A class for elements that solve the polar Navier–Stokes equations, This contains the generic maths – any concrete implementation must be derived from this. More...

#include <polar_navier_stokes_elements.h>

Inheritance diagram for oomph::PolarNavierStokesEquations:

Public Types
typedef void(*	NavierStokesBodyForceFctPt) (const double &time, const Vector< double > &x, Vector< double > &body_force)
	Function pointer to body force function fct(t,x,f(x)) x is a Vector! More...

typedef double(*	NavierStokesSourceFctPt) (const double &time, const Vector< double > &x)
	Function pointer to source function fct(t,x) x is a Vector! More...

Public Types inherited from oomph::FiniteElement
typedef void(*	SteadyExactSolutionFctPt) (const Vector< double > &, Vector< double > &)
	Function pointer for function that computes vector-valued steady "exact solution" ${\bf f}({\bf x})$ as $\mbox{\tt fct}({\bf x}, {\bf f})$ . More...

typedef void(*	UnsteadyExactSolutionFctPt) (const double &, const Vector< double > &, Vector< double > &)
	Function pointer for function that computes Vector-valued time-dependent function ${\bf f}(t,{\bf x})$ as $\mbox{\tt fct}(t, {\bf x}, {\bf f})$ . More...

Public Member Functions
	PolarNavierStokesEquations ()
	Constructor: NULL the body force and source function. More...

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

const double &	alpha () const
	Alpha. More...

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

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

double *&	alpha_pt ()
	Pointer to Alpha. More...

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

const double &	viscosity_ratio () const
	Viscosity ratio for element: Element's viscosity relative to the viscosity used in the definition of the Reynolds number. More...

double *&	viscosity_ratio_pt ()
	Pointer to Viscosity Ratio. More...

const double &	density_ratio () const
	Density ratio for element: Element's density relative to the viscosity used in the definition of the Reynolds number. More...

double *&	density_ratio_pt ()
	Pointer to Density ratio. More...

const double &	re_invfr () const
	Global inverse Froude number. More...

double *&	re_invfr_pt ()
	Pointer to global inverse Froude number. More...

const Vector< double > &	g () const
	Vector of gravitational components. More...

Vector< double > *&	g_pt ()
	Pointer to Vector of gravitational components. More...

NavierStokesBodyForceFctPt &	body_force_fct_pt ()
	Access function for the body-force pointer. More...

NavierStokesBodyForceFctPt	body_force_fct_pt () const
	Access function for the body-force pointer. Const version. More...

NavierStokesSourceFctPt &	source_fct_pt ()
	Access function for the source-function pointer. More...

NavierStokesSourceFctPt	source_fct_pt () const
	Access function for the source-function pointer. Const version. More...

virtual unsigned	npres_pnst () const =0
	Function to return number of pressure degrees of freedom. More...

virtual double	u_pnst (const unsigned &n, const unsigned &i) const =0
	Velocity i at local node n. Uses suitably interpolated value for hanging nodes. More...

virtual double	u_pnst (const unsigned &t, const unsigned &n, const unsigned &i) const =0
	Velocity i at local node n at timestep t (t=0: present; t>0: previous). Uses suitably interpolated value for hanging nodes. More...

virtual unsigned	u_index_pnst (const unsigned &i) const
	Return the index at which the i-th unknown velocity component is stored. The default value, i, is appropriate for single-physics problems. In derived multi-physics elements, this function should be overloaded to reflect the chosen storage scheme. Note that these equations require that the unknowns are always stored at the same indices at each node. More...

double	du_dt_pnst (const unsigned &n, const unsigned &i) const
	i-th component of du/dt at local node n. Uses suitably interpolated value for hanging nodes. More...

virtual double	p_pnst (const unsigned &n_p) const =0
	Pressure at local pressure "node" n_p Uses suitably interpolated value for hanging nodes. More...

virtual void	fix_pressure (const unsigned &p_dof, const double &p_value)=0
	Pin p_dof-th pressure dof and set it to value specified by p_value. More...

virtual int	p_nodal_index_pnst ()
	Which nodal value represents the pressure? (Default: negative, indicating that pressure is not based on nodal interpolation). More...

virtual Node *	pressure_node_pt (const unsigned &n_p)
	Pointer to n_p-th pressure node (Default: NULL, indicating that pressure is not based on nodal interpolation). More...

double	pressure_integral () const
	Integral of pressure over element. More...

double	dissipation () const
	Return integral of dissipation over element. More...

double	dissipation (const Vector< double > &s) const
	Return dissipation at local coordinate s. More...

double	kin_energy () const
	Get integral of kinetic energy over element. More...

void	strain_rate (const Vector< double > &s, DenseMatrix< double > &strain_rate) const
	Strain-rate tensor: Now returns polar strain. More...

void	strain_rate_by_r (const Vector< double > &s, DenseMatrix< double > &strain_rate) const
	Function to return polar strain multiplied by r. More...

void	get_traction (const Vector< double > &s, const Vector< double > &N, Vector< double > &traction)
	Compute traction (on the viscous scale) at local coordinate s for outer unit normal N. More...

void	get_load (const Vector< double > &s, const Vector< double > &xi, const Vector< double > &x, const Vector< double > &N, Vector< double > &load)
	The potential loading on an external SolidElement is always provided by the traction function. More...

void	output (std::ostream &outfile)
	Output functionget_vels(const Vector<double>& x_to_get, Vector<double>& vels): x,y,[z],u,v,[w],p in tecplot format. Default number of plot points. More...

void	output (std::ostream &outfile, const unsigned &nplot)
	Output function: x,y,[z],u,v,[w],p in tecplot format. nplot points in each coordinate direction. More...

void	output (FILE *file_pt)
	C-style output function: x,y,[z],u,v,[w],p in tecplot format. Default number of plot points. More...

void	output (FILE *file_pt, const unsigned &nplot)
	C-style output function: x,y,[z],u,v,[w],p in tecplot format. nplot points in each coordinate direction. More...

void	full_output (std::ostream &outfile)
	Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. Default number of plot points. More...

void	full_output (std::ostream &outfile, const unsigned &nplot)
	Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. nplot points in each coordinate direction. More...

void	output_veloc (std::ostream &outfile, const unsigned &nplot, const unsigned &t)
	Output function: x,y,[z],u,v,[w] in tecplot format. nplot points in each coordinate direction at timestep t (t=0: present; t>0: previous timestep) More...

void	output_fct (std::ostream &outfile, const unsigned &nplot, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
	Output exact solution specified via function pointer at a given number of plot points. Function prints as many components as are returned in solution Vector. More...

void	output_fct (std::ostream &outfile, const unsigned &nplot, const double &time, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
	Output exact solution specified via function pointer at a given time and at a given number of plot points. Function prints as many components as are returned in solution Vector. More...

void	compute_error (std::ostream &outfile, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, double &error, double &norm)
	Validate against exact solution at given time Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element. More...

void	compute_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, double &error, double &norm)
	Validate against exact solution. Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element. More...

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

void	fill_in_contribution_to_jacobian (Vector< double > &residuals, DenseMatrix< double > &jacobian)
	Compute the element's residual Vector and the jacobian matrix Virtual function can be overloaded by hanging-node version. More...

void	fill_in_contribution_to_jacobian_and_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix)
	Compute the element's residual Vector and the jacobian matrix Plus the mass matrix especially for eigenvalue problems. More...

virtual void	fill_in_generic_residual_contribution (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix, unsigned flag)
	Compute the residuals for the Navier–Stokes equations; flag=1(or 0): do (or don't) compute the Jacobian as well. More...

void	interpolated_u_pnst (const Vector< double > &s, Vector< double > &veloc) const
	Compute vector of FE interpolated velocity u at local coordinate s. More...

double	interpolated_u_pnst (const Vector< double > &s, const unsigned &i) const
	Return FE interpolated velocity u[i] at local coordinate s. More...

double	interpolated_p_pnst (const Vector< double > &s) const
	Return FE interpolated pressure at local coordinate s. More...

double	interpolated_dudx_pnst (const Vector< double > &s, const unsigned &i, const unsigned &j) const
	//////////////////////////////////////////////////////////////// My own work: /// Return FE interpolated velocity derivative du[i]/dx[j] /// at local coordinate s /// More...

Public Member Functions inherited from oomph::FiniteElement
void	set_dimension (const unsigned &dim)
	Set the dimension of the element and initially set the dimension of the nodes to be the same as the dimension of the element. More...

void	set_nodal_dimension (const unsigned &nodal_dim)
	Set the dimension of the nodes in the element. This will typically only be required when constructing FaceElements or in beam and shell type elements where a lower dimensional surface is embedded in a higher dimensional space. More...

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)
	Set the number of nodes in the element to n, by resizing the storage for pointers to the Node objects. More...

int	nodal_local_eqn (const unsigned &n, const unsigned &i) const
	Return the local equation number corresponding to the i-th value at the n-th local node. More...

double	dJ_eulerian_at_knot (const unsigned &ipt, Shape &psi, DenseMatrix< double > &djacobian_dX) const
	Compute the geometric shape functions (psi) at integration point ipt. Return the determinant of the jacobian of the mapping (detJ). Additionally calculate the derivatives of "detJ" w.r.t. the nodal coordinates. More...

	FiniteElement ()
	Constructor. More...

virtual	~FiniteElement ()
	The destructor cleans up the static memory allocated for shape function storage. Internal and external data get wiped by the GeneralisedElement destructor; nodes get killed in mesh destructor. More...

	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
	Adjust local coordinates so that they're located inside the element. More...

void	get_centre_of_gravity_and_max_radius_in_terms_of_zeta (Vector< double > &cog, double &max_radius) const
	Compute centre of gravity of all nodes and radius of node that is furthest from it. Used to assess approximately if a point is likely to be contained with an element in locate_zeta-like operations. More...

virtual void	local_coordinate_of_node (const unsigned &j, Vector< double > &s) const
	Get local coordinates of node j in the element; vector sets its own size (broken virtual) More...

virtual void	local_fraction_of_node (const unsigned &j, Vector< double > &s_fraction)
	Get the local fraction of the node j in the element A dumb, but correct default implementation is provided. More...

virtual double	local_one_d_fraction_of_node (const unsigned &n1d, const unsigned &i)
	Get the local fraction of any node in the n-th position in a one dimensional expansion along the i-th local coordinate. More...

virtual void	set_macro_elem_pt (MacroElement *macro_elem_pt)
	Set pointer to macro element – can be overloaded in derived elements to perform additional tasks. More...

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

void	get_x (const Vector< double > &s, Vector< double > &x) const
	Global coordinates as function of local coordinates. Either via FE representation or via macro-element (if Macro_elem_pt!=0) More...

void	get_x (const unsigned &t, const Vector< double > &s, Vector< double > &x)
	Global coordinates as function of local coordinates at previous time "level" t (t=0: present; t>0: previous). Either via FE representation of QElement or via macro-element (if Macro_elem_pt!=0). More...

virtual void	get_x_from_macro_element (const Vector< double > &s, Vector< double > &x) const
	Global coordinates as function of local coordinates using macro element representation. (Broken virtual — this must be overloaded in specific geometric element classes) More...

virtual void	get_x_from_macro_element (const unsigned &t, const Vector< double > &s, Vector< double > &x)
	Global coordinates as function of local coordinates at previous time "level" t (t=0: present; t>0: previous). using macro element representation (Broken virtual – overload in specific geometric element class if you want to use this functionality.) More...

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
	Calculate the geometric shape functions at local coordinate s. This function must be overloaded for each specific geometric element. More...

virtual void	shape_at_knot (const unsigned &ipt, Shape &psi) const
	Return the geometric shape function at the ipt-th integration point. More...

virtual void	dshape_local (const Vector< double > &s, Shape &psi, DShape &dpsids) const
	Function to compute the geometric shape functions and derivatives w.r.t. local coordinates at local coordinate s. This function must be overloaded for each specific geometric element. (Broken virtual function — specifies the interface) More...

virtual void	dshape_local_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsids) const
	Return the geometric shape function and its derivative w.r.t. the local coordinates at the ipt-th integration point. More...

virtual void	d2shape_local (const Vector< double > &s, Shape &psi, DShape &dpsids, DShape &d2psids) const
	Function to compute the geometric shape functions and also first and second derivatives w.r.t. local coordinates at local coordinate s. This function must be overloaded for each specific geometric element (if required). (Broken virtual function — specifies the interface). Numbering: 1D: d2psids(i,0) = $d^2 \psi_j / ds^2$ 2D: d2psids(i,0) = $\partial^2 \psi_j / \partial s_0^2$ d2psids(i,1) = $\partial^2 \psi_j / \partial s_1^2$ d2psids(i,2) = $\partial^2 \psi_j / \partial s_0 \partial s_1$ 3D: d2psids(i,0) = $\partial^2 \psi_j / \partial s_0^2$ d2psids(i,1) = $\partial^2 \psi_j / \partial s_1^2$ d2psids(i,2) = $\partial^2 \psi_j / \partial s_2^2$ d2psids(i,3) = $\partial^2 \psi_j / \partial s_0 \partial s_1$ d2psids(i,4) = $\partial^2 \psi_j / \partial s_0 \partial s_2$ d2psids(i,5) = $\partial^2 \psi_j / \partial s_1 \partial s_2$ . More...

virtual void	d2shape_local_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsids, DShape &d2psids) const
	Return the geometric shape function and its first and second derivatives w.r.t. the local coordinates at the ipt-th integration point. Numbering: 1D: d2psids(i,0) = $d^2 \psi_j / ds^2$ 2D: d2psids(i,0) = $\partial^2 \psi_j / \partial s_0^2$ d2psids(i,1) = $\partial^2 \psi_j / \partial s_1^2$ d2psids(i,2) = $\partial^2 \psi_j / \partial s_0 \partial s_1$ 3D: d2psids(i,0) = $\partial^2 \psi_j / \partial s_0^2$ d2psids(i,1) = $\partial^2 \psi_j / \partial s_1^2$ d2psids(i,2) = $\partial^2 \psi_j / \partial s_2^2$ d2psids(i,3) = $\partial^2 \psi_j / \partial s_0 \partial s_1$ d2psids(i,4) = $\partial^2 \psi_j / \partial s_0 \partial s_2$ d2psids(i,5) = $\partial^2 \psi_j / \partial s_1 \partial s_2$ . More...

virtual double	J_eulerian (const Vector< double > &s) const
	Return the Jacobian of mapping from local to global coordinates at local position s. More...

virtual double	J_eulerian_at_knot (const unsigned &ipt) const
	Return the Jacobian of the mapping from local to global coordinates at the ipt-th integration point. More...

void	check_J_eulerian_at_knots (bool &passed) const
	Check that Jacobian of mapping between local and Eulerian coordinates at all integration points is positive. More...

void	check_jacobian (const double &jacobian) const
	Helper function used to check for singular or negative Jacobians in the transform from local to global or Lagrangian coordinates. More...

double	dshape_eulerian (const Vector< double > &s, Shape &psi, DShape &dpsidx) const
	Compute the geometric shape functions and also first derivatives w.r.t. global coordinates at local coordinate s; Returns Jacobian of mapping from global to local coordinates. More...

virtual double	dshape_eulerian_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsidx) const
	Return the geometric shape functions and also first derivatives w.r.t. global coordinates at the ipt-th integration point. More...

virtual double	dshape_eulerian_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsi, DenseMatrix< double > &djacobian_dX, RankFourTensor< double > &d_dpsidx_dX) const
	Compute the geometric shape functions (psi) and first derivatives w.r.t. global coordinates (dpsidx) at the ipt-th integration point. Return the determinant of the jacobian of the mapping (detJ). Additionally calculate the derivatives of both "detJ" and "dpsidx" w.r.t. the nodal coordinates. More...

double	d2shape_eulerian (const Vector< double > &s, Shape &psi, DShape &dpsidx, DShape &d2psidx) const
	Compute the geometric shape functions and also first and second derivatives w.r.t. global coordinates at local coordinate s; Returns Jacobian of mapping from global to local coordinates. Numbering: 1D: d2psidx(i,0) = $d^2 \psi_j / d x^2$ 2D: d2psidx(i,0) = $\partial^2 \psi_j / \partial x_0^2$ d2psidx(i,1) = $\partial^2 \psi_j / \partial x_1^2$ d2psidx(i,2) = $\partial^2 \psi_j / \partial x_0 \partial x_1$ 3D: d2psidx(i,0) = $\partial^2 \psi_j / \partial x_0^2$ d2psidx(i,1) = $\partial^2 \psi_j / \partial x_1^2$ d2psidx(i,2) = $\partial^2 \psi_j / \partial x_2^2$ d2psidx(i,3) = $\partial^2 \psi_j / \partial x_0 \partial x_1$ d2psidx(i,4) = $\partial^2 \psi_j / \partial x_0 \partial x_2$ d2psidx(i,5) = $\partial^2 \psi_j / \partial x_1 \partial x_2$ . More...

virtual double	d2shape_eulerian_at_knot (const unsigned &ipt, Shape &psi, DShape &dpsidx, DShape &d2psidx) const
	Return the geometric shape functions and also first and second derivatives w.r.t. global coordinates at ipt-th integration point. Numbering: 1D: d2psidx(i,0) = $d^2 \psi_j / d s^2$ 2D: d2psidx(i,0) = $\partial^2 \psi_j / \partial x_0^2$ d2psidx(i,1) = $\partial^2 \psi_j / \partial x_1^2$ d2psidx(i,2) = $\partial^2 \psi_j / \partial x_0 \partial x_1$ 3D: d2psidx(i,0) = $\partial^2 \psi_j / \partial x_0^2$ d2psidx(i,1) = $\partial^2 \psi_j / \partial x_1^2$ d2psidx(i,2) = $\partial^2 \psi_j / \partial x_2^2$ d2psidx(i,3) = $\partial^2 \psi_j / \partial x_0 \partial x_1$ d2psidx(i,4) = $\partial^2 \psi_j / \partial x_0 \partial x_2$ d2psidx(i,5) = $\partial^2 \psi_j / \partial x_1 \partial x_2$ . More...

virtual void	assign_nodal_local_eqn_numbers (const bool &store_local_dof_pt)
	Assign the local equation numbers for Data stored at the nodes Virtual so that it can be overloaded by RefineableFiniteElements. If the boolean is true then the pointers to the degrees of freedom associated with each equation number are stored in Dof_pt. More...

virtual void	describe_local_dofs (std::ostream &out, const std::string &current_string) const
	Function to describe the local dofs of the element[s]. The ostream specifies the output stream to which the description is written; the string stores the currently assembled output that is ultimately written to the output stream by Data::describe_dofs(...); it is typically built up incrementally as we descend through the call hierarchy of this function when called from Problem::describe_dofs(...) More...

virtual void	describe_nodal_local_dofs (std::ostream &out, const std::string &current_string) const
	Function to describe the local dofs of the element[s]. The ostream specifies the output stream to which the description is written; the string stores the currently assembled output that is ultimately written to the output stream by Data::describe_dofs(...); it is typically built up incrementally as we descend through the call hierarchy of this function when called from Problem::describe_dofs(...) More...

virtual void	assign_all_generic_local_eqn_numbers (const bool &store_local_dof_pt)
	Overloaded version of the calculation of the local equation numbers. If the boolean argument is true then pointers to the degrees of freedom associated with each equation number are stored locally in the array Dof_pt. More...

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
	Return the number of nodes along one edge of the element Default is to return zero — must be overloaded by geometric elements. More...

double	raw_nodal_position (const unsigned &n, const unsigned &i) const
	Return the i-th coordinate at local node n. Do not use the hanging node representation. NOTE: Moved to cc file because of a possible compiler bug in gcc (yes, really!). The move to the cc file avoids inlining which appears to cause problems (only) when compiled with gcc and -O3; offensive "illegal read" is in optimised-out section of code and data that is allegedly illegal is readily readable (by other means) just before this function is called so I can't really see how we could possibly be responsible for this... More...

double	raw_nodal_position (const unsigned &t, const unsigned &n, const unsigned &i) const
	Return the i-th coordinate at local node n, at time level t (t=0: present; t>0: previous time level). Do not use the hanging node representation. More...

double	raw_dnodal_position_dt (const unsigned &n, const unsigned &i) const
	Return the i-th component of nodal velocity: dx/dt at local node n. Do not use the hanging node representation. More...

double	raw_dnodal_position_dt (const unsigned &n, const unsigned &j, const unsigned &i) const
	Return the i-th component of j-th derivative of nodal position: d^jx/dt^j at node n. Do not use the hanging node representation. More...

double	raw_nodal_position_gen (const unsigned &n, const unsigned &k, const unsigned &i) const
	Return the value of the k-th type of the i-th positional variable at the local node n. Do not use the hanging node representation. More...

double	raw_nodal_position_gen (const unsigned &t, const unsigned &n, const unsigned &k, const unsigned &i) const
	Return the generalised nodal position (type k, i-th variable) at previous timesteps at local node n. Do not use the hanging node representation. More...

double	raw_dnodal_position_gen_dt (const unsigned &n, const unsigned &k, const unsigned &i) const
	i-th component of time derivative (velocity) of the generalised position, dx(k,i)/dt at local node n. ‘Type’: k; Coordinate direction: i. Do not use the hanging node representation. More...

double	raw_dnodal_position_gen_dt (const unsigned &j, const unsigned &n, const unsigned &k, const unsigned &i) const
	i-th component of j-th time derivative of the generalised position, dx(k,i)/dt at local node n. ‘Type’: k; Coordinate direction: i. Do not use the hanging node representation. More...

double	nodal_position (const unsigned &n, const unsigned &i) const
	Return the i-th coordinate at local node n. If the node is hanging, the appropriate interpolation is handled by the position function in the Node class. More...

double	nodal_position (const unsigned &t, const unsigned &n, const unsigned &i) const
	Return the i-th coordinate at local node n, at time level t (t=0: present; t>0: previous time level) Returns suitably interpolated version for hanging nodes. More...

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
	Return the i-th component of j-th derivative of nodal position: d^jx/dt^j at node n. More...

double	nodal_position_gen (const unsigned &n, const unsigned &k, const unsigned &i) const
	Return the value of the k-th type of the i-th positional variable at the local node n. More...

double	nodal_position_gen (const unsigned &t, const unsigned &n, const unsigned &k, const unsigned &i) const
	Return the generalised nodal position (type k, i-th variable) at previous timesteps at local node n. More...

double	dnodal_position_gen_dt (const unsigned &n, const unsigned &k, const unsigned &i) const
	i-th component of time derivative (velocity) of the generalised position, dx(k,i)/dt at local node n. ‘Type’: k; Coordinate direction: i. More...

double	dnodal_position_gen_dt (const unsigned &j, const unsigned &n, const unsigned &k, const unsigned &i) const
	i-th component of j-th time derivative of the generalised position, dx(k,i)/dt at local node n. ‘Type’: k; Coordinate direction: i. More...

virtual void	get_dresidual_dnodal_coordinates (RankThreeTensor< double > &dresidual_dnodal_coordinates)
	Compute derivatives of elemental residual vector with respect to nodal coordinates. Default implementation by FD can be overwritten for specific elements. dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}. More...

virtual void	disable_ALE ()
	This is an empty function that establishes a uniform interface for all (derived) elements that involve time-derivatives. Such elements are/should be implemented in ALE form to allow mesh motions. The additional expense associated with the computation of the mesh velocities is, of course, superfluous if the elements are used in problems in which the mesh is stationary. This function should therefore be overloaded in all derived elements that are formulated in ALE form to suppress the computation of the mesh velocities. The user disables the ALE functionality at his/her own risk! If the mesh does move after all, then the results will be wrong. Here we simply issue a warning message stating that the empty function has been called. More...

virtual void	enable_ALE ()
	(Re-)enable ALE, i.e. take possible mesh motion into account when evaluating the time-derivative. This function is empty and simply establishes a common interface for all derived elements that are formulated in ALE form. More...

virtual unsigned	required_nvalue (const unsigned &n) const
	Number of values that must be stored at local node n by the element. The default is 0, until over-ridden by a particular element. For example, a Poisson equation requires only one value to be stored at each node; 2D Navier–Stokes equations require two values (velocity components) to be stored at each Node (provided that the pressure interpolation is discontinuous). More...

unsigned	nnodal_position_type () const
	Return the number of coordinate types that the element requires to interpolate the geometry between the nodes. For Lagrange elements it is 1. More...

bool	has_hanging_nodes () const
	Return boolean to indicate if any of the element's nodes are geometrically hanging. More...

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

virtual unsigned	nvertex_node () const
	Return the number of vertex nodes in this element. Broken virtual function in "pure" finite elements. More...

virtual Node *	vertex_node_pt (const unsigned &j) const
	Pointer to the j-th vertex node in the element. Broken virtual function in "pure" finite elements. More...

virtual Node *	construct_node (const unsigned &n)
	Construct the local node n and return a pointer to the newly created node object. More...

virtual Node *	construct_node (const unsigned &n, TimeStepper *const &time_stepper_pt)
	Construct the local node n, including storage for history values required by timestepper, and return a pointer to the newly created node object. More...

virtual Node *	construct_boundary_node (const unsigned &n)
	Construct the local node n as a boundary node; that is a node that MAY be placed on a mesh boundary and return a pointer to the newly created node object. More...

virtual Node *	construct_boundary_node (const unsigned &n, TimeStepper *const &time_stepper_pt)
	Construct the local node n, including storage for history values required by timestepper, as a boundary node; that is a node that MAY be placed on a mesh boundary and return a pointer to the newly created node object. More...

int	get_node_number (Node *const &node_pt) const
	Return the number of the node *node_pt if this node is in the element, else return -1;. More...

virtual Node *	get_node_at_local_coordinate (const Vector< double > &s) const
	If there is a node at this local coordinate, return the pointer to the node. More...

double	raw_nodal_value (const unsigned &n, const unsigned &i) const
	Return the i-th value stored at local node n but do NOT take hanging nodes into account. More...

double	raw_nodal_value (const unsigned &t, const unsigned &n, const unsigned &i) const
	Return the i-th value stored at local node n, at time level t (t=0: present; t>0 previous timesteps), but do NOT take hanging nodes into account. More...

double	nodal_value (const unsigned &n, const unsigned &i) const
	Return the i-th value stored at local node n. Produces suitably interpolated values for hanging nodes. More...

double	nodal_value (const unsigned &t, const unsigned &n, const unsigned &i) const
	Return the i-th value stored at local node n, at time level t (t=0: present; t>0 previous timesteps). Produces suitably interpolated values for hanging nodes. More...

unsigned	dim () const
	Return the spatial dimension of the element, i.e. the number of local coordinates required to parametrise its geometry. More...

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
	Return FE interpolated coordinate x[i] at local coordinate s at previous timestep t (t=0: present; t>0: previous timestep) More...

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
	Return FE interpolated position x[] at local coordinate s at previous timestep t as Vector (t=0: present; t>0: previous timestep) More...

virtual double	interpolated_dxdt (const Vector< double > &s, const unsigned &i, const unsigned &t)
	Return t-th time-derivative of the i-th FE-interpolated Eulerian coordinate at local coordinate s. More...

virtual void	interpolated_dxdt (const Vector< double > &s, const unsigned &t, Vector< double > &dxdt)
	Compte t-th time-derivative of the FE-interpolated Eulerian coordinate vector at local coordinate s. More...

unsigned	ngeom_data () const
	A standard FiniteElement is fixed, so there are no geometric data when viewed in its GeomObject incarnation. More...

Data *	geom_data_pt (const unsigned &j)
	A standard FiniteElement is fixed, so there are no geometric data when viewed in its GeomObject incarnation. More...

void	position (const Vector< double > &zeta, Vector< double > &r) const
	Return the parametrised position of the FiniteElement in its incarnation as a GeomObject, r(zeta). The position is given by the Eulerian coordinate and the intrinsic coordinate (zeta) is the local coordinate of the element (s). More...

void	position (const unsigned &t, const Vector< double > &zeta, Vector< double > &r) const
	Return the parametrised position of the FiniteElement in its GeomObject incarnation: r(zeta). The position is given by the Eulerian coordinate and the intrinsic coordinate (zeta) is the local coordinate of the element (s) This version of the function returns the position as a function of time t=0: current time; t>0: previous timestep. Works for t=0 but needs to be overloaded if genuine time-dependence is required. More...

void	dposition_dt (const Vector< double > &zeta, const unsigned &t, Vector< double > &drdt)
	Return the t-th time derivative of the parametrised position of the FiniteElement in its GeomObject incarnation: $\frac{d^{t} dr(zeta)}{d t^{t}}$ . Call the t-th time derivative of the FE-interpolated Eulerian coordinate. More...

virtual double	zeta_nodal (const unsigned &n, const unsigned &k, const unsigned &i) const
	Specify the values of the "global" intrinsic coordinate, zeta, of a compound geometric object (a mesh of elements) when the element is viewied as a sub-geometric object. The default assumption is that the element will be treated as a sub-geometric object in a bulk Mesh of other elements (geometric objects). The "global" coordinate of the compound geometric object is simply the Eulerian coordinate, x. The second default assumption is that the coordinate zeta will be stored at the nodes and interpolated using the shape functions of the element. This function returns the value of zeta stored at local node n, where k is the type of coordinate and i is the coordinate direction. The function is virtual so that it can be overloaded by different types of element: FaceElements and SolidFiniteElements. More...

void	interpolated_zeta (const Vector< double > &s, Vector< double > &zeta) const
	Calculate the interpolated value of zeta, the intrinsic coordinate of the element when viewed as a compound geometric object within a Mesh as a function of the local coordinate of the element, s. The default assumption is the zeta is interpolated using the shape functions of the element with the values given by zeta_nodal(). A MacroElement representation of the intrinsic coordinate parametrised by the local coordinate s is used if available. Choosing the MacroElement representation of zeta (Eulerian x by default) allows a correspondence to be established between elements on different Meshes covering the same curvilinear domain in cases where one element is much coarser than the other. More...

void	locate_zeta (const Vector< double > &zeta, GeomObject *&geom_object_pt, Vector< double > &s, const bool &use_coordinate_as_initial_guess=false)
	For a given value of zeta, the "global" intrinsic coordinate of a mesh of FiniteElements represented as a compound geometric object, find the local coordinate in this element that corresponds to the requested value of zeta. If zeta cannot be located in this element, geom_object_pt is set to NULL. If zeta is located in this element, we return its "this" pointer. By default don't use any value passed in to the local coordinate s as the initial guess in the Newton method. More...

virtual void	node_update ()
	Update the positions of all nodes in the element using each node update function. The default implementation may be overloaded so that more efficient versions can be written. More...

virtual void	identify_field_data_for_interactions (std::set< std::pair< Data *, unsigned >> &paired_field_data)
	The purpose of this function is to identify all possible Data that can affect the fields interpolated by the FiniteElement. The information will typically be used in interaction problems in which the FiniteElement provides a forcing term for an ElementWithExternalElement. The Data must be provided as `paired_load` data containing. More...

virtual void	identify_geometric_data (std::set< Data * > &geometric_data_pt)
	The purpose of this function is to identify all `Data` objects that affect the elements' geometry. This function is implemented as an empty virtual function since it can only be implemented in conjunction with a node-update strategy. A specific implementation is provided in the ElementWithMovingNodes class. More...

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
	Calculate the size of the element (length, area, volume,...) in Eulerian computational coordinates. Use suitably overloaded compute_physical_size() function to compute the actual size (taking into account factors such as 2pi or radii the integrand) – such function can only be implemented on an equation-by-equation basis. More...

virtual double	compute_physical_size () const
	Broken virtual function to compute the actual size (taking into account factors such as 2pi or radii the integrand) – such function can only be implemented on an equation-by-equation basis. More...

virtual void	point_output_data (const Vector< double > &s, Vector< double > &data)
	Virtual function to write the double precision numbers that appear in a single line of output into the data vector. Empty virtual, can be overloaded for specific elements; used e.g. by LineVisualiser. More...

void	point_output (std::ostream &outfile, const Vector< double > &s)
	Output solution (as defined by point_output_data()) at local cordinates s. More...

virtual unsigned	nplot_points_paraview (const unsigned &nplot) const
	Return the number of actual plot points for paraview plot with parameter nplot. Broken virtual; can be overloaded in specific elements. More...

virtual unsigned	nsub_elements_paraview (const unsigned &nplot) const
	Return the number of local sub-elements for paraview plot with parameter nplot. Broken virtual; can be overloaded in specific elements. More...

void	output_paraview (std::ofstream &file_out, const unsigned &nplot) const
	Paraview output – this outputs the coordinates at the plot points (for parameter nplot) to specified output file. More...

virtual void	write_paraview_output_offset_information (std::ofstream &file_out, const unsigned &nplot, unsigned &counter) const
	Fill in the offset information for paraview plot. Broken virtual. Needs to be implemented for each new geometric element type; see http://www.vtk.org/VTK/img/file-formats.pdf. More...

virtual void	write_paraview_type (std::ofstream &file_out, const unsigned &nplot) const
	Return the paraview element type. Broken virtual. Needs to be implemented for each new geometric element type; see http://www.vtk.org/VTK/img/file-formats.pdf. More...

virtual void	write_paraview_offsets (std::ofstream &file_out, const unsigned &nplot, unsigned &offset_sum) const
	Return the offsets for the paraview sub-elements. Broken virtual. Needs to be implemented for each new geometric element type; see http://www.vtk.org/VTK/img/file-formats.pdf. More...

virtual unsigned	nscalar_paraview () const
	Number of scalars/fields output by this element. Broken virtual. Needs to be implemented for each new specific element type. More...

virtual void	scalar_value_paraview (std::ofstream &file_out, const unsigned &i, const unsigned &nplot) const
	Write values of the i-th scalar field at the plot points. Broken virtual. Needs to be implemented for each new specific element type. More...

virtual void	scalar_value_fct_paraview (std::ofstream &file_out, const unsigned &i, const unsigned &nplot, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) const
	Write values of the i-th scalar field at the plot points. Broken virtual. Needs to be implemented for each new specific element type. More...

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
	Write values of the i-th scalar field at the plot points. Broken virtual. Needs to be implemented for each new specific element type. More...

virtual std::string	scalar_name_paraview (const unsigned &i) const
	Name of the i-th scalar field. Default implementation returns V1 for the first one, V2 for the second etc. Can (should!) be overloaded with more meaningful names in specific elements. More...

virtual void	output (const unsigned &t, std::ostream &outfile, const unsigned &n_plot) const
	Output the element data at time step t. This is const because it is newly added and so can be done easily. Really all the output(...) functions should be const! 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
	Get cector of local coordinates of plot point i (when plotting nplot points in each "coordinate direction"). Generally these plot points will be uniformly spaced across the element. The optional final boolean flag (default: false) allows them to be shifted inwards to avoid duplication of plot point points between elements – useful when they are used in locate_zeta, say. More...

virtual std::string	tecplot_zone_string (const unsigned &nplot) const
	Return string for tecplot zone header (when plotting nplot points in each "coordinate direction") More...

virtual void	write_tecplot_zone_footer (std::ostream &outfile, const unsigned &nplot) const
	Add tecplot zone "footer" to output stream (when plotting nplot points in each "coordinate direction"). Empty by default – can be used, e.g., to add FE connectivity lists to elements that need it. More...

virtual void	write_tecplot_zone_footer (FILE *file_pt, const unsigned &nplot) const
	Add tecplot zone "footer" to C-style output. (when plotting nplot points in each "coordinate direction"). Empty by default – can be used, e.g., to add FE connectivity lists to elements that need it. More...

virtual unsigned	nplot_points (const unsigned &nplot) const
	Return total number of plot points (when plotting nplot points in each "coordinate direction") More...

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)
	Given the exact solution ${\bf f}({\bf x})$ this function calculates the norm of the error and that of the exact solution. Version with vectors of norms and errors so that different variables' norms and errors can be returned individually. More...

virtual void	compute_error (FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, Vector< double > &error, Vector< double > &norm)
	Given the exact solution ${\bf f}({\bf x})$ this function calculates the norm of the error and that of the exact solution. Version with vectors of norms and errors so that different variables' norms and errors can be returned individually. More...

virtual void	compute_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, Vector< double > &error, Vector< double > &norm)
	Plot the error when compared against a given exact solution ${\bf f}({\bf x})$ . Also calculates the norm of the error and that of the exact solution. The version with vectors of norms and errors so that different variables' norms and errors can be returned individually. More...

virtual void	compute_error (std::ostream &outfile, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, Vector< double > &error, Vector< double > &norm)
	Plot the error when compared against a given time-dependent exact solution ${\bf f}(t,{\bf x})$ . Also calculates the norm of the error and that of the exact solution. The version with vectors of norms and errors so that different variables' norms and errors can be returned individually. More...

virtual void	compute_abs_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, double &error)
	Plot the error when compared against a given exact solution ${\bf f}({\bf x})$ . Also calculates the maximum absolute error. More...

void	integrate_fct (FiniteElement::SteadyExactSolutionFctPt integrand_fct_pt, Vector< double > &integral)
	Evaluate integral of a Vector-valued function ${\bf f}({\bf x})$ over the element. More...

void	integrate_fct (FiniteElement::UnsteadyExactSolutionFctPt integrand_fct_pt, const double &time, Vector< double > &integral)
	Evaluate integral of a Vector-valued, time-dependent function ${\bf f}(t,{\bf x})$ over the element. More...

virtual void	build_face_element (const int &face_index, FaceElement *face_element_pt)
	Function for building a lower dimensional FaceElement on the specified face of the FiniteElement. The arguments are the index of the face, an integer whose value depends on the particular element type, and a pointer to the FaceElement. More...

virtual unsigned	self_test ()
	Self-test: Check inversion of element & do self-test for GeneralisedElement. Return 0 if OK. More...

virtual unsigned	get_bulk_node_number (const int &face_index, const unsigned &i) const
	Get the number of the ith node on face face_index (in the bulk node vector). More...

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
	Get a pointer to the function mapping face coordinates to bulk coordinates. More...

virtual BulkCoordinateDerivativesFctPt	bulk_coordinate_derivatives_fct_pt (const int &face_index) const
	Get a pointer to the derivative of the mapping from face to bulk coordinates. More...

Public Member Functions inherited from oomph::GeneralisedElement
GeneralisedElement()	GeneralisedElement (const GeneralisedElement &)=delete
	Constructor: Initialise all pointers and all values to zero. 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
	Return the global equation number corresponding to the ieqn_local-th local equation number. More...

int	local_eqn_number (const unsigned long &ieqn_global) const
	Return the local equation number corresponding to the ieqn_global-th global equation number. Returns minus one (-1) if there is no local degree of freedom corresponding to the chosen global equation number. More...

unsigned	add_external_data (Data *const &data_pt, const bool &fd=true)
	Add a (pointer to an) external data object to the element and return its index (i.e. the index required to obtain it from the access function `external_data_pt(...)`. The optional boolean flag indicates whether the data should be included in the general finite-difference loop when calculating the jacobian. The default value is true, i.e. the data will be included in the finite-differencing. More...

bool	external_data_fd (const unsigned &i) const
	Return the status of the boolean flag indicating whether the external data is included in the finite difference loop. More...

void	exclude_external_data_fd (const unsigned &i)
	Set the boolean flag to exclude the external datum from the the finite difference loop when computing the jacobian matrix. More...

void	include_external_data_fd (const unsigned &i)
	Set the boolean flag to include the external datum in the the finite difference loop when computing the jacobian matrix. More...

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)
	Set the timestepper associated with the i-th internal data object. More...

void	assign_internal_eqn_numbers (unsigned long &global_number, Vector< double * > &Dof_pt)
	Assign the global equation numbers to the internal Data. The arguments are the current highest global equation number (which will be incremented) and a Vector of pointers to the global variables (to which any unpinned values in the internal Data are added). More...

void	describe_dofs (std::ostream &out, const std::string &current_string) const
	Function to describe the dofs of the element. The ostream specifies the output stream to which the description is written; the string stores the currently assembled output that is ultimately written to the output stream by Data::describe_dofs(...); it is typically built up incrementally as we descend through the call hierarchy of this function when called from Problem::describe_dofs(...) More...

void	add_internal_value_pt_to_map (std::map< unsigned, double * > &map_of_value_pt)
	Add pointers to the internal data values to map indexed by the global equation number. More...

void	add_internal_data_values_to_vector (Vector< double > &vector_of_values)
	Add all internal data and time history values to the vector in the internal storage order. More...

void	read_internal_data_values_from_vector (const Vector< double > &vector_of_values, unsigned &index)
	Read all internal data and time history values from the vector starting from index. On return the index will be set to the value at the end of the data that has been read in. More...

void	add_internal_eqn_numbers_to_vector (Vector< long > &vector_of_eqn_numbers)
	Add all equation numbers associated with internal data to the vector in the internal storage order. More...

void	read_internal_eqn_numbers_from_vector (const Vector< long > &vector_of_eqn_numbers, unsigned &index)
	Read all equation numbers associated with internal data from the vector starting from index. On return the index will be set to the value at the end of the data that has been read in. More...

virtual void	assign_local_eqn_numbers (const bool &store_local_dof_pt)
	Setup the arrays of local equation numbers for the element. If the optional boolean argument is true, then pointers to the associated degrees of freedom are stored locally in the array Dof_pt. More...

virtual void	complete_setup_of_dependencies ()
	Complete the setup of any additional dependencies that the element may have. Empty virtual function that may be overloaded for specific derived elements. Used, e.g., for elements with algebraic node update functions to determine the "geometric Data", i.e. the Data that affects the element's shape. This function is called (for all elements) at the very beginning of the equation numbering procedure to ensure that all dependencies are accounted for. More...

virtual void	get_residuals (Vector< double > &residuals)
	Calculate the vector of residuals of the equations in the element. By default initialise the vector to zero and then call the fill_in_contribution_to_residuals() function. Note that this entire function can be overloaded if desired. More...

virtual void	get_jacobian (Vector< double > &residuals, DenseMatrix< double > &jacobian)
	Calculate the elemental Jacobian matrix "d equation / d variable". More...

virtual void	get_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &mass_matrix)
	Calculate the residuals and the elemental "mass" matrix, the matrix that multiplies the time derivative terms in a problem. More...

virtual void	get_jacobian_and_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix)
	Calculate the residuals and jacobian and elemental "mass" matrix, the matrix that multiplies the time derivative terms. More...

virtual void	get_dresiduals_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam)
	Calculate the derivatives of the residuals with respect to a parameter. More...

virtual void	get_djacobian_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam, DenseMatrix< double > &djac_dparam)
	Calculate the derivatives of the elemental Jacobian matrix and residuals with respect to a parameter. More...

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)
	Calculate the derivatives of the elemental Jacobian matrix mass matrix and residuals with respect to a parameter. More...

virtual void	get_hessian_vector_products (Vector< double > const &Y, DenseMatrix< double > const &C, DenseMatrix< double > &product)
	Calculate the product of the Hessian (derivative of Jacobian with respect to all variables) an eigenvector, Y, and other specified vectors, C (d(J_{ij})/d u_{k}) Y_{j} C_{k}. More...

virtual void	get_inner_products (Vector< std::pair< unsigned, unsigned >> const &history_index, Vector< double > &inner_product)
	Return the vector of inner product of the given pairs of history values. More...

virtual void	get_inner_product_vectors (Vector< unsigned > const &history_index, Vector< Vector< double >> &inner_product_vector)
	Compute the vectors that when taken as a dot product with other history values give the inner product over the element. More...

virtual void	compute_norm (Vector< double > &norm)
	Compute norm of solution – broken virtual can be overloaded by element writer to implement whatever norm is desired for the specific element. More...

virtual void	compute_norm (double &norm)
	Compute norm of solution – broken virtual can be overloaded by element writer to implement whatever norm is desired for the specific element. More...

void	set_halo (const unsigned &non_halo_proc_ID)
	Label the element as halo and specify processor that holds non-halo counterpart. More...

void	set_nonhalo ()
	Label the element as not being a halo. More...

bool	is_halo () const
	Is this element a halo? More...

int	non_halo_proc_ID ()
	ID of processor ID that holds non-halo counterpart of halo element; negative if not a halo. More...

void	set_must_be_kept_as_halo ()
	Insist that this element be kept as a halo element during a distribute? More...

void	unset_must_be_kept_as_halo ()
	Do not insist that this element be kept as a halo element during distribution. More...

bool	must_be_kept_as_halo () const
	Test whether the element must be kept as a halo element. More...

virtual unsigned	ndof_types () const
	The number of types of degrees of freedom in this element are sub-divided into. More...

virtual void	get_dof_numbers_for_unknowns (std::list< std::pair< unsigned long, unsigned >> &dof_lookup_list) const
	Create a list of pairs for the unknowns that this element is "in charge of" – ignore any unknowns associated with external `Data`. The first entry in each pair must contain the global equation number of the unknown, while the second one contains the number of the DOF type that this unknown is associated with. (The function can obviously only be called if the equation numbering scheme has been set up.) More...

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

	GeomObject (const unsigned &ndim)
	Constructor: Pass dimension of geometric object (# of Eulerian coords = # of Lagrangian coords; no time history available/needed) More...

	GeomObject (const unsigned &nlagrangian, const unsigned &ndim)
	Constructor: pass # of Eulerian and Lagrangian coordinates. No time history available/needed. More...

	GeomObject (const unsigned &nlagrangian, const unsigned &ndim, TimeStepper *time_stepper_pt)
	Constructor: pass # of Eulerian and Lagrangian coordinates and pointer to time-stepper which is used to handle the position at previous timesteps and allows the evaluation of veloc/acceleration etc. in cases where the GeomData varies with time. More...

	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 ()
	Access function for pointer to time stepper: Null if object is not time-dependent. More...

TimeStepper *	time_stepper_pt () const
	Access function for pointer to time stepper: Null if object is not time-dependent. Const version. More...

virtual void	position (const double &t, const Vector< double > &zeta, Vector< double > &r) const
	Parametrised position on object: r(zeta). Evaluated at the continuous time value, t. More...

virtual void	dposition (const Vector< double > &zeta, DenseMatrix< double > &drdzeta) const
	Derivative of position Vector w.r.t. to coordinates: $\frac{dR_i}{d \zeta_\alpha}$ = drdzeta(alpha,i). Evaluated at current time. More...

virtual void	d2position (const Vector< double > &zeta, RankThreeTensor< double > &ddrdzeta) const
	2nd derivative of position Vector w.r.t. to coordinates: $\frac{d^2R_i}{d \zeta_\alpha d \zeta_\beta}$ = ddrdzeta(alpha,beta,i). Evaluated at current time. More...

virtual void	d2position (const Vector< double > &zeta, Vector< double > &r, DenseMatrix< double > &drdzeta, RankThreeTensor< double > &ddrdzeta) const
	Posn Vector and its 1st & 2nd derivatives w.r.t. to coordinates: $\frac{dR_i}{d \zeta_\alpha}$ = drdzeta(alpha,i). $\frac{d^2R_i}{d \zeta_\alpha d \zeta_\beta}$ = ddrdzeta(alpha,beta,i). Evaluated at current time. More...

Static Public Attributes
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
	Boolean that if set to true allows a negative jacobian in the transform between global and local coordinates (negative surface area = left-handed coordinate system). More...

static bool	Suppress_output_while_checking_for_inverted_elements
	Static boolean to suppress output while checking for inverted elements. More...

Static Public Attributes inherited from oomph::GeneralisedElement
static bool	Suppress_warning_about_repeated_internal_data
	Static boolean to suppress warnings about repeated internal data. Defaults to false. More...

static bool	Suppress_warning_about_repeated_external_data = true
	Static boolean to suppress warnings about repeated external data. Defaults to true. More...

static double	Default_fd_jacobian_step = 1.0e-8
	Double used for the default finite difference step in elemental jacobian calculations. More...

Protected Member Functions
virtual int	p_local_eqn (const unsigned &n)=0
	Access function for the local equation number information for the pressure. p_local_eqn[n] = local equation number or < 0 if pinned. More...

virtual double	dshape_and_dtest_eulerian_pnst (const Vector< double > &s, Shape &psi, DShape &dpsidx, Shape &test, DShape &dtestdx) const =0
	Compute the shape functions and derivatives w.r.t. global coords at local coordinate s. Return Jacobian of mapping between local and global coordinates. More...

virtual double	dshape_and_dtest_eulerian_at_knot_pnst (const unsigned &ipt, Shape &psi, DShape &dpsidx, Shape &test, DShape &dtestdx) const =0
	Compute the shape functions and derivatives w.r.t. global coords at ipt-th integration point Return Jacobian of mapping between local and global coordinates. More...

virtual void	pshape_pnst (const Vector< double > &s, Shape &psi) const =0
	Compute the pressure shape functions at local coordinate s. More...

virtual void	pshape_pnst (const Vector< double > &s, Shape &psi, Shape &test) const =0
	Compute the pressure shape and test functions at local coordinate s. More...

void	get_body_force (double time, const Vector< double > &x, Vector< double > &result)
	Calculate the body force at a given time and Eulerian position. More...

double	get_source_fct (double time, const Vector< double > &x)
	Calculate the source fct at given time and Eulerian position. More...

Protected Member Functions inherited from oomph::FiniteElement
virtual void	assemble_local_to_eulerian_jacobian (const DShape &dpsids, DenseMatrix< double > &jacobian) const
	Assemble the jacobian matrix for the mapping from local to Eulerian coordinates, given the derivatives of the shape function w.r.t the local coordinates. More...

virtual void	assemble_local_to_eulerian_jacobian2 (const DShape &d2psids, DenseMatrix< double > &jacobian2) const
	Assemble the the "jacobian" matrix of second derivatives of the mapping from local to Eulerian coordinates, given the second derivatives of the shape functions w.r.t. local coordinates. More...

virtual void	assemble_eulerian_base_vectors (const DShape &dpsids, DenseMatrix< double > &interpolated_G) const
	Assemble the covariant Eulerian base vectors, assuming that the derivatives of the shape functions with respect to the local coordinates have already been constructed. More...

template<unsigned DIM>
double	invert_jacobian (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	Take the matrix passed as jacobian and return its inverse in inverse_jacobian. This function is templated by the dimension of the element because matrix inversion cannot be written efficiently in a generic manner. More...

virtual double	invert_jacobian_mapping (const DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	A template-free interface that takes the matrix passed as jacobian and return its inverse in inverse_jacobian. By default the function will use the dimension of the element to call the correct invert_jacobian(..) function. This should be overloaded for efficiency (removal of a switch statement) in specific elements. More...

virtual double	local_to_eulerian_mapping (const DShape &dpsids, DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	Calculate the mapping from local to Eulerian coordinates, given the derivatives of the shape functions w.r.t. local coordinates. Returns the determinant of the jacobian, the jacobian and inverse jacobian. More...

double	local_to_eulerian_mapping (const DShape &dpsids, DenseMatrix< double > &inverse_jacobian) const
	Calculate the mapping from local to Eulerian coordinates, given the derivatives of the shape functions w.r.t. local coordinates, Return only the determinant of the jacobian and the inverse of the mapping (ds/dx). More...

virtual double	local_to_eulerian_mapping_diagonal (const DShape &dpsids, DenseMatrix< double > &jacobian, DenseMatrix< double > &inverse_jacobian) const
	Calculate the mapping from local to Eulerian coordinates given the derivatives of the shape functions w.r.t the local coordinates. assuming that the coordinates are aligned in the direction of the local coordinates, i.e. there are no cross terms and the jacobian is diagonal. This function returns the determinant of the jacobian, the jacobian and the inverse jacobian. More...

virtual void	dJ_eulerian_dnodal_coordinates (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const
	A template-free interface that calculates the derivative of the jacobian of a mapping with respect to the nodal coordinates X_ij. To do this it requires the jacobian matrix and the derivatives of the shape functions w.r.t. the local coordinates. By default the function will use the dimension of the element to call the correct dJ_eulerian_dnodal_coordinates_templated_helper(..) function. This should be overloaded for efficiency (removal of a switch statement) in specific elements. More...

template<unsigned DIM>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const
	Calculate the derivative of the jacobian of a mapping with respect to the nodal coordinates X_ij using the jacobian matrix and the derivatives of the shape functions w.r.t. the local coordinates. This function is templated by the dimension of the element. More...

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
	A template-free interface that calculates the derivative w.r.t. the nodal coordinates $X_{pq}$ of the derivative of the shape functions $\psi_j$ w.r.t. the global eulerian coordinates . I.e. this function calculates. More...

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
	Calculate the derivative w.r.t. the nodal coordinates $X_{pq}$ of the derivative of the shape functions w.r.t. the global eulerian coordinates , using the determinant of the jacobian mapping, its derivative w.r.t. the nodal coordinates $X_{pq}$ , the inverse jacobian and the derivatives of the shape functions w.r.t. the local coordinates. The result is returned as a tensor of rank four. Numbering: d_dpsidx_dX(p,q,j,i) = $\frac{\partial}{\partial X_{pq}} \left( \frac{\partial \psi_j}{\partial x_i} \right)$ This function is templated by the dimension of the element. More...

virtual void	transform_derivatives (const DenseMatrix< double > &inverse_jacobian, DShape &dbasis) const
	Convert derivative w.r.t.local coordinates to derivatives w.r.t the coordinates used to assemble the inverse_jacobian passed in the mapping. On entry, dbasis must contain the basis function derivatives w.r.t. the local coordinates; it will contain the derivatives w.r.t. the new coordinates on exit. This is virtual so that it may be overloaded if desired for efficiency reasons. More...

void	transform_derivatives_diagonal (const DenseMatrix< double > &inverse_jacobian, DShape &dbasis) const
	Convert derivative w.r.t local coordinates to derivatives w.r.t the coordinates used to assemble the inverse jacobian passed in the mapping, assuming that the coordinates are aligned in the direction of the local coordinates. On entry dbasis must contain the derivatives of the basis functions w.r.t. the local coordinates; it will contain the derivatives w.r.t. the new coordinates. are converted into the new using the mapping inverse_jacobian. More...

virtual void	transform_second_derivatives (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const
	Convert derivatives and second derivatives w.r.t. local coordiantes to derivatives and second derivatives w.r.t. the coordinates used to assemble the jacobian, inverse jacobian and jacobian2 passed to the function. By default this function will call transform_second_derivatives_template<>(...) using the dimension of the element as the template parameter. It is virtual so that it can be overloaded by a specific element to save using a switch statement. Optionally, the element writer may wish to use the transform_second_derivatives_diagonal<>(...) function On entry dbasis and d2basis must contain the derivatives w.r.t. the local coordinates; on exit they will be the derivatives w.r.t. the transformed coordinates. More...

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
	Convert derivatives and second derivatives w.r.t. local coordinates to derivatives and second derivatives w.r.t. the coordinates used to asssmble the jacobian, inverse jacobian and jacobian2 passed in the mapping. This is templated by dimension because the method of calculation varies significantly with the dimension. On entry dbasis and d2basis must contain the derivatives w.r.t. the local coordinates; on exit they will be the derivatives w.r.t. the transformed coordinates. More...

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
	Convert derivatives and second derivatives w.r.t. local coordinates to derivatives and second derivatives w.r.t. the coordinates used to asssmble the jacobian, inverse jacobian and jacobian2 passed in the mapping. This version of the function assumes that the local coordinates are aligned with the global coordinates, i.e. the jacobians are diagonal On entry dbasis and d2basis must contain the derivatives w.r.t. the local coordinates; on exit they will be the derivatives w.r.t. the transformed coordinates. More...

virtual void	fill_in_jacobian_from_nodal_by_fd (Vector< double > &residuals, DenseMatrix< double > &jacobian)
	Calculate the contributions to the jacobian from the nodal degrees of freedom using finite differences. This version of the function assumes that the residuals vector has already been calculated. More...

void	fill_in_jacobian_from_nodal_by_fd (DenseMatrix< double > &jacobian)
	Calculate the contributions to the jacobian from the nodal degrees of freedom using finite differences. This version computes the residuals vector before calculating the jacobian terms. More...

virtual void	update_before_nodal_fd ()
	Function that is called before the finite differencing of any nodal data. This may be overloaded to update any dependent data before finite differencing takes place. More...

virtual void	reset_after_nodal_fd ()
	Function that is call after the finite differencing of the nodal data. This may be overloaded to reset any dependent variables that may have changed during the finite differencing. More...

virtual void	update_in_nodal_fd (const unsigned &i)
	Function called within the finite difference loop for nodal data after a change in the i-th nodal value. More...

virtual void	reset_in_nodal_fd (const unsigned &i)
	Function called within the finite difference loop for nodal data after the i-th nodal values is reset. The default behaviour is to call the update function. More...

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
	Three-d specialisation of function to calculate inverse of jacobian mapping. More...

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const
	Zero-d specialisation of function to calculate the derivative of the jacobian of a mapping with respect to the nodal coordinates X_ij. More...

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const
	One-d specialisation of function to calculate the derivative of the jacobian of a mapping with respect to the nodal coordinates X_ij. More...

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const
	Two-d specialisation of function to calculate the derivative of the jacobian of a mapping with respect to the nodal coordinates X_ij. More...

template<>
void	dJ_eulerian_dnodal_coordinates_templated_helper (const DenseMatrix< double > &jacobian, const DShape &dpsids, DenseMatrix< double > &djacobian_dX) const
	Three-d specialisation of function to calculate the derivative of the jacobian of a mapping with respect to the nodal coordinates X_ij. More...

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
	Zero-d specialisation of function to calculate the derivative w.r.t. the nodal coordinates $X_{pq}$ of the derivative of the shape functions w.r.t. the global eulerian coordinates . More...

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
	One-d specialisation of function to calculate the derivative w.r.t. the nodal coordinates $X_{pq}$ of the derivative of the shape functions w.r.t. the global eulerian coordinates . More...

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
	Two-d specialisation of function to calculate the derivative w.r.t. the nodal coordinates $X_{pq}$ of the derivative of the shape functions w.r.t. the global eulerian coordinates . More...

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
	Three-d specialisation of function to calculate the derivative w.r.t. the nodal coordinates $X_{pq}$ of the derivative of the shape functions w.r.t. the global eulerian coordinates . More...

template<>
void	transform_second_derivatives_template (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const
	Convert derivatives and second derivatives w.r.t local coordinates to derivatives w.r.t. the coordinates used to assemble the jacobian, inverse_jacobian and jacobian 2 passed. This must be specialised for each dimension, otherwise it gets very ugly Specialisation to one dimension. More...

template<>
void	transform_second_derivatives_template (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const
	Convert derivatives and second derivatives w.r.t local coordinates to derivatives w.r.t. the coordinates used to assemble the jacobian, inverse_jacobian and jacobian 2 passed. This must be specialised for each dimension, otherwise it gets very ugly. Specialisation to two spatial dimensions. More...

template<>
void	transform_second_derivatives_diagonal (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const
	Convert derivatives and second derivatives w.r.t local coordinates to derivatives w.r.t. the coordinates used to assemble the jacobian, inverse_jacobian and jacobian 2 passed. This must be specialised for each dimension, otherwise it gets very ugly Specialisation to one dimension. More...

template<>
void	transform_second_derivatives_diagonal (const DenseMatrix< double > &jacobian, const DenseMatrix< double > &inverse_jacobian, const DenseMatrix< double > &jacobian2, DShape &dbasis, DShape &d2basis) const
	Convert second derivatives w.r.t. local coordinates to second derivatives w.r.t. the coordinates passed in the tensor coordinate. Specialised to two spatial dimension. More...

Protected Member Functions inherited from oomph::GeneralisedElement
unsigned	add_internal_data (Data *const &data_pt, const bool &fd=true)
	Add a (pointer to an) internal data object to the element and return the index required to obtain it from the access function `internal_data_pt()`. The boolean indicates whether the datum should be included in the general finite-difference loop when calculating the jacobian. The default value is true, i.e. the data will be included in the finite differencing. More...

bool	internal_data_fd (const unsigned &i) const
	Return the status of the boolean flag indicating whether the internal data is included in the finite difference loop. More...

void	exclude_internal_data_fd (const unsigned &i)
	Set the boolean flag to exclude the internal datum from the finite difference loop when computing the jacobian matrix. More...

void	include_internal_data_fd (const unsigned &i)
	Set the boolean flag to include the internal datum in the finite difference loop when computing the jacobian matrix. More...

void	clear_global_eqn_numbers ()
	Clear the storage for the global equation numbers and pointers to dofs (if stored) More...

void	add_global_eqn_numbers (std::deque< unsigned long > const &global_eqn_numbers, std::deque< double * > const &global_dof_pt)
	Add the contents of the queue global_eqn_numbers to the local storage for the local-to-global translation scheme. It is essential that the entries in the queue are added IN ORDER i.e. from the front. More...

virtual void	assign_internal_and_external_local_eqn_numbers (const bool &store_local_dof_pt)
	Assign the local equation numbers for the internal and external Data This must be called after the global equation numbers have all been assigned. It is virtual so that it can be overloaded by ElementWithExternalElements so that any external data from the external elements in included in the numbering scheme. If the boolean argument is true then pointers to the dofs will be stored in Dof_pt. More...

virtual void	assign_additional_local_eqn_numbers ()
	Setup any additional look-up schemes for local equation numbers. Examples of use include using local storage to refer to explicit degrees of freedom. The additional memory cost of such storage may or may not be offset by fast local access. More...

int	internal_local_eqn (const unsigned &i, const unsigned &j) const
	Return the local equation number corresponding to the j-th value stored at the i-th internal data. More...

int	external_local_eqn (const unsigned &i, const unsigned &j)
	Return the local equation number corresponding to the j-th value stored at the i-th external data. More...

void	fill_in_jacobian_from_internal_by_fd (Vector< double > &residuals, DenseMatrix< double > &jacobian, const bool &fd_all_data=false)
	Calculate the contributions to the jacobian from the internal degrees of freedom using finite differences. This version of the function assumes that the residuals vector has already been calculated. If the boolean argument is true, the finite differencing will be performed for all internal data, irrespective of the information in Data_fd. The default value (false) uses the information in Data_fd to selectively difference only certain data. More...

void	fill_in_jacobian_from_internal_by_fd (DenseMatrix< double > &jacobian, const bool &fd_all_data=false)
	Calculate the contributions to the jacobian from the internal degrees of freedom using finite differences. This version computes the residuals vector before calculating the jacobian terms. If the boolean argument is true, the finite differencing will be performed for all internal data, irrespective of the information in Data_fd. The default value (false) uses the information in Data_fd to selectively difference only certain data. More...

void	fill_in_jacobian_from_external_by_fd (Vector< double > &residuals, DenseMatrix< double > &jacobian, const bool &fd_all_data=false)
	Calculate the contributions to the jacobian from the external degrees of freedom using finite differences. This version of the function assumes that the residuals vector has already been calculated. If the boolean argument is true, the finite differencing will be performed for all external data, irrespective of the information in Data_fd. The default value (false) uses the information in Data_fd to selectively difference only certain data. More...

void	fill_in_jacobian_from_external_by_fd (DenseMatrix< double > &jacobian, const bool &fd_all_data=false)
	Calculate the contributions to the jacobian from the external degrees of freedom using finite differences. This version computes the residuals vector before calculating the jacobian terms. If the boolean argument is true, the finite differencing will be performed for all internal data, irrespective of the information in Data_fd. The default value (false) uses the information in Data_fd to selectively difference only certain data. More...

virtual void	update_before_internal_fd ()
	Function that is called before the finite differencing of any internal data. This may be overloaded to update any dependent data before finite differencing takes place. More...

virtual void	reset_after_internal_fd ()
	Function that is call after the finite differencing of the internal data. This may be overloaded to reset any dependent variables that may have changed during the finite differencing. More...

virtual void	update_in_internal_fd (const unsigned &i)
	Function called within the finite difference loop for internal data after a change in any values in the i-th internal data object. More...

virtual void	reset_in_internal_fd (const unsigned &i)
	Function called within the finite difference loop for internal data after the values in the i-th external data object are reset. The default behaviour is to call the update function. More...

virtual void	update_before_external_fd ()
	Function that is called before the finite differencing of any external data. This may be overloaded to update any dependent data before finite differencing takes place. More...

virtual void	reset_after_external_fd ()
	Function that is call after the finite differencing of the external data. This may be overloaded to reset any dependent variables that may have changed during the finite differencing. More...

virtual void	update_in_external_fd (const unsigned &i)
	Function called within the finite difference loop for external data after a change in any values in the i-th external data object. More...

virtual void	reset_in_external_fd (const unsigned &i)
	Function called within the finite difference loop for external data after the values in the i-th external data object are reset. The default behaviour is to call the update function. More...

virtual void	fill_in_contribution_to_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &mass_matrix)
	Add the elemental contribution to the mass matrix matrix. and the residuals vector. Note that this function should NOT initialise the residuals vector or the mass matrix. It must be called after the residuals vector and jacobian matrix have been initialised to zero. The default is deliberately broken. More...

virtual void	fill_in_contribution_to_dresiduals_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam)
	Add the elemental contribution to the derivatives of the residuals with respect to a parameter. This function should NOT initialise any entries and must be called after the entries have been initialised to zero The default implementation is to use finite differences to calculate the derivatives. More...

virtual void	fill_in_contribution_to_djacobian_dparameter (double *const &parameter_pt, Vector< double > &dres_dparam, DenseMatrix< double > &djac_dparam)
	Add the elemental contribution to the derivatives of the elemental Jacobian matrix and residuals with respect to a parameter. This function should NOT initialise any entries and must be called after the entries have been initialised to zero The default implementation is to use finite differences to calculate the derivatives. More...

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)
	Add the elemental contribution to the derivative of the jacobian matrix, mass matrix and the residuals vector with respect to the passed parameter. Note that this function should NOT initialise any entries. It must be called after the residuals vector and matrices have been initialised to zero. More...

virtual void	fill_in_contribution_to_hessian_vector_products (Vector< double > const &Y, DenseMatrix< double > const &C, DenseMatrix< double > &product)
	Fill in contribution to the product of the Hessian (derivative of Jacobian with respect to all variables) an eigenvector, Y, and other specified vectors, C (d(J_{ij})/d u_{k}) Y_{j} C_{k}. More...

virtual void	fill_in_contribution_to_inner_products (Vector< std::pair< unsigned, unsigned >> const &history_index, Vector< double > &inner_product)
	Fill in the contribution to the inner products between given pairs of history values. More...

virtual void	fill_in_contribution_to_inner_product_vectors (Vector< unsigned > const &history_index, Vector< Vector< double >> &inner_product_vector)
	Fill in the contributions to the vectors that when taken as dot product with other history values give the inner product over the element. More...

Protected Attributes
double *	Viscosity_Ratio_pt
	Pointer to the viscosity ratio (relative to the viscosity used in the definition of the Reynolds number) More...

double *	Density_Ratio_pt
	Pointer to the density ratio (relative to the density used in the definition of the Reynolds number) More...

double *	Alpha_pt
	Pointer to the angle alpha. More...

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

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

double *	ReInvFr_pt
	Pointer to global Reynolds number x inverse Froude number (= Bond number / Capillary number) More...

Vector< double > *	G_pt
	Pointer to global gravity Vector. More...

NavierStokesBodyForceFctPt	Body_force_fct_pt
	Pointer to body force function. More...

NavierStokesSourceFctPt	Source_fct_pt
	Pointer to volumetric source function. More...

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::GeneralisedElement
int	Non_halo_proc_ID
	Non-halo processor ID for Data; -1 if it's not a halo. More...

bool	Must_be_kept_as_halo
	Does this element need to be kept as a halo element during a distribute? 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
	Timestepper (used to handle access to geometry at previous timesteps) More...

Static Private Attributes
static int	Pressure_not_stored_at_node = -100
	Static "magic" number that indicates that the pressure is not stored at a node. More...

static double	Default_Physical_Constant_Value = 0.0
	Static default value for the physical constants (all initialised to zero) More...

static double	Default_Physical_Ratio_Value = 1.0
	Static default value for the physical ratios (all are initialised to one) More...

static Vector< double >	Default_Gravity_vector
	Static default value for the gravity vector. More...

Additional Inherited Members
Static Protected Attributes inherited from oomph::FiniteElement
static const unsigned	Default_Initial_Nvalue = 0
	Default return value for required_nvalue(n) which gives the number of "data" values required by the element at node n; for example, solving a Poisson equation would required only one "data" value at each node. The defaults is set to zero, because a general element is problem-less. More...

static const double	Node_location_tolerance = 1.0e-14
	Default value for the tolerance to be used when locating nodes via local coordinates. More...

static const unsigned	N2deriv [] = {0, 1, 3, 6}
	Static array that holds the number of second derivatives as a function of the dimension of the element. More...

Static Protected Attributes inherited from oomph::GeneralisedElement
static DenseMatrix< double >	Dummy_matrix
	Empty dense matrix used as a dummy argument to combined residual and jacobian functions in the case when only the residuals are being assembled. More...

static std::deque< double * >	Dof_pt_deque
	Static storage for deque used to add_global_equation_numbers when pointers to the dofs in each element are not required. More...

Detailed Description

A class for elements that solve the polar Navier–Stokes equations, This contains the generic maths – any concrete implementation must be derived from this.

Definition at line 107 of file polar_navier_stokes_elements.h.

Member Typedef Documentation

◆ NavierStokesBodyForceFctPt

typedef void(* oomph::PolarNavierStokesEquations::NavierStokesBodyForceFctPt) (const double &time, const Vector< double > &x, Vector< double > &body_force)

Function pointer to body force function fct(t,x,f(x)) x is a Vector!

Definition at line 112 of file polar_navier_stokes_elements.h.

◆ NavierStokesSourceFctPt

typedef double(* oomph::PolarNavierStokesEquations::NavierStokesSourceFctPt) (const double &time, const Vector< double > &x)

Function pointer to source function fct(t,x) x is a Vector!

Definition at line 118 of file polar_navier_stokes_elements.h.

Constructor & Destructor Documentation

◆ PolarNavierStokesEquations()

oomph::PolarNavierStokesEquations::PolarNavierStokesEquations ( )

inline

Constructor: NULL the body force and source function.

Definition at line 245 of file polar_navier_stokes_elements.h.

References Default_Gravity_vector, Default_Physical_Constant_Value, Default_Physical_Ratio_Value, Density_Ratio_pt, G_pt, Re_pt, ReInvFr_pt, ReSt_pt, and Viscosity_Ratio_pt.

Member Function Documentation

◆ alpha()

const double& oomph::PolarNavierStokesEquations::alpha ( ) const

inline

Alpha.

Definition at line 272 of file polar_navier_stokes_elements.h.

References Alpha_pt.

Referenced by fill_in_generic_residual_contribution(), oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution(), output(), strain_rate(), and strain_rate_by_r().

◆ alpha_pt()

double*& oomph::PolarNavierStokesEquations::alpha_pt ( )

inline

Pointer to Alpha.

Definition at line 290 of file polar_navier_stokes_elements.h.

References Alpha_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ body_force_fct_pt() [1/2]

NavierStokesBodyForceFctPt& oomph::PolarNavierStokesEquations::body_force_fct_pt ( )

inline

Access function for the body-force pointer.

Definition at line 352 of file polar_navier_stokes_elements.h.

References Body_force_fct_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ body_force_fct_pt() [2/2]

NavierStokesBodyForceFctPt oomph::PolarNavierStokesEquations::body_force_fct_pt ( ) const

inline

Access function for the body-force pointer. Const version.

Definition at line 358 of file polar_navier_stokes_elements.h.

References Body_force_fct_pt.

◆ compute_error() [1/2]

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

virtual

Validate against exact solution. Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element.

Validate against exact velocity solution Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element.

Reimplemented from oomph::FiniteElement.

Definition at line 139 of file polar_navier_stokes_elements.cc.

References i, oomph::FiniteElement::integral_pt(), interpolated_u_pnst(), oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::J_eulerian(), oomph::Integral::knot(), oomph::Integral::nweight(), s, oomph::QuadTreeNames::W, and oomph::Integral::weight().

◆ compute_error() [2/2]

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

virtual

Validate against exact solution at given time Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element.

Validate against exact velocity solution at given time. Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element.

Reimplemented from oomph::FiniteElement.

Definition at line 61 of file polar_navier_stokes_elements.cc.

References i, oomph::FiniteElement::integral_pt(), interpolated_u_pnst(), oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::J_eulerian(), oomph::Integral::knot(), oomph::Integral::nweight(), s, oomph::QuadTreeNames::W, and oomph::Integral::weight().

◆ density_ratio()

const double& oomph::PolarNavierStokesEquations::density_ratio ( ) const

inline

Density ratio for element: Element's density relative to the viscosity used in the definition of the Reynolds number.

Definition at line 316 of file polar_navier_stokes_elements.h.

References Density_Ratio_pt.

◆ density_ratio_pt()

double*& oomph::PolarNavierStokesEquations::density_ratio_pt ( )

inline

Pointer to Density ratio.

Definition at line 322 of file polar_navier_stokes_elements.h.

References Density_Ratio_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ dissipation() [1/2]

double oomph::PolarNavierStokesEquations::dissipation ( ) const

Return integral of dissipation over element.

Definition at line 721 of file polar_navier_stokes_elements.cc.

References i, oomph::FiniteElement::integral_pt(), oomph::FiniteElement::J_eulerian(), oomph::Integral::knot(), oomph::Integral::nweight(), s, strain_rate(), and oomph::Integral::weight().

Referenced by full_output().

◆ dissipation() [2/2]

double oomph::PolarNavierStokesEquations::dissipation ( const Vector< double > & s ) const

Return dissipation at local coordinate s.

Definition at line 798 of file polar_navier_stokes_elements.cc.

References i, s, and strain_rate().

◆ dshape_and_dtest_eulerian_at_knot_pnst()

virtual double oomph::PolarNavierStokesEquations::dshape_and_dtest_eulerian_at_knot_pnst	(	const unsigned &	ipt,
		Shape &	psi,
		DShape &	dpsidx,
		Shape &	test,
		DShape &	dtestdx
	)		const

protectedpure virtual

Compute the shape functions and derivatives w.r.t. global coords at ipt-th integration point Return Jacobian of mapping between local and global coordinates.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Referenced by fill_in_generic_residual_contribution(), and oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution().

◆ dshape_and_dtest_eulerian_pnst()

virtual double oomph::PolarNavierStokesEquations::dshape_and_dtest_eulerian_pnst	(	const Vector< double > &	s,
		Shape &	psi,
		DShape &	dpsidx,
		Shape &	test,
		DShape &	dtestdx
	)		const

protectedpure virtual

Compute the shape functions and derivatives w.r.t. global coords at local coordinate s. Return Jacobian of mapping between local and global coordinates.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

◆ du_dt_pnst()

double oomph::PolarNavierStokesEquations::du_dt_pnst	(	const unsigned &	n,
		const unsigned &	i
	)		const

inline

i-th component of du/dt at local node n. Uses suitably interpolated value for hanging nodes.

Definition at line 401 of file polar_navier_stokes_elements.h.

References i, oomph::FiniteElement::node_pt(), oomph::TimeStepper::ntstorage(), t, oomph::GeomObject::time_stepper_pt(), oomph::Data::time_stepper_pt(), oomph::TimeStepper::type(), u_pnst(), and oomph::TimeStepper::weight().

Referenced by full_output().

◆ fill_in_contribution_to_jacobian()

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

inlinevirtual

Compute the element's residual Vector and the jacobian matrix Virtual function can be overloaded by hanging-node version.

Reimplemented from oomph::FiniteElement.

Definition at line 575 of file polar_navier_stokes_elements.h.

References oomph::GeneralisedElement::Dummy_matrix, and fill_in_generic_residual_contribution().

◆ fill_in_contribution_to_jacobian_and_mass_matrix()

void oomph::PolarNavierStokesEquations::fill_in_contribution_to_jacobian_and_mass_matrix	(	Vector< double > &	residuals,
		DenseMatrix< double > &	jacobian,
		DenseMatrix< double > &	mass_matrix
	)

inlinevirtual

Compute the element's residual Vector and the jacobian matrix Plus the mass matrix especially for eigenvalue problems.

Reimplemented from oomph::GeneralisedElement.

Definition at line 643 of file polar_navier_stokes_elements.h.

References fill_in_generic_residual_contribution().

◆ fill_in_contribution_to_residuals()

void oomph::PolarNavierStokesEquations::fill_in_contribution_to_residuals ( Vector< double > & residuals )

inlinevirtual

Compute the element's residual Vector.

Reimplemented from oomph::GeneralisedElement.

Definition at line 564 of file polar_navier_stokes_elements.h.

References oomph::GeneralisedElement::Dummy_matrix, and fill_in_generic_residual_contribution().

◆ fill_in_generic_residual_contribution()

void oomph::PolarNavierStokesEquations::fill_in_generic_residual_contribution	(	Vector< double > &	residuals,
		DenseMatrix< double > &	jacobian,
		DenseMatrix< double > &	mass_matrix,
		unsigned	flag
	)

virtual

Compute the residuals for the Navier–Stokes equations; flag=1(or 0): do (or don't) compute the Jacobian as well.

/////////////////////////////////////////////////////////////////// / The finished version of the new equations /////

Compute the residuals for the Navier–Stokes equations; flag=1(or 0): do (or don't) compute the Jacobian as well. flag=2 for Residuals, Jacobian and mass_matrix

This is now my new version with Jacobian and dimensionless phi

Reimplemented in oomph::RefineablePolarNavierStokesEquations.

Definition at line 1100 of file polar_navier_stokes_elements.cc.

References alpha(), dshape_and_dtest_eulerian_at_knot_pnst(), Gamma, i, oomph::FiniteElement::integral_pt(), oomph::FiniteElement::interpolated_x(), oomph::Integral::knot(), oomph::FiniteElement::nnode(), oomph::FiniteElement::nodal_local_eqn(), oomph::FiniteElement::nodal_position(), oomph::FiniteElement::nodal_value(), npres_pnst(), oomph::Integral::nweight(), p_local_eqn(), p_pnst(), pshape_pnst(), re(), re_st(), s, u_index_pnst(), oomph::QuadTreeNames::W, and oomph::Integral::weight().

Referenced by fill_in_contribution_to_jacobian(), fill_in_contribution_to_jacobian_and_mass_matrix(), and fill_in_contribution_to_residuals().

◆ fix_pressure()

virtual void oomph::PolarNavierStokesEquations::fix_pressure	(	const unsigned &	p_dof,
		const double &	p_value
	)

pure virtual

Pin p_dof-th pressure dof and set it to value specified by p_value.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

◆ full_output() [1/2]

void oomph::PolarNavierStokesEquations::full_output ( std::ostream & outfile )

inline

Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. Default number of plot points.

Definition at line 510 of file polar_navier_stokes_elements.h.

Referenced by oomph::PolarCrouzeixRaviartElement::full_output().

◆ full_output() [2/2]

void oomph::PolarNavierStokesEquations::full_output	(	std::ostream &	outfile,
		const unsigned &	nplot
	)

Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. nplot points in each coordinate direction.

Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. Specified number of plot points in each coordinate direction.

Definition at line 607 of file polar_navier_stokes_elements.cc.

References dissipation(), oomph::FiniteElement::dnodal_position_dt(), oomph::FiniteElement::dshape_eulerian(), du_dt_pnst(), oomph::FiniteElement::get_s_plot(), i, interpolated_p_pnst(), interpolated_u_pnst(), oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::nnode(), oomph::FiniteElement::nplot_points(), s, oomph::FiniteElement::tecplot_zone_string(), and u_pnst().

◆ g()

const Vector<double>& oomph::PolarNavierStokesEquations::g ( ) const

inline

Vector of gravitational components.

Definition at line 340 of file polar_navier_stokes_elements.h.

References G_pt.

◆ g_pt()

Vector<double>*& oomph::PolarNavierStokesEquations::g_pt ( )

inline

Pointer to Vector of gravitational components.

Definition at line 346 of file polar_navier_stokes_elements.h.

References G_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ get_body_force()

void oomph::PolarNavierStokesEquations::get_body_force	(	double	time,
		const Vector< double > &	x,
		Vector< double > &	result
	)

inlineprotected

Calculate the body force at a given time and Eulerian position.

Definition at line 207 of file polar_navier_stokes_elements.h.

References Body_force_fct_pt, and i.

◆ get_load()

void oomph::PolarNavierStokesEquations::get_load	(	const Vector< double > &	s,
		const Vector< double > &	xi,
		const Vector< double > &	x,
		const Vector< double > &	N,
		Vector< double > &	load
	)

inline

The potential loading on an external SolidElement is always provided by the traction function.

Definition at line 472 of file polar_navier_stokes_elements.h.

References get_traction(), oomph::QuadTreeNames::N, and s.

◆ get_source_fct()

double oomph::PolarNavierStokesEquations::get_source_fct	(	double	time,
		const Vector< double > &	x
	)

inlineprotected

Calculate the source fct at given time and Eulerian position.

Definition at line 229 of file polar_navier_stokes_elements.h.

References Source_fct_pt.

◆ get_traction()

void oomph::PolarNavierStokesEquations::get_traction	(	const Vector< double > &	s,
		const Vector< double > &	N,
		Vector< double > &	traction
	)

Compute traction (on the viscous scale) at local coordinate s for outer unit normal N.

Definition at line 772 of file polar_navier_stokes_elements.cc.

References i, interpolated_p_pnst(), oomph::QuadTreeNames::N, s, and strain_rate().

Referenced by get_load().

◆ interpolated_dudx_pnst()

double oomph::PolarNavierStokesEquations::interpolated_dudx_pnst	(	const Vector< double > &	s,
		const unsigned &	i,
		const unsigned &	j
	)		const

inline

//////////////////////////////////////////////////////////////// My own work: /// Return FE interpolated velocity derivative du[i]/dx[j] /// at local coordinate s ///

Definition at line 731 of file polar_navier_stokes_elements.h.

References oomph::FiniteElement::dshape_eulerian(), i, oomph::FiniteElement::nnode(), s, and u_pnst().

◆ interpolated_p_pnst()

double oomph::PolarNavierStokesEquations::interpolated_p_pnst ( const Vector< double > & s ) const

inline

Return FE interpolated pressure at local coordinate s.

Definition at line 706 of file polar_navier_stokes_elements.h.

References npres_pnst(), p_pnst(), pshape_pnst(), and s.

Referenced by full_output(), oomph::RefineablePolarCrouzeixRaviartElement::further_build(), oomph::RefineablePolarTaylorHoodElement::get_interpolated_values(), get_traction(), output(), and pressure_integral().

◆ interpolated_u_pnst() [1/2]

double oomph::PolarNavierStokesEquations::interpolated_u_pnst	(	const Vector< double > &	s,
		const unsigned &	i
	)		const

inline

Return FE interpolated velocity u[i] at local coordinate s.

Definition at line 685 of file polar_navier_stokes_elements.h.

References i, oomph::FiniteElement::nnode(), s, oomph::FiniteElement::shape(), and u_pnst().

◆ interpolated_u_pnst() [2/2]

void oomph::PolarNavierStokesEquations::interpolated_u_pnst	(	const Vector< double > &	s,
		Vector< double > &	veloc
	)		const

inline

Compute vector of FE interpolated velocity u at local coordinate s.

Definition at line 662 of file polar_navier_stokes_elements.h.

References i, oomph::FiniteElement::nnode(), s, oomph::FiniteElement::shape(), and u_pnst().

Referenced by compute_error(), full_output(), oomph::RefineablePolarTaylorHoodElement::get_interpolated_values(), oomph::RefineablePolarCrouzeixRaviartElement::get_interpolated_values(), kin_energy(), and output().

◆ kin_energy()

double oomph::PolarNavierStokesEquations::kin_energy ( ) const

Get integral of kinetic energy over element.

Get integral of kinetic energy over element:

Definition at line 1006 of file polar_navier_stokes_elements.cc.

References i, oomph::FiniteElement::integral_pt(), interpolated_u_pnst(), oomph::FiniteElement::J_eulerian(), oomph::Integral::knot(), oomph::Integral::nweight(), s, and oomph::Integral::weight().

◆ npres_pnst()

virtual unsigned oomph::PolarNavierStokesEquations::npres_pnst ( ) const

pure virtual

Function to return number of pressure degrees of freedom.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Referenced by fill_in_generic_residual_contribution(), oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution(), and interpolated_p_pnst().

◆ output() [1/4]

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

inlinevirtual

C-style output function: x,y,[z],u,v,[w],p in tecplot format. Default number of plot points.

Reimplemented from oomph::FiniteElement.

Reimplemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Definition at line 497 of file polar_navier_stokes_elements.h.

References output().

◆ output() [2/4]

void oomph::PolarNavierStokesEquations::output	(	FILE *	file_pt,
		const unsigned &	nplot
	)

virtual

C-style output function: x,y,[z],u,v,[w],p in tecplot format. nplot points in each coordinate direction.

C-style output function: x,y,[z],u,v,[w],p in tecplot format. Specified number of plot points in each coordinate direction.

Reimplemented from oomph::FiniteElement.

Reimplemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Definition at line 563 of file polar_navier_stokes_elements.cc.

References oomph::FiniteElement::get_s_plot(), i, interpolated_p_pnst(), interpolated_u_pnst(), oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::nplot_points(), s, and oomph::FiniteElement::tecplot_zone_string().

◆ output() [3/4]

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

inlinevirtual

Output functionget_vels(const Vector<double>& x_to_get, Vector<double>& vels): x,y,[z],u,v,[w],p in tecplot format. Default number of plot points.

Reimplemented from oomph::FiniteElement.

Reimplemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Definition at line 485 of file polar_navier_stokes_elements.h.

Referenced by output(), oomph::PolarCrouzeixRaviartElement::output(), and oomph::PolarTaylorHoodElement::output().

◆ output() [4/4]

void oomph::PolarNavierStokesEquations::output	(	std::ostream &	outfile,
		const unsigned &	nplot
	)

virtual

Output function: x,y,[z],u,v,[w],p in tecplot format. nplot points in each coordinate direction.

Output function: x,y,[z],u,v,[w],p in tecplot format. Specified number of plot points in each coordinate direction.

Reimplemented from oomph::FiniteElement.

Reimplemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Definition at line 394 of file polar_navier_stokes_elements.cc.

References alpha(), oomph::FiniteElement::get_s_plot(), interpolated_p_pnst(), interpolated_u_pnst(), oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::nplot_points(), s, and oomph::FiniteElement::tecplot_zone_string().

◆ output_fct() [1/2]

void oomph::PolarNavierStokesEquations::output_fct	(	std::ostream &	outfile,
		const unsigned &	nplot,
		const double &	time,
		FiniteElement::UnsteadyExactSolutionFctPt	exact_soln_pt
	)

virtual

Output exact solution specified via function pointer at a given time and at a given number of plot points. Function prints as many components as are returned in solution Vector.

Output "exact" solution at a given time Solution is provided via function pointer. Plot at a given number of plot points. Function prints as many components as are returned in solution Vector.

Reimplemented from oomph::FiniteElement.

Definition at line 271 of file polar_navier_stokes_elements.cc.

References oomph::FiniteElement::get_s_plot(), i, oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::nplot_points(), s, oomph::FiniteElement::tecplot_zone_string(), and oomph::FiniteElement::write_tecplot_zone_footer().

◆ output_fct() [2/2]

void oomph::PolarNavierStokesEquations::output_fct	(	std::ostream &	outfile,
		const unsigned &	nplot,
		FiniteElement::SteadyExactSolutionFctPt	exact_soln_pt
	)

virtual

Output exact solution specified via function pointer at a given number of plot points. Function prints as many components as are returned in solution Vector.

Output "exact" solution Solution is provided via function pointer. Plot at a given number of plot points. Function prints as many components as are returned in solution Vector.

Reimplemented from oomph::FiniteElement.

Definition at line 216 of file polar_navier_stokes_elements.cc.

References oomph::FiniteElement::get_s_plot(), i, oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::nplot_points(), s, oomph::FiniteElement::tecplot_zone_string(), and oomph::FiniteElement::write_tecplot_zone_footer().

◆ output_veloc()

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

Output function: x,y,[z],u,v,[w] in tecplot format. nplot points in each coordinate direction at timestep t (t=0: present; t>0: previous timestep)

Output function: Velocities only x,y,[z],u,v,[w] in tecplot format at specified previous timestep (t=0: present; t>0: previous timestep). Specified number of plot points in each coordinate direction.

Definition at line 328 of file polar_navier_stokes_elements.cc.

References oomph::FiniteElement::get_s_plot(), i, oomph::FiniteElement::interpolated_x(), oomph::FiniteElement::nnode(), oomph::FiniteElement::nodal_position(), oomph::FiniteElement::nplot_points(), s, oomph::FiniteElement::shape(), t, oomph::FiniteElement::tecplot_zone_string(), u_pnst(), and oomph::FiniteElement::write_tecplot_zone_footer().

◆ p_local_eqn()

virtual int oomph::PolarNavierStokesEquations::p_local_eqn ( const unsigned & n )

protectedpure virtual

Access function for the local equation number information for the pressure. p_local_eqn[n] = local equation number or < 0 if pinned.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Referenced by fill_in_generic_residual_contribution(), and oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution().

◆ p_nodal_index_pnst()

virtual int oomph::PolarNavierStokesEquations::p_nodal_index_pnst ( )

inlinevirtual

Which nodal value represents the pressure? (Default: negative, indicating that pressure is not based on nodal interpolation).

Reimplemented in oomph::PolarTaylorHoodElement.

Definition at line 432 of file polar_navier_stokes_elements.h.

References Pressure_not_stored_at_node.

Referenced by oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution().

◆ p_pnst()

virtual double oomph::PolarNavierStokesEquations::p_pnst ( const unsigned & n_p ) const

pure virtual

Pressure at local pressure "node" n_p Uses suitably interpolated value for hanging nodes.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Referenced by fill_in_generic_residual_contribution(), oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution(), and interpolated_p_pnst().

◆ pressure_integral()

double oomph::PolarNavierStokesEquations::pressure_integral ( ) const

Integral of pressure over element.

Return pressure integrated over the element.

Definition at line 1048 of file polar_navier_stokes_elements.cc.

References i, oomph::FiniteElement::integral_pt(), interpolated_p_pnst(), oomph::FiniteElement::J_eulerian(), oomph::Integral::knot(), oomph::Integral::nweight(), s, oomph::QuadTreeNames::W, and oomph::Integral::weight().

◆ pressure_node_pt()

virtual Node* oomph::PolarNavierStokesEquations::pressure_node_pt ( const unsigned & n_p )

inlinevirtual

Pointer to n_p-th pressure node (Default: NULL, indicating that pressure is not based on nodal interpolation).

Reimplemented in oomph::RefineablePolarTaylorHoodElement, oomph::RefineablePolarNavierStokesEquations, and oomph::PolarTaylorHoodElement.

Definition at line 439 of file polar_navier_stokes_elements.h.

◆ pshape_pnst() [1/2]

virtual void oomph::PolarNavierStokesEquations::pshape_pnst	(	const Vector< double > &	s,
		Shape &	psi
	)		const

protectedpure virtual

Compute the pressure shape functions at local coordinate s.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Referenced by fill_in_generic_residual_contribution(), oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution(), and interpolated_p_pnst().

◆ pshape_pnst() [2/2]

virtual void oomph::PolarNavierStokesEquations::pshape_pnst	(	const Vector< double > &	s,
		Shape &	psi,
		Shape &	test
	)		const

protectedpure virtual

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

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

◆ re()

const double& oomph::PolarNavierStokesEquations::re ( ) const

inline

Reynolds number.

Definition at line 266 of file polar_navier_stokes_elements.h.

References Re_pt.

Referenced by fill_in_generic_residual_contribution(), and oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution().

◆ re_invfr()

const double& oomph::PolarNavierStokesEquations::re_invfr ( ) const

inline

Global inverse Froude number.

Definition at line 328 of file polar_navier_stokes_elements.h.

References ReInvFr_pt.

◆ re_invfr_pt()

double*& oomph::PolarNavierStokesEquations::re_invfr_pt ( )

inline

Pointer to global inverse Froude number.

Definition at line 334 of file polar_navier_stokes_elements.h.

References ReInvFr_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ re_pt()

double*& oomph::PolarNavierStokesEquations::re_pt ( )

inline

Pointer to Reynolds number.

Definition at line 284 of file polar_navier_stokes_elements.h.

References Re_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ re_st()

const double& oomph::PolarNavierStokesEquations::re_st ( ) const

inline

Product of Reynolds and Strouhal number (=Womersley number)

Definition at line 278 of file polar_navier_stokes_elements.h.

References ReSt_pt.

Referenced by fill_in_generic_residual_contribution(), and oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution().

◆ re_st_pt()

double*& oomph::PolarNavierStokesEquations::re_st_pt ( )

inline

Pointer to product of Reynolds and Strouhal number (=Womersley number)

Definition at line 296 of file polar_navier_stokes_elements.h.

References ReSt_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ source_fct_pt() [1/2]

NavierStokesSourceFctPt& oomph::PolarNavierStokesEquations::source_fct_pt ( )

inline

Access function for the source-function pointer.

Definition at line 364 of file polar_navier_stokes_elements.h.

References Source_fct_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

◆ source_fct_pt() [2/2]

NavierStokesSourceFctPt oomph::PolarNavierStokesEquations::source_fct_pt ( ) const

inline

Access function for the source-function pointer. Const version.

Definition at line 370 of file polar_navier_stokes_elements.h.

References Source_fct_pt.

◆ strain_rate()

void oomph::PolarNavierStokesEquations::strain_rate	(	const Vector< double > &	s,
		DenseMatrix< double > &	strain_rate
	)		const

Strain-rate tensor: Now returns polar strain.

Get strain-rate tensor: Slightly more complicated in polar coordinates (see eg. Aris)

Definition at line 822 of file polar_navier_stokes_elements.cc.

References alpha(), oomph::FiniteElement::dshape_eulerian(), i, oomph::FiniteElement::interpolated_x(), oomph::DenseMatrix< T >::ncol(), oomph::FiniteElement::nnode(), oomph::FiniteElement::nodal_position(), oomph::FiniteElement::nodal_value(), oomph::DenseMatrix< T >::nrow(), s, and u_index_pnst().

Referenced by dissipation(), and get_traction().

◆ strain_rate_by_r()

void oomph::PolarNavierStokesEquations::strain_rate_by_r	(	const Vector< double > &	s,
		DenseMatrix< double > &	strain_rate
	)		const

Function to return polar strain multiplied by r.

Return polar strain-rate tensor multiplied by r Slightly more complicated in polar coordinates (see eg. Aris)

Definition at line 909 of file polar_navier_stokes_elements.cc.

References alpha(), oomph::FiniteElement::dshape_eulerian(), i, oomph::FiniteElement::interpolated_x(), oomph::DenseMatrix< T >::ncol(), oomph::FiniteElement::nnode(), oomph::FiniteElement::nodal_position(), oomph::FiniteElement::nodal_value(), oomph::DenseMatrix< T >::nrow(), s, and u_index_pnst().

Referenced by oomph::RefineablePolarNavierStokesEquations::get_Z2_flux().

◆ u_index_pnst()

virtual unsigned oomph::PolarNavierStokesEquations::u_index_pnst ( const unsigned & i ) const

inlinevirtual

Return the index at which the i-th unknown velocity component is stored. The default value, i, is appropriate for single-physics problems. In derived multi-physics elements, this function should be overloaded to reflect the chosen storage scheme. Note that these equations require that the unknowns are always stored at the same indices at each node.

Definition at line 394 of file polar_navier_stokes_elements.h.

References i.

Referenced by fill_in_generic_residual_contribution(), oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution(), oomph::RefineablePolarTaylorHoodElement::get_interpolated_values(), oomph::RefineablePolarCrouzeixRaviartElement::get_interpolated_values(), oomph::RefineablePolarTaylorHoodElement::insert_load_data(), oomph::RefineablePolarCrouzeixRaviartElement::insert_load_data(), strain_rate(), and strain_rate_by_r().

◆ u_pnst() [1/2]

virtual double oomph::PolarNavierStokesEquations::u_pnst	(	const unsigned &	n,
		const unsigned &	i
	)		const

pure virtual

Velocity i at local node n. Uses suitably interpolated value for hanging nodes.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

Referenced by du_dt_pnst(), full_output(), interpolated_dudx_pnst(), interpolated_u_pnst(), and output_veloc().

◆ u_pnst() [2/2]

virtual double oomph::PolarNavierStokesEquations::u_pnst	(	const unsigned &	t,
		const unsigned &	n,
		const unsigned &	i
	)		const

pure virtual

Velocity i at local node n at timestep t (t=0: present; t>0: previous). Uses suitably interpolated value for hanging nodes.

Implemented in oomph::PolarTaylorHoodElement, and oomph::PolarCrouzeixRaviartElement.

◆ viscosity_ratio()

const double& oomph::PolarNavierStokesEquations::viscosity_ratio ( ) const

inline

Viscosity ratio for element: Element's viscosity relative to the viscosity used in the definition of the Reynolds number.

Definition at line 303 of file polar_navier_stokes_elements.h.

References Viscosity_Ratio_pt.

◆ viscosity_ratio_pt()

double*& oomph::PolarNavierStokesEquations::viscosity_ratio_pt ( )

inline

Pointer to Viscosity Ratio.

Definition at line 309 of file polar_navier_stokes_elements.h.

References Viscosity_Ratio_pt.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build().

Member Data Documentation

◆ Alpha_pt

double* oomph::PolarNavierStokesEquations::Alpha_pt

protected

Pointer to the angle alpha.

Definition at line 151 of file polar_navier_stokes_elements.h.

Referenced by alpha(), alpha_pt(), and oomph::RefineablePolarNavierStokesEquations::further_build().

◆ Body_force_fct_pt

NavierStokesBodyForceFctPt oomph::PolarNavierStokesEquations::Body_force_fct_pt

protected

Pointer to body force function.

Definition at line 167 of file polar_navier_stokes_elements.h.

Referenced by body_force_fct_pt(), oomph::RefineablePolarNavierStokesEquations::further_build(), and get_body_force().

◆ Default_Gravity_vector

Vector< double > oomph::PolarNavierStokesEquations::Default_Gravity_vector

staticprivate

Static default value for the gravity vector.

Navier-Stokes equations default gravity vector.

Definition at line 135 of file polar_navier_stokes_elements.h.

Referenced by PolarNavierStokesEquations().

◆ Default_Physical_Constant_Value

double oomph::PolarNavierStokesEquations::Default_Physical_Constant_Value = 0.0

staticprivate

Static default value for the physical constants (all initialised to zero)

Navier–Stokes equations static data.

Definition at line 128 of file polar_navier_stokes_elements.h.

Referenced by PolarNavierStokesEquations().

◆ Default_Physical_Ratio_Value

double oomph::PolarNavierStokesEquations::Default_Physical_Ratio_Value = 1.0

staticprivate

Static default value for the physical ratios (all are initialised to one)

Navier–Stokes equations static data.

Definition at line 132 of file polar_navier_stokes_elements.h.

Referenced by PolarNavierStokesEquations().

◆ Density_Ratio_pt

double* oomph::PolarNavierStokesEquations::Density_Ratio_pt

protected

Pointer to the density ratio (relative to the density used in the definition of the Reynolds number)

Definition at line 146 of file polar_navier_stokes_elements.h.

Referenced by density_ratio(), density_ratio_pt(), oomph::RefineablePolarNavierStokesEquations::further_build(), and PolarNavierStokesEquations().

◆ G_pt

Vector<double>* oomph::PolarNavierStokesEquations::G_pt

protected

Pointer to global gravity Vector.

Definition at line 164 of file polar_navier_stokes_elements.h.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build(), g(), g_pt(), and PolarNavierStokesEquations().

◆ Gamma

Vector< double > oomph::PolarNavierStokesEquations::Gamma

static

Vector to decide whether the stress-divergence form is used or not.

Start of what would've been navier_stokes_elements.cc //.

Navier–Stokes equations static data

Definition at line 261 of file polar_navier_stokes_elements.h.

Referenced by fill_in_generic_residual_contribution(), and oomph::RefineablePolarNavierStokesEquations::fill_in_generic_residual_contribution().

◆ Pressure_not_stored_at_node

int oomph::PolarNavierStokesEquations::Pressure_not_stored_at_node = -100

staticprivate

Static "magic" number that indicates that the pressure is not stored at a node.

"Magic" negative number that indicates that the pressure is not stored at a node. This cannot be -1 because that represents the positional hanging scheme in the hanging_pt object of nodes

Definition at line 124 of file polar_navier_stokes_elements.h.

Referenced by p_nodal_index_pnst().

◆ Re_pt

double* oomph::PolarNavierStokesEquations::Re_pt

protected

Pointer to global Reynolds number.

Definition at line 154 of file polar_navier_stokes_elements.h.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build(), PolarNavierStokesEquations(), re(), and re_pt().

◆ ReInvFr_pt

double* oomph::PolarNavierStokesEquations::ReInvFr_pt

protected

Pointer to global Reynolds number x inverse Froude number (= Bond number / Capillary number)

Definition at line 161 of file polar_navier_stokes_elements.h.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build(), PolarNavierStokesEquations(), re_invfr(), and re_invfr_pt().

◆ ReSt_pt

double* oomph::PolarNavierStokesEquations::ReSt_pt

protected

Pointer to global Reynolds number x Strouhal number (=Womersley)

Definition at line 157 of file polar_navier_stokes_elements.h.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build(), PolarNavierStokesEquations(), re_st(), and re_st_pt().

◆ Source_fct_pt

NavierStokesSourceFctPt oomph::PolarNavierStokesEquations::Source_fct_pt

protected

Pointer to volumetric source function.

Definition at line 170 of file polar_navier_stokes_elements.h.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build(), get_source_fct(), and source_fct_pt().

◆ Viscosity_Ratio_pt

double* oomph::PolarNavierStokesEquations::Viscosity_Ratio_pt

protected

Pointer to the viscosity ratio (relative to the viscosity used in the definition of the Reynolds number)

Definition at line 142 of file polar_navier_stokes_elements.h.

Referenced by oomph::RefineablePolarNavierStokesEquations::further_build(), PolarNavierStokesEquations(), viscosity_ratio(), and viscosity_ratio_pt().

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

Public Types

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Static Private Attributes

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ NavierStokesBodyForceFctPt

◆ NavierStokesSourceFctPt

Constructor & Destructor Documentation

◆ PolarNavierStokesEquations()

Member Function Documentation

◆ alpha()

◆ alpha_pt()

◆ body_force_fct_pt() [1/2]

◆ body_force_fct_pt() [2/2]

◆ compute_error() [1/2]

◆ compute_error() [2/2]

◆ density_ratio()

◆ density_ratio_pt()

◆ dissipation() [1/2]

◆ dissipation() [2/2]

◆ dshape_and_dtest_eulerian_at_knot_pnst()

◆ dshape_and_dtest_eulerian_pnst()

◆ du_dt_pnst()

◆ fill_in_contribution_to_jacobian()

◆ fill_in_contribution_to_jacobian_and_mass_matrix()

◆ fill_in_contribution_to_residuals()

◆ fill_in_generic_residual_contribution()

◆ fix_pressure()

◆ full_output() [1/2]

◆ full_output() [2/2]

◆ g()

◆ g_pt()

◆ get_body_force()

◆ get_load()

◆ get_source_fct()

◆ get_traction()

◆ interpolated_dudx_pnst()

◆ interpolated_p_pnst()

◆ interpolated_u_pnst() [1/2]

◆ interpolated_u_pnst() [2/2]

◆ kin_energy()

◆ npres_pnst()

◆ output() [1/4]

◆ output() [2/4]

◆ output() [3/4]

◆ output() [4/4]

◆ output_fct() [1/2]

◆ output_fct() [2/2]

◆ output_veloc()

◆ p_local_eqn()

◆ p_nodal_index_pnst()

◆ p_pnst()

◆ pressure_integral()

◆ pressure_node_pt()

◆ pshape_pnst() [1/2]

◆ pshape_pnst() [2/2]

◆ re()

◆ re_invfr()

◆ re_invfr_pt()

◆ re_pt()

◆ re_st()

◆ re_st_pt()

◆ source_fct_pt() [1/2]

◆ source_fct_pt() [2/2]

◆ strain_rate()

◆ strain_rate_by_r()

◆ u_index_pnst()

◆ u_pnst() [1/2]

◆ u_pnst() [2/2]

◆ viscosity_ratio()

◆ viscosity_ratio_pt()

Member Data Documentation

◆ Alpha_pt

◆ Body_force_fct_pt

◆ Default_Gravity_vector

◆ Default_Physical_Constant_Value