Represents a reference triangular element for nodal Discontinuous Galerkin methods. More...
#include <ref_triangle.hpp>
Public Member Functions | |
| RefTriangle (unsigned order) | |
| Constructs a RefTriangle object of a specific polynomial order. | |
| Public Member Functions inherited from oiseau::dg::nodal::RefElement | |
| const xt::xarray< double > & | v () const |
| const xt::xarray< double > & | gv () const |
| const xt::xarray< double > & | d () const |
| const xt::xarray< double > & | r () const |
| unsigned | order () const |
| unsigned | number_of_nodes () const |
| unsigned | number_of_face_nodes () const |
Static Public Member Functions | |
| static xt::xarray< double > | basis_function (const xt::xarray< double > &ab, int i, int j) |
| Static member function that evaluates a 2D orthonormal basis function on the reference triangle in collapsed (a, b) coordinates. | |
| static xt::xarray< double > | grad_basis_function (const xt::xarray< double > &ab, int i, int j) |
| Static member function that computes the gradient of a 2D orthonormal basis function on the reference triangle in (r, s) coordinates. | |
Additional Inherited Members | |
| Protected Member Functions inherited from oiseau::dg::nodal::RefElement | |
| RefElement (unsigned order) | |
| Protected Attributes inherited from oiseau::dg::nodal::RefElement | |
| unsigned | m_order |
| unsigned | m_np {} |
| unsigned | m_nfp {} |
| xt::xarray< double > | m_v |
| xt::xarray< double > | m_gv |
| xt::xarray< double > | m_d |
| xt::xarray< double > | m_r |
Detailed Description
Represents a reference triangular element for nodal Discontinuous Galerkin methods.
The reference triangle is defined in (r, s) coordinates with vertices at (-1, -1), (1, -1), and (-1, 1). This class computes and stores properties such as nodal distribution, Vandermonde matrix, and differentiation matrices for a specified polynomial order.
Constructor & Destructor Documentation
◆ RefTriangle()
|
explicit |
Constructs a RefTriangle object of a specific polynomial order.
- Parameters
-
order The polynomial order for the basis functions on the triangle. Determines the number of nodes and approximation accuracy.
Member Function Documentation
◆ basis_function()
|
static |
Static member function that evaluates a 2D orthonormal basis function on the reference triangle in collapsed (a, b) coordinates.
The basis function is defined by: \(\sqrt{2}\,P_i^{0,0}(a)\,P_j^{2i+1,0}(b)\,(1 - b)^i\) where \(P_n^{\alpha,\beta}\) are Jacobi polynomials.
- Parameters
-
ab 2D array (shape: N_points × 2) of (a, b) coordinates where to evaluate the basis. i Polynomial degree index for the a-direction component. j Polynomial degree index for the b-direction component.
- Returns
- 1D array (length N_points) of basis function values at each (a, b) point.
◆ grad_basis_function()
|
static |
Static member function that computes the gradient of a 2D orthonormal basis function on the reference triangle in (r, s) coordinates.
Uses the chain rule to transform derivatives from collapsed (a, b) to (r, s).
- Parameters
-
ab 2D array (shape: N_points × 2) of (a, b) coordinates from rs_to_ab. i Polynomial degree index for the a-component. j Polynomial degree index for the b-component.
- Returns
- 2D array (shape: N_points × 2) containing (∂/∂r, ∂/∂s) for each point.
The documentation for this class was generated from the following files:
- /__w/oiseau/oiseau/src/oiseau/dg/nodal/ref_triangle.hpp
- /__w/oiseau/oiseau/src/oiseau/dg/nodal/ref_triangle.cpp
