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 are integration weights and are integration points. It is also called Gauss point. In the following table, you can find Gauss-Legendre quadrature points and weights.

1
2

3



Thus, if the two-point integration rule is used, then

For example, if , 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 , we usually use the product rule with integration points, thus

The integration points and weights are given in the following table.

1
2
3
4

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 product rule with integration points, thus

The integration points and weights are given in the following table.

1
2
3
4
5
6
7
8

This integration scheme is often used in the calculation of stiffness matrix for eight-node hexahedral elements.

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