Numerical integration over simplexes

Gauss Legendre Quadrature Formulae for Tetrahedra

“…where v is the tetrahedron in (X, Y, Z ) space with vertices spanning the points (5, 5, 0), (10, 10, 0), (8,7,8), (10,5,0) .…”

“…The basic problem of integrating an arbitrary function of two variables over the surface of the triangle was first given by Hammer, Marlowe and Stroud [4], and Hammer and Stroud [5,6]. Cowper [7] provided a table of Gaussian quadrature formulae with symmetrically placed integration points.…”

“…Reddy [15] and Reddy and Shippy [16] derived three-point, four-point, six-point, seven-point of precision 3, 4, 6 and 7 respectively, which gave improved accuracy. Since the precision of all the formulae derived by the authors [4][5][6][7][8][9][10][11][12][13][14][15][16] is limited to a precision of degree ten and it is not likely that the techniques can be extended much further to give a greater accuracy, which may be demanded in future, Lague and Baldur [17] proposed product formulae based only on the roots and weight co-efficients of Gauss Legendre quadrature rules. By the proposed method, this restriction is removed and one can now obtain numerical integration of very high degree of precision as the derivations now rely on standard Gauss Legendre quadarature rules [17].…”

On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

“…The basic problem of integrating an arbitrary function of two variables over the surface of the triangle were first given by Hammer et al [4], and Hammer and Stroud [5,6]. Cowper [7] provided a table of Gaussian quadrature formulae with symmetrically placed integration points.…”

“…Reddy [17] and Reddy and Shippy [18] derived the 3-point, 4-point, 6-point and 7-point rules of precision 3, 4, 6 and 7 respectively which gave improved accuracy. Since the precision of all the formulae derived by the authors [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] is limited to a precision of degree ten and it is not likely that the techniques can be extended much further to give a greater accuracy which may be demanded in future, Lague and Baldur [19] proposed the product formulae based only on the sampling points and weight coefficients of Gauss-Legendre quadrature rules. By the proposed method of [19] this restriction is removed and one can now obtain numerical integration rules of very high degree of precision as the derivation now rely on standard Gauss-Legendre Quadarature rules.…”