Moderate-degree tetrahedral quadrature formulas
Computer Methods in Applied Mechanics and Engineering
On the construction of good fully symmetric integration rules
SIAM Journal on Numerical Analysis
On zeros of multivariate quasi-orthogonal polynomials and Gaussian cubature formulae
SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Rotation invariant cubature formulas over the n-dimensional unit cube
Journal of Computational and Applied Mathematics
Lower bound for the number of nodes of cubature formulae on the unit ball
Journal of Complexity
Asymmetric cubature formulas for polynomial integration in the triangle and square
Journal of Computational and Applied Mathematics
Higher-order Finite Elements for Hybrid Meshes Using New Nodal Pyramidal Elements
Journal of Scientific Computing
Simple universal bounds for Chebyshev-type quadratures
Journal of Approximation Theory
Numerical Analysis
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In this paper, new efficient nonproduct numerical integration, or multidimensional quadrature, formulas for pyramidal elements are derived and presented. The nonproduct formulas are developed using the method of polynomial moment fitting, where the weights and points of the formulas are determined by a system of coupled, highly nonlinear equations. Given that the number of equations quickly becomes prohibitively large in three dimensions, the symmetry of the pyramid is used to reduce the number of equations and unknowns of the resulting systems. The new formulas, which in some cases are optimal (that is, minimal-point), are the most efficient means available for numerically computing volume integrals over pyramidal elements in that they require fewer points than any other presently available formulas of the same polynomial degree. By comparison, conventional approaches using products of one-dimensional Gaussian formulas require, on average, more than twice as many points and weights as the new formulas derived here.