1D line element defined by two boundary nodes, with interior nodes placed at Gauss-Lobatto-Legendre points More...

#include <segment.h>

Inheritance diagram for hummingbird::Segment:

Collaboration diagram for hummingbird::Segment:

Public Member Functions
	Segment (const std::array< size_t, 2 > boundary_node_ids, const int material_id, const int source_id, const Mesh &mesh)
	Construct a new Segment object.
std::vector< Node >	CreateInteriorNodes (const std::vector< Node > &existing_nodes, const GaussLobattoLegendre &gll_quadrature) override
	Create interior Node objects using the Gauss-Lobatto-Legendre quadrature set. The mapping from the reference to the real domain was based on the equations provided by the book Computational Seismology by Heiner Igel (https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.geokniga.org/bookfiles/geokniga-computationalseismologyapracticalintroductionbyigelheinerz-liborg.pdf&ved=2ahUKEwiMuanq7_OVAxXVN4YAHVgAD1QQFnoECBUQAQ&usg=AOvVaw35g1Tl3ZSFUMusz8LKQxHk). For a line segment defined by two endpoints, the mapping is:
arma::Col< double >	LocalForcingVector (const GaussLobattoLegendre &gll_quad, const Mesh &mesh, const MaterialBank &material_bank, const Ordinate &ordinate, const size_t ordinate_index) override
	Construct the local forcing vector, which is:
arma::Mat< double >	LocalStiffnessMatrix (const GaussLobattoLegendre &gll_quad, const MaterialBank &material_bank, const Ordinate &ordinate) override
	Construct the dense local stiffness matrix using spectral elements on a Gauss-Lobatto-Legendre grid, defined as:
arma::SpMat< double >	LocalMassMatrix (const GaussLobattoLegendre &gll_quad, const MaterialBank &material_bank) override
	Construct the sparse diagonal local mass matrix using spectral elements on a Gauss-Lobatto-Legendre grid, defined as:
unsigned int	dimension () const override
	Get the spatial dimension of a Segment (always 1).
Public Member Functions inherited from hummingbird::Element
	Element (const int material_id, const int source_id)
	Construct a new Element object.
virtual	~Element ()=default
	Destroy the Element object. Virtual because Element is used polymorphically through std::unique_ptr<Element> (see Mesh::elements_).
std::vector< size_t >	node_ids () const
	Get node IDs.
int	material_id () const
	Get Material ID.
int	source_id () const
	Get Source ID.
void	SetMaterialID (const int material_id)
	Set the Material ID, e.g. once it has been resolved from a raw gmsh tag to a MaterialBank-assigned ID (see Mesh::ResolveIDs).
void	SetSourceID (const int source_id)
	Set the Source ID, e.g. once it has been resolved from a raw gmsh tag to a SourceBank-assigned ID (see Mesh::ResolveIDs).
void	SetNewNodeID (const size_t prev_id, const size_t new_id)
	Set the new Node ID.
void	SetNodeIDs (std::vector< size_t > node_ids)
	Replace all node IDs at once, e.g. when renumbering. Unlike SetNewNodeID, this does not search by value, so it is safe to use when remapped IDs could otherwise collide with not-yet-remapped IDs.
void	SetNodeSourceFluxes (Mesh &mesh, const MaterialBank &material_bank, const SourceBank &source_bank, const QuadratureBase< Ordinate > &angular_quadrature)
	Compute and set the source flux values on this element's nodes using this element's material and source IDs.

Private Member Functions
double	ComputeLength (const std::array< size_t, 2 > boundary_node_ids, const Mesh &mesh)
	Compute the length of the element on construction.

Private Attributes
std::array< size_t, 2 >	boundary_node_ids_
	Node IDs defining the Segment bounds.
const double	length_
	Length of the element.

Additional Inherited Members
Protected Attributes inherited from hummingbird::Element
int	material_id_ = -1
	Material ID to access Material Bank. Auto initialized to -1. Value must be set to a positive value (including 0) to map into the MaterialBank.
int	source_id_ = 0
	Source ID to access Source Bank. ID of 0 is always NONE, meaning the element has no external source.
std::vector< size_t >	node_ids_
	IDs of nodes defining the element.

Detailed Description

1D line element defined by two boundary nodes, with interior nodes placed at Gauss-Lobatto-Legendre points

Definition at line 20 of file segment.h.

Constructor & Destructor Documentation

◆ Segment()

hummingbird::Segment::Segment	(	const std::array< size_t, 2 >	boundary_node_ids,
		const int	material_id,
		const int	source_id,
		const Mesh &	mesh )

Construct a new Segment object.

Parameters

boundary_node_ids	Node IDs defining the Segment bounds
material_id	Material ID
source_id	Source ID
mesh	Mesh

Definition at line 4 of file segment.cc.

Member Function Documentation

◆ ComputeLength()

double hummingbird::Segment::ComputeLength	(	const std::array< size_t, 2 >	boundary_node_ids,
		const Mesh &	mesh )

private

Compute the length of the element on construction.

Parameters

boundary_node_ids	Boundary node IDs
mesh	Mesh

Returns: double

Definition at line 10 of file segment.cc.

◆ CreateInteriorNodes()

std::vector< Node > hummingbird::Segment::CreateInteriorNodes	(	const std::vector< Node > &	existing_nodes,
		const GaussLobattoLegendre &	gll_quadrature )

overridevirtual

Create interior Node objects using the Gauss-Lobatto-Legendre quadrature set. The mapping from the reference to the real domain was based on the equations provided by the book Computational Seismology by Heiner Igel (https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.geokniga.org/bookfiles/geokniga-computationalseismologyapracticalintroductionbyigelheinerz-liborg.pdf&ved=2ahUKEwiMuanq7_OVAxXVN4YAHVgAD1QQFnoECBUQAQ&usg=AOvVaw35g1Tl3ZSFUMusz8LKQxHk). For a line segment defined by two endpoints, the mapping is:

\[\mathbf{r}(\xi) = \mathbf{r}_1+\left( \frac{\xi+1}{2} \right)\left( \mathbf{r}_2-\mathbf{r}_1 \right) \]

where \(x\in[-1,1]\).

Parameters

existing_nodes	Nodes already existing in Mesh
gll_quadrature	Gauss-Lobatto-Legendre quadrature set

Returns: Additional nodes to be defined along the GLL quadrature set. Will be of length N-2 (the two endpoint account for the remaining two nodes)

Implements hummingbird::Element.

Definition at line 18 of file segment.cc.

◆ dimension()

unsigned int hummingbird::Segment::dimension ( ) const

overridevirtual

Get the spatial dimension of a Segment (always 1).

Returns: unsigned int

Implements hummingbird::Element.

Definition at line 107 of file segment.cc.

◆ LocalForcingVector()

arma::Col< double > hummingbird::Segment::LocalForcingVector	(	const GaussLobattoLegendre &	gll_quad,
		const Mesh &	mesh,
		const MaterialBank &	material_bank,
		const Ordinate &	ordinate,
		const size_t	ordinate_index )

overridevirtual

Construct the local forcing vector, which is:

\[f_i = \frac{h_e}{2}\int_{-1}^1Q_n(\xi)\ell_i (\xi)d\xi + \frac{\hat{\Omega}_x}{\Sigma_t^e}\int_{-1}^1Q_n(\xi)\frac{d\ell_i}{d\xi}d\xi \]

which the integrals are again approximated using the GLL quadrature set:

\[f_i \approx \sum_{k=0}^{N_x}\rho_k Q_n(\xi_k)\left[ \frac{h_e}{2}\ell_i (\xi_k) +\frac{\hat{\Omega}_x}{\Sigma_t^e}\frac{d\ell_i}{d\xi}(\xi_k)\right] \]

Parameters

gll_quad	GaussLobattoLegendre quadrature set
mesh	Mesh
material_bank	Material bank
ordinate	Ordinate (direction)
ordinate_index	Index of the ordinate in the angular quadrature

Returns: arma::Col<double>

Implements hummingbird::Element.

Definition at line 52 of file segment.cc.

◆ LocalMassMatrix()

arma::SpMat< double > hummingbird::Segment::LocalMassMatrix	(	const GaussLobattoLegendre &	gll_quad,
		const MaterialBank &	material_bank )

overridevirtual

Construct the sparse diagonal local mass matrix using spectral elements on a Gauss-Lobatto-Legendre grid, defined as:

\[M_{ij}=\Sigma_t^e\frac{h_e}{2}\sum_{k=0}^{N_x}\rho_k\ell_i(\xi_k)\ell_j(\xi_k) \]

Which, using the cardinality of Lagrange polynomials, greatly simplifies to:

\[\Sigma_t^eM_{ij}=\begin{cases} 0, & i\neq j \\ \Sigma_t^e\frac{h_e}{2}\rho_i, &i=j \end{cases} \]

Parameters

gll_quad	GaussLobattoLegendre set
material_bank	MaterialBank

Returns: arma::SpMat<double>

Implements hummingbird::Element.

Definition at line 97 of file segment.cc.

◆ LocalStiffnessMatrix()

arma::Mat< double > hummingbird::Segment::LocalStiffnessMatrix	(	const GaussLobattoLegendre &	gll_quad,
		const MaterialBank &	material_bank,
		const Ordinate &	ordinate )

overridevirtual

Construct the dense local stiffness matrix using spectral elements on a Gauss-Lobatto-Legendre grid, defined as:

\[K_{ij}=\frac{\hat{\Omega}_x^2}{\Sigma_t^e}\frac{2}{h_e}\sum_{k=0}^{N_x}\rho_k \frac{d\ell_i}{d\xi}(\xi_k)\frac{d\ell_j}{d\xi}(\xi_k) \]

Parameters

gll_quad	GaussLobattoLegendre set
material_bank	MaterialBank
ordinate	Ordinate (direction)

Returns: arma::Mat<double>

Todo: See about cutting the loops in half by leveraging symmetry.

Implements hummingbird::Element.

Definition at line 77 of file segment.cc.

Member Data Documentation

◆ boundary_node_ids_

std::array<size_t, 2> hummingbird::Segment::boundary_node_ids_

private

Node IDs defining the Segment bounds.

Definition at line 134 of file segment.h.

◆ length_

const double hummingbird::Segment::length_

private

Length of the element.

Definition at line 137 of file segment.h.

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

/home/runner/work/hummingbird/hummingbird/include/mesh/segment.h
/home/runner/work/hummingbird/hummingbird/src/mesh/segment.cc

	hummingbird Solving the self-adjoint angular flux transport equation using spectral elements on Cartesian geometry

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ Segment()

Member Function Documentation

◆ ComputeLength()

◆ CreateInteriorNodes()

◆ dimension()

◆ LocalForcingVector()

◆ LocalMassMatrix()

◆ LocalStiffnessMatrix()

Member Data Documentation

◆ boundary_node_ids_

◆ length_