New computationally efficient quadrature formulas for pyramidal elements

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.