDiagJacobian(), and sumSquareGradEta().

◆ _gradarg

std::vector<const VariableGradient *> ACInterface::_gradarg

protectedinherited

Gradients for all coupled variables.

Definition at line 68 of file ACInterface.h.

Referenced by ACInterface::ACInterface(), ACInterface::computeQpJacobian(), computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpOffDiagJacobian(), computeQpOffDiagJacobian(), ACInterface2DMultiPhase2::computeQpOffDiagJacobian(), and ACInterface::gradL().

◆ _kappa

const MaterialProperty<Real>& ACInterface::_kappa

protectedinherited

Interfacial parameter.

Definition at line 45 of file ACInterface.h.

Referenced by ACInterfaceCleavageFracture::betaNablaPsi(), ACInterface::computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpOffDiagJacobian(), ACInterface2DMultiPhase2::computeQpOffDiagJacobian(), and ACInterface::kappaNablaLPsi().

◆ _L

const MaterialProperty<Real>& ACInterface::_L

protectedinherited

Mobility.

Definition at line 43 of file ACInterface.h.

Referenced by ACInterfaceCleavageFracture::betaNablaPsi(), ACInterface::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpOffDiagJacobian(), and ACInterface::nablaLPsi().

◆ _num_etas

unsigned int ACInterface2DMultiPhase1::_num_etas

protected

Order parameters.

Definition at line 41 of file ACInterface2DMultiPhase1.h.

Referenced by sumSquareGradEta().

◆ _variable_L

const bool ACInterface::_variable_L

protectedinherited

flag set if L is a function of non-linear variables in args

Definition at line 48 of file ACInterface.h.

Referenced by ACInterface::computeQpJacobian(), computeQpJacobian(), ACInterface2DMultiPhase2::computeQpJacobian(), ACInterfaceCleavageFracture::computeQpJacobian(), ACInterface::computeQpOffDiagJacobian(), computeQpOffDiagJacobian(), ACInterface2DMultiPhase2::computeQpOffDiagJacobian(), and ACInterface::nablaLPsi().

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

phase_field/include/kernels/ACInterface2DMultiPhase1.h
phase_field/src/kernels/ACInterface2DMultiPhase1.C

Public Types

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ ACInterface2DMultiPhase1()

Member Function Documentation

◆ computeQpJacobian()

◆ computeQpOffDiagJacobian()

◆ computeQpResidual()

◆ gradKappa()

◆ gradL()

◆ initialSetup()

◆ kappaNablaLPsi()

◆ nablaLPsi()

◆ sumSquareGradEta()

◆ validParams()

Member Data Documentation

◆ _d2kappadgrad_etaa

◆ _d2Ldarg2

◆ _d2Ldargdop

◆ _d2Ldop2

◆ _dkappadarg

◆ _dkappadgrad_etaa

◆ _dkappadop

◆ _dLdarg

◆ _dLdop

◆ _eta

◆ _grad_eta

◆ _gradarg

◆ _kappa

◆ _L

◆ _num_etas

◆ _variable_L