Class to solve the monodomain equation with Fitzhugh-Nagumo ionic model for the cardiac electrophysiology using the discontinuous Galerkin method. More...

#include <monodomain_fhn_dg.hpp>

Inheritance diagram for DUBeat::models::MonodomainFHNDG< basis >:

[legend]

Collaboration diagram for DUBeat::models::MonodomainFHNDG< basis >:

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Public Member Functions
	MonodomainFHNDG ()
	Constructor. More...

Public Member Functions inherited from ModelDG_t< basis >
	ModelDG_t (std::string model_name)
	Constructor. More...

	ModelDG_t (ModelDG_t< basis > &ModelDG_t)=default
	Default copy constructor. More...

	ModelDG_t (const ModelDG_t< basis > &ModelDG_t)=default
	Default const copy constructor. More...

	ModelDG_t (ModelDG_t< basis > &&ModelDG_t)=default
	Default move constructor. More...

virtual	~ModelDG_t ()=default
	Destructor. More...

Public Member Functions inherited from ModelDG< basis >
	ModelDG (std::string model_name)
	Constructor. More...

	ModelDG (ModelDG< basis > &ModelDG)=default
	Default copy constructor. More...

	ModelDG (const ModelDG< basis > &ModelDG)=default
	Default const copy constructor. More...

	ModelDG (ModelDG< basis > &&ModelDG)=default
	Default move constructor. More...

unsigned int	get_dofs_per_cell () const
	Return the number of degrees of freedom per element. More...

virtual	~ModelDG ()=default
	Destructor. More...

Private Member Functions
virtual void	declare_parameters (lifex::ParamHandler &params) const override
	Override for declaration of additional parameters. More...

virtual void	parse_parameters (lifex::ParamHandler &params) override
	Override to parse additional parameters. More...

void	run () override
	Override for the simulation run. More...

void	update_time () override
	Override to update time for both u and w. More...

void	time_initialization () override
	Override to initialize both u and w. More...

void	assemble_system () override
	Assembly of the Monodomain system. More...

Private Attributes
double	ChiM
	Monodomain equation parameter. More...

double	Sigma
	Diffusion scalar parameter. More...

double	Cm
	Membrane capacity. More...

double	kappa
	Factor for the nonlinear reaction in Fitzhugh Nagumo model. More...

double	epsilon
	ODE parameter. More...

double	gamma
	ODE parameter. More...

double	a
	ODE parameter. More...

lifex::LinAlg::MPI::Vector	solution_owned_w
	Solution gating variable, without ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_w
	Solution gating variable, with ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_ex_owned_w
	Solution exact gating variable, without ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_ex_w
	Solution exact gating variable, without ghost entries. More...

std::shared_ptr< lifex::utils::FunctionDirichlet >	w_ex
	dealii::Pointer to exact solution function gating variable. More...

std::shared_ptr< dealii::Function< lifex::dim > >	grad_w_ex
	dealii::Pointer to exact gradient solution Function gating variable More...

lifex::utils::BDFHandler< lifex::LinAlg::MPI::Vector >	bdf_handler_w
	BDF time advancing handler. More...

lifex::LinAlg::MPI::Vector	solution_bdf_w
	BDF solution, with ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_ext_w
	BDF extrapolated solution, with ghost entries. More...

lifex::LinAlg::MPI::SparseMatrix	matrix_t0
	Component of the system matrix that does not depend on time. More...

Additional Inherited Members
Protected Member Functions inherited from ModelDG_t< basis >
virtual void	intermediate_error_print (const lifex::LinAlg::MPI::Vector &solution_owned, const lifex::LinAlg::MPI::Vector &solution_ex_owned, const std::shared_ptr< dealii::Function< lifex::dim >> &u_ex, const char solution_name=(char )"u")
	Computation of the \(L^\infty\) error at an intermediate time-step. More...

void	run () override
	Override for the simulation run. More...

Protected Member Functions inherited from ModelDG< basis >
virtual void	setup_system ()
	Setup of the problem before the resolution. More...

void	make_sparsity_pattern (const DoFHandlerDG< basis > &dof, dealii::DynamicSparsityPattern &sparsity, const dealii::AffineConstraints< double > &constraints=dealii::AffineConstraints< double >(), const bool keep_constrained_dofs=true, const dealii::types::subdomain_id subdomain_id=dealii::numbers::invalid_subdomain_id)
	Creation of the sparsity pattern to assign to the system matrix before assembling. More...

void	initialize_solution (lifex::LinAlg::MPI::Vector &solution_owned, lifex::LinAlg::MPI::Vector &solution)
	To inizialize the solutions using the deal.II reinit. More...

void	create_mesh ()
	Load the mesh from the default path. More...

void	create_mesh (std::string mesh_path)
	Load the mesh from a user-defined path. More...

void	solve_system ()
	System solving. More...

void	compute_errors (const lifex::LinAlg::MPI::Vector &solution_owned, const lifex::LinAlg::MPI::Vector &solution_ex_owned, const std::shared_ptr< dealii::Function< lifex::dim >> &u_ex, const std::shared_ptr< dealii::Function< lifex::dim >> &grad_u_ex, const char solution_name=(char )"u") const
	To compute the \(L^\infty\) error, the \(L^2\) error, the \(H^1\) error and the \(DG\) error at the end of system solving, it exploits the DGComputeErrors<basis> class. More...

virtual void	output_results (std::string output_name="solution") const
	Output of results. More...

virtual void	conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned)
	To convert a discretized solution from modal to nodal basis (does nothing if problem is already in nodal basis), in-place version. More...

virtual lifex::LinAlg::MPI::Vector	conversion_to_fem (const lifex::LinAlg::MPI::Vector &sol_owned) const
	To convert a discretized solution from modal to nodal basis (does nothing if problem is already in nodal basis), const version. More...

virtual void	conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem=1, const double scaling_factor=1) const

virtual void	conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned)
	To convert a discretized solution in Dubiner basis (only for problems using Dubiner basis). More...

virtual lifex::LinAlg::MPI::Vector	conversion_to_dub (const lifex::LinAlg::MPI::Vector &sol_owned) const
	To convert a discretized solution in Dubiner basis (only for problems using Dubiner basis). More...

virtual void	conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem=1, const double scaling_factor=1) const

void	discretize_analytical_solution (const std::shared_ptr< dealii::Function< lifex::dim >> &u_analytical, lifex::LinAlg::MPI::Vector &sol_owned)
	Conversion of an analytical solution from FEM to basis coefficients. More...

void	conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned)
	Conversion of a discretized solution from Dubiner coefficients to FEM coefficients. More...

lifex::LinAlg::MPI::Vector	conversion_to_fem (const lifex::LinAlg::MPI::Vector &sol_owned) const
	Conversion to FEM coefficients, const version. More...

void	conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem, const double scaling_factor) const

void	conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned)
	Conversion of a discretized solution from FEM coefficients to Dubiner coefficients. More...

lifex::LinAlg::MPI::Vector	conversion_to_dub (const lifex::LinAlg::MPI::Vector &sol_owned) const
	Conversion to DUB coefficients, const version. More...

void	conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem, const double scaling_factor) const

void	discretize_analytical_solution (const std::shared_ptr< dealii::Function< lifex::dim >> &u_analytical, lifex::LinAlg::MPI::Vector &sol_owned)
	Conversion of an analytical solution from FEM to basis coefficients. More...

void	discretize_analytical_solution (const std::shared_ptr< dealii::Function< lifex::dim >> &u_analytical, lifex::LinAlg::MPI::Vector &sol_owned)
	Conversion of an analytical solution from FEM to basis coefficients. More...

Protected Attributes inherited from ModelDG_t< basis >
double	prm_time_init
	Initial time. More...

double	prm_time_final
	Final time. More...

double	prm_time_step
	Time-step amplitude. More...

unsigned int	prm_bdf_order
	BDF order. More...

double	time
	Current time. More...

unsigned int	timestep_number
	Current time-step number. More...

lifex::utils::BDFHandler< lifex::LinAlg::MPI::Vector >	bdf_handler
	BDF time advancing handler. More...

lifex::LinAlg::MPI::Vector	solution_bdf
	BDF solution, with ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_ext
	BDF extrapolated solution, with ghost entries. More...

Protected Attributes inherited from ModelDG< basis >
const std::string	model_name
	Name of the class/problem. More...

unsigned int	prm_fe_degree
	Polynomials degree. More...

unsigned int	prm_n_refinements
	Mesh refinement level (>=1). More...

double	prm_penalty_coeff
	DG Penalty coefficient. More...

double	prm_stability_coeff
	DG stabilty coefficient. More...

double	scaling_factor = 1
	Scaling factor. More...

std::shared_ptr< lifex::utils::MeshHandler >	triangulation
	Triangulation (internal use for useful already implemented methods). More...

unsigned int	dofs_per_cell
	Number of degrees of freedom per cell. More...

DoFHandlerDG< basis >	dof_handler
	DoFHandler (internal use for useful already implemented methods). More...

std::shared_ptr< DUBFEMHandler< basis > >	dub_fem_values
	Member used for conversions between analytical, nodal and modal representations of the solutions. More...

std::unique_ptr< AssembleDG< basis > >	assemble
	Matrix assembler. More...

lifex::utils::LinearSolverHandler< lifex::LinAlg::MPI::Vector >	linear_solver
	Linear solver handler. More...

lifex::utils::PreconditionerHandler	preconditioner
	Preconditioner handler. More...

lifex::LinAlg::MPI::SparseMatrix	matrix
	Distributed matrix of the linear system. More...

lifex::LinAlg::MPI::Vector	rhs
	Distributed right hand side vector of the linear system. More...

lifex::LinAlg::MPI::Vector	solution_owned
	Distributed solution vector, without ghost entries. More...

lifex::LinAlg::MPI::Vector	solution
	Distributed solution vector, with ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_ex_owned
	Distributed exact solution vector, without ghost entries. More...

lifex::LinAlg::MPI::Vector	solution_ex
	Distributed exact solution vector, without ghost entries. More...

std::shared_ptr< lifex::utils::FunctionDirichlet >	u_ex
	Pointer to exact solution function. More...

std::shared_ptr< dealii::Function< lifex::dim > >	grad_u_ex
	Pointer to exact gradient solution Function. More...

std::shared_ptr< dealii::Function< lifex::dim > >	f_ex
	Known forcing term. More...

std::shared_ptr< dealii::Function< lifex::dim > >	g_n
	Neumann boundary conditions. More...

Detailed Description

template<class basis>
class DUBeat::models::MonodomainFHNDG< basis >

Class to solve the monodomain equation with Fitzhugh-Nagumo ionic model for the cardiac electrophysiology using the discontinuous Galerkin method.

\[ \begin{aligned} \chi_m C_m \frac{\partial V_m}{\partial t} -\nabla \cdot (\Sigma \nabla V_m)+\chi_m I_{ion}(V_m, w) &= I_{ext} & \quad & \text{in } \Omega = (-1, 1)^d \times (0, T], \\ I_{ion}(V_m, w) &= k V_m (V_m - a)(V_m - 1) + w & \quad & \text{in } \Omega \times (0, T], \\ \frac{\partial w}{\partial t} &= \epsilon (V_m - \gamma w) & \quad & \text{in } \Omega \times (0, T], \\ \Sigma \frac{\partial V_m}{\partial n} &= g & \quad & \text{on } \partial\Omega \times (0, T], \\ V_m &= V_{m_\mathrm{ex}} & \quad & \text{in } \Omega \times \{ t = 0 \}, \\ w &= w_\mathrm{ex} & \quad & \text{in } \Omega \times \{ t = 0 \} \end{aligned} \]

where \( I_{ion}(V_m, w)\) is defined through the FitzHugh-Nagumo model.

In particular, it can be solved using the Lagrangian basis (basis=dealii::FE_SimplexDGP<lifex::dim>) or the Dubiner basis (basis=DUBValues<lifex::dim>).

The problem is time-discretized using the implicit finite difference scheme \(\mathrm{BDF}\sigma\) (where \(\sigma = 1,2,...\) is the order of the BDF formula) as follows:

\[ \begin{aligned} \chi_m C_m ( \frac{\alpha_{\mathrm{BDF}\sigma} V_m^{n+1} - V_{m_{\mathrm{BDF}\sigma}}^n}{\Delta t}) -\nabla \cdot (\Sigma \nabla V_m^{n+1})+\chi_m I_{ion}(V_m^{n+1}, w^n) &= I_{ext}^{n+1} & \quad & \text{in } \Omega = (-1, 1)^d \times \{n=0, \dots, N \}, \\ I_{ion}(V_m^{n+1}, w^n) &= k V_m^{n+1} (V_m^n - a)(V_m^n - 1) + w^n & \quad & \text{in } \Omega \times \{n=0, \dots, N \}, \\ \frac{\alpha_{\mathrm{BDF}\sigma} w^{n+1} - w_{\mathrm{BDF}\sigma}^n}{\Delta t} &= \epsilon (V_m^{n+1} - \gamma w^n) & \quad & \text{in } \Omega \times \{n=0, \dots, N \}, \\ \Sigma \frac{\partial V_m^{n+1}}{\partial n} &= g & \quad & \text{on } \partial\Omega \times \{n=0, \dots, N \}, \\ V_m^0 &= V_{m_\mathrm{ex}} & \quad & \text{in } \Omega \times \{ n = 0 \}, \\ w^0 &= w_\mathrm{ex} & \quad & \text{in } \Omega \times \{ n = 0 \} \end{aligned} \]

where \(\Delta t = t^{n+1}-t^{n}\) is the time step.

Boundary conditions, initial conditions and source terms are provided assuming that the exact solution is:

\[ \begin{aligned} d=2: \: &V_{m_\mathrm{ex}}(x,y) &= \sin(2\pi x)\sin(2\pi y)e^{-5t}, \hspace{6mm} &(x,y) &\in \Omega=(1,1)^2&, t \in [0,T], \\ &w_{\mathrm{ex}}(x,y) &= \frac{\epsilon}{\epsilon\cdot\gamma -5}\sin(2\pi x)\sin(2\pi y)e^{-5t}, \hspace{6mm} &(x,y) &\in \Omega=(1,1)^2&, t \in [0,T], \\ d=3: \: &V_{m_\mathrm{ex}}(x,y,z) &= \sin\left(2\pi x + \frac{\pi}{4}\right)\sin\left(2\pi y + \frac{\pi}{4}\right)\sin\left(2\pi z + \frac{\pi}{4}\right) e^{-5t}, \hspace{6mm} &(x,y,z) &\in \Omega=(1,1)^3&, t \in [0,T], \\ &w_{\mathrm{ex}}(x,y,z) &= \frac{\epsilon}{\epsilon\cdot\gamma -5} \sin(2\pi x)\sin(2\pi y)\sin(2\pi z) e^{-5t}, \hspace{6mm} &(x,y,z) &\in \Omega=(1,1)^3&, t \in [0,T]. \end{aligned} \]

Finally, \(d\) is specified in the lifex configuration and \(T\) as well as the monodomain scalar parameters in the .prm parameter file.

Definition at line 446 of file monodomain_fhn_dg.hpp.

Constructor & Destructor Documentation

◆ MonodomainFHNDG()

template<class basis >

DUBeat::models::MonodomainFHNDG< basis >::MonodomainFHNDG ( )

inline

Constructor.

Definition at line 340 of file monodomain_fhn_dg.hpp.

Member Function Documentation

◆ assemble_system()

template<class basis >

void DUBeat::models::MonodomainFHNDG< basis >::assemble_system

overrideprivatevirtual

Assembly of the Monodomain system.

Implements ModelDG< basis >.

Definition at line 780 of file monodomain_fhn_dg.hpp.

◆ declare_parameters()

template<class basis >

void DUBeat::models::MonodomainFHNDG< basis >::declare_parameters ( lifex::ParamHandler & params ) const

overrideprivatevirtual

Override for declaration of additional parameters.

Reimplemented from ModelDG_t< basis >.

Definition at line 518 of file monodomain_fhn_dg.hpp.

◆ parse_parameters()

template<class basis >

void DUBeat::models::MonodomainFHNDG< basis >::parse_parameters ( lifex::ParamHandler & params )

overrideprivatevirtual

Override to parse additional parameters.

Reimplemented from ModelDG_t< basis >.

Definition at line 612 of file monodomain_fhn_dg.hpp.

◆ run()

template<class basis >

void DUBeat::models::MonodomainFHNDG< basis >::run

overrideprivatevirtual

Override for the simulation run.

Reimplemented from ModelDG< basis >.

Definition at line 662 of file monodomain_fhn_dg.hpp.

◆ time_initialization()

template<class basis >

void DUBeat::models::MonodomainFHNDG< basis >::time_initialization

overrideprivatevirtual

Override to initialize both u and w.

Reimplemented from ModelDG_t< basis >.

Definition at line 746 of file monodomain_fhn_dg.hpp.

◆ update_time()

template<class basis >

void DUBeat::models::MonodomainFHNDG< basis >::update_time

overrideprivatevirtual

Override to update time for both u and w.

Reimplemented from ModelDG_t< basis >.

Definition at line 724 of file monodomain_fhn_dg.hpp.

Member Data Documentation

◆ a

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::a

private

ODE parameter.

Definition at line 468 of file monodomain_fhn_dg.hpp.

◆ bdf_handler_w

template<class basis >

lifex::utils::BDFHandler<lifex::LinAlg::MPI::Vector> DUBeat::models::MonodomainFHNDG< basis >::bdf_handler_w

private

BDF time advancing handler.

Definition at line 482 of file monodomain_fhn_dg.hpp.

◆ ChiM

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::ChiM

private

Monodomain equation parameter.

Definition at line 456 of file monodomain_fhn_dg.hpp.

◆ Cm

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::Cm

private

Membrane capacity.

Definition at line 460 of file monodomain_fhn_dg.hpp.

◆ epsilon

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::epsilon

private

ODE parameter.

Definition at line 464 of file monodomain_fhn_dg.hpp.

◆ gamma

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::gamma

private

ODE parameter.

Definition at line 466 of file monodomain_fhn_dg.hpp.

◆ grad_w_ex

template<class basis >

std::shared_ptr<dealii::Function<lifex::dim> > DUBeat::models::MonodomainFHNDG< basis >::grad_w_ex

private

dealii::Pointer to exact gradient solution Function gating variable

Definition at line 480 of file monodomain_fhn_dg.hpp.

◆ kappa

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::kappa

private

Factor for the nonlinear reaction in Fitzhugh Nagumo model.

Definition at line 462 of file monodomain_fhn_dg.hpp.

◆ matrix_t0

template<class basis >

lifex::LinAlg::MPI::SparseMatrix DUBeat::models::MonodomainFHNDG< basis >::matrix_t0

private

Component of the system matrix that does not depend on time.

Definition at line 489 of file monodomain_fhn_dg.hpp.

◆ Sigma

template<class basis >

double DUBeat::models::MonodomainFHNDG< basis >::Sigma

private

Diffusion scalar parameter.

Definition at line 458 of file monodomain_fhn_dg.hpp.

◆ solution_bdf_w

template<class basis >

lifex::LinAlg::MPI::Vector DUBeat::models::MonodomainFHNDG< basis >::solution_bdf_w

private

BDF solution, with ghost entries.

Definition at line 484 of file monodomain_fhn_dg.hpp.

◆ solution_ex_owned_w

template<class basis >

lifex::LinAlg::MPI::Vector DUBeat::models::MonodomainFHNDG< basis >::solution_ex_owned_w

private

Solution exact gating variable, without ghost entries.

Definition at line 474 of file monodomain_fhn_dg.hpp.

◆ solution_ex_w

template<class basis >

lifex::LinAlg::MPI::Vector DUBeat::models::MonodomainFHNDG< basis >::solution_ex_w

private

Solution exact gating variable, without ghost entries.

Definition at line 476 of file monodomain_fhn_dg.hpp.

◆ solution_ext_w

template<class basis >

lifex::LinAlg::MPI::Vector DUBeat::models::MonodomainFHNDG< basis >::solution_ext_w

private

BDF extrapolated solution, with ghost entries.

Definition at line 486 of file monodomain_fhn_dg.hpp.

◆ solution_owned_w

template<class basis >

lifex::LinAlg::MPI::Vector DUBeat::models::MonodomainFHNDG< basis >::solution_owned_w

private

Solution gating variable, without ghost entries.

Definition at line 470 of file monodomain_fhn_dg.hpp.

◆ solution_w

template<class basis >

lifex::LinAlg::MPI::Vector DUBeat::models::MonodomainFHNDG< basis >::solution_w

private

Solution gating variable, with ghost entries.

Definition at line 472 of file monodomain_fhn_dg.hpp.

◆ w_ex

template<class basis >

std::shared_ptr<lifex::utils::FunctionDirichlet> DUBeat::models::MonodomainFHNDG< basis >::w_ex

private

dealii::Pointer to exact solution function gating variable.

Definition at line 478 of file monodomain_fhn_dg.hpp.

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

models/monodomain_fhn_dg.hpp

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<class basis> class DUBeat::models::MonodomainFHNDG< basis >

Constructor & Destructor Documentation

◆ MonodomainFHNDG()

Member Function Documentation

◆ assemble_system()

◆ declare_parameters()

◆ parse_parameters()

◆ run()

◆ time_initialization()

◆ update_time()

Member Data Documentation

◆ a

◆ bdf_handler_w

◆ ChiM

◆ Cm

◆ epsilon

◆ gamma

◆ grad_w_ex

◆ kappa

◆ matrix_t0

◆ Sigma

◆ solution_bdf_w

◆ solution_ex_owned_w

◆ solution_ex_w

◆ solution_ext_w

◆ solution_owned_w

◆ solution_w

◆ w_ex

template<class basis>
class DUBeat::models::MonodomainFHNDG< basis >