oiseau::dg::nodal::RefTetrahedron Class Reference

Represents a reference tetrahedron element for nodal Discontinuous Galerkin methods. More...

#include <ref_tetrahedron.hpp>

Inheritance diagram for oiseau::dg::nodal::RefTetrahedron:

Public Member Functions
	RefTetrahedron (unsigned order)
	Constructs a RefTetrahedron object of a specific polynomial order.
xt::xarray< double >	basis_function (const xt::xarray< double > &rst, int i, int j, int k) const
	Evaluates a 3D basis function on the reference tetrahedron.
xt::xarray< double >	grad_basis_function (const xt::xarray< double > &rst, int i, int j, int k) const
	Computes the gradient of a 3D basis function on the reference tetrahedron.
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

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 tetrahedron element for nodal Discontinuous Galerkin methods.

The reference tetrahedron is defined in a canonical (r, s, t) coordinate system, typically mapping to the region { (r,s,t) | r,s,t >= -1, r+s+t <= -1 }. This class computes properties like nodal distribution, Vandermonde matrices, and differentiation operators for a specified polynomial order.

Constructor & Destructor Documentation

RefTetrahedron()

oiseau::dg::nodal::RefTetrahedron::RefTetrahedron ( unsigned order )

explicit

Constructs a RefTetrahedron object of a specific polynomial order.

Parameters

order

The polynomial order for the basis functions on the tetrahedron. Determines the number of nodes and approximation accuracy. The total degree of basis polynomials will be at most order.

Member Function Documentation

basis_function()

xt::xarray< double > oiseau::dg::nodal::RefTetrahedron::basis_function	(	const xt::xarray< double > &	rst,
		int	i,
		int	j,
		int	k ) const

Evaluates a 3D basis function on the reference tetrahedron.

The basis functions are typically orthogonal polynomials (e.g., a product of Jacobi polynomials) defined in a transformed coordinate system (a,b,c) derived from the reference (r,s,t) coordinates. The specific form and the range of indices i,j,k may depend on this->order(). Typically, \(i+j+k \le \text{order}\).

Parameters

rst	2D array (shape: N_points × 3) of (r, s, t) coordinates where to evaluate.
i	Polynomial degree index (e.g., for the first transformed coordinate).
j	Polynomial degree index (e.g., for the second transformed coordinate).
k	Polynomial degree index (e.g., for the third transformed coordinate).

Returns

1D array (length N_points) of basis function values at each (r, s, t) point.

grad_basis_function()

xt::xarray< double > oiseau::dg::nodal::RefTetrahedron::grad_basis_function	(	const xt::xarray< double > &	rst,
		int	i,
		int	j,
		int	k ) const

Computes the gradient of a 3D basis function on the reference tetrahedron.

Computes derivatives of the basis function \(\phi_{ijk}\) with respect to the reference coordinates (r, s, t).

Parameters

rst	2D array (shape: N_points × 3) of (r, s, t) coordinates.
i	Polynomial degree index.
j	Polynomial degree index.
k	Polynomial degree index.

Returns

2D array (shape: N_points × 3) containing (∂/∂r, ∂/∂s, ∂/∂t) for each point.

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

• /__w/oiseau/oiseau/src/oiseau/dg/nodal/ref_tetrahedron.hpp
• /__w/oiseau/oiseau/src/oiseau/dg/nodal/ref_tetrahedron.cpp