Ngô Quốc Anh

February 19, 2010

Filed under: Giải tích 9 (MA5265) — Ngô Quốc Anh @ 21:17

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.

$\displaystyle I = \frac{1}{A} \iint_S F(r_i) ds \approx \sum_i w_i F(r_i)$

where $F(r_i)$ is the function sampled at quadrature point $r_i$ on the triangle, and $w_i$ 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 $\frac{1}{A}$ 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 $v_1$, $v_2$ and $v_3$, any point $r$ inside that triangle can be written in terms of the barycentric coordinates $\beta_1$, $\beta_2$ and $\beta_3$ as

$\displaystyle r = \beta_1 v_1 + \beta_2 v_2 + \beta_3 v_3$.

Given a set of barycentric coordinates (quadrature points), each location $r_i$ 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.

————————————————————————————————————————————————————————–
i     $\beta_1$            $\beta_2$            $\beta_3$            $w_i$
————————————————————————————————————————————————————————–
1    0.33333333    0.33333333    0.33333333    1.00000000

————————————————————————————————————————————————————————–
i     $\beta_1$            $\beta_2$            $\beta_3$            $w_i$
————————————————————————————————————————————————————————–
1    0.50000000    0.00000000    0.50000000    1.00000000
2    0.00000000    0.50000000    0.50000000    1.00000000
3    0.50000000    0.50000000    0.00000000    1.00000000

————————————————————————————————————————————————————————–
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

————————————————————————————————————————————————————————–
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

1. would you give the derivation of these formula, I’m interest in these. thanks.

Comment by liyqxtu — April 21, 2010 @ 20:40

• Thanks for your interest. ‘Cos I am busy this month so I will do it next month.

Comment by Ngô Quốc Anh — April 21, 2010 @ 20:42

2. Noting > month later. Heh.

Any idea where these formulas come from? Can try some standard formulas (like from Abramowitz/Stegun) and transform them to barycentric coordinates but they do not agree (at least for 7 pt which is only one I can find in any book).

If anyone knows where these come from it be great to know.

Thanks.

Comment by interested person — November 25, 2010 @ 6:27