ref_triangle.hpp File Reference

Defines the reference triangle element used in nodal Discontinuous Galerkin methods. More...

#include "oiseau/dg/nodal/ref_element.hpp"
#include "xtensor/core/xtensor_forward.hpp"

Go to the source code of this file.

Classes
class	oiseau::dg::nodal::RefTriangle
	Represents a reference triangular element for nodal Discontinuous Galerkin methods. More...

Functions
xt::xarray< double >	oiseau::dg::nodal::detail::rs_to_ab (const xt::xarray< double > &rs)
	Converts (r, s) coordinates to collapsed (a, b) coordinates for the triangle.
xt::xarray< double >	oiseau::dg::nodal::detail::generate_triangle_nodes (unsigned order)
	Generates nodal points on an equilateral triangle via Warp & Blend.
xt::xarray< double >	oiseau::dg::nodal::detail::equilateral_xy_to_rs (const xt::xarray< double > &xy)
	Maps equilateral triangle (x, y) to reference (r, s) coordinates.
xt::xarray< double >	oiseau::dg::nodal::detail::warp_factor (unsigned order, const xt::xarray< double > &rs)
	Computes 1D warp factor for node distribution on triangle edges.

Detailed Description

Defines the reference triangle element used in nodal Discontinuous Galerkin methods.

This file contains the declaration of the RefTriangle class, which represents a reference triangular element, and associated internal helper functions in the detail namespace for generating nodes, basis functions, and performing coordinate transformations.

ASCII representation of the reference triangle node numbering (order=4):

(14) | \ (12) (13) | \ (9) (10) (11) | \ (5) (6) (7) (8) | \ (0)–(1)–(2)–(3)–(4)

Function Documentation

equilateral_xy_to_rs()

xt::xarray< double > oiseau::dg::nodal::detail::equilateral_xy_to_rs

(

const xt::xarray< double > &

xy

)

Maps equilateral triangle (x, y) to reference (r, s) coordinates.

Parameters

xy	2D array (N_points × 2) of equilateral (x, y).

Returns

2D array (N_points × 2) of reference (r, s).

generate_triangle_nodes()

xt::xarray< double > oiseau::dg::nodal::detail::generate_triangle_nodes

(

unsigned

order

)

Generates nodal points on an equilateral triangle via Warp & Blend.

The result is then mapped to the (r, s) reference triangle.

Parameters

order

Polynomial order (number of points = (order+1)*(order+2)/2).

Returns

2D array (N_points × 2) of (x, y) on the equilateral triangle.

rs_to_ab()

xt::xarray< double > oiseau::dg::nodal::detail::rs_to_ab

(

const xt::xarray< double > &

rs

)

Converts (r, s) coordinates to collapsed (a, b) coordinates for the triangle.

a = 2*(1+r)/(1-s) - 1, b = s; with special handling when s == 1.

Parameters

rs	2D array (shape: N_points × 2) of (r, s) coordinates.

Returns

2D array (shape: N_points × 2) of (a, b) coordinates.

warp_factor()

xt::xarray< double > oiseau::dg::nodal::detail::warp_factor	(	unsigned	order,
		const xt::xarray< double > &	rs )

Computes 1D warp factor for node distribution on triangle edges.

Parameters

order	Polynomial order.
rs	1D array (length = order+1) of coordinates along an edge.

Returns

1D array (length = order+1) of warp offsets.