# Computational Biomechanics

Dr. Kewei Li

## Numerical Integration

### One-dimensional Integration

Numerical integration is a method to approximate a definite integral up to a given degree of accuracy.

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

For example, if $f(x) = x^2$, then

The exact solution is

For higher order functions, you may need to use more integration points, see the Gaussian Quadrature page on Wikipedia for more information.

### Two-dimensional Integration

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

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.

### Three-dimensional Integration

For a three-dimensional domain such as,

we usually use the $2 \times 2 \times 2$ product rule with $p=8$ integration points, thus

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.

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