Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

A family of quadrature rules for integration over tetrahedral volumes is developed. The underlying structure of the rules is based on the cubic close-packed (CCP) lattice arrangement using 1, 4, 10, 20, 35, and 56 quadrature points. The rules are characterized by rapid convergence, positive weights, and symmetry. Each rule is an optimal approximation in the sense that lower-order terms have zero contribution to the truncation error and the leading-order error term is minimized. Quadrature formulas up to order 9 are presented with relevant numerical examples.