#include <linearised_navier_stokes_elements.h>

Inheritance diagram for oomph::LinearisedQTaylorHoodElement:

Public Member Functions
	LinearisedQTaylorHoodElement ()
	Constructor, no internal data points. More...

virtual unsigned	required_nvalue (const unsigned &n) const

virtual int	p_index_linearised_nst (const unsigned &i) const

double	p_linearised_nst (const unsigned &n_p, const unsigned &i) const

void	pin_pressure_normalisation_dofs ()
	Pin the normalisation dofs. More...

virtual void	pin_real_or_imag (const unsigned &real)

virtual void	unpin_real_or_imag (const unsigned &real)

unsigned	npres_linearised_nst () const

void	fix_pressure (const unsigned &n_p, const double &pvalue)

int	p_local_eqn (const unsigned &n, const unsigned &i)

void	output (std::ostream &outfile)
	Redirect output to NavierStokesEquations output. More...

void	output (std::ostream &outfile, const unsigned &n_plot)
	Redirect output to NavierStokesEquations output. More...

void	output (FILE *file_pt)
	Redirect output to NavierStokesEquations output. More...

void	output (FILE *file_pt, const unsigned &n_plot)
	Redirect output to NavierStokesEquations output. More...

unsigned	ndof_types () const

Public Member Functions inherited from oomph::LinearisedNavierStokesEquations
	LinearisedNavierStokesEquations ()

const double &	re () const
	Reynolds number. More...

const double &	re_st () const
	Product of Reynolds and Strouhal number (=Womersley number) More...

const double &	lambda () const

const double &	omega () const

double *&	re_pt ()
	Pointer to Reynolds number. More...

double *&	re_st_pt ()
	Pointer to product of Reynolds and Strouhal number (=Womersley number) More...

double *&	lambda_pt ()
	Pointer to lambda. More...

double *&	omega_pt ()
	Pointer to frequency. More...

LinearisedNavierStokesEigenfunctionNormalisationElement *	normalisation_element_pt ()
	Pointer to normalisation element. More...

void	set_eigenfunction_normalisation_element (LinearisedNavierStokesEigenfunctionNormalisationElement *const &normalisation_el_pt)
	the boolean flag check_nodal_data is set to false. More...

const double &	viscosity_ratio () const

double *&	viscosity_ratio_pt ()
	Pointer to the viscosity ratio. More...

const double &	density_ratio () const

double *&	density_ratio_pt ()
	Pointer to the density ratio. More...

virtual unsigned	u_index_linearised_nst (const unsigned &i) const

double	du_dt_linearised_nst (const unsigned &n, const unsigned &i) const

void	disable_ALE ()

void	enable_ALE ()

void	strain_rate (const Vector< double > &s, DenseMatrix< double > &strain_rate, const unsigned &real) const

void	output_veloc (std::ostream &outfile, const unsigned &nplot, const unsigned &t)

void	fill_in_contribution_to_residuals (Vector< double > &residuals)
	Compute the element's residual Vector. More...

double	interpolated_u_linearised_nst (const Vector< double > &s, const unsigned &i) const

double	interpolated_p_linearised_nst (const Vector< double > &s, const unsigned &i) const

Public Member Functions inherited from oomph::FiniteElement
void	set_dimension (const unsigned &dim)

void	set_nodal_dimension (const unsigned &nodal_dim)

void	set_nnodal_position_type (const unsigned &nposition_type)
	Set the number of types required to interpolate the coordinate. More...

void	set_n_node (const unsigned &n)

int	nodal_local_eqn (const unsigned &n, const unsigned &i) const

double	dJ_eulerian_at_knot (const unsigned &ipt, Shape &psi, DenseMatrix< double > &djacobian_dX) const

	FiniteElement ()
	Constructor. More...

virtual	~FiniteElement ()

	FiniteElement (const FiniteElement &)=delete
	Broken copy constructor. More...

virtual bool	local_coord_is_valid (const Vector< double > &s)
	Broken assignment operator. More...

virtual void	move_local_coord_back_into_element (Vector< double > &s) const

void	get_centre_of_gravity_and_max_radius_in_terms_of_zeta (Vector< double > &cog, double &max_radius) const

virtual void	local_coordinate_of_node (const unsigned &j, Vector< double > &s) const

virtual void	local_fraction_of_node (const unsigned &j, Vector< double > &s_fraction)

virtual double	local_one_d_fraction_of_node (const unsigned &n1d, const unsigned &i)

virtual void	set_macro_elem_pt (MacroElement *macro_elem_pt)

MacroElement *	macro_elem_pt ()
	Access function to pointer to macro element. More...

void	get_x (const Vector< double > &s, Vector< double > &x) const

void	get_x (const unsigned &t, const Vector< double > &s, Vector< double > &x)

virtual void	get_x_from_macro_element (const Vector< double > &s, Vector< double > &x) const

virtual void	get_x_from_macro_element (const unsigned &t, const Vector< double > &s, Vector< double > &x)

virtual void	set_integration_scheme (Integral *const &integral_pt)
	Set the spatial integration scheme. More...

Integral *const &	integral_pt () const
	Return the pointer to the integration scheme (const version) More...

virtual void	shape (const Vector< double > &s, Shape &psi) const =0

virtual void	shape_at_knot (const unsigned &ipt, Shape &psi) const

virtual void	dshape_local (const Vector< double > &s, Shape &psi, DShape &dpsids) const

virtual void	dshape_local_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsids) const

virtual void	d2shape_local (const Vector< double > &s, Shape &psi, DShape &dpsids, DShape &d2psids) const

virtual void	d2shape_local_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsids, DShape &d2psids) const

virtual double	J_eulerian (const Vector< double > &s) const

virtual double	J_eulerian_at_knot (const unsigned &ipt) const

void	check_J_eulerian_at_knots (bool &passed) const

void	check_jacobian (const double &jacobian) const

double	dshape_eulerian (const Vector< double > &s, Shape &psi, DShape &dpsidx) const

virtual double	dshape_eulerian_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsidx) const

virtual double	dshape_eulerian_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsi, DenseMatrix< double > &djacobian_dX, RankFourTensor< double > &d_dpsidx_dX) const

double	d2shape_eulerian (const Vector< double > &s, Shape &psi, DShape &dpsidx, DShape &d2psidx) const

virtual double	d2shape_eulerian_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsidx, DShape &d2psidx) const

virtual void	assign_nodal_local_eqn_numbers (const bool &store_local_dof_pt)

virtual void	describe_local_dofs (std::ostream &out, const std::string &current_string) const

virtual void	describe_nodal_local_dofs (std::ostream &out, const std::string &current_string) const

virtual void	assign_all_generic_local_eqn_numbers (const bool &store_local_dof_pt)

Node *&	node_pt (const unsigned &n)
	Return a pointer to the local node n. More...

Node *const &	node_pt (const unsigned &n) const
	Return a pointer to the local node n (const version) More...

unsigned	nnode () const
	Return the number of nodes. More...

virtual unsigned	nnode_1d () const

double	raw_nodal_position (const unsigned &n, const unsigned &i) const

double	raw_nodal_position (const unsigned &t, const unsigned &n, const unsigned &i) const

double	raw_dnodal_position_dt (const unsigned &n, const unsigned &i) const

double	raw_dnodal_position_dt (const unsigned &n, const unsigned &j, const unsigned &i) const

double	raw_nodal_position_gen (const unsigned &n, const unsigned &k, const unsigned &i) const

double	raw_nodal_position_gen (const unsigned &t, const unsigned &n, const unsigned &k, const unsigned &i) const

double	raw_dnodal_position_gen_dt (const unsigned &n, const unsigned &k, const unsigned &i) const

double	raw_dnodal_position_gen_dt (const unsigned &j, const unsigned &n, const unsigned &k, const unsigned &i) const

double	nodal_position (const unsigned &n, const unsigned &i) const

double	nodal_position (const unsigned &t, const unsigned &n, const unsigned &i) const

double	dnodal_position_dt (const unsigned &n, const unsigned &i) const
	Return the i-th component of nodal velocity: dx/dt at local node n. More...

double	dnodal_position_dt (const unsigned &n, const unsigned &j, const unsigned &i) const

double	nodal_position_gen (const unsigned &n, const unsigned &k, const unsigned &i) const

double	nodal_position_gen (const unsigned &t, const unsigned &n, const unsigned &k, const unsigned &i) const

double	dnodal_position_gen_dt (const unsigned &n, const unsigned &k, const unsigned &i) const

double	dnodal_position_gen_dt (const unsigned &j, const unsigned &n, const unsigned &k, const unsigned &i) const

virtual void	get_dresidual_dnodal_coordinates (RankThreeTensor< double > &dresidual_dnodal_coordinates)

unsigned	nnodal_position_type () const

bool	has_hanging_nodes () const

unsigned	nodal_dimension () const
	Return the required Eulerian dimension of the nodes in this element. More...

virtual unsigned	nvertex_node () const

virtual Node *	vertex_node_pt (const unsigned &j) const

virtual Node *	construct_node (const unsigned &n)

virtual Node *	construct_node (const unsigned &n, TimeStepper *const &time_stepper_pt)

virtual Node *	construct_boundary_node (const unsigned &n)

virtual Node *	construct_boundary_node (const unsigned &n, TimeStepper *const &time_stepper_pt)

int	get_node_number (Node *const &node_pt) const

virtual Node *	get_node_at_local_coordinate (const Vector< double > &s) const

double	raw_nodal_value (const unsigned &n, const unsigned &i) const

double	raw_nodal_value (const unsigned &t, const unsigned &n, const unsigned &i) const

double	nodal_value (const unsigned &n, const unsigned &i) const

double	nodal_value (const unsigned &t, const unsigned &n, const unsigned &i) const

unsigned	dim () const

virtual ElementGeometry::ElementGeometry	element_geometry () const
	Return the geometry type of the element (either Q or T usually). More...

virtual double	interpolated_x (const Vector< double > &s, const unsigned &i) const
	Return FE interpolated coordinate x[i] at local coordinate s. More...

virtual double	interpolated_x (const unsigned &t, const Vector< double > &s, const unsigned &i) const

virtual void	interpolated_x (const Vector< double > &s, Vector< double > &x) const
	Return FE interpolated position x[] at local coordinate s as Vector. More...

virtual void	interpolated_x (const unsigned &t, const Vector< double > &s, Vector< double > &x) const

virtual double	interpolated_dxdt (const Vector< double > &s, const unsigned &i, const unsigned &t)

virtual void	interpolated_dxdt (const Vector< double > &s, const unsigned &t, Vector< double > &dxdt)

unsigned	ngeom_data () const

Data *	geom_data_pt (const unsigned &j)

void	position (const Vector< double > &zeta, Vector< double > &r) const

void	position (const unsigned &t, const Vector< double > &zeta, Vector< double > &r) const

void	dposition_dt (const Vector< double > &zeta, const unsigned &t, Vector< double > &drdt)

virtual double	zeta_nodal (const unsigned &n, const unsigned &k, const unsigned &i) const

void	interpolated_zeta (const Vector< double > &s, Vector< double > &zeta) const

void	locate_zeta (const Vector< double > &zeta, GeomObject *&geom_object_pt, Vector< double > &s, const bool &use_coordinate_as_initial_guess=false)

virtual void	node_update ()

virtual void	identify_field_data_for_interactions (std::set< std::pair< Data *, unsigned >> &paired_field_data)

virtual void	identify_geometric_data (std::set< Data * > &geometric_data_pt)

virtual double	s_min () const
	Min value of local coordinate. More...

virtual double	s_max () const
	Max. value of local coordinate. More...

double	size () const

virtual double	compute_physical_size () const

virtual void	point_output_data (const Vector< double > &s, Vector< double > &data)

void	point_output (std::ostream &outfile, const Vector< double > &s)

virtual unsigned	nplot_points_paraview (const unsigned &nplot) const

virtual unsigned	nsub_elements_paraview (const unsigned &nplot) const

void	output_paraview (std::ofstream &file_out, const unsigned &nplot) const

virtual void	write_paraview_output_offset_information (std::ofstream &file_out, const unsigned &nplot, unsigned &counter) const

virtual void	write_paraview_type (std::ofstream &file_out, const unsigned &nplot) const

virtual void	write_paraview_offsets (std::ofstream &file_out, const unsigned &nplot, unsigned &offset_sum) const

virtual unsigned	nscalar_paraview () const

virtual void	scalar_value_paraview (std::ofstream &file_out, const unsigned &i, const unsigned &nplot) const

virtual void	scalar_value_fct_paraview (std::ofstream &file_out, const unsigned &i, const unsigned &nplot, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) const

virtual void	scalar_value_fct_paraview (std::ofstream &file_out, const unsigned &i, const unsigned &nplot, const double &time, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt) const

virtual std::string	scalar_name_paraview (const unsigned &i) const

virtual void	output (const unsigned &t, std::ostream &outfile, const unsigned &n_plot) const

virtual void	output_fct (std::ostream &outfile, const unsigned &n_plot, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
	Output an exact solution over the element. More...

virtual void	output_fct (std::ostream &outfile, const unsigned &n_plot, const double &time, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
	Output a time-dependent exact solution over the element. More...

virtual void	output_fct (std::ostream &outfile, const unsigned &n_plot, const double &time, const SolutionFunctorBase &exact_soln) const
	Output a time-dependent exact solution over the element. More...

virtual void	get_s_plot (const unsigned &i, const unsigned &nplot, Vector< double > &s, const bool &shifted_to_interior=false) const

virtual std::string	tecplot_zone_string (const unsigned &nplot) const

virtual void	write_tecplot_zone_footer (std::ostream &outfile, const unsigned &nplot) const

virtual void	write_tecplot_zone_footer (FILE *file_pt, const unsigned &nplot) const

virtual unsigned	nplot_points (const unsigned &nplot) const

virtual void	compute_error (FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, double &error, double &norm)
	Calculate the norm of the error and that of the exact solution. More...

virtual void	compute_error (FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, double &error, double &norm)
	Calculate the norm of the error and that of the exact solution. More...

virtual void	compute_error (FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, Vector< double > &error, Vector< double > &norm)

virtual void	compute_error (FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, Vector< double > &error, Vector< double > &norm)

virtual void	compute_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, double &error, double &norm)

virtual void	compute_error (std::ostream &outfile, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, double &error, double &norm)

virtual void	compute_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, Vector< double > &error, Vector< double > &norm)

virtual void	compute_error (std::ostream &outfile, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, Vector< double > &error, Vector< double > &norm)

virtual void	compute_abs_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, double &error)

void	integrate_fct (FiniteElement::SteadyExactSolutionFctPt integrand_fct_pt, Vector< double > &integral)
	Integrate Vector-valued function over element. More...

void	integrate_fct (FiniteElement::UnsteadyExactSolutionFctPt integrand_fct_pt, const double &time, Vector< double > &integral)
	Integrate Vector-valued time-dep function over element. More...

virtual void	build_face_element (const int &face_index, FaceElement *face_element_pt)

virtual unsigned	self_test ()

virtual unsigned	get_bulk_node_number (const int &face_index, const unsigned &i) const

virtual int	face_outer_unit_normal_sign (const int &face_index) const
	Get the sign of the outer unit normal on the face given by face_index. More...

virtual unsigned	nnode_on_face () const

void	face_node_number_error_check (const unsigned &i) const
	Range check for face node numbers. More...

virtual CoordinateMappingFctPt	face_to_bulk_coordinate_fct_pt (const int &face_index) const

virtual BulkCoordinateDerivativesFctPt	bulk_coordinate_derivatives_fct_pt (const int &face_index) const

Public Member Functions inherited from oomph::GeneralisedElement
	GeneralisedElement ()
	Constructor: Initialise all pointers and all values to zero. More...

virtual	~GeneralisedElement ()
	Virtual destructor to clean up any memory allocated by the object. More...

	GeneralisedElement (const GeneralisedElement &)=delete
	Broken copy constructor. More...

void	operator= (const GeneralisedElement &)=delete
	Broken assignment operator. More...

Data *&	internal_data_pt (const unsigned &i)
	Return a pointer to i-th internal data object. More...

Data *const &	internal_data_pt (const unsigned &i) const
	Return a pointer to i-th internal data object (const version) More...

Data *&	external_data_pt (const unsigned &i)
	Return a pointer to i-th external data object. More...

Data *const &	external_data_pt (const unsigned &i) const
	Return a pointer to i-th external data object (const version) More...

unsigned long	eqn_number (const unsigned &ieqn_local) const

int	local_eqn_number (const unsigned long &ieqn_global) const

unsigned	add_external_data (Data *const &data_pt, const bool &fd=true)

bool	external_data_fd (const unsigned &i) const

void	exclude_external_data_fd (const unsigned &i)

void	include_external_data_fd (const unsigned &i)

void	flush_external_data ()
	Flush all external data. More...

void	flush_external_data (Data *const &data_pt)
	Flush the object addressed by data_pt from the external data array. More...

unsigned	ninternal_data () const
	Return the number of internal data objects. More...

unsigned	nexternal_data () const
	Return the number of external data objects. More...

unsigned	ndof () const
	Return the number of equations/dofs in the element. More...

void	dof_vector (const unsigned &t, Vector< double > &dof)
	Return the vector of dof values at time level t. More...

void	dof_pt_vector (Vector< double * > &dof_pt)
	Return the vector of pointers to dof values. More...

void	set_internal_data_time_stepper (const unsigned &i, TimeStepper *const &time_stepper_pt, const bool &preserve_existing_data)

void	assign_internal_eqn_numbers (unsigned long &global_number, Vector< double * > &Dof_pt)

void	describe_dofs (std::ostream &out, const std::string &current_string) const

void	add_internal_value_pt_to_map (std::map< unsigned, double * > &map_of_value_pt)

virtual void	assign_local_eqn_numbers (const bool &store_local_dof_pt)

virtual void	complete_setup_of_dependencies ()

virtual void	get_residuals (Vector< double > &residuals)

virtual void	get_jacobian (Vector< double > &residuals, DenseMatrix< double > &jacobian)

virtual void	get_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &mass_matrix)

virtual void	get_jacobian_and_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix)

virtual void	get_dresiduals_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam)

virtual void	get_djacobian_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam, DenseMatrix< double > &djac_dparam)

virtual void	get_djacobian_and_dmass_matrix_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam, DenseMatrix< double > &djac_dparam, DenseMatrix< double > &dmass_matrix_dparam)

virtual void	get_hessian_vector_products (Vector< double > const &Y, DenseMatrix< double > const &C, DenseMatrix< double > &product)

virtual void	get_inner_products (Vector< std::pair< unsigned, unsigned >> const &history_index, Vector< double > &inner_product)

virtual void	get_inner_product_vectors (Vector< unsigned > const &history_index, Vector< Vector< double >> &inner_product_vector)

virtual void	compute_norm (Vector< double > &norm)

virtual void	compute_norm (double &norm)

virtual void	get_dof_numbers_for_unknowns (std::list< std::pair< unsigned long, unsigned >> &dof_lookup_list) const

Public Member Functions inherited from oomph::GeomObject
	GeomObject ()
	Default constructor. More...

	GeomObject (const unsigned &ndim)

	GeomObject (const unsigned &nlagrangian, const unsigned &ndim)

	GeomObject (const unsigned &nlagrangian, const unsigned &ndim, TimeStepper *time_stepper_pt)

	GeomObject (const GeomObject &dummy)=delete
	Broken copy constructor. More...

void	operator= (const GeomObject &)=delete
	Broken assignment operator. More...

virtual	~GeomObject ()
	(Empty) destructor More...

unsigned	nlagrangian () const
	Access function to # of Lagrangian coordinates. More...

unsigned	ndim () const
	Access function to # of Eulerian coordinates. More...

void	set_nlagrangian_and_ndim (const unsigned &n_lagrangian, const unsigned &n_dim)
	Set # of Lagrangian and Eulerian coordinates. More...

TimeStepper *&	time_stepper_pt ()

TimeStepper *	time_stepper_pt () const

virtual void	position (const double &t, const Vector< double > &zeta, Vector< double > &r) const

virtual void	dposition (const Vector< double > &zeta, DenseMatrix< double > &drdzeta) const

virtual void	d2position (const Vector< double > &zeta, RankThreeTensor< double > &ddrdzeta) const

virtual void	d2position (const Vector< double > &zeta, Vector< double > &r, DenseMatrix< double > &drdzeta, RankThreeTensor< double > &ddrdzeta) const

Protected Member Functions
double	dshape_and_dtest_eulerian_linearised_nst (const Vector< double > &s, Shape &psi, DShape &dpsidx, Shape &test, DShape &dtestdx) const

double	dshape_and_dtest_eulerian_at_knot_linearised_nst (const unsigned &ipt, Shape &psi, DShape &dpsidx, Shape &test, DShape &dtestdx) const

void	pshape_linearised_nst (const Vector< double > &s, Shape &psi) const
	Compute the pressure shape functions at local coordinate s. More...

void	pshape_linearised_nst (const Vector< double > &s, Shape &psi, Shape &test) const
	Compute the pressure shape and test functions at local coordinte s. More...

Protected Member Functions inherited from oomph::LinearisedNavierStokesEquations
virtual void	get_base_flow_u (const double &time, const unsigned &ipt, const Vector< double > &x, Vector< double > &result) const

virtual void	get_base_flow_dudx (const double &time, const unsigned &ipt, const Vector< double > &x, DenseMatrix< double > &result) const

int	eigenvalue_local_eqn (const unsigned &i)

virtual void	fill_in_generic_residual_contribution_linearised_nst (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix, unsigned flag)

Protected Member Functions inherited from oomph::FiniteElement
virtual void	assemble_local_to_eulerian_jacobian (const DShape &dpsids, DenseMatrix< double > &jacobian) const

virtual void	assemble_local_to_eulerian_jacobian2 (const DShape &d2psids, DenseMatrix< double > &jacobian2) const

virtual void	assemble_eulerian_base_vectors (const DShape &dpsids, DenseMatrix< double > &interpolated_G) const

template<unsigned DIM>
double	invert_jacobian (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const

virtual double	invert_jacobian_mapping (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const

virtual double	local_to_eulerian_mapping (const DShape &dpsids, DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const

double	local_to_eulerian_mapping (const DShape &dpsids, DenseMatrix< double > &inverse_jacobian) const

virtual double	local_to_eulerian_mapping_diagonal (const DShape &dpsids, DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const

virtual void	dJ_eulerian_dnodal_coordinates (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const

template<unsigned DIM>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const

virtual void	d_dshape_eulerian_dnodal_coordinates (const double &det_jacobian, const DenseMatrix< double > &jacobian, const DenseMatrix< double > &djacobian_dX, const DenseMatrix< double > &inverse_jacobian, const DShape &dpsids, RankFourTensor< double > &d_dpsidx_dX) const

template<unsigned DIM>
void	d_dshape_eulerian_dnodal_coordinates_templated_helper (const double &det_jacobian, const DenseMatrix< double > &jacobian, const DenseMatrix< double > &djacobian_dX, const DenseMatrix< double > &inverse_jacobian, const DShape &dpsids, RankFourTensor< double > &d_dpsidx_dX) const

virtual void	transform_derivatives (const DenseMatrix< double > &inverse_jacobian, DShape &dbasis) const

void	transform_derivatives_diagonal (const DenseMatrix< double > &inverse_jacobian, DShape &dbasis) const

virtual void	transform_second_derivatives (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

template<unsigned DIM>
void	transform_second_derivatives_template (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

template<unsigned DIM>
void	transform_second_derivatives_diagonal (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

virtual void	fill_in_jacobian_from_nodal_by_fd (Vector< double > &residuals, DenseMatrix< double > &jacobian)

void	fill_in_jacobian_from_nodal_by_fd (DenseMatrix< double > &jacobian)

virtual void	update_before_nodal_fd ()

virtual void	reset_after_nodal_fd ()

virtual void	update_in_nodal_fd (const unsigned &i)

virtual void	reset_in_nodal_fd (const unsigned &i)

void	fill_in_contribution_to_jacobian (Vector< double > &residuals, DenseMatrix< double > &jacobian)

template<>
double	invert_jacobian (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	Zero-d specialisation of function to calculate inverse of jacobian mapping. More...

template<>
double	invert_jacobian (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	One-d specialisation of function to calculate inverse of jacobian mapping. More...

template<>
double	invert_jacobian (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	Two-d specialisation of function to calculate inverse of jacobian mapping. More...

template<>
double	invert_jacobian (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const

template<>
void	d_dshape_eulerian_dnodal_coordinates_templated_helper (const double &det_jacobian, const DenseMatrix< double > &jacobian, const DenseMatrix< double > &djacobian_dX, const DenseMatrix< double > &inverse_jacobian, const DShape &dpsids, RankFourTensor< double > &d_dpsidx_dX) const

template<>
void	d_dshape_eulerian_dnodal_coordinates_templated_helper (const double &det_jacobian, const DenseMatrix< double > &jacobian, const DenseMatrix< double > &djacobian_dX, const DenseMatrix< double > &inverse_jacobian, const DShape &dpsids, RankFourTensor< double > &d_dpsidx_dX) const

template<>
void	d_dshape_eulerian_dnodal_coordinates_templated_helper (const double &det_jacobian, const DenseMatrix< double > &jacobian, const DenseMatrix< double > &djacobian_dX, const DenseMatrix< double > &inverse_jacobian, const DShape &dpsids, RankFourTensor< double > &d_dpsidx_dX) const

template<>
void	d_dshape_eulerian_dnodal_coordinates_templated_helper (const double &det_jacobian, const DenseMatrix< double > &jacobian, const DenseMatrix< double > &djacobian_dX, const DenseMatrix< double > &inverse_jacobian, const DShape &dpsids, RankFourTensor< double > &d_dpsidx_dX) const

template<>
void	transform_second_derivatives_template (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

template<>
void	transform_second_derivatives_template (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

template<>
void	transform_second_derivatives_diagonal (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

template<>
void	transform_second_derivatives_diagonal (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const

Protected Member Functions inherited from oomph::GeneralisedElement
unsigned	add_internal_data (Data *const &data_pt, const bool &fd=true)

bool	internal_data_fd (const unsigned &i) const

void	exclude_internal_data_fd (const unsigned &i)

void	include_internal_data_fd (const unsigned &i)

void	clear_global_eqn_numbers ()

void	add_global_eqn_numbers (std::deque< unsigned long > const &global_eqn_numbers, std::deque< double * > const &global_dof_pt)

virtual void	assign_internal_and_external_local_eqn_numbers (const bool &store_local_dof_pt)

virtual void	assign_additional_local_eqn_numbers ()

int	internal_local_eqn (const unsigned &i, const unsigned &j) const

int	external_local_eqn (const unsigned &i, const unsigned &j)

void	fill_in_jacobian_from_internal_by_fd (Vector< double > &residuals, DenseMatrix< double > &jacobian, const bool &fd_all_data=false)

void	fill_in_jacobian_from_internal_by_fd (DenseMatrix< double > &jacobian, const bool &fd_all_data=false)

void	fill_in_jacobian_from_external_by_fd (Vector< double > &residuals, DenseMatrix< double > &jacobian, const bool &fd_all_data=false)

void	fill_in_jacobian_from_external_by_fd (DenseMatrix< double > &jacobian, const bool &fd_all_data=false)

virtual void	update_before_internal_fd ()

virtual void	reset_after_internal_fd ()

virtual void	update_in_internal_fd (const unsigned &i)

virtual void	reset_in_internal_fd (const unsigned &i)

virtual void	update_before_external_fd ()

virtual void	reset_after_external_fd ()

virtual void	update_in_external_fd (const unsigned &i)

virtual void	reset_in_external_fd (const unsigned &i)

virtual void	fill_in_contribution_to_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &mass_matrix)

virtual void	fill_in_contribution_to_jacobian_and_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix)

virtual void	fill_in_contribution_to_dresiduals_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam)

virtual void	fill_in_contribution_to_djacobian_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam, DenseMatrix< double > &djac_dparam)

virtual void	fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam, DenseMatrix< double > &djac_dparam, DenseMatrix< double > &dmass_matrix_dparam)

virtual void	fill_in_contribution_to_hessian_vector_products (Vector< double > const &Y, DenseMatrix< double > const &C, DenseMatrix< double > &product)

virtual void	fill_in_contribution_to_inner_products (Vector< std::pair< unsigned, unsigned >> const &history_index, Vector< double > &inner_product)

virtual void	fill_in_contribution_to_inner_product_vectors (Vector< unsigned > const &history_index, Vector< Vector< double >> &inner_product_vector)

Static Protected Attributes
static const unsigned	Pconv [] = {0, 2, 6, 8}
	Set the data for the pressure conversion array. More...

Static Protected Attributes inherited from oomph::FiniteElement
static const unsigned	Default_Initial_Nvalue = 0
	Default value for the number of values at a node. More...

static const double	Node_location_tolerance = 1.0e-14

static const unsigned	N2deriv [] = {0, 1, 3, 6}

Static Protected Attributes inherited from oomph::GeneralisedElement
static DenseMatrix< double >	Dummy_matrix

static std::deque< double * >	Dof_pt_deque

Static Private Attributes
static const unsigned	Initial_Nvalue []
	Static array of ints to hold number of variables at node. More...

Additional Inherited Members
Public Types inherited from oomph::FiniteElement
typedef void(*	SteadyExactSolutionFctPt) (const Vector< double > &, Vector< double > &)

typedef void(*	UnsteadyExactSolutionFctPt) (const double &, const Vector< double > &, Vector< double > &)

Public Attributes inherited from oomph::LinearisedNavierStokesEquations
void(*&)(const double &time, const Vector< double > &x, Vector< double > &f)	base_flow_u_fct_pt ()
	Access function for the base flow solution pointer. More...

void(*&)(const double &time, const Vector< double > &x, DenseMatrix< double > &f)	base_flow_dudx_fct_pt ()

Static Public Attributes inherited from oomph::LinearisedNavierStokesEquations
static Vector< double >	Gamma
	Vector to decide whether the stress-divergence form is used or not. More...

Static Public Attributes inherited from oomph::FiniteElement
static double	Tolerance_for_singular_jacobian = 1.0e-16
	Tolerance below which the jacobian is considered singular. More...

static bool	Accept_negative_jacobian = false

static bool	Suppress_output_while_checking_for_inverted_elements

Static Public Attributes inherited from oomph::GeneralisedElement
static bool	Suppress_warning_about_repeated_internal_data

static bool	Suppress_warning_about_repeated_external_data = true

static double	Default_fd_jacobian_step = 1.0e-8

Protected Attributes inherited from oomph::LinearisedNavierStokesEquations
double *	Viscosity_Ratio_pt

double *	Density_Ratio_pt

double *	Re_pt
	Pointer to global Reynolds number. More...

double *	ReSt_pt
	Pointer to global Reynolds number x Strouhal number (=Womersley) More...

double *	Lambda_pt
	Pointer to eigenvalue. More...

double *	Omega_pt
	Pointer to frequency. More...

LinearisedNavierStokesEigenfunctionNormalisationElement *	Normalisation_element_pt
	Pointer to the normalisation element. More...

unsigned	Data_number_of_eigenvalue
	Index of datum where eigenvalue is stored. More...

unsigned	Index_of_eigenvalue

void(*	Base_flow_u_fct_pt )(const double &time, const Vector< double > &x, Vector< double > &result)
	Pointer to base flow solution (velocity components) function. More...

void(*	Base_flow_dudx_fct_pt )(const double &time, const Vector< double > &x, DenseMatrix< double > &result)

bool	ALE_is_disabled

Protected Attributes inherited from oomph::FiniteElement
MacroElement *	Macro_elem_pt
	Pointer to the element's macro element (NULL by default) More...

Protected Attributes inherited from oomph::GeomObject
unsigned	NLagrangian
	Number of Lagrangian (intrinsic) coordinates. More...

unsigned	Ndim
	Number of Eulerian coordinates. More...

TimeStepper *	Geom_object_time_stepper_pt

Detailed Description

/////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////////// Taylor–Hood elements are Navier–Stokes elements with quadratic interpolation for velocities and positions and continuous linear pressure interpolation

Constructor & Destructor Documentation

◆ LinearisedQTaylorHoodElement()

oomph::LinearisedQTaylorHoodElement::LinearisedQTaylorHoodElement ( )

inline

Constructor, no internal data points.

       : QElement<2, 3>(), LinearisedNavierStokesEquations()
     {
     }

Member Function Documentation

◆ dshape_and_dtest_eulerian_at_knot_linearised_nst()

double oomph::LinearisedQTaylorHoodElement::dshape_and_dtest_eulerian_at_knot_linearised_nst	(	const unsigned &	ipt,
		Shape &	psi,
		DShape &	dpsidx,
		Shape &	test,
		DShape &	dtestdx
	)		const

inlineprotectedvirtual

Velocity shape and test functions and their derivatives w.r.t. global coordinates the ipt-th integation point (taken from geometry). Return Jacobian of mapping between local and global coordinates.

Derivatives of the shape functions and test functions w.r.t global (Eulerian) coordinates at the ipt-th integration point. Return Jacobian of mapping between local and global coordinates.

Implements oomph::LinearisedNavierStokesEquations.

   {
     // Call the geometrical shape functions and derivatives
     const double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
  
     test = psi;
     dtestdx = dpsidx;
  
     // Return the Jacobian
     return J;
   }

References oomph::FiniteElement::dshape_eulerian_at_knot(), and J.

◆ dshape_and_dtest_eulerian_linearised_nst()

double oomph::LinearisedQTaylorHoodElement::dshape_and_dtest_eulerian_linearised_nst	(	const Vector< double > &	s,
		Shape &	psi,
		DShape &	dpsidx,
		Shape &	test,
		DShape &	dtestdx
	)		const

inlineprotectedvirtual

Velocity shape and test functions and their derivatives w.r.t. global coordinates at local coordinate s (taken from geometry). Return Jacobian of mapping between local and global coordinates.

Derivatives of the shape functions and test functions w.r.t global (Eulerian) coordinates at local coordinate s. Return Jacobian of mapping between local and global coordinates.

Implements oomph::LinearisedNavierStokesEquations.

   {
     // Call the geometrical shape functions and derivatives
     const double J = this->dshape_eulerian(s, psi, dpsidx);
  
     test = psi;
     dtestdx = dpsidx;
  
     // Return the Jacobian
     return J;
   }

References oomph::FiniteElement::dshape_eulerian(), and J.

◆ fix_pressure()

void oomph::LinearisedQTaylorHoodElement::fix_pressure	(	const unsigned &	n_p,
		const double &	pvalue
	)

inline

Fix both components of the pressure at local pressure node n_p to pvalue

     {
       // Loop over the two pressure components
       for (unsigned i = 0; i < 2; i++)
       {
         this->node_pt(Pconv[n_p])->pin(p_index_linearised_nst(i));
         this->node_pt(Pconv[n_p])->set_value(p_index_linearised_nst(i), pvalue);
       }
     }

References i, oomph::FiniteElement::node_pt(), p_index_linearised_nst(), oomph::Data::pin(), and oomph::Data::set_value().

◆ ndof_types()

unsigned oomph::LinearisedQTaylorHoodElement::ndof_types ( ) const

inlinevirtual

Returns the number of "dof-blocks" that degrees of freedom in this element are sub-divided into: Velocity and pressure.

Reimplemented from oomph::GeneralisedElement.

     {
       return 8;
     }

◆ npres_linearised_nst()

unsigned oomph::LinearisedQTaylorHoodElement::npres_linearised_nst ( ) const

inlinevirtual

Return number of pressure values corresponding to a single pressure component

Implements oomph::LinearisedNavierStokesEquations.

     {
       return 4;
     }

◆ output() [1/4]

void oomph::LinearisedQTaylorHoodElement::output ( FILE * file_pt )

inlinevirtual

Redirect output to NavierStokesEquations output.

Reimplemented from oomph::LinearisedNavierStokesEquations.

     {
       LinearisedNavierStokesEquations::output(file_pt);
     }

References oomph::LinearisedNavierStokesEquations::output().

◆ output() [2/4]

void oomph::LinearisedQTaylorHoodElement::output	(	FILE *	file_pt,
		const unsigned &	n_plot
	)

inlinevirtual

Redirect output to NavierStokesEquations output.

Reimplemented from oomph::LinearisedNavierStokesEquations.

     {
       LinearisedNavierStokesEquations::output(file_pt, n_plot);
     }

References oomph::LinearisedNavierStokesEquations::output().

◆ output() [3/4]

void oomph::LinearisedQTaylorHoodElement::output ( std::ostream & outfile )

inlinevirtual

Redirect output to NavierStokesEquations output.

Reimplemented from oomph::LinearisedNavierStokesEquations.

     {
       LinearisedNavierStokesEquations::output(outfile);
     }

References oomph::LinearisedNavierStokesEquations::output().

◆ output() [4/4]

void oomph::LinearisedQTaylorHoodElement::output	(	std::ostream &	outfile,
		const unsigned &	n_plot
	)

inlinevirtual

Redirect output to NavierStokesEquations output.

Reimplemented from oomph::LinearisedNavierStokesEquations.

     {
       LinearisedNavierStokesEquations::output(outfile, n_plot);
     }

References oomph::LinearisedNavierStokesEquations::output().

◆ p_index_linearised_nst()

virtual int oomph::LinearisedQTaylorHoodElement::p_index_linearised_nst ( const unsigned & i ) const

inlinevirtual

Which nodal value represents the pressure? Overload version in base class which returns static int "Pressure_not_stored_at_node"

Reimplemented from oomph::LinearisedNavierStokesEquations.

     {
       return (2 * DIM + i);
     }

References DIM, and i.

Referenced by fix_pressure(), p_linearised_nst(), and p_local_eqn().

◆ p_linearised_nst()

double oomph::LinearisedQTaylorHoodElement::p_linearised_nst	(	const unsigned &	n_p,
		const unsigned &	i
	)		const

inlinevirtual

Access function for the i-th component of pressure at local pressure node n_p (const version).

Implements oomph::LinearisedNavierStokesEquations.

     {
       return nodal_value(Pconv[n_p], p_index_linearised_nst(i));
     }

References i, oomph::FiniteElement::nodal_value(), p_index_linearised_nst(), and Pconv.

◆ p_local_eqn()

int oomph::LinearisedQTaylorHoodElement::p_local_eqn	(	const unsigned &	n,
		const unsigned &	i
	)

inlinevirtual

Overload the access function for the i-th component of the pressure's local equation numbers

Implements oomph::LinearisedNavierStokesEquations.

     {
       return nodal_local_eqn(Pconv[n], p_index_linearised_nst(i));
     }

References i, n, oomph::FiniteElement::nodal_local_eqn(), p_index_linearised_nst(), and Pconv.

◆ pin_pressure_normalisation_dofs()

void oomph::LinearisedQTaylorHoodElement::pin_pressure_normalisation_dofs ( )

inlinevirtual

Pin the normalisation dofs.

Implements oomph::LinearisedNavierStokesEquations.

     {
       throw OomphLibError("This is not implemented yet\n",
                           OOMPH_CURRENT_FUNCTION,
                           OOMPH_EXCEPTION_LOCATION);
     }

References OOMPH_CURRENT_FUNCTION, and OOMPH_EXCEPTION_LOCATION.

◆ pin_real_or_imag()

virtual void oomph::LinearisedQTaylorHoodElement::pin_real_or_imag ( const unsigned & real )

inlinevirtual

Pin the real or imaginary part of the problem Input integer 0 for real 1 for imaginary

Implements oomph::LinearisedNavierStokesEquations.

     {
       throw OomphLibError("This is not implemented yet\n",
                           OOMPH_CURRENT_FUNCTION,
                           OOMPH_EXCEPTION_LOCATION);
     }

References OOMPH_CURRENT_FUNCTION, and OOMPH_EXCEPTION_LOCATION.

◆ pshape_linearised_nst() [1/2]

void oomph::LinearisedQTaylorHoodElement::pshape_linearised_nst	(	const Vector< double > &	s,
		Shape &	psi
	)		const

inlineprotectedvirtual

Compute the pressure shape functions at local coordinate s.

Pressure shape functions.

Implements oomph::LinearisedNavierStokesEquations.

   {
     // Allocate local storage for the pressure shape functions
     double psi1[2], psi2[2];
  
     // Call the one-dimensional shape functions
     OneDimLagrange::shape<2>(s[0], psi1);
     OneDimLagrange::shape<2>(s[1], psi2);
  
     // Now let's loop over the nodal points in the element
     // s1 is the "r" coordinate, s2 the "z"
     for (unsigned i = 0; i < 2; i++)
     {
       for (unsigned j = 0; j < 2; j++)
       {
         // Multiply the two 1D functions together to get the 2D function
         psi[2 * i + j] = psi2[i] * psi1[j];
       }
     }
   }

References i, j, s, and oomph::OneDimLagrange::shape< 2 >().

Referenced by pshape_linearised_nst().

◆ pshape_linearised_nst() [2/2]

void oomph::LinearisedQTaylorHoodElement::pshape_linearised_nst	(	const Vector< double > &	s,
		Shape &	psi,
		Shape &	test
	)		const

inlineprotectedvirtual

Compute the pressure shape and test functions at local coordinte s.

Pressure shape and test functions.

Implements oomph::LinearisedNavierStokesEquations.

   {
     // Call the pressure shape functions
     pshape_linearised_nst(s, psi);
  
     // Loop over the test functions and set them equal to the shape functions
     for (unsigned i = 0; i < 4; i++)
     {
       test[i] = psi[i];
     }
   }

References i, pshape_linearised_nst(), and s.

◆ required_nvalue()

virtual unsigned oomph::LinearisedQTaylorHoodElement::required_nvalue ( const unsigned & n ) const

inlinevirtual

Number of values (pinned or dofs) required at node n. Can be overwritten for hanging node version

Reimplemented from oomph::FiniteElement.

Reimplemented in oomph::RefineableLinearisedQTaylorHoodElement.

     {
       return Initial_Nvalue[n];
     }

References Initial_Nvalue, and n.

◆ unpin_real_or_imag()

virtual void oomph::LinearisedQTaylorHoodElement::unpin_real_or_imag ( const unsigned & real )

inlinevirtual

Implements oomph::LinearisedNavierStokesEquations.

     {
       throw OomphLibError("This is not implemented yet\n",
                           OOMPH_CURRENT_FUNCTION,
                           OOMPH_EXCEPTION_LOCATION);
     }

References OOMPH_CURRENT_FUNCTION, and OOMPH_EXCEPTION_LOCATION.

Member Data Documentation

◆ Initial_Nvalue

const unsigned oomph::LinearisedQTaylorHoodElement::Initial_Nvalue

staticprivate

Initial value:

= {

12, 8, 12, 8, 8, 8, 12, 8, 12}

Static array of ints to hold number of variables at node.

Set the data for the number of variables at each node.

//////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////////

Linearised axisymmetric Taylor-Hood elements

Referenced by required_nvalue().

◆ Pconv

const unsigned oomph::LinearisedQTaylorHoodElement::Pconv = {0, 2, 6, 8}

staticprotected

Set the data for the pressure conversion array.

Static array of ints to hold conversion from pressure node numbers to actual node numbers

Referenced by p_linearised_nst(), and p_local_eqn().

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

Public Member Functions

Protected Member Functions

Static Protected Attributes

Static Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ LinearisedQTaylorHoodElement()

Member Function Documentation

◆ dshape_and_dtest_eulerian_at_knot_linearised_nst()

◆ dshape_and_dtest_eulerian_linearised_nst()

◆ fix_pressure()

◆ ndof_types()

◆ npres_linearised_nst()

◆ output() [1/4]

◆ output() [2/4]

◆ output() [3/4]

◆ output() [4/4]

◆ p_index_linearised_nst()

◆ p_linearised_nst()

◆ p_local_eqn()

◆ pin_pressure_normalisation_dofs()

◆ pin_real_or_imag()

◆ pshape_linearised_nst() [1/2]

◆ pshape_linearised_nst() [2/2]

◆ required_nvalue()

◆ unpin_real_or_imag()

Member Data Documentation

◆ Initial_Nvalue

Linearised axisymmetric Taylor-Hood elements

◆ Pconv