Gauss quadrature numerical integration

Gauss quadrature

In isoparametric elements, the stiffness integral $\int [B]^T[D][B]\det J\,d\xi\,d\eta$ is rational in $\xi$ and $\eta$, so it is evaluated numerically over $-1 \le \xi \le 1$:

$$\int_{-1}^{1} f(\xi)\,d\xi \approx \sum_{i=1}^{n} w_i\,f(\xi_i)$$

An $n$-point rule has sampling points $\xi_i$ and weights $w_i$ chosen to integrate exactly a polynomial of degree up to $2n - 1$.

For two dimensions, the rule is applied in both directions, $\sum\sum w_iw_jf(\xi_i, \eta_j)$ ($2\times2$ gives 4 points for a 4-node element).

Numerical example

Change of limits: $x = 1 + \xi$, so $dx = d\xi$ and $f(\xi) = (1 + \xi)^4 + 3(1 + \xi)$.

Exact value:

$$I = \left[\frac{x^5}{5} + \frac{3x^2}{2}\right]_0^2 = \frac{32}{5} + 6 = 12.4$$

2-point rule: $\xi = \pm0.57735$

$$I_2 = 1.29986 + 10.92236 = 12.222$$

The error is $-1.4\%$; the rule is not exact because $x^4$ is of degree 4 > 3.

3-point rule: $\xi = -0.77460,\ 0,\ +0.77460$

$$I_3 = \tfrac{5}{9}(0.67879 + 15.24121) + \tfrac{8}{9}(4.0) = 8.8444 + 3.5556 = 12.400$$

This equals the exact value, since the polynomial is of degree 4 ≤ 5.

Answer: 2-point: $I = 12.222$ (error 1.4 %); 3-point: $I = 12.400$ (exact).