This class is used to solve a system of equations similar to the one presented in the journal article M. More...

#include <app_cathode.h>

Inheritance diagram for FuelCell::Application::AppCathode< dim >:

Collaboration diagram for FuelCell::Application::AppCathode< dim >:

Public Member Functions
Constructors, destructor, and initialization
	AppCathode (boost::shared_ptr< FuelCell::ApplicationCore::ApplicationData > data=boost::shared_ptr< FuelCell::ApplicationCore::ApplicationData >())
	Constructor. More...

	~AppCathode ()
	Destructor. More...

virtual void	declare_parameters (ParameterHandler &param)
	Declare parameters. More...

virtual void	initialize (ParameterHandler &param)
	Initialize parameters. More...

virtual void	initialize_solution (FEVector &initial_guess, std::shared_ptr< Function< dim > > initial_function=std::shared_ptr< Function< dim > >())
	The nonlinear solution initial guess along with the appropriate BCs is formed here. More...

Local CG FEM based assemblers
virtual void	cell_matrix (MatrixVector &cell_matrices, const typename DoFApplication< dim >::CellInfo &cell_info)
	Assemble local cell matrix. More...

virtual void	cell_residual (FuelCell::ApplicationCore::FEVector &cell_res, const typename DoFApplication< dim >::CellInfo &cell_info)
	Assemble local cell residual. More...

Other functions
virtual void	dirichlet_bc (std::map< unsigned int, double > &boundary_values) const
	BCs. More...

virtual double	evaluate (const FuelCell::ApplicationCore::FEVectors &src)
	Returns the current density. More...

virtual void	data_out (const std::string &filename, const FEVectors &src)
	Output results. More...

Post-processing
virtual void	cell_responses (std::vector< double > &dst, const typename DoFApplication< dim >::CellInfo &cell_info, const FEVector &)
	Compute some functionals. More...

void	global_responses (std::vector< double > &resp, const FuelCell::ApplicationCore::FEVector &)
	This class is used to evaluate all responses that do not require looping over cells. More...

Public Member Functions inherited from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >
	OptimizationBlockMatrixApplication (FuelCell::ApplicationCore::DoFApplication< dim > &, bool triangulation_only)
	Constructor for an object, owning its own mesh and dof handler. More...

	OptimizationBlockMatrixApplication (boost::shared_ptr< FuelCell::ApplicationCore::ApplicationData > data=boost::shared_ptr< FuelCell::ApplicationCore::ApplicationData >())
	Constructor for an object, owning its own mesh and dof handler. More...

	~OptimizationBlockMatrixApplication ()
	Destructor. More...

virtual void	responses (std::vector< double > &f, const FuelCell::ApplicationCore::FEVectors &vectors)
	Post-processing. More...

virtual void	check_responses ()
	This class is called by responses to make sure that all responses requested are implemented in either cell_responses, global_responses or face_responses. More...

virtual void	bdry_responses (std::vector< double > &resp, const typename DoFApplication< dim >::FaceInfo &info, const FuelCell::ApplicationCore::FEVector &sol)
	This class is used to evaluate all the functionals that require looping over boundaries. More...

virtual void	print_responses (std::vector< double > &resp)
	This function is used to print the responses to screen (FcstUtilities::log) More...

virtual void	dresponses_dl (std::vector< std::vector< double > > &df_dl, const FuelCell::ApplicationCore::FEVectors &src)
	Post-processing. More...

virtual void	cell_dresponses_dl (std::vector< std::vector< double > > &cell_df_dl, const typename DoFApplication< dim >::CellInfo &info, const FuelCell::ApplicationCore::FEVector &src)
	This class is used to evaluate the derivative of all the functionals that require looping over cells with respect to the design variables. More...

virtual void	global_dresponses_dl (std::vector< std::vector< double > > &df_dl, const FuelCell::ApplicationCore::FEVector &src)
	This class is used to evaluate the sensitivities of all responses that do not require looping over cells with respect of the design variables. More...

virtual void	dresponses_du (std::vector< FuelCell::ApplicationCore::FEVector > &dst, const FuelCell::ApplicationCore::FEVectors &src)
	Loop over all cells and assemble the vector $\frac{\partial f_m}{\partial u_i}$ that is used to solve the sensitivity equations by using the local matrix integration functions cell_dfunctional_du(), bdry_dfunctional_dlu() and edge_dfunctinal_du() provided by the derived class. More...

virtual void	cell_dresponses_du (std::vector< FuelCell::ApplicationCore::FEVector > &df_du, const typename DoFApplication< dim >::CellInfo &info, std::vector< std::vector< double > > &src)
	Integration of local bilinear form. More...

virtual void	global_dresponses_du (std::vector< FuelCell::ApplicationCore::FEVector > &df_du, const FuelCell::ApplicationCore::FEVector &src)
	This class is used to evaluate the sensitivities of all responses that do not require looping over cells with respect of the design variables. More...

virtual void	dresidual_dlambda (std::vector< FuelCell::ApplicationCore::FEVector > &dst, const FuelCell::ApplicationCore::FEVectors &src)
	Loop over all cells and assemble the vector $\frac{\partial R_n}{\partial \lambda_k}$ that is used to solve the sensitivity equations by using the local matrix integration functions cell_dresidual_dlambda(), bdry_dresidual_dlambda() and edge_dresidual_dlambda() provided by the derived class. More...

virtual void	cell_dresidual_dlambda (std::vector< FuelCell::ApplicationCore::FEVector > &cell_vector, const typename DoFApplication< dim >::CellInfo &cell, std::vector< std::vector< double > > &src)
	Integration of local bilinear form. More...

void	solve_direct (std::vector< std::vector< double > > &df_dl, const FuelCell::ApplicationCore::FEVectors &sol)
	Solver in order to obtain the analytical sensitivities for the given objective and constraints using the direct method. More...

void	solve_adjoint (std::vector< std::vector< double > > &df_dl, const FuelCell::ApplicationCore::FEVector &sol)
	Solver in order to obtain the analytical sensitivities for the given objective and constraints using the adjoint method. More...

unsigned int	get_n_resp () const
	Member function that returns the number of responses. More...

unsigned int	get_n_dvar () const
	Member function that returns the number of design variables. More...

std::vector< std::string >	get_name_dvar () const
	Member function that returns the name of the design variables. More...

std::vector< std::string >	get_name_responses () const
	Member function that returns the name of the responses. More...

const std::vector< std::string >	get_all_responses_names () const
	Function used to return all possible response names in OpenFCST. More...

void	print_default_parameter_file ()
	Print the default parameter handler file. More...

bool	get_bool_transfer_solution ()
	Function to see if we are transferring a solution on a refined grid to the initial coarse grid. More...

void	set_optimization_parameters (unsigned int &n_dvar, unsigned int &n_resp, std::vector< std::string > &name_design_var, std::vector< std::string > &name_responses)
	This routine assigns the value of n_dvar, n_resp, name_design_var and name_responses to the internal variables n_dvar, n_resp, name_design_var and name_response. More...

void	set_output_variables (std::vector< std::string > &dakota_name_responses)

virtual void	compute_L1_L2_error_and_convergence_rate (const FuelCell::ApplicationCore::FEVector &solution, const unsigned int &refinement_cycle, std::vector< ConvergenceTable > &convergence_tables) const
	If the exact or analytical solution is available, then this function computes L1 and L2 norms of the error and convergence rates for each particular component of the numerical solution. More...

Public Member Functions inherited from FuelCell::ApplicationCore::BlockMatrixApplication< dim >
void	serial_assemble (const FEVectors &)
	serial and PETSc assemble functions called by assemble(). More...

virtual void	post_cell_assemble ()
	A call back function for assemble(), called after all cell matrices have been entered, but before the face matrices are computed. More...

virtual void	solve (FEVector &dst, const FEVectors &src)
	Solve the system assembled with right hand side in FEVectors `src` and return the result in FEVector `dst`. More...

void	serial_solve (FuelCell::ApplicationCore::FEVector system_rhs, FEVector &solution)

	BlockMatrixApplication (boost::shared_ptr< ApplicationData > data=boost::shared_ptr< ApplicationData >())
	Constructor for an object, owning its own mesh and dof handler. More...

	BlockMatrixApplication (DoFApplication< dim > &, bool triangulation_only)
	Constructor for an object, borrowing mesh and dof handler from another object. More...

void	_initialize (ParameterHandler &param)
	Initialize data of this class. More...

void	remesh_matrices ()
	Initialize sparsity patterns and matrices for the new mesh. More...

virtual void	remesh ()
	Refine grid accordingly to the Refinement options under "Grid Generation" in the parameter file. More...

void	assemble (const FEVectors &)
	Loop over all cells and assemble the system #matrices by using the local matrix integration functions cell_matrix(), bdry_matrix() and face_matrix() provided by the derived class. More...

void	assemble_numerically (const FEVectors &src, const double delta=1e-6)
	Compute the Jacobian of the system of equations, $J(i,j) = \frac{\partial R_i}{\partial u_j}$ by using forward differences. More...

void	residual_constraints (FEVector &dst) const
	Redefinition of residual_constraints() in DoFHandler. More...

virtual void	bdry_matrix (MatrixVector &face_matrices, const typename DoFApplication< dim >::FaceInfo &face)
	Integration of local bilinear form. More...

virtual void	face_matrix (MatrixVector &matrices11, MatrixVector &matrices12, MatrixVector &matrices21, MatrixVector &matrices22, const typename DoFApplication< dim >::FaceInfo &face1, const typename DoFApplication< dim >::FaceInfo &face2)
	Integration of local bilinear form. More...

Public Member Functions inherited from FuelCell::ApplicationCore::DoFApplication< dim >
	DoFApplication (boost::shared_ptr< ApplicationData > data=boost::shared_ptr< ApplicationData >())
	Constructor for an object owning its own mesh and dof handler. More...

	DoFApplication ()
	Constructor for an object owning its own mesh and dof handler and creating new ApplicationData. More...

	DoFApplication (DoFApplication< dim > &dof_app, bool triangulation_only)
	Constructor for an object, borrowing mesh and dof handler from another object. More...

	~DoFApplication ()
	Destructor which deletes owned objects. More...

virtual void	remesh_dofs ()
	Initialize dof handler, count the dofs in each block and renumber the dofs. More...

virtual void	init_vector (FEVector &dst) const
	Reinitialize the BlockVector dst such that it contains block_size.size() blocks. More...

virtual double	estimate (const FEVectors &src)
	Estimate the error. More...

virtual double	residual (FEVector &dst, const FEVectors &src, bool apply_boundaries=true)
	Loop over all cells to compute the residual. More...

void	store_triangulation (Triangulation< dim > &new_tr)
	Function to copy a triangulation object for use after refinement. More...

void	add_vector_for_transfer (FEVector *src)
	Add the vector to be transfered from one mesh to the next. More...

void	delete_vector_for_transfer ()
	Delete the vector to be transfered from one mesh to the next. More...

void	transfer_solution_to_coarse_mesh (Triangulation< dim > &tr_coarse, FEVector &coarse_solution, FEVector &refined_solution)
	Function to perform the transfer of a solution on a refined grid to the initial coarse grid. More...

unsigned int	memory_consumption () const
	Compute the amount of memory needed by this object. More...

virtual void	grid_out (const std::string &basename)
	Output the grid used to solve the problem. More...

void	print (const std::string &basename, const FEVector &src, const std::vector< unsigned int > &src_indices=std::vector< unsigned int >()) const
	This function prints `FEVector` `src` to a text file `basename`. More...

Public Member Functions inherited from FuelCell::ApplicationCore::ApplicationBase
	ApplicationBase (boost::shared_ptr< ApplicationData > data=boost::shared_ptr< ApplicationData >())
	Constructor for an application. More...

	ApplicationBase (const ApplicationBase &other)
	Copy constructor. More...

virtual	~ApplicationBase ()
	Virtual destructor. More...

void	print_parameters_to_file (ParameterHandler &param, const std::string &file_name, const ParameterHandler::OutputStyle &style)
	Print default parameters for the application to a file. More...

virtual void	start_vector (FEVector &dst, std::string) const
	Initialize vector to problem size. More...

virtual void	Tsolve (FEVector &, const FEVectors &)
	Solve the dual system assembled with right hand side `rhs` and return the result in `start`. More...

boost::shared_ptr < ApplicationData >	get_data ()
	Get access to the protected variable data. More...

const boost::shared_ptr < ApplicationData >	get_data () const
	Get read-only access to the protected variable data. More...

virtual std::string	id () const
	Return a unique identification string for this application. More...

virtual void	notify (const Event &reason)
	Add a reason for assembling. More...

virtual void	clear ()
	All `true` in `notifications`. More...

virtual void	clear_events ()
	All `false` in `notifications`. More...

virtual unsigned int	get_solution_index ()
	Returns solution index. More...

Protected Attributes
Pre-processor object
boost::shared_ptr < FuelCellShop::Geometry::GridBase < dim > >	grid
	Grid. More...

Gases and operating conditions data
FuelCell::OperatingConditions	OC
	Operating conditions. More...

FuelCellShop::Material::Oxygen	solute
	Solute. More...

FuelCellShop::Material::Hydrogen	solute_anode

FuelCellShop::Material::Nitrogen	solvent
	Solvent. More...

Layer objects
boost::shared_ptr < FuelCellShop::Layer::GasDiffusionLayer < dim > >	CGDL
	Cathode GDL. More...

boost::shared_ptr < FuelCellShop::Layer::MicroPorousLayer < dim > >	CMPL
	Cathode MPL. More...

boost::shared_ptr < FuelCellShop::Layer::CatalystLayer < dim > >	CCL
	Cathode CL. More...

bool	anode
	Run an anode instead of a cathode: More...

Equation objects
FuelCellShop::Equation::FicksTransportEquation < dim > *	ficks_transport_equation
	This object describes the equations that we are going to solve here. More...

FuelCellShop::Equation::FicksTransportEquation < dim >	ficks_transport_equation_cathode

FuelCellShop::Equation::FicksTransportEquation < dim >	ficks_transport_equation_anode

FuelCellShop::Equation::ElectronTransportEquation < dim >	electron_transport_equation
	This object describes the equations that we are going to solve here. More...

FuelCellShop::Equation::ProtonTransportEquation < dim >	proton_transport_equation
	This object describes the equations that we are going to solve here. More...

FuelCellShop::Equation::ReactionSourceTerms < dim >	reaction_source_terms
	The reaction source terms for all underlying equations. More...

Post-processing objects (Functional evaluation)
FuelCellShop::PostProcessing::ORRCurrentDensityResponse < dim >	ORRCurrent

FuelCellShop::PostProcessing::HORCurrentDensityResponse < dim >	HORCurrent

Protected Attributes inherited from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >
unsigned int	n_dvar
	Number of design variables. More...

unsigned int	n_obj
	Number of objective functions. More...

unsigned int	n_resp
	Number of responses = n_obj + n_con. More...

std::vector< std::string >	name_design_var
	Member that stores the name of the design variables. More...

std::vector< std::string >	name_responses
	Member that stores the name of the responses, i.e. More...

std::vector< std::string >	name_output_var
	Member that stores the name of the output variables, These names will be written to name_responses if optimization is not being used and ignored otherwise. More...

bool	output_coarse_solution
	Decision variable for whether the solution is to be output for transfer. More...

bool	optimization
	Decision variable for whether the application is being used for optimization. More...

bool	boundary_responses
	`true`, if boundary responses are supposed to be computed. More...

std::vector< std::string >	all_response_names
	Variable that holds a list of response names. More...

std::vector< unsigned int >	user_input_bdry
	Boundary id on which the boundary response is computed. More...

Protected Attributes inherited from FuelCell::ApplicationCore::BlockMatrixApplication< dim >
boost::shared_ptr< Quadrature < dim > >	quadrature_assemble_cell
	Quadrature rule for matrix assembling on cells. More...

boost::shared_ptr< Quadrature < dim-1 > >	quadrature_assemble_face
	Quadrature rule for matrix assembling on faces. More...

bool	repair_diagonal
	Bool determining whether or not to repair diagonal before solving. More...

std::map< unsigned int, double >	boundary_values
	Variable to store boundary values, so they only need to be computed once per mesh refinement. More...

BlockSparseMatrix< double >	matrix
	Storage for the actual matrix. More...

SolverControl	solver_control
	Solver control object. More...

bool	assemble_numerically_flag
	Variable used to select if assembly should be done analytically or numerically. More...

bool	mumps_additional_mem
	Variable used for configuring MUMPS to use more memory in solve() function. More...

bool	symmetric_matrix_flag
	Variable used to specify if the matrix is symmetric to the linear solver. More...

bool	print_debug
	Flag specifying if the matrix and rhs should be printed at each iteration. More...

Protected Attributes inherited from FuelCell::ApplicationCore::DoFApplication< dim >
GridOut	g_out
	The object for writing grids. More...

DataOut< dim, DoFHandler< dim > >	d_out
	The object for writing data. More...

FuelCell::SystemManagement	system_management
	This object knows everything about FCST equations, variables, couplings, etc. More...

std::map< unsigned int, double >	boundary_constraints
	List of all nodes constrained by a strong boundary condition, together with a value to be assigned. More...

ConstraintMatrix	hanging_node_constraints
	Constraint Matrix object. More...

boost::shared_ptr< Mapping< dim > >	mapping
	The mapping used for the transformation of mesh cells. More...

unsigned int	mapping_degree
	Degree used for polynomial mapping of arbitrary order. More...

boost::shared_ptr < FiniteElement< dim > >	element
	The finite element used in dof. More...

boost::shared_ptr< DoFHandler < dim > >	dof
	Pointer to the DoFHandler object. More...

BlockInfo	block_info

Vector< float >	cell_errors
	The result of error estimation by cell. More...

Vector< float >	face_errors
	The result of error estimation by face. More...

std::string	refinement
	Refinement parameter from parameter file. More...

unsigned int	initial_refinement
	Initial refinement from parameter file. More...

double	refinement_threshold
	Refinement threshold for adaptive method from parameter file. More...

double	coarsening_threshold
	Coarsening threshold for adaptive method from parameter file. More...

bool	sort_cuthill
	Flag for sorting with Cuthill McKee algorithm. More...

Point< dim >	sort_direction
	Direction for downstream sorting. More...

std::vector< FEVector * >	transfer_vectors
	List of vector names to be transfered from one grid to the next. More...

Quadrature< dim >	quadrature_residual_cell
	Quadrature rule for residual computation on cells. More...

Quadrature< dim-1 >	quadrature_residual_bdry
	Quadrature rule for residual computation on boundary faces. More...

Quadrature< dim-1 >	quadrature_residual_face
	Quadrature rule for residual computation on faces. More...

Table< 2, DoFTools::Coupling >	cell_couplings
	Couplings through cell bilinear forms. More...

Table< 2, DoFTools::Coupling >	flux_couplings
	Couplings through flux bilinear forms. More...

bool	boundary_fluxes
	Extend the integration loops in assemble() and residual() also to boundary faces. More...

bool	interior_fluxes
	Extend the integration loops in assemble() and residual() also to interior faces. More...

unsigned int	verbosity
	Controls verbosity of certain functions. More...

std::vector < component_materialID_value_map >	component_materialID_value_maps
	Each entry of this std::vector reflects the following structure (see FuelCell::InitialAndBoundaryData namespace docs): More...

std::vector < component_boundaryID_value_map >	component_boundaryID_value_maps
	Each entry of this std::vector reflects the following structure (see FuelCell::InitialAndBoundaryData namespace docs): More...

boost::shared_ptr < FuelCellShop::Geometry::GridBase < dim > >	mesh_generator
	Grid. More...

boost::shared_ptr < Triangulation< dim > >	tr
	Pointer to the Triangulation object. More...

Protected Attributes inherited from FuelCell::ApplicationCore::ApplicationBase
boost::shared_ptr < ApplicationData >	data
	Object for auxiliary data. More...

Event	notifications
	Accumulate reasons for assembling here. More...

Private Member Functions
virtual void	cell_responses_aux (std::vector< double > &dst, const typename DoFApplication< dim >::CellInfo &cell_info, const FEVector &)
	Compute some functionals that are not needed for most applications (this section is not necessary in most cases.) More...

void	set_default_parameters_for_application (ParameterHandler &param)
	Function to modify the default values of the data file in order to make sure that the equations match those needed in the application. More...

Additional Inherited Members
Public Types inherited from FuelCell::ApplicationCore::DoFApplication< dim >
typedef FuelCell::ApplicationCore::IntegrationInfo < dim, FEValuesBase< dim > >	CellInfo
	Shortcut. More...

typedef FuelCell::ApplicationCore::IntegrationInfo < dim, FEFaceValuesBase< dim > >	FaceInfo
	Shortcut. More...

Public Attributes inherited from FuelCell::ApplicationCore::DoFApplication< dim >
boost::shared_ptr< Boundary < dim > >	curved_boundary
	Curved boundary. More...

types::boundary_id	curved_bdry_id
	Curved boundary ID. More...

std::string	filename_initial_sol
	Filename where to output the initial grid. More...

bool	output_initial_sol
	Flag to output the initial solution used to start the solving process. More...

bool	read_in_initial_solution
	Bool flag used to specify if the initial solution to the problem, specially important for non-linear problems, should be read from file. More...

bool	use_predefined_solution
	Use user pre-defined initial solution. More...

bool	output_coarse_solution
	Bool flag used to specify if the final solution should be stored in the coarse mesh in order to be used later as an initial solution to solve another problem using the flag read_in_initial_solution. More...

std::vector < DataComponentInterpretation::DataComponentInterpretation >	solution_interpretations
	`solution_interpretations` identifies whether some `solution_names` are scalars or parts of a vector. More...

std::vector < DataComponentInterpretation::DataComponentInterpretation >	postprocessing_interpretations
	`postprocessing_interpretations` identifies whether some `postprocessing_names` are scalars or parts of a vector. More...

std::vector < DataComponentInterpretation::DataComponentInterpretation >	data_interpretation

bool	output_materials_and_levels
	`output_materials_and_levels` if `true` then visualized, otherwise suppressed. More...

Vector< double >	output_materials
	Vector that will be used by `data_out` to store the material ids. More...

Vector< double >	output_levels
	Vector that will be used by `data_out` to store the refinement levels at each cell. More...

bool	output_actual_degree
	`true`, if you want to output solution and postprocessing data using actual finite element fields Q_n with n >= 1. More...

bool	print_solution
	`true`, if you want to print `FEVector` `solution` to a text file. More...

bool	print_postprocessing
	`true`, if you want to print `FEVector` `postprocessing` to a text file. More...

std::vector< unsigned int >	solution_printing_indices
	The indices of the `FEVector` `solution` to be printed to a text file. More...

std::vector< unsigned int >	postprocessing_printing_indices
	The indices of the `FEVector` `postprocessing` to be printed to a text file. More...

bool	print_blocks_instead_of_indices
	`true`, if the whole blocks of `FEVector` `solution` or `FEVector` `postprocessing` to be printed to a text file instead of separate indices. More...

bool	output_matrices_and_rhs
	If true, all cell matrices and right hand sides will be output. More...

Static Public Attributes inherited from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >
static const FuelCell::ApplicationCore::Event	sensitivity_analysis
	The Event set by OptimizationMatrixApplication if if we want to compute the sensitivities and a new matrix should be assembled. More...

Protected Types inherited from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >
typedef void(*	CELL_Dvalues )(std::vector< FuelCell::ApplicationCore::FEVector > &, const typename DoFApplication< dim >::CellInfo &, std::vector< std::vector< double > > &)
	Definition of a pointer to a function that is called in a loop over cells to compute the derivatives of either residual or responses with respect to design variables. More...

Protected Member Functions inherited from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >
void	print_dresponses_dl (std::vector< std::vector< double > > pdf_pdl)
	Auxiliary routine to print the values df_dl This routine should be called once df_df is assembled. More...

void	print_dresponses_du (std::vector< FuelCell::ApplicationCore::FEVector > df_du)
	Auxiliary routine to print the values of df_du This routine should be called once df_du is assembled. More...

const std::string	extend_filename (const std::string &, const int precision=3) const
	Member function used in order to extend the name of a file with the design variable name and value. More...

void	dfunction (std::vector< FuelCell::ApplicationCore::FEVector > &dst, const FuelCell::ApplicationCore::FEVectors &src, bool dfunctional_du, bool dresidual_dlambda)
	This is an auxiliary function called by dresidual_dlambda and dfunctional_du. More...

void	set_all_response_names ()
	Initialize the data member `all_response_names` with all the available Responses in OpenFCST. More...

Protected Member Functions inherited from FuelCell::ApplicationCore::BlockMatrixApplication< dim >
template<typename Vector >
void	print_matrix_and_rhs (Vector &sys_rhs) const
	Internal routine to print matrix and rhs. More...

Protected Member Functions inherited from FuelCell::ApplicationCore::DoFApplication< dim >
virtual void	data_out (const std::string &basename, const FEVector &solution, const std::vector< std::string > &solution_names, const std::vector< DataPostprocessor< dim > * > &PostProcessing)
	This routine is used to write data in the format specified by the ParameterHandler. More...

virtual void	data_out (const std::string &basename, const FEVector &solution, const std::vector< std::string > &solution_names, const FEVector &postprocessing=FEVector(), const std::vector< std::string > &postprocessing_names=std::vector< std::string >())
	This function outputs the results of a computation. More...

void	constrain_boundary (FEVector &v, bool homogeneous) const
	Apply either homogeneous or inhomogeneous boundary_constraints. More...

void	_initialize (ParameterHandler &param)
	Initialize from parameter values. More...

virtual void	initialize_triangulation (ParameterHandler &param)
	Function used to read in a mesh and hand it over to the boost::shared_ptr<Triangulation<dim> > tr object. More...

void	read_init_solution (FEVector &dst, bool &good_solution) const
	Create a mesh and assign it to object tr. More...

virtual void	cell_residual (FEVector &cell_vector, const CellInfo &cell)
	Local integration. More...

virtual void	bdry_residual (FEVector &face_vector, const FaceInfo &face)
	Local integration. More...

virtual void	face_residual (FEVector &face_vector1, FEVector &face_vector2, const FaceInfo &face1, const FaceInfo &face2)
	Local integration. More...

virtual double	cell_estimate (const CellInfo &src)
	Local estimation. More...

virtual double	bdry_estimate (const FaceInfo &src)
	Local estimation. More...

virtual double	face_estimate (const FaceInfo &src1, const FaceInfo &src2)
	Local estimation. More...

Protected Member Functions inherited from FuelCell::ApplicationCore::ApplicationBase
void	print_caller_name (const std::string &caller_name) const
	Print caller name. More...

Detailed Description

template<int dim>
class FuelCell::Application::AppCathode< dim >

This class is used to solve a system of equations similar to the one presented in the journal article M.

Secanell et al. "Numerical Optimization of Proton Exchange Membrane Fuel Cell Cathode Electrodes", Electrochimica Acta, 52(7):2668-2682, February 2007 and also presented at the ECCM 2006 conference under the abstract titled M. Secanell et al, "A PEM fuel cell electrode model for gradient-based optimization", June 2006.

The cathode catalyst layer and the gas diffusion layer are modelled using a set of equations found in the journal article:

M. Secanell et al. "Numerical Optimization of Proton Exchange Membrane Fuel Cell Cathode Electrodes", Electrochimica Acta, 52(7):2668-2682, February 2007.

The simulation is isothermal, isotropic, models the electrochemical reactions using tafel kinetics, oxygen diffusion using Ficks Law and electronic and protonic transport using Ohms Law.

The equations solved are written as follows:

$R_1(\vec{u}) = \nabla \cdot (c_{total}D^{eff}_{O_2} \nabla x_{O_2} ) = \frac{1}{nF}\nabla\cdot i$

$R_2(\vec{u}) = \nabla \cdot (\sigma^{eff}_{m}\nabla\phi_m) = \nabla \cdot i$

$R_3(\vec{u}) = \nabla \cdot (\sigma^{eff}_{s}\nabla\phi_s) = -\nabla \cdot i$

where, for the case of a macro-homogeneous catalyst layer,

$\nabla \cdot i = A_v i^{ref}_0 \left( \frac{c_{O_2}} {c^{ref}_{O_2}} \right) \mbox{exp} \left( \frac{\alpha F}{RT}(\phi_m - \phi_s) \right)$

The solution variables, $\vec{u}$ are the protonic potential, $\phi_m$ , the electronic potential, $\phi_s$ and, instead of the oxygen concentration, we use the oxygen molar fraction, $x_{O_2}$ , that also accounts for the oxygen dissolving in the ionomer by using Henrys Law.

The govering equations above are nonlinear and therefore they cannot be solved directly.

In OpenFCST, we have decided to solve the system of equations using a nonlinear Newton solver. Therefore, instead of implemeting the equations above, cell_matrix and cell_residual implement a linearization of the governing equations above, i.e.

$\frac{\partial R_i(\vec{u^{n-1}})}{\partial u_j} u_j = R_i(\vec{u^{n-1}})$

For more information on the governing equations, discretization and solution methodology please read the following references:

M. Secanell et al. "Numerical Optimization of Proton Exchange Membrane Fuel Cell Cathode Electrodes", Electrochimica Acta, 52(7):2668-2682, February 2007.

Also, the folder /data/cathode/Secanell_EA07_Numerical_Optimization_PEMFC_Cathode_Electrodes contains a sample datafile to run this application with a macro-homogeneous model.

The application can be used to solve a cathode with and without an MPL. You can select the appropriate geometry by changing the following in the input file:

* subsection Grid generation
*   set Type of mesh = Cathode OR CathodeMPL
*   subsection Internal mesh generator parameters
*      subsection Dimensions
*       set Cathode current collector width [cm] = 0.1 # Thickness of the rib of the bipolar plates (BPP)
*       set Cathode channel width [cm] = 0.1           # Thickness of the channel on the BPP
*       set Cathode CL thickness [cm] = 1E-3           # Thickness of the cathode catalyst layer [cm]
*       set Cathode GDL thickness [cm]  = 2E-2         # Thickness of the cathode gas diffusion layer [cm]
*       set Cathode MPL thickness [cm] = 2E-3          # Thickness of the cathode microporous layer [cm]
*      end
*   end
* end
* 

The model can also be used to solve an agglomerate catalyst layer model. The governing equations are similar to the ones outlined above, however, the volumetric current density source, i.e. $\nabla \cdot i$ is obtained as specified in the following article:

M. Secanell, K. Karan, A. Suleman and N. Djilali, "Multi-Variable Optimization of PEMFC Cathodes using an Agglomerate Model ", Electrochimica Acta, 52(22):6318-6337, 2007.

A sample dataset used to solve an agglomerate cathode catalyst layer model is given in data/cathode/Secanell_EA07_MultiVariable_Optimization_PEMFC_Cathodes_Agglomerate_Model

If using this function please cite the articles above as references.

Author: Marc Secanell and Valentin N. Zingan, 2009-2014

Constructor & Destructor Documentation

template<int dim>

FuelCell::Application::AppCathode< dim >::AppCathode ( boost::shared_ptr< FuelCell::ApplicationCore::ApplicationData > data = boost::shared_ptr< FuelCell::ApplicationCore::ApplicationData >() )

Constructor.

template<int dim>

FuelCell::Application::AppCathode< dim >::~AppCathode ( )

Destructor.

Member Function Documentation

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::cell_matrix	(	MatrixVector &	cell_matrices,
		const typename DoFApplication< dim >::CellInfo &	cell_info
	)

virtual

Assemble local cell matrix.

Reimplemented from FuelCell::ApplicationCore::BlockMatrixApplication< dim >.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::cell_residual	(	FuelCell::ApplicationCore::FEVector &	cell_res,
		const typename DoFApplication< dim >::CellInfo &	cell_info
	)

virtual

Assemble local cell residual.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::cell_responses	(	std::vector< double > &	dst,
		const typename DoFApplication< dim >::CellInfo &	cell_info,
		const FEVector &
	)

virtual

Compute some functionals.

Reimplemented from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::cell_responses_aux	(	std::vector< double > &	dst,
		const typename DoFApplication< dim >::CellInfo &	cell_info,
		const FEVector &
	)

privatevirtual

Compute some functionals that are not needed for most applications (this section is not necessary in most cases.)

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::data_out	(	const std::string &	filename,
		const FEVectors &	src
	)

virtual

Output results.

Reimplemented from FuelCell::ApplicationCore::DoFApplication< dim >.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::declare_parameters ( ParameterHandler & param )

virtual

Declare parameters.

Reimplemented from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::dirichlet_bc ( std::map< unsigned int, double > & boundary_values ) const

virtual

BCs.

Reimplemented from FuelCell::ApplicationCore::BlockMatrixApplication< dim >.

template<int dim>

virtual double FuelCell::Application::AppCathode< dim >::evaluate ( const FuelCell::ApplicationCore::FEVectors & src )

virtual

Returns the current density.

Reimplemented from FuelCell::ApplicationCore::DoFApplication< dim >.

template<int dim>

void FuelCell::Application::AppCathode< dim >::global_responses	(	std::vector< double > &	resp,
		const FuelCell::ApplicationCore::FEVector &	sol
	)

virtual

This class is used to evaluate all responses that do not require looping over cells.

An example of one of this types of constraints is the solid volume fraction.

Reimplemented from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::initialize ( ParameterHandler & param )

virtual

Initialize parameters.

Reimplemented from FuelCell::ApplicationCore::OptimizationBlockMatrixApplication< dim >.

template<int dim>

virtual void FuelCell::Application::AppCathode< dim >::initialize_solution	(	FEVector &	initial_guess,
		std::shared_ptr< Function< dim > >	initial_function = `std::shared_ptr< Function< dim > >()`
	)