In engineering analysis it is often necessary to numerically evaluate surface integrals over a particular geometric shape. The triangle is one of the most widely used shapes used to evaluate surface integrals arising in techniques such as the Finite Element Method (FEM) and the Method of Moments (MoM) in electromagnetics.

Numerical Quadrature Over Triangles is typically done by sampling the function being integrated at discrete points on the triangle, multiplying each samples by specially chosen quadrature weights, and summing these up, i.e.

I = (1/A) ∫s F(ri) ds ~ Σi wi F(ri)

where F(ri) is the function sampled at quadrature point ri on the triangle, and wi is the quadrature weight for that point. The quadrature operation results in a quantity that is normalized by the triangle's area A, hence the (1/A) coefficient.

I will present a set of quadrature points and weights useful for performing typical quadratures. These quadrature points are generalized into what are called barycentric coordinates, also commonly referred to as area coordinates. Given a triangle defined by the three vertexes v1, v2 and v3, any point r inside that triangle can be written in terms of the barycentric coordinates β1, β2 and β3 as

r = β1 v1 + β2 v2 + β3 v3

Given a set of barycentric coordinates (quadrature points), each location ri on the triangle may be found and the integrated function sampled. The weights to be presented are what are called Gauss-Legendre quadrature weights, the derivation of which I will not show here. Read numerical integration for a more in-depth discussion on this.

Below are tables of sampling points and weights. The points are chosen to lie in the interior of the triangle. There exist quadrature formulas that have points on the triangle edge, however they are not suitable for the analysis that I do so I don't have them on hand.

One-point quadrature (center of triangle):

i    beta_1       beta_2      beta_3       w_i
1   0.33333333  0.33333333  0.33333333  1.00000000

Four-point quadrature:

i    beta_1       beta_2      beta_3       w_i
1   0.33333333  0.33333333  0.33333333  0.28125000
2   0.73333333  0.13333333  0.13333333  0.26041666
3   0.13333333  0.73333333  0.13333333  0.26041666
4   0.13333333  0.13333333  0.73333333  0.26041666

Seven-point quadrature:

i    beta_1       beta_2      beta_3       w_i
1   0.33333333  0.33333333  0.33333333  0.22500000
2   0.05961587  0.47014206  0.47014206  0.13239415
3   0.47014206  0.05961587  0.47014206  0.13239415
4   0.47014206  0.47014206  0.05961587  0.13239415
5   0.79742699  0.10128651  0.10128651  0.12593918
6   0.10128651  0.79742699  0.10128651  0.12593918
7   0.10128651  0.10128651  0.79742699  0.12593918

May the Force Be With You. Go evaluate some integrals over triangles!