Numerical Integration
Numerical integration refers to all the procedures, algorithms and techniques in the numerical analysis to obtain an approximate solution to a definite integral.
That is, how to obtain a numerical value of:
where can be a 1D, 2D or 3D domain.
For our interest in the Finite Element Method, the purpose is to describe how the element matrices can be integrated numerically.
Gauss-Legendre Numerical Integration
To fix the most basic concepts on Numerical Integration, we will focus our description on a one dimensional integration using the Gauss-Legendre quadrature, that is, to solve:
The Gauss-Legendre quadrature establish that the definite integral of a function can be approximate by using a weighted sum of function values at specified points within the domain of integration. An p-point Gaussian quadrature rule is constructed to yield an exact result for polynomials of degree 2p − 1 or less by a suitable choice of the points and weights for .
The coordinates and related weights are:
Number of points, p | Points, ±ξ_{i } | Weights, w_{i} |
---|---|---|
or, using numerical values:
Number of points, p | Points, ±ξ_{i } | Weights, w_{i} |
---|---|---|
1 | 0.0 | 2.0 |
2 | 0.5773502692 | 1.0 |
3 | 0.0 | 0.8888888889 |
0.774596697 | 0.5555555556 | |
4 | 0.3399810436 | 0.6521451549 |
0.8611363116 | 0.3478548451 | |
5 | 0.0 | 0.5688888889 |
0.5384693101 | 0.4786286705 | |
0.9061798459 | 0.2369268851 |
References
- Carlos A. Felippa, "A compendium of FEM integration formulas for symbolic work", Engineering Computations, Vol. 21 No. 8, 2004, pp. 867-890, (c) Emerald Group Publishing Limited [1]
- Numerical Integration
- Gaussian Quadrature