#include <CellQuadrature.hpp>

Public Member Functions
	CellQuadrature (const UnstructuredGrid &grid, const int cell, const int degree)

int	numQuadPts () const

void	quadPtCoord (const int index, double *coord) const

double	quadPtWeight (const int index) const

Detailed Description

A class providing numerical quadrature for cells. In general: {cell} g(x) dx = {i=0}^{n-1} w_i g(x_i). Note that this class does multiply weights by cell volume, so weights always sum to cell volume.

Degree 1 method: Midpoint (centroid) method. n = 1, w_0 = cell volume, x_0 = cell centroid

Degree 2 method for 2d (but see the note): Based on subdivision of the cell into triangles, with the centroid as a common vertex, and the triangle edge midpoint rule. Triangle i consists of the centroid C, nodes N_i and N_{i+1}. Its area is A_i. n = 2 * nn (nn = num nodes in face) For i = 0..(nn-1): w_i = 1/3 A_i. w_{nn+i} = 1/3 A_{i-1} + 1/3 A_i x_i = (N_i + N_{i+1})/2 x_{nn+i} = (C + N_i)/2 All N and A indices are interpreted cyclic, modulus nn. Note: for simplicity of implementation, we currently use n = 3 * nn For i = 0..(nn-1): w_{3*i + {0,1,2}} = 1/3 A_i x_{3*i} = (N_i + N_{i+1})/2 x_{3*i + {1,2}} = (C + N_{i,i+1})/2 This is simpler, because we can implement it easily based on iteration over faces without requiring any particular (cyclic) ordering.

Degree 2 method for 3d: Based on subdivision of each cell face into triangles with the face centroid as a common vertex, and then subdividing the cell into tetrahedra with the cell centroid as a common vertex. Then apply the tetrahedron rule with the following 4 nodes (uniform weights): a = 0.138196601125010515179541316563436 x_i has all barycentric coordinates = a, except for the i'th coordinate which is = 1 - 3a. This rule is from http://nines.cs.kuleuven.be/ecf, it is the second degree 2 4-point rule for tets, referenced to Stroud(1971). The tetrahedra are numbered T_{i,j}, and are given by the cell centroid C, the face centroid FC_i, and two nodes of face i: FN_{i,j}, FN_{i,j+1}.