wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts Class Reference

This class solves: \(\boldsymbol{v} + \nabla\cdot \bar{\bar{F}} = \boldsymbol{S}\) in weak form for auxiliary variable \(\boldsymbol{v}\) by parts in same manner as with DG. More...

#include <divergence_integral_by_parts.h>

Inheritance diagram for wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts:

Detailed Description

This class solves: \(\boldsymbol{v} + \nabla\cdot \bar{\bar{F}} = \boldsymbol{S}\) in weak form for auxiliary variable \(\boldsymbol{v}\) by parts in same manner as with DG.

Useful for solving terms of Ohm's Law for electric field, which is a state equation, as opposed to a time advancing pde.

For warpy class see warpy.variable_adjusters.auxiliary_variables.auxiliary_variables.divergence_integral_by_parts

Explicitly, it calculates:

\begin{align*} v_{ij}^{\lambda} = {J_{lm}^{\lambda}}^{-1} \Upsilon_{jlk} F_{imk}^{\lambda} - \sum_{\gamma \in \bar{\Gamma}_{\lambda}} G_{\lambda\gamma}^{-1} \Xi_{jk}^{\lambda\gamma} F_{ik}^{\lambda\gamma} + S_{ij} \end{align*}

For the internal flux part \({J_{lm}^{\lambda}}^{-1} \Upsilon_{jlk} F_{imk}^{\lambda}\):

• \({J_{lm}^{\lambda}}^{-1} = \frac{\partial \xi_{l}}{\partial x_{m}}\) is the inverse jacobian for the element.
• \(\Upsilon_{jlk}\) is the internal flux matrix constructed as \(\Upsilon_{jlk} = M_{jp}^{-1}A_{klp}\).
• \(M_{jp}\) is mass matrix \(\int\limits_{\lambda} \psi_{j} \psi_{p}d\boldsymbol{\xi}\).
• \(A_{klp}\) is the advection matrix \(\int\limits_{\lambda} \psi_{k} \frac{\partial}{\partial\xi_{l}} \psi_{p} d\boldsymbol{\xi}\).
• \(F_{imk}^{\lambda}\) is the analytical flux.

For the numerical flux part \(\sum_{\gamma \in \bar{\Gamma}_{\lambda}} G_{\lambda\gamma}^{-1}\Xi_{jk}^{\lambda\gamma} F_{ik}^{\lambda\gamma}\):

• \(G_{\lambda\gamma}^{-1} = \left|\left|{J_{ji}^{\lambda}}^{-1} m_{j}^{\lambda\gamma}\right|\right|\) where \(m\) is the normal of the isoparametric element for the face of element \(\lambda\) facing element \(\gamma\).
• \(\Xi_{jk}^{\lambda\gamma}\) is the numerical flux matrix constructed as \(\Xi_{jk}^{\lambda\gamma} = M_{jl}^{-1}\Gamma_{lk}^{\lambda\gamma}\).
• \(\Gamma_{lk}^{\lambda\gamma}\) is the area mass matrix \(\int\limits_{\partial\lambda^{\gamma}} \psi_{l} \psi_{k} d\boldsymbol{\xi}\) integrated along the simplex face between elements \(\lambda\) and \(\gamma\).
• \(F_{ik}^{\lambda\gamma}\) is the numerical flux on the the simplex face between elements \(\lambda\) and \(\gamma\).

For the source part \(S_{ij}\):

• \(S_{ij}\) is calulated and accumulated into \(v_{ij}\) for LGL node source calculation.
• For GQ node source calculation, we need \(^{3}\Upsilon_{jk} S_{ik}\) where
• \(^{3}\Upsilon_{jk} = M_{jl}^{-1}\Theta_{lk}\) where
• \(\Theta_{lk}\) is the source mass matrix \(w_{l}\psi_{k}(\boldsymbol{\xi}_{l})\) where
• \(w_{l}\) is the gaussian quadrature basis function weight for its node located at \(\boldsymbol{\xi}_{l}\) and
• \(\psi_{k}\) is the LGL basis function used for the DG scheme.

Note that in general:

• \(\lambda\) = "inside" element
• \(\gamma\) = "outside" element
• i = auxiliary variable component
• j = local node
• k = cross local node
• l = isoparametric space dimension index
• m = real space dimension index

Public Member Functions
	DivergenceIntegralByParts ()=default
void	setup (const WxCryptSet &wxc) override
	Setup the spatial solver using the cryptset.
void	solve (real time, variables_type &input) override
	Solves the spatial system and puts result in _rhs.
Public Member Functions inherited from wxm::dfem::variable_adjuster::variable_adjuster_t
	variable_adjuster_t ()
	Constructor.
void	setup (const WxCryptSet &wxc) override
	Setup the spatial solver using the cryptset.
virtual void	solve (real time, variables_type &input)=0
	applies the variable adjuster to input
std::vector< wxm::array::patch_array_t * >	get_patch_arrays (const variables_type &in)
void	process () override
virtual const std::vector< size_t > &	get_input_indices () const
int	priority () const
	Getter function for priority.
virtual void	Barrier (const WxMsgBase &msg, const real time, variables_type &input)
	An opportunity for the variable adjuster to perform any MPI operations it needs to after solve is called.
Public Member Functions inherited from WmPatchProcess
	WmPatchProcess ()
virtual	~WmPatchProcess ()=default
void	step ()
virtual void	process ()=0
void	setPatch (const WmUnstructuredPatch *patch)
void	setParentTaskProcessor (WmPatchProcessor *taskProcessor)
virtual std::string	name (std::string prefix="patchedProcessor.") const
virtual void	setup (const WxCryptSet &wxc)
void	setParentSolver (const WmSolverBase *solver)
void	setDt (WxStepper::time_t dt)
void	setTime (WxStepper::time_t time)
WxStepper::time_t	getSuggestedDt () const
const WmUnstructuredPatch &	get_patch () const
const WmSolver &	get_solver () const

Protected Member Functions
void	rhs (real time, const WxRange &element_scope_range, std::vector< wxm::array::patch_array_t * > input)
	Actually calculates the right hand side for Auxiliary Variable \(v_{ij}^{\lambda}\).

Protected Attributes
std::string	_basisSetName
	Spatial basis set name.
WmBasisArraySet	_basis_set
	Spatial basis set.
std::unique_ptr< WmUDGGeometry >	_dg_geometry
	WmUDGGeometry.
wxm::dfem::tools::scope_t	_element_flux
	Scope.
real	_penalty_beta
	Penalties.
real	_penalty_eta
std::string	_source_quad_style
	Source quadrature style.
bool	_has_fluxes
bool	_has_sources
bool	_sources_lgl
bool	_sources_gq
std::vector< size_t >	_rhs_idcs
	checking of indexes
std::vector< size_t >	_all_idcs
Protected Attributes inherited from wxm::dfem::variable_adjuster::variable_adjuster_t
std::vector< size_t >	_var_idcs
std::vector< std::unique_ptr< WmApplication > >	_apps
bool	_grab_all = false
int	_priority
Protected Attributes inherited from WmPatchProcess
std::string	_ppName
WmPatchProcessor *	_parentTaskProcessor
const WmUnstructuredPatch *	_patch
std::vector< std::string >	_onSubdomains
WxStepper::time_t	_time
WxStepper::time_t	_dt
WxStepper::time_t	_suggested_dt
WxLogStream	_debStrm = WxLogger::get("warpx-root.console")->getDebugStream()
const WmSolverBase *	_parentSolver = nullptr

Additional Inherited Members
Public Types inherited from wxm::dfem::variable_adjuster::variable_adjuster_t
typedef wxm::temporal_solver::variables_type	variables_type

Constructor & Destructor Documentation

DivergenceIntegralByParts()

wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::DivergenceIntegralByParts ( )

default

Member Function Documentation

rhs()

void wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::rhs ( real  time, const WxRange &  element_scope_range, std::vector< wxm::array::patch_array_t * >  input  )

protected

Actually calculates the right hand side for Auxiliary Variable \(v_{ij}^{\lambda}\).

setup()

void wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::setup ( const WxCryptSet &  wxc )

overridevirtual

Setup the spatial solver using the cryptset.

Parameters

wxc	The cryptset

Reimplemented from WmPatchProcess.

solve()

void wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::solve ( real  time, variables_type &  input  )

overridevirtual

Solves the spatial system and puts result in _rhs.

Implements wxm::dfem::variable_adjuster::variable_adjuster_t.

Member Data Documentation

_all_idcs

std::vector<size_t> wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_all_idcs

protected

_basis_set

WmBasisArraySet wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_basis_set

protected

Spatial basis set.

_basisSetName

std::string wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_basisSetName

protected

Spatial basis set name.

_dg_geometry

std::unique_ptr<WmUDGGeometry> wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_dg_geometry

protected

WmUDGGeometry.

_element_flux

wxm::dfem::tools::scope_t wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_element_flux

protected

Scope.

_has_fluxes

bool wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_has_fluxes

protected

_has_sources

bool wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_has_sources

protected

_penalty_beta

real wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_penalty_beta

protected

Penalties.

_penalty_eta

real wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_penalty_eta

protected

_rhs_idcs

std::vector<size_t> wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_rhs_idcs

protected

checking of indexes

_source_quad_style

std::string wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_source_quad_style

protected

Source quadrature style.

_sources_gq

bool wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_sources_gq

protected

_sources_lgl

bool wxm::dfem::variable_adjuster::auxiliary_variables::DivergenceIntegralByParts::_sources_lgl

protected

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

• src/dfem/variable_adjusters/auxiliary_variables/divergence_integral_by_parts.h