Numerical integration is a method to approximate a definite integral up to a given degree of accuracy.
\[I = \int_{-1}^{1} f(x) \mathrm{d} x \approx \sum_{i=1}^{p} f(x_i) w_i\]where \(w_i\) are integration weights and \(x_i\) are integration points. It is also called Gauss point. In the following table, you can find Gauss-Legendre quadrature points and weights.
| \(p\) | \(x_i\) | \(w_i\) |
|---|---|---|
| 1 | \(x_1 = 0.0\) | \(w_1=2.0\) |
| 2 | \(x_1 = -1/\sqrt{3}\) \(x_2 = 1/\sqrt{3}\) |
\(w_1=1.0\) \(w_2=1.0\) |
| 3 | \(x_1=-\sqrt{0.6}\) \(x_2 = 0.0\) \(x_3 = \sqrt{0.6}\) |
\(w_1=5.0/9.0\) \(w_2 = 8.0/9.0\) \(w_3=5.0/9.0\) |
Thus, if the two-point integration rule is used, then
\[I \approx f(x_1 = -\frac{1}{\sqrt{3}} ) + f(x_2 =\frac{1}{\sqrt{3}})\]For example, if \(f(x) = x^2\), then
\[\int_{-1}^{1} x^2 \mathrm{d} x \approx f(x_1 = -\frac{1}{\sqrt{3}} ) + f(x_2 =\frac{1}{\sqrt{3}}) = \frac{2}{3}\]The exact solution is
\[\int_{-1}^{1} x^2 \mathrm{d} x = \left[ \frac{x^3}{3} \right]^{1}_{-1} = \frac{2}{3}\]For higher order functions, you may need to use more integration points, see the Gaussian Quadrature page on Wikipedia for more information.
For two-dimensional domain \(\{(x, y) \mid -1 \le x \le 1, -1 \le y \le 1\}\), we usually use the \(2 \times 2\) product rule with \(p=4\) integration points, thus
\[I = \int_{-1}^{1}\int_{-1}^{1} f(x,y) \mathrm{d} x \mathrm{d} y \approx \sum_{i=1}^{p} f(x_i, y_i) w_i\]The integration points and weights are given in the following table.
| \(i\) | \(x_i\) | \(y_i\) | \(w_i\) |
|---|---|---|---|
| 1 | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1.0\) |
| 2 | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1.0\) |
| 3 | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1.0\) |
| 4 | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1.0\) |
This integration scheme is often used in the calculation of stiffness matrix for four-node quadrilateral elements.
For a three-dimensional domain such as,
\[\{(x, y, z) \mid -1 \le x \le 1, -1 \le y \le 1, -1 \le z \le 1\}\]we usually use the \(2 \times 2 \times 2\) product rule with \(p=8\) integration points, thus
\[I = \int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1} f(x,y, z) \mathrm{d} x \mathrm{d} y \mathrm{d} z \approx \sum_{i=1}^{p} f(x_i, y_i, z_i) w_i\]The integration points and weights are given in the following table.
| \(i\) | \(x_i\) | \(y_i\) | \(z_i\) | \(w_i\) |
|---|---|---|---|---|
| 1 | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1.0\) |
| 2 | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1.0\) |
| 3 | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1.0\) |
| 4 | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1.0\) |
| 5 | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1.0\) |
| 6 | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1.0\) |
| 7 | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1.0\) |
| 8 | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1.0\) |
This integration scheme is often used in the calculation of stiffness matrix for eight-node hexahedral elements.