NonMatching::QuadratureGenerator< dim > Class Template Reference

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

Inheritance diagram for NonMatching::QuadratureGenerator< dim >:

Public Types
using	AdditionalData = AdditionalQGeneratorData

Public Member Functions
	QuadratureGenerator (const hp::QCollection< 1 > &quadratures1D, const AdditionalData &additional_data=AdditionalData())
void	generate (const Function< dim > &level_set, const BoundingBox< dim > &box)
const Quadrature< dim > &	get_inside_quadrature () const
const Quadrature< dim > &	get_outside_quadrature () const
const ImmersedSurfaceQuadrature< dim > &	get_surface_quadrature () const
void	set_1D_quadrature (const unsigned int q_index)

Private Attributes
internal::QuadratureGeneratorImplementation::QGenerator< dim, dim >	q_generator

Detailed Description

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

This class creates immersed quadrature rules over a BoundingBox, \(B \subset \mathbb{R}^{dim}\), when the domain is described by a level set function, \(\psi\).

This class creates quadrature rules for the intersections between the box and the three different regions defined by the level set function. That is, it creates quadrature rules to integrate over the following regions

\[ N = \{x \in B : \psi(x) < 0 \}, \\ P = \{x \in B : \psi(x) > 0 \}, \\ S = \{x \in B : \psi(x) = 0 \}. \]

When working with level set functions, the most common is to describe a domain, \(\Omega\), as

\[ \Omega = \{ x \in \mathbb{R}^{dim} : \psi(x) < 0 \}. \]

Given this, we shall use the name convention that \(N\) is the "inside" region (i.e. inside \(\Omega\)), \(P\) is the "outside" region and \(S\) is the "surface" region. The "inside" and "outside" quadratures will also be referred to as the "bulk"-quadratures.

The underlying algorithm use a 1-dimensional quadrature rule as base for creating the immersed quadrature rules. Gauss-Legendre quadrature (QGauss) is recommended. The constructor takes an hp::QCollection<1>. One can select which 1d-quadrature in the collection should be used through the set_1d_quadrature() function. The number of quadrature points in the constructed quadratures will vary depending on the level set function. More quadrature points will be created if the intersection is "bad", for example, if the zero-contour has a high curvature compared to the size of the box. However, if the number of points in the 1d quadrature is \(n\) the number of points will be proportional to \(n^{dim}\) in the bulk quadratures and to \(n^{dim-1}\) in the surface quadrature. For example, in the 2d-example in the above figure, there are 2 points in the 1d-quadrature. If the 1d-quadrature is a Gauss-Legendre quadrature and the grid has size \(h\), the immersed quadratures typically give global errors proportional to \(h^{2n}\), both for the bulk and surface integrals. If the 1d-quadrature has positive weights, the weights of the immersed quadratures will also be positive.

A detailed description of the underlying algorithm can be found in "High-Order %Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles, R. I. Saye, SIAM J. Sci. Comput., 37(2), <a href="http://www.dx.doi.org/10.1137/140966290"> doi:10.1137/140966290</a>". This implementation has some modifications compared to the algorithm description in the paper. In particular, it builds the three different types of quadratures (inside, outside and surface) simultaneously. Further, the so-called "pruning" step is not yet implemented.

Definition at line 188 of file quadrature_generator.h.

Member Typedef Documentation

AdditionalData

template<int dim>

using NonMatching::QuadratureGenerator< dim >::AdditionalData = AdditionalQGeneratorData

Definition at line 191 of file quadrature_generator.h.

Constructor & Destructor Documentation

QuadratureGenerator()

template<int dim>

NonMatching::QuadratureGenerator< dim >::QuadratureGenerator	(	const hp::QCollection< 1 > &	quadratures1D,
		const AdditionalData &	additional_data = AdditionalData()
	)

Constructor. Each Quadrature<1> in quadratures1d can be chosen as base for generating the immersed quadrature rules.

Note

It is important that each 1d-quadrature rule in the hp::QCollection does not contain the points 0 and 1.

Definition at line 1623 of file quadrature_generator.cc.

Member Function Documentation

generate()

template<int dim>

void NonMatching::QuadratureGenerator< dim >::generate	(	const Function< dim > &	level_set,
		const BoundingBox< dim > &	box
	)

Construct immersed quadratures rules for the incoming level set function over the BoundingBox.

To get the constructed quadratures, use the functions get_inside_quadrature(), get_outside_quadrature(), get_surface_quadrature().

Note

Both value, gradient and hessian need to be implemented on the incoming function.

Definition at line 1636 of file quadrature_generator.cc.

get_inside_quadrature()

template<int dim>

const Quadrature< dim > & NonMatching::QuadratureGenerator< dim >::get_inside_quadrature

Return the quadrature rule for the region \(\{x \in B : \psi(x) < 0 \}\) created in the previous call to generate(). Here, \(B\) is BoundingBox passed to generate().

Definition at line 1668 of file quadrature_generator.cc.

get_outside_quadrature()

template<int dim>

const Quadrature< dim > & NonMatching::QuadratureGenerator< dim >::get_outside_quadrature

Return the quadrature rule for the region \(\{x \in B : \psi(x) > 0 \}\) created in the previous call to generate(). Here, \(B\) is BoundingBox passed to generate().

Definition at line 1677 of file quadrature_generator.cc.

get_surface_quadrature()

template<int dim>

const ImmersedSurfaceQuadrature< dim > & NonMatching::QuadratureGenerator< dim >::get_surface_quadrature

Return the quadrature rule for the region \(\{x \in B : \psi(x) = 0 \}\) created in the previous call to generate(). Here, \(B\) is BoundingBox passed to generate().

Note

The normal at the quadrature points will be parallel to \(\nabla \psi\).

Definition at line 1686 of file quadrature_generator.cc.

set_1D_quadrature()

template<int dim>

void NonMatching::QuadratureGenerator< dim >::set_1D_quadrature

(

const unsigned int

q_index

)

Set which 1d-quadrature in the collection passed to the constructor should be used to create the immersed quadratures.

Definition at line 1694 of file quadrature_generator.cc.

Member Data Documentation

q_generator

template<int dim>

internal::QuadratureGeneratorImplementation::QGenerator<dim, dim> NonMatching::QuadratureGenerator< dim >::q_generator

private

QuadratureGenerator is mainly used to start up the recursive algorithm. This is the object that actually generates the quadratures.

Definition at line 261 of file quadrature_generator.h.

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

• include/deal.II/non_matching/quadrature_generator.h
• source/non_matching/quadrature_generator.cc