#include <deal.II/non_matching/immersed_surface_quadrature.h>

Inheritance diagram for NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >:

Public Types
using	SubQuadrature = Quadrature< dim==0 ? 0 :dim - 1 >

Public Member Functions
	ImmersedSurfaceQuadrature ()=default

	ImmersedSurfaceQuadrature (const std::vector< Point< dim > > &points, const std::vector< double > &weights, const std::vector< Tensor< 1, spacedim > > &normals)

void	push_back (const Point< dim > &point, const double weight, const Tensor< 1, spacedim > &normal)

const Tensor< 1, spacedim > &	normal_vector (const unsigned int i) const

const std::vector< Tensor< 1, spacedim > > &	get_normal_vectors () const

bool	operator== (const Quadrature< dim > &p) const

void	initialize (const std::vector< Point< dim > > &points, const std::vector< double > &weights)

unsigned int	size () const

const Point< dim > &	point (const unsigned int i) const

const std::vector< Point< dim > > &	get_points () const

double	weight (const unsigned int i) const

const std::vector< double > &	get_weights () const

std::size_t	memory_consumption () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

bool	is_tensor_product () const

const std::array< Quadrature< 1 >, dim > &	get_tensor_basis () const

Protected Attributes
std::vector< Tensor< 1, spacedim > >	normals

std::vector< Point< dim > >	quadrature_points

std::vector< double >	weights

bool	is_tensor_product_flag

std::unique_ptr< std::array< Quadrature< 1 >, dim > >	tensor_basis

Subscriptor functionality
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.
void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

using	map_value_type = decltype(counter_map)::value_type

using	map_iterator = decltype(counter_map)::iterator

std::atomic< unsigned int >	counter

std::map< std::string, unsigned int >	counter_map

std::vector< std::atomic< bool > * >	validity_pointers

const std::type_info *	object_info

static std::mutex	mutex

static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

void	check_no_subscribers () const noexcept

Detailed Description

template<int dim, int spacedim = dim>
class NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >

This class defines a quadrature formula to integrate over the intersection between an oriented surface, \(\hat{S}\), and a cell or face. The word "immersed" in the class name reflects that the surface may intersect the cell/face in an arbitrary way.

The spacedim template parameter of this class is the dimension that the (spacedim-1)-dimensional surface is embedded in: \(\hat{S} \subset \mathbb{R}^{\text{spacedim}}\). The dim parameter describes the dimension of the "object" that the surface intersects. That is, dim = spacedim corresponds to the surface intersecting a cell and dim = spacedim - 1 corresponds to the surface intersecting a face. The quadrature formula is described by a set of quadrature points, \(\hat{x}_q \in \mathbb{R}^{\text{dim}}\), weights, \(w_q\), and normalized surface normals, \(\hat{n}_q \in \mathbb{R}^{\text{spacedim}}\).

Consider first the case dim = spacedim. We typically want to compute integrals in real space. A surface, \(S\), intersecting a cell, \(K\), in real space can be mapped onto a surface, \(\hat{S}\), intersecting the unit cell, \(\hat{K}\). Thus an integral over \(S\cap K\) in real space can be transformed to an integral over \(\hat{S} \cap \hat{K}\) according to

\[ \int_{S\cap K} f dS = \int_{S\cap K} f |d\bar{S}| = \int_{\hat{S}\cap\hat{K}} f \circ F_{K} \det(J) |\left( J^{-1} \right )^T d\hat{S}|, \]

where \(F_K\) is the mapping from reference to real space and \(J\) is its Jacobian matrix. This transformation is possible since the continuous surface elements are vectors: \(d\bar{S}, d\hat{S} \in \mathbb{R}^{spacedim}\), which are parallel to the normals of \(S\) and \(\hat{S}\). That is, the normal is needed to do the transformation. Thus, in addition to storing points and weights, this quadrature stores also the normalized normal for each quadrature point. This can be viewed as storing a discrete surface element,

\[ \Delta \hat{S}_q \dealcoloneq w_q \hat{n}_q \approx d\hat{S}(\hat{x}_q), \]

for each quadrature point. The surface integral in real space would then be approximated as

\[ \int_{S\cap K} f dS \approx \sum_{q} f \left(F_{K}(\hat{x}_{q}) \right) \det(J_q) |\left( J_q^{-1} \right)^T \hat{n}_q| w_q. \]

When dim = spacedim - 1, this class represents a (spacedim-2)-dimensional integral. That is, if spacedim = 3 we have a line integral immersed in a face. Here, the transformation between the face, \(F\), and reference face, \(\hat{F}\), reads

\[ \int_{S\cap F} f dr = \int_{S\cap F} f |d\bar{r}| = \int_{\hat{S}\cap\hat{F}} f \circ F_{K} | J d\hat{r}| \approx \sum_{q} f \left(F_{K}(\hat{x}_{q}) \right) |J_q \hat{t}_q| w_q, \]

where \(\hat{t}_q = \hat{n}_q \times \hat{n}_F\) is the tangent to the curve at \(\hat{x}_q\) and \(\hat{n}_F\) is the face normal. It would be possible to compute the tangent by only knowing the normal to the curve in the face plane (i.e. the dim-dimensional normal). However, when these quadratures are used, the weak form typically involves the so-called conormal, which can not be computed without knowing the surface normal in \(\mathbb{R}^{\text{spacedim}}\). The conormal is the unit vector parallel to the projection of the face normal into the surface plane. This is essentially the same thing as the normalized boundary form.

Definition at line 101 of file immersed_surface_quadrature.h.

Member Typedef Documentation

◆ SubQuadrature

template<int dim>

using Quadrature< dim >::SubQuadrature = Quadrature<dim == 0 ? 0 : dim - 1>

inherited

Define an alias for a quadrature that acts on an object of one dimension less. For cells, this would then be a face quadrature. A sub quadrature of a 0-dimensional quadrature is defined as still being 0-dimensional.

Definition at line 94 of file quadrature.h.

Constructor & Destructor Documentation

◆ ImmersedSurfaceQuadrature() [1/2]

template<int dim, int spacedim = dim>

NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::ImmersedSurfaceQuadrature ( )

default

Default constructor to initialize the quadrature with no quadrature points.

◆ ImmersedSurfaceQuadrature() [2/2]

template<int dim, int spacedim>

NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::ImmersedSurfaceQuadrature	(	const std::vector< Point< dim > > &	points,
		const std::vector< double > &	weights,
		const std::vector< Tensor< 1, spacedim > > &	normals
	)

Construct a quadrature formula from vectors of points, weights and surface normals. The points, weights and normals should be with respect to reference space, and the normals should be normalized.

Definition at line 22 of file immersed_surface_quadrature.cc.

Member Function Documentation

◆ push_back()

template<int dim, int spacedim>

void NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::push_back	(	const Point< dim > &	point,
		const double	weight,
		const Tensor< 1, spacedim > &	normal
	)

Extend the given formula by an additional quadrature point. The point, weight and normal should be with respect to reference space, and the normal should be normalized.

This function exists since immersed quadrature rules can be rather complicated to construct. Often the construction is done by partitioning the cell into regions and constructing points on each region separately. This can make it cumbersome to create the quadrature from the constructor since all quadrature points have to be known at time of creation of the object.

Note: This function should only be used during construction of the quadrature formula.

Definition at line 43 of file immersed_surface_quadrature.cc.

◆ normal_vector()

template<int dim, int spacedim>

const Tensor< 1, spacedim > & NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::normal_vector ( const unsigned int i ) const

Return a reference to the ith surface normal.

Definition at line 59 of file immersed_surface_quadrature.cc.

◆ get_normal_vectors()

template<int dim, int spacedim>

const std::vector< Tensor< 1, spacedim > > & NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::get_normal_vectors

Return a reference to the whole vector of normals.

Definition at line 70 of file immersed_surface_quadrature.cc.

◆ operator==()

template<int dim>

bool Quadrature< dim >::operator== ( const Quadrature< dim > & p ) const

inherited

Test for equality of two quadratures.

Definition at line 304 of file quadrature.cc.

◆ initialize()

template<int dim>

void Quadrature< dim >::initialize	(	const std::vector< Point< dim > > &	points,
		const std::vector< double > &	weights
	)

std::size_t Quadrature< dim >::memory_consumption

inherited

Determine an estimate for the memory consumption (in bytes) of this object.

Definition at line 313 of file quadrature.cc.

◆ serialize()

template<int dim>

template<class Archive >

void Quadrature< dim >::serialize	(	Archive &	ar,
		const unsigned int	version
	)

inherited

Write or read the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

◆ is_tensor_product()

template<int dim>

bool Quadrature< dim >::is_tensor_product ( ) const

inherited

This function returns true if the quadrature object is a tensor product of one-dimensional formulas and the quadrature points are sorted lexicographically.

◆ get_tensor_basis()

template<int dim>

std::conditional< dim==1, std::array< Quadrature< 1 >, dim >, conststd::array< Quadrature< 1 >, dim > & >::type Quadrature< dim >::get_tensor_basis

inherited

In case the quadrature formula is a tensor product, this function returns the dim one-dimensional basis objects. Otherwise, calling this function is not allowed.

For dim equal to one, we can not return the std::array as a const reference and have to return it by value. In this case, the array will always contain a single element (this).

Note: The actual return type of this function is
std::conditional<dim == 1,

std::array<Quadrature<1>, dim>,

const std::array<Quadrature<1>, dim> &>::type

The type is abbreviated in the online documentation to improve readability of this page.

Stores the one-dimensional tensor basis objects in case this object can be represented by a tensor product.

Definition at line 311 of file quadrature.h.

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

include/deal.II/non_matching/immersed_surface_quadrature.h
source/non_matching/immersed_surface_quadrature.cc

Public Types

Public Member Functions

Protected Attributes

Subscriptor functionality

Detailed Description

Member Typedef Documentation

◆ SubQuadrature

Constructor & Destructor Documentation

◆ ImmersedSurfaceQuadrature() [1/2]

◆ ImmersedSurfaceQuadrature() [2/2]

Member Function Documentation

◆ push_back()

◆ normal_vector()

◆ get_normal_vectors()

◆ operator==()

◆ initialize()

◆ size()

◆ point()

◆ get_points()

◆ weight()

◆ get_weights()

◆ memory_consumption()

◆ serialize()

◆ is_tensor_product()

◆ get_tensor_basis()

Member Data Documentation

◆ normals

◆ quadrature_points

◆ weights

◆ is_tensor_product_flag

◆ tensor_basis