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

Inheritance diagram for NonMatching::ImmersedSurfaceQuadrature< dim >:

Public Member Functions
	ImmersedSurfaceQuadrature ()=default

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

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

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

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

Public Member Functions inherited from Quadrature< dim >
	Quadrature (const unsigned int n_quadrature_points=0)

	Quadrature (const SubQuadrature &, const Quadrature< 1 > &)

	Quadrature (const Quadrature< dim !=1 ? 1 :0 > &quadrature_1d)

	Quadrature (const Quadrature< dim > &q)

	Quadrature (Quadrature< dim > &&) noexcept=default

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

	Quadrature (const std::vector< Point< dim > > &points)

	Quadrature (const Point< dim > &point)

virtual	~Quadrature ()=default

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

Quadrature &	operator= (Quadrature< dim > &&)=default

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

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

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (const char *identifier=nullptr) const

void	unsubscribe (const char *identifier=nullptr) const

unsigned int	n_subscriptions () const

void	list_subscribers () const

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

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

Protected Attributes inherited from Quadrature< dim >
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

Additional Inherited Members
Public Types inherited from Quadrature< dim >
typedef Quadrature< dim-1 >	SubQuadrature

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

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

Detailed Description

template<int dim>
class NonMatching::ImmersedSurfaceQuadrature< dim >

Defines a quadrature formula for integration over an oriented surface, \(\hat{S}\), immersed in the unit cell. By immersed it is meant that the surface may intersect the unit cell in an arbitrary way. The quadrature formula is described by a set of quadrature points, \(\hat{x}_q\), weights, \(w_q\), and normalized surface normals, \(\hat{n}_q\).

We typically want to compute surface 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 a surface integral over \(S\cap K\) in real space can be transformed to a surface integral over \(\hat{S} \cap \hat{K}\) according to

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

where \(F_K\) is the mapping from reference to real space and \(J\) is its Jacobian. This transformation is possible since the continuous surface elements are vectors: \(d\bar{S}, d\hat{S} \in \mathbb{R}^{dim}\) which are parallel to the normals of \(S\) and \(\hat{S}\). So in order to compute the integral in real space one needs information about the normal 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,

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

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

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

Author: Simon Sticko, 2017

Definition at line 74 of file immersed_surface_quadrature.h.

Constructor & Destructor Documentation

◆ ImmersedSurfaceQuadrature() [1/2]

template<int dim>

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

default

Default constructor to initialize the quadrature with no quadrature points.

◆ ImmersedSurfaceQuadrature() [2/2]

template<int dim>

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

Member Function Documentation

◆ push_back()

template<int dim>

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