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Reference documentation for deal.II version 8.5.1
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Reference documentation for deal.II version 8.5.1
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#include <deal.II/base/quadrature_lib.h>
Public Member Functions | |
| QTelles (const Quadrature< 1 > &base_quad, const Point< dim > &singularity) | |
| QTelles (const unsigned int n, const Point< dim > &singularity) | |
| template<> | |
| QTelles (const Quadrature< 1 > &base_quad, const Point< 1 > &singularity) | |
 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 > &&)=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 () |
| Quadrature & | operator= (const Quadrature< dim > &) |
| 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) |
| Public Member Functions inherited from Subscriptor | |
| Subscriptor () | |
| Subscriptor (const Subscriptor &) | |
| Subscriptor (Subscriptor &&) | |
| virtual | ~Subscriptor () |
| Subscriptor & | operator= (const Subscriptor &) |
| Subscriptor & | operator= (Subscriptor &&) |
| void | subscribe (const char *identifier=0) const |
| void | unsubscribe (const char *identifier=0) const |
| unsigned int | n_subscriptions () const |
| void | list_subscribers () const |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
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, char *arg2, std::string &arg3) |
| static ::ExceptionBase & | ExcNoSubscriber (char *arg1, char *arg2) |
| Protected Attributes inherited from Quadrature< dim > | |
| std::vector< Point< dim > > | quadrature_points |
| std::vector< double > | weights |
Telles quadrature of arbitrary order.
The coefficients of these quadrature rules are computed using a non linear change of variables starting from a Gauss-Legendre quadrature formula. This is done using a cubic polynomial, \(n = a x^3 + b x^2 + c x + d\) in order to integrate a singular integral, with singularity at a given point x_0.
We start from a Gauss Quadrature Formula with arbitrary function. Then we apply the cubic variable change. In the paper, J.C.F.Telles:A Self-Adaptive Co-ordinate Transformation For Efficient Numerical Evaluation of General Boundary Element Integrals. International Journal for Numerical Methods in Engineering, vol 24, pages 959–973. year 1987, the author applies the transformation on the reference cell \([-1, 1]\) getting
\begin{align*} n(1) &= 1, \\ n(-1) &= -1, \\ \frac{dn}{dx} &= 0 \text{ at } x = x_0, \\ \frac{d^2n}{dx^2} &= 0 \text{ at } x = x_0 \end{align*}
We get
\begin{align*} a &= \frac{1}{q}, \\ b &= -3 \frac{\bar{\Gamma}}{q}, \\ c &= 3 \frac{\bar{\Gamma}}{q}, \\ d &= -b, \end{align*}
with
\begin{align*} \eta^{*} &= \bar{\eta}^2 - 1, \\ \bar{\Gamma} &= \sqrt[3]{\bar{\eta} \eta^{*} + |\eta^{*} | } + \sqrt[3]{ \bar{\eta} \eta^{*} - |\eta^{*} | } + \bar{\eta}, \\ q &= (\Gamma-\bar{\Gamma})^3 + \bar{\Gamma} \frac{\bar{\Gamma}^2+3}{1+3\bar{\Gamma}^2} \end{align*}
Since the library assumes \([0,1]\) as reference interval, we will map these values on the proper reference interval in the implementation.
This variable change can be used to integrate singular integrals. One example is \(f(x)/|x-x_0|\) on the reference interval \([0,1]\), where \(x_0\) is given at construction time, and is the location of the singularity \(x_0\), and \(f(x)\) is a smooth non singular function.
Singular quadrature formula are rather expensive, nevertheless Telles' quadrature formula are much easier to compute with respect to other singular integration techniques as Lachat-Watson.
We have implemented the case for \(dim = 1\). When we deal the case \(dim >1\) we have computed the quadrature formula has a tensorial product of one dimensional Telles' quadrature formulas considering the different components of the singularity.
The weights and functions for Gauss Legendre formula have been tabulated up to order 12.
Definition at line 512 of file quadrature_lib.h.
| QTelles< dim >::QTelles | ( | const Quadrature< 1 > & | base_quad, |
| const Point< dim > & | singularity | ||
| ) |
A constructor that takes a quadrature formula and a singular point as argument. The quadrature formula will be mapped using Telles' rule. Make sure that the order of the quadrature rule is appropriate for the singularity in question.
We need the explicit implementation if dim == 1. If dim > 1 we use the former implementation and apply a tensorial product to obtain the higher dimensions.
Definition at line 1014 of file quadrature_lib.cc.
| QTelles< dim >::QTelles | ( | const unsigned int | n, |
| const Point< dim > & | singularity | ||
| ) |
A variant of above constructor that takes as parameters the order n and location of a singularity. A Gauss Legendre quadrature of order n will be used
In this case we map the standard Gauss Legendre formula using the given singularity point coordinates.
Definition at line 1037 of file quadrature_lib.cc.
| QTelles< 1 >::QTelles | ( | const Quadrature< 1 > & | base_quad, |
| const Point< 1 > & | singularity | ||
| ) |
We explicitly implement the Telles' variable change if dim == 1.
We define all the constants to be used in the implementation of Telles' rule
Definition at line 1050 of file quadrature_lib.cc.
1.8.14