The Tetrahedron Domain

The reference domain for a tetrahedral element is the right tetrahedron $\hat{T}$ with vertices $(0,0,0)$ , $(1,0,0)$ , $(0,1,0)$ , and $(0,0,1)$ . It has volume $\tfrac{1}{6}$ , and every tetrahedral element in a mesh is the image of $\hat{T}$ under an affine map $F_e$ .

As with the triangle, there is no canonical quadrature family for the tetrahedron. One uses tabulated symmetric rules from Keast (1986) and other standard sources, supporting orders 1 through 3 with 1, 4, and 5 quadrature points respectively.

The figure below shows the point locations for each order. Marker size is proportional to the absolute weight, and colour indicates sign — blue for positive, red for negative.

Quadrature points on the reference tetrahedron for orders 1–3

The order-1 rule places a single point at the centroid $(\tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4})$ with weight $\tfrac{1}{6}$ . The order-2 rule uses 4 points arranged symmetrically near the four vertices, all with equal weight $\tfrac{1}{24}$ . The order-3 rule adds a fifth point at the centroid with a negative weight, in the same spirit as the order-3 triangle rule. All weights sum to $\tfrac{1}{6}$ , the volume of the reference tetrahedron.

In the library, these rules are provided by TetrahedronDomain; its quadrature(order) method returns (points, weights) for the lowest-order rule that integrates polynomials of the requested degree exactly, up to a maximum of order 3.