This is a base class for transient solvers. More...

#include <transient_base.h>

Inheritance diagram for FuelCell::ApplicationCore::TransientBase:

Collaboration diagram for FuelCell::ApplicationCore::TransientBase:

Public Member Functions
	TransientBase (ApplicationBase &app)
	Constructor, receiving an application with transient model and a data object. More...

virtual void	declare_parameters (ParameterHandler &param)
	Declare input parameters in section "Transient" of parameter file. More...

virtual void	initialize (ParameterHandler &param)
	Read the parameters of TransientBase. More...

virtual void	solve (FuelCell::ApplicationCore::FEVector &u, const FuelCell::ApplicationCore::FEVectors &in_vectors)=0
	The actual transient solver. More...

Public Member Functions inherited from FuelCell::ApplicationCore::ApplicationWrapper
	ApplicationWrapper (ApplicationBase &app)
	Constructor for a derived application. More...

	~ApplicationWrapper ()
	Destructor. More...

virtual void	remesh ()
	Generate the next mesh depending on the mesh generation parameters. More...

virtual void	init_vector (FEVector &dst) const
	Initialize vector to problem size. More...

virtual double	residual (FEVector &dst, const FEVectors &src, bool apply_boundaries=true)
	Compute residual of `src` and store it into `dst`. More...

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

virtual double	estimate (const FEVectors &src)
	Estimate cell-wise errors. More...

virtual double	evaluate (const FEVectors &src)
	Evaluate a functional. More...

virtual void	grid_out (const std::string &filename) const

virtual void	data_out (const std::string &filename, const FEVectors &src)
	Write data in the format specified by the ParameterHandler. 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...

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	grid_out (const std::string &)
	Write the mesh in the format specified by the ParameterHandler. 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 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 Member Functions
void	debug_output (const FEVector &sol)
	Function used to output any necessary information at each time layer for debugging purposes. More...

void	responses_output ()
	Function used to output boundary responses computed by application. More...

void	error_output (const double &err)
	Function used to output the solution error on each layer. More...

void	tau_output ()
	Function used to output time step on each time layer. More...

void	initialize_u_old (const FEVector u, const double n_ref, const FEVector old_solution, const bool force=true)
	If we are inside the adaptive refinement loop, then we receive the old solution from class since it was transferred to a finer mesh and has now a different size. More...

void	ats_Richardson (FEVector &u, const FEVector u1, const FEVector u2, int scheme_order, bool &solution_accepted)
	This routine estimates solution error and changes the time step using Richardson extrapolation. More...

bool	file_exists (const std::string &file)

Protected Member Functions inherited from FuelCell::ApplicationCore::ApplicationWrapper
SmartPointer< ApplicationBase >	get_wrapped_application ()
	Gain access to the inner application. More...

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

Protected Attributes
bool	print_last_layer_only
	If this variable is set to true, then only the last time layer data will be printed. More...

bool	debug_solution
	Print updated solution after each step into file `solution_NNNNN.dat`. More...

unsigned int	debug
	Write debug output to FcstUtilities::log; the higher the number, the more output. More...

unsigned int	step
	Number of the current time layer. More...

std::string	ats_method
	Decision varible for selecting the adaptive time stepping method. More...

double	tau
	Initial time step size. More...

double	max_tau
	Maximal time step size. More...

double	min_tau
	Minimal time step size. More...

int	max_ats_it
	Maximal number of adaptive time step iterations used to resolve each time layer. More...

FEVector	u_old
	A vector to store the solution from the previous time layer in. More...

int	ar_freq
	If adaptive refinement is selected, then it will be performed at each `ar_freq`-th layer. More...

double	total_time
	Total time of simulation in seconds. More...

double	time
	Current time of simulation. More...

double	abs_err_estimate

double	rel_err_estimate

double	abs_error_tolerance

double	rel_error_tolerance

bool	output_error

bool	output_tau

std::ofstream	responses_outfile
	Variable used in printing responses data into a file. More...

std::ofstream	error_outfile
	Variable used in printing solution error data into a file. More...

std::ofstream	tau_outfile
	Variable used in printing time step into a file. More...

const std::string	fname_err ="Solution_error.dat"
	File name for solution error output. More...

const std::string	fname_tau ="Time_step.dat"
	File name for solution error output. More...

std::vector< double >	all_time_values
	A vector of all time values. More...

bool	everything_is_fine =true

unsigned int	n_ref

Protected Attributes inherited from FuelCell::ApplicationCore::ApplicationWrapper
SmartPointer< ApplicationBase >	app
	Pointer to the application this one depends upon. 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
const std::string	get_time_stepping_method_names () const
	This function returns the string of names for the adaptive time stepping methods. More...

Private Attributes
std::string	fname
	File name for debug output. More...

std::ofstream	streamOut
	Variable for streaming to file. More...

double	precaution_factor
	Precaution factor in optimal time step computation. More...

Detailed Description

This is a base class for transient solvers.

All it does is sets some basic parameters common for all numerical methods implementing marching in time (such as time step and its value's limits, debug level for output, etc.) and outputs solution to a .dat file if required.

Default parameters in data input file are as follows:

subsection Transient
 set Initial time step, [s] = 0.1
 set Maximal time step, [s] = 1
 set Minimal time step, [s] = 0.01
 set Debug level = 0
 set Debug solution = false
end

Author: Aslan Kosakian, 2015

Constructor & Destructor Documentation

FuelCell::ApplicationCore::TransientBase::TransientBase ( ApplicationBase & app )

Constructor, receiving an application with transient model and a data object.

Member Function Documentation

void FuelCell::ApplicationCore::TransientBase::ats_Richardson	(	FEVector &	u,
		const FEVector	u1,
		const FEVector	u2,
		int	scheme_order,
		bool &	solution_accepted
	)

protected

This routine estimates solution error and changes the time step using Richardson extrapolation.

Assuming $ m $ -th order method,

$u^*=u_1+c\tau^m+\mathcal{O}\left(\tau^{m+1}\right),$

$u^*=u_2+c\frac{\tau^m}{2^m}+\mathcal{O}\left(\tau^{m+1}\right),$

where $ u^* $ is the exact solution, $ u_1 $ is the solution obtained from previous time layer with a single time step of the size $\tau$ and $ u_2 $ is the solution obtained from the same previous time layer with two time steps of the size $\tau/2$ . From these two relations, we can find constant $ c $ :

$c = \frac{u_2-u_1}{\tau^m}\frac{2^m}{2^m-1}.$

Decision on time step change is made based on the absolute and relative difference between solutions $ u_1 $ and $ u_2 $ in some norm, here - in the l2 norm defined as L2 norm of the solution divided by the square root of number of DoFs minus one. Suppose that we want to find such step $\hat{\tau}$ that the solution $ u_2 $ (computed with that new time step) satisfies $\lVert u^* - u_2 \rVert \approx \mathrm{TOL}_\mathrm{abs}$ , where $\mathrm{TOL}_\mathrm{abs}$ is absolute solution error tolerance prescribed by the user.

Then, given

$u^*=u_2+c\frac{\hat{\tau}^m}{2^m}+\mathcal{O}\left(\hat{\tau}^{m+1}\right),$

we can assume

$\lVert c \rVert \frac{\hat{\tau}^m}{2^m} \approx \mathrm{TOL}_\mathrm{abs},$

from where we get

$\hat{\tau}^m \approx \frac{\mathrm{TOL}_\mathrm{abs} \tau^m \left( 2^m-1 \right)}{\lVert u_2 - u_1 \rVert}.$

In practice, a precaution factor $\omega$ is introduced:

$\hat{\tau} = \omega \tau \sqrt[m]{ \frac{\mathrm{TOL}_\mathrm{abs} \left( 2^m-1 \right)}{\lVert u_2 - u_1 \rVert} }.$

For most of the cases, value $\omega=0.9$ can be safely used [1].

To satisfy the relative error tolerance $\mathrm{TOL}_\mathrm{rel}$ as well, we want

$\frac{\lVert u^* - u_2 \rVert}{\lVert u_2 \rVert} \approx \frac{\lVert c \rVert }{\lVert u_2 \rVert}\frac{\hat{\tau}^m}{2^m} \approx \mathrm{TOL}_\mathrm{rel}.$

In this case, we get

$\hat{\tau} = \omega \tau \sqrt[m]{ \frac{\mathrm{TOL}_\mathrm{rel} \lVert u_2 \rVert \left( 2^m-1 \right)}{\lVert u_2 - u_1 \rVert} }.$

The resulting time step is chosen as minimum between the two values obtained based on absolute and relative error estimation.

The resulting solution

$u^{n+1}=u_2+\frac{u_2-u_1}{2^m-1}+\mathcal{O}\left(\tau^{m+1}\right)$

has accuracy of order $ m+1 $ .

Note: This implementation is guaranteed to work only with Euler methods $(\theta=0,~\theta=1)$ and Crank-Nicolson scheme $(\theta=0.5)$ . It is done this way because otherwise the user has to perform a numerical analysis of the scheme and provide the order of it in the parameter file. If a different value of $\theta$ is specified, a constant time step of initial value will be used.

References

Faragó, István, Ágnes Havasi, and Zahari Zlatev. "Efficient implementation of stable Richardson extrapolation algorithms." Computers & Mathematics with Applications 60.8 (2010): 2309-2325.

Note: In [1], the ratio under the square root in the equation for the optimal step size is mistakenly flipped over. It makes sense to use smaller step if $\mathrm{ERROR}_n>\mathrm{TOL}$ in the paper and larger step if $\mathrm{ERROR}_n<\mathrm{TOL}$ .

void FuelCell::ApplicationCore::TransientBase::debug_output ( const FEVector & sol )

protected

Function used to output any necessary information at each time layer for debugging purposes.

virtual void FuelCell::ApplicationCore::TransientBase::declare_parameters ( ParameterHandler & param )

virtual

Declare input parameters in section "Transient" of parameter file.

Reimplemented from FuelCell::ApplicationCore::ApplicationWrapper.

Reimplemented in FuelCell::ApplicationCore::ThetaScheme.

void FuelCell::ApplicationCore::TransientBase::error_output ( const double & err )

protected

Function used to output the solution error on each layer.

Data can be used to plot solution error vs. time and see if there were any oscillations.

This is the most convenient way to check stability of the numerical scheme (after theoretical analysis, of course, if it is not too hard to do). If the plot is not monotonous, then you might need to use a smaller time step, a finer mesh, or try adaptive time stepping algorithms.

bool FuelCell::ApplicationCore::TransientBase::file_exists ( const std::string & file )

inlineprotected

const std::string FuelCell::ApplicationCore::TransientBase::get_time_stepping_method_names ( ) const

inlineprivate

This function returns the string of names for the adaptive time stepping methods.

virtual void FuelCell::ApplicationCore::TransientBase::initialize ( ParameterHandler & param )

virtual

Read the parameters of TransientBase.

Those are general parameters for transient loop, e.g. time step.

Reimplemented from FuelCell::ApplicationCore::ApplicationWrapper.

Reimplemented in FuelCell::ApplicationCore::ThetaScheme.

void FuelCell::ApplicationCore::TransientBase::initialize_u_old	(	const FEVector	u,
		const double	n_ref,
		const FEVector	old_solution,
		const bool	force = `true`
	)

protected

If we are inside the adaptive refinement loop, then we receive the old solution from class since it was transferred to a finer mesh and has now a different size.

The transferred solution will be stored in the local variable u_old. If no adaptive refinement is used, then u_old is initialized with u. Variable force is used in adaptive time stepping algorithm. Forced initialization, i.e., initialization with force=true, sets u_old to u, which is previous iterate of the model - it can be different from the old solution u^n in transient sense. For example, when we do two half-steps in Richardson extrapolation, the first half-step uses the old_solution solution from class and the second half-step - the old solution in a global sense, u, which was obtained from the first half-step.

void FuelCell::ApplicationCore::TransientBase::responses_output ( )

protected

Function used to output boundary responses computed by application.

virtual void FuelCell::ApplicationCore::TransientBase::solve	(	FuelCell::ApplicationCore::FEVector &	u,
		const FuelCell::ApplicationCore::FEVectors &	in_vectors
	)

pure virtual

The actual transient solver.

Performs loop over time.

This function's definition is used in classes derived from TransientBase (such as ThetaScheme), but not in current class itself.

Reimplemented from FuelCell::ApplicationCore::ApplicationWrapper.

Implemented in FuelCell::ApplicationCore::ThetaScheme.

void FuelCell::ApplicationCore::TransientBase::tau_output ( )

protected

Function used to output time step on each time layer.

Data can be used to analyze the adaptive time stepping algorithm that was used in the simulation.

Member Data Documentation

double FuelCell::ApplicationCore::TransientBase::abs_err_estimate

protected

double FuelCell::ApplicationCore::TransientBase::abs_error_tolerance

protected

std::vector<double> FuelCell::ApplicationCore::TransientBase::all_time_values

protected

A vector of all time values.

Used in the funciton.

int FuelCell::ApplicationCore::TransientBase::ar_freq

protected

If adaptive refinement is selected, then it will be performed at each ar_freq-th layer.

std::string FuelCell::ApplicationCore::TransientBase::ats_method

protected

Decision varible for selecting the adaptive time stepping method.

This variable is set in the input file in:

* subsection Transient
*    set Adaptive stepping algorithm = 
* end
* 

Possible options are:

None: Constant time step will be used.
Richardson: Time stepping algorithm based on Richardson extrapolation will be used.

unsigned int FuelCell::ApplicationCore::TransientBase::debug

protected

Write debug output to FcstUtilities::log; the higher the number, the more output.

bool FuelCell::ApplicationCore::TransientBase::debug_solution

protected

Print updated solution after each step into file solution_NNNNN.dat.

std::ofstream FuelCell::ApplicationCore::TransientBase::error_outfile

protected

Variable used in printing solution error data into a file.

bool FuelCell::ApplicationCore::TransientBase::everything_is_fine =true

protected

std::string FuelCell::ApplicationCore::TransientBase::fname

private

File name for debug output.

const std::string FuelCell::ApplicationCore::TransientBase::fname_err ="Solution_error.dat"

protected

File name for solution error output.

const std::string FuelCell::ApplicationCore::TransientBase::fname_tau ="Time_step.dat"

protected

File name for solution error output.

int FuelCell::ApplicationCore::TransientBase::max_ats_it

protected

Maximal number of adaptive time step iterations used to resolve each time layer.

No convergence error will be thrown if the number of iterations exceeds this value.

double FuelCell::ApplicationCore::TransientBase::max_tau

protected

Maximal time step size.

Used in adaptive time stepping.

double FuelCell::ApplicationCore::TransientBase::min_tau

protected

Minimal time step size.

Used in adaptive time stepping.

unsigned int FuelCell::ApplicationCore::TransientBase::n_ref

protected

bool FuelCell::ApplicationCore::TransientBase::output_error

protected

bool FuelCell::ApplicationCore::TransientBase::output_tau

protected

double FuelCell::ApplicationCore::TransientBase::precaution_factor

private

Precaution factor in optimal time step computation.

bool FuelCell::ApplicationCore::TransientBase::print_last_layer_only

protected

If this variable is set to true, then only the last time layer data will be printed.

double FuelCell::ApplicationCore::TransientBase::rel_err_estimate

protected

double FuelCell::ApplicationCore::TransientBase::rel_error_tolerance

protected

std::ofstream FuelCell::ApplicationCore::TransientBase::responses_outfile

protected

Variable used in printing responses data into a file.

unsigned int FuelCell::ApplicationCore::TransientBase::step

protected

Number of the current time layer.

Time layer 0 contains initial solution.

std::ofstream FuelCell::ApplicationCore::TransientBase::streamOut

private

Variable for streaming to file.

double FuelCell::ApplicationCore::TransientBase::tau

protected

Initial time step size.

Note: Can be modified by the transient solver if an adaptive time stepping algorithm is used.

std::ofstream FuelCell::ApplicationCore::TransientBase::tau_outfile

protected

Variable used in printing time step into a file.

double FuelCell::ApplicationCore::TransientBase::time

protected

Current time of simulation.

double FuelCell::ApplicationCore::TransientBase::total_time

protected

Total time of simulation in seconds.

FEVector FuelCell::ApplicationCore::TransientBase::u_old

protected

A vector to store the solution from the previous time layer in.

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

transient_base.h

Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation