DUBeat::models::HeatDG< basis > Class Template Reference

Class to solve the heat equation using the Discontinuous Galerkin method. More...

#include <heat_dg.hpp>

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

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

Public Member Functions
	HeatDG ()
	Constructor.
Public Member Functions inherited from ModelDG_t< basis >
	ModelDG_t (std::string model_name)
	Constructor.
	ModelDG_t (ModelDG_t< basis > &ModelDG_t)=default
	Default copy constructor.
	ModelDG_t (const ModelDG_t< basis > &ModelDG_t)=default
	Default const copy constructor.
	ModelDG_t (ModelDG_t< basis > &&ModelDG_t)=default
	Default move constructor.
virtual	~ModelDG_t ()=default
	Destructor.
Public Member Functions inherited from ModelDG< basis >
	ModelDG (std::string model_name)
	Constructor.
	ModelDG (ModelDG< basis > &ModelDG)=default
	Default copy constructor.
	ModelDG (const ModelDG< basis > &ModelDG)=default
	Default const copy constructor.
	ModelDG (ModelDG< basis > &&ModelDG)=default
	Default move constructor.
unsigned int	get_dofs_per_cell () const
	Return the number of degrees of freedom per element.
virtual	~ModelDG ()=default
	Destructor.

Private Member Functions
void	assemble_system () override
	Assembly of the linear system.

Additional Inherited Members
Protected Member Functions inherited from ModelDG_t< basis >
virtual void	declare_parameters (lifex::ParamHandler &params) const override
	Override for declaration of additional parameters.
virtual void	parse_parameters (lifex::ParamHandler &params) override
	Override to parse additional parameters.
virtual void	time_initialization ()
	Setup for the time-dependent problems at the initial time-step.
virtual void	update_time ()
	To perform the time increment.
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.
void	run () override
	Override for the simulation run.
Protected Member Functions inherited from ModelDG< basis >
virtual void	setup_system ()
	Setup of the problem before the resolution.
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.
void	initialize_solution (lifex::LinAlg::MPI::Vector &solution_owned, lifex::LinAlg::MPI::Vector &solution)
	To inizialize the solutions using the deal.II reinit.
void	create_mesh ()
	Load the mesh from the default path.
void	create_mesh (std::string mesh_path)
	Load the mesh from a user-defined path.
void	solve_system ()
	System solving.
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.
virtual void	output_results (std::string output_name="solution") const
	Output of results.
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.
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.
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).
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).
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.
void	conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned)
	Conversion of a discretized solution from Dubiner coefficients to FEM coefficients.
lifex::LinAlg::MPI::Vector	conversion_to_fem (const lifex::LinAlg::MPI::Vector &sol_owned) const
	Conversion to FEM coefficients, const version.
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.
lifex::LinAlg::MPI::Vector	conversion_to_dub (const lifex::LinAlg::MPI::Vector &sol_owned) const
	Conversion to DUB coefficients, const version.
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.
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.
Protected Attributes inherited from ModelDG_t< basis >
double	prm_time_init
	Initial time.
double	prm_time_final
	Final time.
double	prm_time_step
	Time-step amplitude.
unsigned int	prm_bdf_order
	BDF order.
double	time
	Current time.
unsigned int	timestep_number
	Current time-step number.
lifex::utils::BDFHandler< lifex::LinAlg::MPI::Vector >	bdf_handler
	BDF time advancing handler.
lifex::LinAlg::MPI::Vector	solution_bdf
	BDF solution, with ghost entries.
lifex::LinAlg::MPI::Vector	solution_ext
	BDF extrapolated solution, with ghost entries.
Protected Attributes inherited from ModelDG< basis >
const std::string	model_name
	Name of the class/problem.
unsigned int	prm_fe_degree
	Polynomials degree.
unsigned int	prm_n_refinements
	Mesh refinement level (>=1).
double	prm_penalty_coeff
	DG Penalty coefficient.
double	prm_stability_coeff
	DG stabilty coefficient.
double	scaling_factor = 1
	Scaling factor.
std::shared_ptr< lifex::utils::MeshHandler >	triangulation
	Triangulation (internal use for useful already implemented methods).
unsigned int	dofs_per_cell
	Number of degrees of freedom per cell.
DoFHandlerDG< basis >	dof_handler
	DoFHandler (internal use for useful already implemented methods).
std::shared_ptr< DUBFEMHandler< basis > >	dub_fem_values
	Member used for conversions between analytical, nodal and modal representations of the solutions.
std::unique_ptr< AssembleDG< basis > >	assemble
	Matrix assembler.
lifex::utils::LinearSolverHandler< lifex::LinAlg::MPI::Vector >	linear_solver
	Linear solver handler.
lifex::utils::PreconditionerHandler	preconditioner
	Preconditioner handler.
lifex::LinAlg::MPI::SparseMatrix	matrix
	Distributed matrix of the linear system.
lifex::LinAlg::MPI::Vector	rhs
	Distributed right hand side vector of the linear system.
lifex::LinAlg::MPI::Vector	solution_owned
	Distributed solution vector, without ghost entries.
lifex::LinAlg::MPI::Vector	solution
	Distributed solution vector, with ghost entries.
lifex::LinAlg::MPI::Vector	solution_ex_owned
	Distributed exact solution vector, without ghost entries.
lifex::LinAlg::MPI::Vector	solution_ex
	Distributed exact solution vector, without ghost entries.
std::shared_ptr< lifex::utils::FunctionDirichlet >	u_ex
	Pointer to exact solution function.
std::shared_ptr< dealii::Function< lifex::dim > >	grad_u_ex
	Pointer to exact gradient solution Function.
std::shared_ptr< dealii::Function< lifex::dim > >	f_ex
	Known forcing term.
std::shared_ptr< dealii::Function< lifex::dim > >	g_n
	Neumann boundary conditions.

Detailed Description

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

Class to solve the heat equation using the Discontinuous Galerkin method.

\[ \begin{aligned} \frac{\partial u}{\partial t} -\Delta u &= f, & \quad & \text{in } \Omega = (-1, 1)^d \times (0, T], \\ \frac{\partial u}{\partial n} &= g, & \quad & \text{on } \partial\Omega \times (0, T], \\ u &= u_\mathrm{ex}, & \quad & \text{in } \Omega \times \{ t = 0 \}. \end{aligned} \]

In particular, it can be solved using the basis functions the FEM 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:

\[ \frac{\alpha_{\mathrm{BDF}\sigma} u^{n+1} - u_{\mathrm{BDF}\sigma}^n}{\Delta t} - \Delta u^{n+1} + u^{n+1} = f^{n+1}, \]

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

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

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

Finally, \(d\) is specified in the lifex configuration and \(T\) in the .prm parameter file.

Definition at line 259 of file heat_dg.hpp.

Constructor & Destructor Documentation

HeatDG()

template<class basis >

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

inline

Constructor.

Definition at line 181 of file heat_dg.hpp.

Member Function Documentation

assemble_system()

template<class basis >

void DUBeat::models::HeatDG< basis >::assemble_system ( )

overrideprivatevirtual

Assembly of the linear system.

Implements ModelDG< basis >.

Definition at line 280 of file heat_dg.hpp.

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

• models/heat_dg.hpp