Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements

• A general approach for computing optimal symmetric quadrature rules on simplexes is identified.
• A new family of optimal symmetric quadrature rules on the 2-simplex (triangle) is identified.
• A new optimal symmetric quadrature rule on the 3-simplex (tetrahedron) is identified.

Sphere close packed (SCP) lattice arrangements of points are well-suited for formulating symmetric quadrature rules on simplexes, as they are symmetric under affine transformations of the simplex unto itself in 2D and 3D. As a result, SCP lattice arrangements have been utilized to formulate symmetric quadrature rules with Np=1Np=1, 4, 10, 20, 35, and 56 points on the 3-simplex (Shunn and Ham, 2012). In what follows, the work on the 3-simplex is extended, and SCP lattices are employed to identify symmetric quadrature rules with Np=1Np=1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points on the 2-simplex and Np=84Np=84 points on the 3-simplex. These rules are found to be capable of exactly integrating polynomials of up to degree 1717 in 2D and up to degree 99 in 3D.