Simple 2D triangular mesh for TElements. More...

#include <simple_rectangular_tri_mesh.template.h>

Inheritance diagram for oomph::SimpleRectangularTriMesh< ELEMENT >:

Public Member Functions
	SimpleRectangularTriMesh (const unsigned &n_x, const unsigned &n_y, const double &l_x, const double &l_y, TimeStepper *time_stepper_pt=&Mesh::Default_TimeStepper)

const unsigned &	nx () const
	Access function for number of elements in x directions. More...

const unsigned &	ny () const
	Access function for number of elements in y directions. More...

Public Member Functions inherited from oomph::TriangleMeshBase
	TriangleMeshBase ()
	Constructor. More...

	TriangleMeshBase (const TriangleMeshBase &node)=delete
	Broken copy constructor. More...

virtual	~TriangleMeshBase ()
	Broken assignment operator. More...

void	setup_boundary_element_info ()

void	setup_boundary_element_info (std::ostream &outfile)

virtual void	load_balance (const Vector< unsigned > &target_domain_for_local_non_halo_element)
	Virtual function to perform the load balance rutines. More...

virtual void	reset_boundary_element_info (Vector< unsigned > &ntmp_boundary_elements, Vector< Vector< unsigned >> &ntmp_boundary_elements_in_region, Vector< FiniteElement * > &deleted_elements)
	Virtual function to perform the reset boundary elements info rutines. More...

Public Member Functions inherited from oomph::UnstructuredTwoDMeshGeometryBase
	UnstructuredTwoDMeshGeometryBase ()
	Empty constructor. More...

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

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

	~UnstructuredTwoDMeshGeometryBase ()
	Empty destructor. More...

unsigned	nregion ()
	Return the number of regions specified by attributes. More...

unsigned	nregion_element (const unsigned &i)
	Return the number of elements in the i-th region. More...

FiniteElement *	region_element_pt (const unsigned &i, const unsigned &e)
	Return the e-th element in the i-th region. More...

unsigned	nregion_attribute ()
	Return the number of attributes used in the mesh. More...

double	region_attribute (const unsigned &i)
	Return the attribute associated with region i. More...

GeomObject *	boundary_geom_object_pt (const unsigned &b)

std::map< unsigned, GeomObject * > &	boundary_geom_object_pt ()
	Return direct access to the geometric object storage. More...

std::map< unsigned, Vector< double > > &	boundary_coordinate_limits ()

Vector< double > &	boundary_coordinate_limits (const unsigned &b)

unsigned	nboundary_element_in_region (const unsigned &b, const unsigned &r) const
	Return the number of elements adjacent to boundary b in region r. More...

FiniteElement *	boundary_element_in_region_pt (const unsigned &b, const unsigned &r, const unsigned &e) const
	Return pointer to the e-th element adjacent to boundary b in region r. More...

int	face_index_at_boundary_in_region (const unsigned &b, const unsigned &r, const unsigned &e) const
	Return face index of the e-th element adjacent to boundary b in region r. More...

TriangleMeshPolyLine *	boundary_polyline_pt (const unsigned &b)

std::map< unsigned, std::set< Node * > > &	nodes_on_boundary_pt ()

const bool	get_connected_vertex_number_on_destination_polyline (TriangleMeshPolyLine *dst_polyline_pt, Vector< double > &vertex_coordinates, unsigned &vertex_number)

void	check_contiguousness_on_polylines_helper (Vector< TriangleMeshPolyLine * > &polylines_pt, unsigned &index)

void	check_contiguousness_on_polylines_helper (Vector< TriangleMeshPolyLine * > &polylines_pt, unsigned &index_halo_start, unsigned &index_halo_end)

bool	is_point_inside_polygon_helper (Vector< Vector< double >> polygon_vertices, Vector< double > point)
	Helper function that checks if a given point is inside a polygon. More...

void	enable_automatic_creation_of_vertices_on_boundaries ()

void	disable_automatic_creation_of_vertices_on_boundaries ()

bool	is_automatic_creation_of_vertices_on_boundaries_allowed ()

template<class ELEMENT >
void	setup_boundary_coordinates (const unsigned &b)

template<class ELEMENT >
void	setup_boundary_coordinates (const unsigned &b, std::ofstream &outfile)

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

	Mesh (const Vector< Mesh * > &sub_mesh_pt)

void	merge_meshes (const Vector< Mesh * > &sub_mesh_pt)

template<class BULK_ELEMENT >
void	doc_boundary_coordinates (const unsigned &b, std::ofstream &the_file)

virtual void	scale_mesh (const double &factor)

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

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

virtual	~Mesh ()
	Virtual Destructor to clean up all memory. More...

void	flush_element_and_node_storage ()

void	flush_element_storage ()

void	flush_node_storage ()

Node *&	node_pt (const unsigned long &n)
	Return pointer to global node n. More...

Node *	node_pt (const unsigned long &n) const
	Return pointer to global node n (const version) More...

GeneralisedElement *&	element_pt (const unsigned long &e)
	Return pointer to element e. More...

GeneralisedElement *	element_pt (const unsigned long &e) const
	Return pointer to element e (const version) More...

const Vector< GeneralisedElement * > &	element_pt () const
	Return reference to the Vector of elements. More...

Vector< GeneralisedElement * > &	element_pt ()
	Return reference to the Vector of elements. More...

FiniteElement *	finite_element_pt (const unsigned &e) const

Node *&	boundary_node_pt (const unsigned &b, const unsigned &n)
	Return pointer to node n on boundary b. More...

Node *	boundary_node_pt (const unsigned &b, const unsigned &n) const
	Return pointer to node n on boundary b. More...

void	set_nboundary (const unsigned &nbound)
	Set the number of boundaries in the mesh. More...

void	remove_boundary_nodes ()
	Clear all pointers to boundary nodes. More...

void	remove_boundary_nodes (const unsigned &b)

void	remove_boundary_node (const unsigned &b, Node *const &node_pt)
	Remove a node from the boundary b. More...

void	add_boundary_node (const unsigned &b, Node *const &node_pt)
	Add a (pointer to) a node to the b-th boundary. More...

void	copy_boundary_node_data_from_nodes ()

bool	boundary_coordinate_exists (const unsigned &i) const
	Indicate whether the i-th boundary has an intrinsic coordinate. More...

unsigned long	nelement () const
	Return number of elements in the mesh. More...

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

unsigned	ndof_types () const
	Return number of dof types in mesh. More...

unsigned	elemental_dimension () const
	Return number of elemental dimension in mesh. More...

unsigned	nodal_dimension () const
	Return number of nodal dimension in mesh. More...

void	add_node_pt (Node *const &node_pt)
	Add a (pointer to a) node to the mesh. More...

void	add_element_pt (GeneralisedElement *const &element_pt)
	Add a (pointer to) an element to the mesh. More...

virtual void	node_update (const bool &update_all_solid_nodes=false)

virtual void	reorder_nodes (const bool &use_old_ordering=true)

virtual void	get_node_reordering (Vector< Node * > &reordering, const bool &use_old_ordering=true) const

template<class BULK_ELEMENT , template< class > class FACE_ELEMENT>
void	build_face_mesh (const unsigned &b, Mesh *const &face_mesh_pt)

unsigned	self_test ()
	Self-test: Check elements and nodes. Return 0 for OK. More...

void	max_and_min_element_size (double &max_size, double &min_size)

double	total_size ()

void	check_inverted_elements (bool &mesh_has_inverted_elements, std::ofstream &inverted_element_file)

void	check_inverted_elements (bool &mesh_has_inverted_elements)

unsigned	check_for_repeated_nodes (const double &epsilon=1.0e-12)

Vector< Node * >	prune_dead_nodes ()

unsigned	nboundary () const
	Return number of boundaries. More...

unsigned long	nboundary_node (const unsigned &ibound) const
	Return number of nodes on a particular boundary. More...

FiniteElement *	boundary_element_pt (const unsigned &b, const unsigned &e) const
	Return pointer to e-th finite element on boundary b. More...

Node *	get_some_non_boundary_node () const

unsigned	nboundary_element (const unsigned &b) const
	Return number of finite elements that are adjacent to boundary b. More...

int	face_index_at_boundary (const unsigned &b, const unsigned &e) const

virtual void	dump (std::ofstream &dump_file, const bool &use_old_ordering=true) const
	Dump the data in the mesh into a file for restart. More...

void	dump (const std::string &dump_file_name, const bool &use_old_ordering=true) const
	Dump the data in the mesh into a file for restart. More...

virtual void	read (std::ifstream &restart_file)
	Read solution from restart file. More...

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

void	output_fct_paraview (std::ofstream &file_out, const unsigned &nplot, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) const

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

void	output (std::ostream &outfile)
	Output for all elements. More...

void	output (std::ostream &outfile, const unsigned &n_plot)
	Output at f(n_plot) points in each element. More...

void	output (FILE *file_pt)
	Output for all elements (C-style output) More...

void	output (FILE *file_pt, const unsigned &nplot)
	Output at f(n_plot) points in each element (C-style output) More...

void	output (const std::string &output_filename)
	Output for all elements. More...

void	output (const std::string &output_filename, const unsigned &n_plot)
	Output at f(n_plot) points in each element. More...

void	output_fct (std::ostream &outfile, const unsigned &n_plot, FiniteElement::SteadyExactSolutionFctPt)
	Output a given Vector function at f(n_plot) points in each element. More...

void	output_fct (std::ostream &outfile, const unsigned &n_plot, const double &time, FiniteElement::UnsteadyExactSolutionFctPt)

void	output_boundaries (std::ostream &outfile)
	Output the nodes on the boundaries (into separate tecplot zones) More...

void	output_boundaries (const std::string &output_filename)

void	assign_initial_values_impulsive ()
	Assign initial values for an impulsive start. More...

void	shift_time_values ()

void	calculate_predictions ()

void	set_nodal_and_elemental_time_stepper (TimeStepper *const &time_stepper_pt, const bool &preserve_existing_data)

virtual void	set_mesh_level_time_stepper (TimeStepper *const &time_stepper_pt, const bool &preserve_existing_data)

void	set_consistent_pinned_values_for_continuation (ContinuationStorageScheme *const &continuation_stepper_pt)
	Set consistent values for pinned data in continuation. More...

bool	does_pointer_correspond_to_mesh_data (double *const &parameter_pt)
	Does the double pointer correspond to any mesh data. More...

void	set_nodal_time_stepper (TimeStepper *const &time_stepper_pt, const bool &preserve_existing_data)
	Set the timestepper associated with the nodal data in the mesh. More...

void	set_elemental_internal_time_stepper (TimeStepper *const &time_stepper_pt, const bool &preserve_existing_data)

virtual void	compute_norm (double &norm)

virtual void	compute_norm (Vector< double > &norm)

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

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

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

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

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

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

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

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

bool	is_mesh_distributed () const
	Boolean to indicate if Mesh has been distributed. More...

OomphCommunicator *	communicator_pt () const

void	delete_all_external_storage ()
	Wipe the storage for all externally-based elements. More...

Private Attributes
unsigned	Nx
	Number of elements in x direction. More...

unsigned	Ny
	Number of elements in y directions. More...

double	Lx
	Length of mesh in x-direction. More...

double	Ly
	Length of mesh in y-direction. More...

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

typedef void(FiniteElement::*	UnsteadyExactSolutionFctPt) (const double &time, const Vector< double > &x, Vector< double > &soln)

Static Public Attributes inherited from oomph::UnstructuredTwoDMeshGeometryBase
static bool	Suppress_warning_about_regions_and_boundaries
	Public static flag to suppress warning; defaults to false. More...

Static Public Attributes inherited from oomph::Mesh
static Steady< 0 >	Default_TimeStepper
	The Steady Timestepper. More...

static bool	Suppress_warning_about_empty_mesh_level_time_stepper_function
	Static boolean flag to control warning about mesh level timesteppers. More...

Protected Member Functions inherited from oomph::UnstructuredTwoDMeshGeometryBase
void	snap_nodes_onto_geometric_objects ()

void	copy_connection_information (TriangleMeshCurveSection input_curve_pt, TriangleMeshCurveSection output_curve_pt)

void	copy_connection_information_to_sub_polylines (TriangleMeshCurveSection input_curve_pt, TriangleMeshCurveSection output_curve_pt)

Protected Member Functions inherited from oomph::Mesh
unsigned long	assign_global_eqn_numbers (Vector< double * > &Dof_pt)
	Assign (global) equation numbers to the nodes. More...

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

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

void	assign_local_eqn_numbers (const bool &store_local_dof_pt)
	Assign local equation numbers in all elements. More...

void	convert_to_boundary_node (Node &node_pt, const Vector< FiniteElement > &finite_element_pt)

void	convert_to_boundary_node (Node *&node_pt)

Protected Attributes inherited from oomph::UnstructuredTwoDMeshGeometryBase
bool	Allow_automatic_creation_of_vertices_on_boundaries

std::map< unsigned, Vector< FiniteElement * > >	Region_element_pt

Vector< double >	Region_attribute
	Vector of attributes associated with the elements in each region. More...

std::map< unsigned, GeomObject * >	Boundary_geom_object_pt
	Storage for the geometric objects associated with any boundaries. More...

std::map< unsigned, Vector< double > >	Boundary_coordinate_limits

Vector< TriangleMeshPolygon * >	Outer_boundary_pt
	Polygon that defines outer boundaries. More...

Vector< TriangleMeshPolygon * >	Internal_polygon_pt
	Vector of polygons that define internal polygons. More...

Vector< TriangleMeshOpenCurve * >	Internal_open_curve_pt
	Vector of open polylines that define internal curves. More...

Vector< Vector< double > >	Extra_holes_coordinates
	Storage for extra coordinates for holes. More...

std::map< unsigned, Vector< double > >	Regions_coordinates

std::map< unsigned, TriangleMeshCurveSection * >	Boundary_curve_section_pt

Vector< std::map< unsigned, Vector< FiniteElement * > > >	Boundary_region_element_pt
	Storage for elements adjacent to a boundary in a particular region. More...

Vector< std::map< unsigned, Vector< int > > >	Face_index_region_at_boundary
	Storage for the face index adjacent to a boundary in a particular region. More...

std::map< unsigned, Vector< std::pair< double, double > > >	Polygonal_vertex_arclength_info

std::map< unsigned, std::set< Node * > >	Nodes_on_boundary_pt

std::set< TriangleMeshCurveSection * >	Free_curve_section_pt

std::set< TriangleMeshPolygon * >	Free_polygon_pt

std::set< TriangleMeshOpenCurve * >	Free_open_curve_pt

Protected Attributes inherited from oomph::Mesh
Vector< Vector< Node * > >	Boundary_node_pt

bool	Lookup_for_elements_next_boundary_is_setup

Vector< Vector< FiniteElement * > >	Boundary_element_pt

Vector< Vector< int > >	Face_index_at_boundary

Vector< Node * >	Node_pt
	Vector of pointers to nodes. More...

Vector< GeneralisedElement * >	Element_pt
	Vector of pointers to generalised elements. More...

std::vector< bool >	Boundary_coordinate_exists

Detailed Description

template<class ELEMENT>
class oomph::SimpleRectangularTriMesh< ELEMENT >

Simple 2D triangular mesh for TElements.

Constructor & Destructor Documentation

◆ SimpleRectangularTriMesh()

template<class ELEMENT >

oomph::SimpleRectangularTriMesh< ELEMENT >::SimpleRectangularTriMesh	(	const unsigned &	n_x,
		const unsigned &	n_y,
		const double &	l_x,
		const double &	l_y,
		TimeStepper *	time_stepper_pt = `&Mesh::Default_TimeStepper`
	)

Constructor n_x : number of elements in the x direction; n_y : number of elements in the y direction; l_x : length in the x direction l_y : length in the y direction Ordering of elements: 'lower left' to 'lower right' then 'upwards'

Constructor for simple 2D triangular mesh class:

n_x : number of elements in the x direction

n_y : number of elements in the y direction

l_x : length in the x direction

l_y : length in the y direction

Ordering of elements: 'lower left' to 'lower right' then 'upwards'

     : Nx(n_x), Ny(n_y), Lx(l_x), Ly(l_y)
   {
     using namespace MathematicalConstants;
  
     // Mesh can only be built with 2D Telements.
     MeshChecker::assert_geometric_element<TElementGeometricBase, ELEMENT>(2);
  
     // Set number of boundaries
     this->set_nboundary(4);
  
     // Allocate the store for the elements
     unsigned n_element = (Nx) * (Ny)*2;
     Element_pt.resize(n_element, 0);
  
     // Create first element
     Element_pt[0] = new ELEMENT;
  
     // Currently this mesh only works for 3 and 6 noded triangles
     if ((finite_element_pt(0)->nnode_1d() != 2) &&
         (finite_element_pt(0)->nnode_1d() != 3) &&
         (finite_element_pt(0)->nnode_1d() != 4))
     {
       throw OomphLibError(
         "Currently this mesh only works for 3, 6 & 10-noded triangles",
         OOMPH_CURRENT_FUNCTION,
         OOMPH_EXCEPTION_LOCATION);
     }
  
     unsigned n_node = 0;
     // Unless nnode_1d returned as !=2 default no extra nodes
     unsigned additional_nnode = 0;
  
     // Allocate storage if no extra nodes
     if (finite_element_pt(0)->nnode_1d() == 2)
     {
       n_node = (Nx + 1) * (Ny + 1);
     }
  
     if (finite_element_pt(0)->nnode_1d() == 3)
     {
       additional_nnode = 3;
       // Allocate storage for nodes (including extra)
       n_node = (2 * Nx + 1) * (2 * Ny + 1);
     }
  
     if (finite_element_pt(0)->nnode_1d() == 4)
     {
       additional_nnode = 7;
       // Allocate storage for notes (including extra)
       n_node = (3 * Nx + 1) * (3 * Ny + 1);
     }
  
     Node_pt.resize(n_node);
  
     // Set up geometrical data
     //------------------------
     unsigned long node_count = 0;
     unsigned long element_count = 0;
     double xinit = 0.0, yinit = 0.0;
     // Set the values of the increments
     double xstep = Lx / (Nx);
     double ystep = Ly / (Ny);
  
  
     // FIRST NODE (lower left corner)
     //-------------------------------
  
     // Set the corner node
  
     // Allocate memory for the corner node of the first element
     //(which is not created loop as all subsequent bottom corners already exist)
     Node_pt[node_count] =
       finite_element_pt(0)->construct_node(1, time_stepper_pt);
  
     // Set the pointer from the element
     finite_element_pt(0)->node_pt(1) = Node_pt[node_count];
  
     // Don't increment node_count yet as we still need its containing box
     //(position and boundaries for first node are allocated in the main loop)
  
     // CREATE THE ELEMENTS (each has 3 local nodes)
     //--------------------------------------------------
     // Elements are created two at a time, the first being lower triangular
     // and the second upper triangular, so that they form a containing box.
     // Local nodes are numbered anti-clockwise with node_pt(1) being the
     // right-angle corner of the triangle.
     // The global node number, node_count, is considered to define
     // a containing box, with node_count defined as the node number
     // of the bottom left corner of the box.
     for (unsigned j = 0; j < Ny + 1; ++j)
     {
       for (unsigned i = 0; i < Nx + 1; ++i)
       {
         // CURRENT BOX
         //(nodes on RHS or top edge of domain do not define a box)
         if (j < Ny && i < Nx)
         {
           // Create two elements
           //------------------------------
           if (element_count != 0) // 0th element already exists
           {
             Element_pt[element_count] = new ELEMENT;
           }
  
           Element_pt[element_count + 1] = new ELEMENT;
  
  
           // Allocate memory for nodes in the current box
           //--------------------------------------------
           // If node on LHS of domain, allocate the top left corner node
           if (i == 0)
           {
             Node_pt[node_count + Nx + 1] =
               finite_element_pt(element_count)
                 ->construct_node(0, time_stepper_pt);
           }
  
           // If on bottom row, allocate the bottom right corner node
           if (j == 0)
           {
             Node_pt[node_count + 1] = finite_element_pt(element_count)
                                         ->construct_node(2, time_stepper_pt);
           }
  
           // Always need to allocate top right corner node
           Node_pt[node_count + Nx + 2] = finite_element_pt(element_count + 1)
                                            ->construct_node(1, time_stepper_pt);
  
           // Bottom left corner node is already allocated
  
  
           // Set the pointers from the elements
           //----------------------------------
           finite_element_pt(element_count)->node_pt(0) =
             Node_pt[node_count + Nx + 1];
           finite_element_pt(element_count)->node_pt(1) = Node_pt[node_count];
           finite_element_pt(element_count)->node_pt(2) =
             Node_pt[node_count + 1];
           finite_element_pt(element_count + 1)->node_pt(0) =
             Node_pt[node_count + 1];
           finite_element_pt(element_count + 1)->node_pt(1) =
             Node_pt[node_count + Nx + 2];
           finite_element_pt(element_count + 1)->node_pt(2) =
             Node_pt[node_count + Nx + 1];
  
           element_count += 2;
         }
  
         // CURRENT (GLOBAL) NODE
         // Set the position
         Node_pt[node_count]->x(0) = xinit + i * xstep;
         Node_pt[node_count]->x(1) = yinit + j * ystep;
  
         // Add node to any relevant boundaries
         if (j == 0)
         {
           this->convert_to_boundary_node(Node_pt[node_count]);
           add_boundary_node(0, Node_pt[node_count]);
         }
         if (j == Ny)
         {
           this->convert_to_boundary_node(Node_pt[node_count]);
           add_boundary_node(2, Node_pt[node_count]);
         }
         if (i == 0)
         {
           this->convert_to_boundary_node(Node_pt[node_count]);
           add_boundary_node(3, Node_pt[node_count]);
         }
         if (i == Nx)
         {
           this->convert_to_boundary_node(Node_pt[node_count]);
           add_boundary_node(1, Node_pt[node_count]);
         }
  
         // Increment counter
         node_count++;
       }
     }
  
  
     if (additional_nnode == 3)
     {
       // Reset element counter to allow looping over existing elements
       // to add extra nodes.
       // Note that the node_count is not reset so no entries are overwritten
       element_count = 0;
       for (unsigned j = 0; j < Ny + 1; ++j)
       {
         // Note: i counter reduced by 1 since i axis runs through middle of
         // elements on x-axis
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The additional nodes will be added in stages for ease of
           // application. Note: local numbering follows the anti-clockwise
           // pattern, node 3 halfway between  nodes 0-1, 4 between 1-2 and the
           // 5th local node between 2-0.
  
           // Restricted to stop creation of additional nodes outside the mesh
           if (j < Ny)
           {
             // If on the bottom row middle-node at bottom needs allocating
             if (j == 0)
             {
               Node_pt[node_count] = finite_element_pt(element_count)
                                       ->construct_node(4, time_stepper_pt);
             }
  
             // Due to directions of iteration node at middle of top box edge
             // (in next element) always needs allocating
             Node_pt[node_count + Nx] = finite_element_pt(element_count + 1)
                                          ->construct_node(4, time_stepper_pt);
  
             // Set pointers to the top/bottom middle nodes
             // Bottom node
             finite_element_pt(element_count)->node_pt(4) = Node_pt[node_count];
             // Top node
             finite_element_pt(element_count + 1)->node_pt(4) =
               Node_pt[node_count + Nx];
  
             // Increase the element counter to add top/bottom nodes to
             // next pair of element on next pass
             element_count += 2;
           } // End extra top/bottom node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 0.5) * xstep;
           Node_pt[node_count]->x(1) = yinit + j * ystep;
  
           // Add node to any applicable boundaries (node 4's can only be top
           // or bottom boundaries)
           if (j == 0)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(0, Node_pt[node_count]);
           }
           if (j == Ny)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(2, Node_pt[node_count]);
           }
  
           // Update node_count
           node_count++;
         }
       }
  
       // Next stage of additional node implementation involes the middle left
       // and right nodes (local number 3 on each triangle)
  
       // Element counter reset for second loop over existing elements
       element_count = 0;
       // Note: j counter reduced by 1 since j axis runs through middle of
       // elements on y-axis
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx + 1; ++i)
         {
           if (j < Ny && i < Nx)
           {
             // If on the left hand boundary the node at middle of the left
             // side needs allocating
             if (i == 0)
             {
               Node_pt[node_count] = finite_element_pt(element_count)
                                       ->construct_node(3, time_stepper_pt);
             }
  
             // The mid node on the right hand side always needs constructing
             // within the bounds of the mesh
             Node_pt[node_count + 1] = finite_element_pt(element_count + 1)
                                         ->construct_node(3, time_stepper_pt);
  
             // Set pointers from the elements to new nodes
             finite_element_pt(element_count)->node_pt(3) = Node_pt[node_count];
             finite_element_pt(element_count + 1)->node_pt(3) =
               Node_pt[node_count + 1];
  
             // Increase element_count by 2
             element_count += 2;
           } // End extra left/right node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + i * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 0.5) * ystep;
  
           // Add node to any applicable boundaries again - only be left/right
           if (i == 0)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(3, Node_pt[node_count]);
           }
           if (i == Nx)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(1, Node_pt[node_count]);
           }
  
           // Update node_count
           node_count++;
         }
       }
  
       // Final stage of inserting extra nodes - inclusion of the local
       // number 5 (middle of hypot. edge)
  
       element_count = 0;
       // Note: both i,j counters reduced by 1 since j axis runs through middle
       // of elements in both x,y
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The middle node always needs constructing
           Node_pt[node_count] = finite_element_pt(element_count)
                                   ->construct_node(5, time_stepper_pt);
  
           // Set pointers from the elements to new nodes
           finite_element_pt(element_count)->node_pt(5) = Node_pt[node_count];
           finite_element_pt(element_count + 1)->node_pt(5) =
             Node_pt[node_count];
  
           // Increase element_count by 2
           element_count += 2;
           // End extra left/right node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 0.5) * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 0.5) * ystep;
  
           // All nodes are internal in this structure so no boundaries
           // applicable
  
           // Update node_count
           node_count++;
         }
       }
     }
     // End of extra nodes for 6 noded trianglur elements
  
  
     if (additional_nnode == 7)
     {
       // Reset element counter to allow looping over existing elements
       // to add extra nodes.
       // Note that the node_count is not reset so no entries are overwritten
       element_count = 0;
       for (unsigned j = 0; j < Ny + 1; ++j)
       {
         // Note: i counter reduced by 1 for implementation of lower element
         // node 5 and upper element node 6's.
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The additional nodes will be added in stages for ease of
           // application. Note: local numbering follows the anti-clockwise
           // pattern, nodes 3 and 4 equidistant between nodes 0-1, 5 and 6
           // between 1-2, 7 and 8  between 2-0 and last node 9 located
           // (internally) at the centroid of the triangle.
  
           // Restricted to stop creation of additional nodes outside the mesh
           if (j < Ny)
           {
             // If on the bottom row middle-left-node at bottom needs allocating
             if (j == 0)
             {
               Node_pt[node_count] = finite_element_pt(element_count)
                                       ->construct_node(5, time_stepper_pt);
             }
  
             // Due to directions of iteration node at middle left of top box
             // edge (in next element) always needs allocating
             Node_pt[node_count + Nx] = finite_element_pt(element_count + 1)
                                          ->construct_node(6, time_stepper_pt);
  
             // Set pointers to the top/bottom middle nodes
             // Bottom node
             finite_element_pt(element_count)->node_pt(5) = Node_pt[node_count];
             // Top node
             finite_element_pt(element_count + 1)->node_pt(6) =
               Node_pt[node_count + Nx];
  
             // Increase the element counter to add top/bottom nodes to
             // next pair of element on next pass
             element_count += 2;
           } // End extra top/bottom node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 1.0 / 3.0) * xstep;
           Node_pt[node_count]->x(1) = yinit + j * ystep;
  
           // Add node to any applicable boundaries (exactly as previous)
           if (j == 0)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(0, Node_pt[node_count]);
           }
           if (j == Ny)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(2, Node_pt[node_count]);
           }
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next stage: bottom-middle-right node (node 6 in lower tri.el.)
       // and top-middle-right node (node 5 in upper tri.el.)
  
       element_count = 0;
       for (unsigned j = 0; j < Ny + 1; ++j)
       {
         // Note: i counter as for above 5/6's
         for (unsigned i = 0; i < Nx; ++i)
         {
           // Restricted to stop creation of additional nodes outside the mesh
           if (j < Ny)
           {
             // If on the bottom row middle-right-node at bottom needs allocating
             if (j == 0)
             {
               Node_pt[node_count] = finite_element_pt(element_count)
                                       ->construct_node(6, time_stepper_pt);
             }
  
             // Due to directions of iteration node at middle left of top box
             // edge (in next element) always needs allocating
             Node_pt[node_count + Nx] = finite_element_pt(element_count + 1)
                                          ->construct_node(5, time_stepper_pt);
  
             // Set pointers to the top/bottom middle nodes
             // Bottom node
             finite_element_pt(element_count)->node_pt(6) = Node_pt[node_count];
             // Top node
             finite_element_pt(element_count + 1)->node_pt(5) =
               Node_pt[node_count + Nx];
  
             // Increase the element counter to add top/bottom nodes to
             // next pair of element on next pass
             element_count += 2;
           } // End extra top/bottom node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 2.0 / 3.0) * xstep;
           Node_pt[node_count]->x(1) = yinit + j * ystep;
  
           // Add node to any applicable boundaries (exactly as previous)
           if (j == 0)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(0, Node_pt[node_count]);
           }
           if (j == Ny)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(2, Node_pt[node_count]);
           }
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next stage of additional node implementation involes node 4 on lower
       // tri. el. and node 3 on upper tri. el.
  
       // Element counter reset for next loop over existing elements
       element_count = 0;
       // Note: j counter reduced by 1 similar to others above
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx + 1; ++i)
         {
           if (j < Ny && i < Nx)
           {
             // If on the left hand boundary the lower middle node of the left
             // side needs allocating
             if (i == 0)
             {
               Node_pt[node_count] = finite_element_pt(element_count)
                                       ->construct_node(4, time_stepper_pt);
             }
  
             // The mid node on the right hand side always needs constructing
             // within the bounds of the mesh
             Node_pt[node_count + 1] = finite_element_pt(element_count + 1)
                                         ->construct_node(3, time_stepper_pt);
  
             // Set pointers from the elements to new nodes
             finite_element_pt(element_count)->node_pt(4) = Node_pt[node_count];
             finite_element_pt(element_count + 1)->node_pt(3) =
               Node_pt[node_count + 1];
  
             // Increase element_count by 2
             element_count += 2;
           } // End extra left/right node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + i * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 1.0 / 3.0) * ystep;
  
           // Add node to any applicable boundaries again - only be left/right
           if (i == 0)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(3, Node_pt[node_count]);
           }
           if (i == Nx)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(1, Node_pt[node_count]);
           }
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next stage of additional node implementation involes node 3 on lower
       // tri. el. and node 4 on upper tri. el.
  
       // Element counter reset for next loop over existing elements
       element_count = 0;
       // Note: j counter reduced by 1 similar to others above
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx + 1; ++i)
         {
           if (j < Ny && i < Nx)
           {
             // If on the left hand boundary the lower middle node of the left
             // side needs allocating
             if (i == 0)
             {
               Node_pt[node_count] = finite_element_pt(element_count)
                                       ->construct_node(3, time_stepper_pt);
             }
  
             // The mid node on the right hand side always needs constructing
             // within the bounds of the mesh
             Node_pt[node_count + 1] = finite_element_pt(element_count + 1)
                                         ->construct_node(4, time_stepper_pt);
  
             // Set pointers from the elements to new nodes
             finite_element_pt(element_count)->node_pt(3) = Node_pt[node_count];
             finite_element_pt(element_count + 1)->node_pt(4) =
               Node_pt[node_count + 1];
  
             // Increase element_count by 2
             element_count += 2;
           } // End extra left/right node construction
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + i * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 2.0 / 3.0) * ystep;
  
           // Add node to any applicable boundaries again - only be left/right
           if (i == 0)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(3, Node_pt[node_count]);
           }
           if (i == Nx)
           {
             this->convert_to_boundary_node(Node_pt[node_count]);
             add_boundary_node(1, Node_pt[node_count]);
           }
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next is the lower tri. el. totally internal node with local number 9
       element_count = 0;
       // Note: both i,j counters reduced by 1
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The middle node always needs constructing
           Node_pt[node_count] = finite_element_pt(element_count)
                                   ->construct_node(9, time_stepper_pt);
  
           // Set pointers from the elements to new nodes
           finite_element_pt(element_count)->node_pt(9) = Node_pt[node_count];
  
           // Increase element_count by 2
           element_count += 2;
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 1.0 / 3.0) * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 1.0 / 3.0) * ystep;
  
           // All nodes are internal in this structure so no boundaries
           // applicable
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next is the bottom hyp. node -
       // lower tri. el. number 7, upper tri. el. number 8
       element_count = 0;
       // Note: both i,j counters reduced by 1
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The node always needs constructing
           Node_pt[node_count] = finite_element_pt(element_count)
                                   ->construct_node(7, time_stepper_pt);
  
           // Set pointers from the elements to new nodes
           finite_element_pt(element_count)->node_pt(7) = Node_pt[node_count];
           finite_element_pt(element_count + 1)->node_pt(8) =
             Node_pt[node_count];
  
           // Increase element_count by 2
           element_count += 2;
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 2.0 / 3.0) * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 1.0 / 3.0) * ystep;
  
           // All nodes are internal in this structure so no boundaries
           // applicable
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next is the upper hyp. node -
       // lower tri. el. number 8, upper tri. el. number 7
       element_count = 0;
       // Note: both i,j counters reduced by 1
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The node always needs constructing
           Node_pt[node_count] = finite_element_pt(element_count)
                                   ->construct_node(8, time_stepper_pt);
  
           // Set pointers from the elements to new nodes
           finite_element_pt(element_count)->node_pt(8) = Node_pt[node_count];
           finite_element_pt(element_count + 1)->node_pt(7) =
             Node_pt[node_count];
  
           // Increase element_count by 2
           element_count += 2;
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 1.0 / 3.0) * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 2.0 / 3.0) * ystep;
  
           // All nodes are internal in this structure so no boundaries
           // applicable
  
           // Update node_count
           node_count++;
         }
       }
  
  
       // Next is the upper tri. el. totally internal node with local number 9
       element_count = 0;
       // Note: both i,j counters reduced by 1
       for (unsigned j = 0; j < Ny; ++j)
       {
         for (unsigned i = 0; i < Nx; ++i)
         {
           // The middle node always needs constructing
           Node_pt[node_count] = finite_element_pt(element_count + 1)
                                   ->construct_node(9, time_stepper_pt);
  
           // Set pointers from the elements to new nodes
           finite_element_pt(element_count + 1)->node_pt(9) =
             Node_pt[node_count];
  
           // Increase element_count by 2
           element_count += 2;
  
           // Set position of current (Global) Node
           Node_pt[node_count]->x(0) = xinit + double(i + 2.0 / 3.0) * xstep;
           Node_pt[node_count]->x(1) = yinit + double(j + 2.0 / 3.0) * ystep;
  
           // All nodes are internal in this structure so no boundaries
           // applicable
  
           // Update node_count
           node_count++;
         }
       }
     }
   }