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deal.II version GIT relicensing-6917-gb466a1809f 2026-10-16 06:30:01+00:00
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deal.II version GIT relicensing-6917-gb466a1809f 2026-10-16 06:30:01+00:00
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#include <deal.II/base/quadrature_lib.h>
Stroud quadrature rules for simplex entities.
Like QGauss, users should specify a number n_points_1d as an indication of what polynomial degree to be integrated exactly. The quadrature rule returns \(n^d\) quadrature points.
For 1d, the quadrature rule degenerates to a QGauss<1>(n_points_1d).
The integral over the reference triangle \(\int_{0}^{1} \int_{0}^{1-y} f(x,y) dx dy\) is transformed using the Duffy transformation
\[ \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} \hat \xi (1- \nu)\\ \hat \nu \end{pmatrix} \]
to the integral over the reference quadrilateral \(\int_{0}^{1} \int_{0}^{1} f(x(\xi, \nu),y(\xi, \nu)) (1-\nu) d\xi d\nu\). The integral can be calculated using standard Gauss integration rules. By choosing a Gauss-Jacobi quadrature rule in \(y\)-direction with \(\alpha = 1\) and \(\beta = 0\), the additional factor \(1-\nu\) is absorbed into the quadrature weights.
The quadrature points on the quadrilateral are transformed back to the triangle using the Duffy transformation. Thus, the function returns quadrature points and weights on the reference simplex. A similar approach is used in 3d.
Definition at line 1027 of file quadrature_lib.h.
Public Member Functions | |
| QStroudSimplex (const unsigned int n_points_1D) | |
| template<int spacedim = dim> | |
| Quadrature< spacedim > | compute_affine_transformation (const std::array< Point< spacedim >, dim+1 > &vertices) const |
| template<int spacedim = dim> | |
| Quadrature< spacedim > | mapped_quadrature (const std::vector< std::array< Point< spacedim >, dim+1 > > &simplices) const |
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explicit |
Constructor taking the equivalent number of quadrature points in 1d n_points_1d.
Definition at line 2276 of file quadrature_lib.cc.
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inherited |
Return an affine transformation of this quadrature, that can be used to integrate on the simplex identified by vertices.
Both the quadrature point locations and the weights are transformed, so that you can effectively use the resulting quadrature to integrate on the simplex.
The transformation is defined as
\[ x = v_0 + B \hat x \]
where the matrix \(B\) is given by \(B_{ij} = v[j][i]-v[0][i]\).
The weights are scaled with the absolute value of the determinant of \(B\), that is \(J \dealcoloneq |\text{det}(B)|\). If \(J\) is zero, an empty quadrature is returned. This may happen, in two dimensions, if the three vertices are aligned, or in three dimensions if the four vertices are on the same plane. The present function works also in the codimension one and codimension two case. For instance, when dim=2 and spacedim=3, we can map the quadrature points so that they live on the physical triangle embedded in the three dimensional space. In such a case, the matrix \(B\) is not square anymore.
| [in] | vertices | The vertices of the simplex you wish to integrate on |
Definition at line 1471 of file quadrature_lib.cc.
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inherited |
Given a collection of simplices, this function creates a global quadrature rule on the area covered by the simplices, by mapping the current quadrature on each simplex. A simplex is identified by its vertices, which are stored into an array of Points. Hence, this function can provide quadrature rules on polygons (or polyhedra), as they can be split into simplices.
| simplices | A std::vector where each entry is an array of dim+1 points, which identifies the vertices of a simplex. |
Definition at line 1505 of file quadrature_lib.cc.
1.9.8