Shape function space for linear triangle on sphere, obtained from the canonical linear (affine) functions. It is mapped using gnomonic projection to a plane tangent at the first vertex It works well for edges that are great circle arcs; RLL meshes do not have this property, but HOMME or MPAS meshes do have it. More...

#include <ElemUtil.hpp>

Inheritance diagram for moab::Element::SphericalTri:

Collaboration diagram for moab::Element::SphericalTri:

Public Member Functions
	SphericalTri (const std::vector< CartVect > &vertices)

virtual	~SphericalTri ()

virtual bool	inside_box (const CartVect &pos, double &tol) const

CartVect	ievaluate (const CartVect &x, double tol=1e-6, const CartVect &x0=CartVect(0.0)) const
	Evaluate the inverse map (calculate $\vec \xi = F^-1($\vec x)$ to given tolerance) More...

Public Member Functions inherited from moab::Element::LinearTri
	LinearTri (const std::vector< CartVect > &vertices)

	LinearTri ()

virtual	~LinearTri ()

virtual CartVect	evaluate (const CartVect &xi) const
	Evaluate the map on $x_i$ (calculate $\vec x = F($\vec \xi)$ ) More...

virtual Matrix3	jacobian (const CartVect &) const
	Evaluate the map's Jacobi matrix. More...

virtual Matrix3	ijacobian (const CartVect &) const
	Evaluate the inverse of the Jacobi matrix. More...

virtual double	det_jacobian (const CartVect &) const
	Evaluate the determinate of the Jacobi matrix. More...

virtual double	det_ijacobian (const CartVect &) const
	Evaluate the determinate of the inverse Jacobi matrix. More...

virtual void	set_vertices (const std::vector< CartVect > &v)
	Set vertices. More...

virtual bool	inside_nat_space (const CartVect &xi, double &tol) const
	decide if within the natural param space, with a tolerance More...

virtual double	evaluate_scalar_field (const CartVect &xi, const double *field_vertex_values) const
	Evaluate a scalar field at a point given field values at the vertices. More...

virtual double	integrate_scalar_field (const double *field_vertex_values) const
	Integrate a scalar field over the element given field values at the vertices. More...

Public Member Functions inherited from moab::Element::Map
	Map (const std::vector< CartVect > &v)
	Construct a Map defined by the given std::vector of vertices. More...

	Map (const unsigned int n)
	Construct a Map defined by n vertices. More...

virtual	~Map ()

unsigned int	size ()
	Size of the vertices vector. More...

const std::vector< CartVect > &	get_vertices ()
	Retrieve vertices. More...

Protected Attributes
CartVect	v1

Matrix3	transf

Protected Attributes inherited from moab::Element::LinearTri
Matrix3	T

Matrix3	T_inverse

double	det_T

double	det_T_inverse

Protected Attributes inherited from moab::Element::Map
std::vector< CartVect >	vertex

Additional Inherited Members
Static Protected Attributes inherited from moab::Element::LinearTri
static const double	corner [3][3] = { { 0, 0, 0 }, { 1, 0, 0 }, { 0, 1, 0 } }

Detailed Description

Shape function space for linear triangle on sphere, obtained from the canonical linear (affine) functions. It is mapped using gnomonic projection to a plane tangent at the first vertex It works well for edges that are great circle arcs; RLL meshes do not have this property, but HOMME or MPAS meshes do have it.

Definition at line 382 of file ElemUtil.hpp.

Constructor & Destructor Documentation

◆ SphericalTri()

moab::Element::SphericalTri::SphericalTri ( const std::vector< CartVect > & vertices )

Definition at line 553 of file ElemUtil.cpp.

  {
  vertex.resize( vertices.size() );
  vertex = vertices;
  // project the vertices to the plane tangent at first vertex
  v1 = vertex[0]; // member data
  double v1v1 = v1 % v1; // this is 1, in general, for unit sphere meshes
  for( int j = 1; j < 3; j++ )
  {
  // first, bring all vertices in the gnomonic plane
  // the new vertex will intersect the plane at vnew
  // so that (vnew-v1)%v1 is 0 ( vnew is in the tangent plane, i.e. normal to v1 )
  // pos is the old position of the vertex, and it is in general on the sphere
  // vnew = alfa*pos; (alfa*pos-v1)%v1 = 0 <=> alfa*(pos%v1)=v1%v1 <=> alfa =
  // v1v1/(pos%v1)
  // <=> vnew = ( v1v1/(pos%v1) )*pos
  CartVect vnew = v1v1 / ( vertex[j] % v1 ) * vertex[j];
  vertex[j] = vnew;
  }
  // will compute a transf matrix, such that a new point will be transformed with
  // newpos = transf * (vnew-v1), and it will be a point in the 2d plane
  // the transformation matrix will be oriented in such a way that orientation will be
  // positive
  CartVect vx = vertex[1] - v1; // this will become Ox axis
  // z axis will be along v1, in such a way that orientation of the quad is positive
  // look at the first 2 edges
  CartVect vz = vx * ( vertex[2] - vertex[1] );
  vz = vz / vz.length();
  
  vx = vx / vx.length();
  
  CartVect vy = vz * vx;
  transf = Matrix3( vx[0], vx[1], vx[2], vy[0], vy[1], vy[2], vz[0], vz[1], vz[2] );
  vertex[0] = CartVect( 0. );
  for( int j = 1; j < 3; j++ )
  vertex[j] = transf * ( vertex[j] - v1 );
  
  LinearTri::set_vertices( vertex );
  }

References moab::CartVect::length(), moab::Element::LinearTri::set_vertices(), transf, and v1.

◆ ~SphericalTri()

virtual moab::Element::SphericalTri::~SphericalTri ( )

inlinevirtual

Definition at line 386 of file ElemUtil.hpp.

386 {};

Member Function Documentation

◆ ievaluate()

CartVect moab::Element::SphericalTri::ievaluate	(	const CartVect &	x,
		double	tol = `1e-6`,
		const CartVect &	x0 = `CartVect( 0.0 )`
	)		const

virtual

Evaluate the inverse map (calculate $\vec \xi = F^-1($\vec x)$ to given tolerance)

Reimplemented from moab::Element::LinearTri.

Definition at line 593 of file ElemUtil.cpp.

  {
  // project to the plane tangent at first vertex (gnomonic projection)
  double v1v1 = v1 % v1;
  CartVect vnew = v1v1 / ( x % v1 ) * x; // so that (vnew-v1)%v1 is 0
  vnew = transf * ( vnew - v1 );
  // det will be positive now
  return LinearTri::ievaluate( vnew );
  }

References moab::Element::LinearTri::ievaluate(), transf, and v1.

Referenced by moab::Coupler::nat_param(), and test_spherical_tri().

◆ inside_box()

bool moab::Element::SphericalTri::inside_box	(	const CartVect &	pos,
		double &	tol
	)		const

virtual

Reimplemented from moab::Element::Map.

Definition at line 603 of file ElemUtil.cpp.

  {
  // project to the plane tangent at first vertex
  // CartVect v1=vertex[0];
  double v1v1 = v1 % v1;
  CartVect vnew = v1v1 / ( pos % v1 ) * pos; // so that (x-v1)%v1 is 0
  vnew = transf * ( vnew - v1 );
  return Map::inside_box( vnew, tol );
  }