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.
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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 | |
6 | 0.2386191861 | 0.4679139346 |
0.6612093865 | 0.3607615730 | |
0.9324695142 | 0.1713244924 | |
7 | 0.0 | 0.4179591837 |
0.4058451514 | 0.3818300505 | |
0.7415311856 | 0.2797053915 | |
0.9491079123 | 0.1294849662 | |
8 | 0.1834346425 | 0.3626837834 |
0.5255324099 | 0.3137066459 | |
0.7966664774 | 0.2223810345 | |
0.9602898565 | 0.1012285636 |
Example of a one dimensional integration
For the function: , the exact integration in [-1,+1] is:
Numerically:
- First order Gauss-Legendre Quadrature:
- Second order Gauss-Legendre Quadrature:
- Third order Gauss-Legendre Quadrature:
- That is the exact value, because for any polynomial function of p^{th} order it is enough to use p-1 integration points.
Two Dimensional Numerical Integration
By using isoparametric formulation we can use natural coordinates to compute any integration. In addition, we can still use the Gauss-Legendre quadrature.
Numerical Integration for Isoparametric Triangular Domains
A general integral expression form for two dimensional domains can be written in terms of the area coordinates and, therefore, computed by using the Gauss quadrature:
with:
- the number of integration points;
- the value of the area coordinates;
- the weight in the integration point p;
The following table and picture shows the integration points and weights for triangles obtained from the Gaussian quadrature (precision means the degree of polynomial for exact integration):
Number of points, n | precision | Points | L_{1} | L_{2} | L_{3} | W_{i} |
---|---|---|---|---|---|---|
1 | Linear | a | 1/3 | 1/3 | 1/3 | 1/2 |
3 | Quadratic | a | 1/2 | 1/2 | 0 | 1/6 |
b | 0 | 1/2 | 1/2 | 1/6 | ||
c | 1/2 | 0 | 1/2 | 1/6 | ||
4 | Cubic | a | 1/3 | 1/3 | 1/3 | -9/32 |
b | 0.6 | 0.2 | 0.2 | 25/96 | ||
c | 0.2 | 0.6 | 0.2 | 25/96 | ||
d | 0.2 | 0.2 | 0.6 | 25/96 | ||
7 | Quartic | a | 0 | 0 | 1 | 1/40 |
b | 1/2 | 0 | 1/2 | 1/15 | ||
c | 1 | 0 | 0 | 1/40 | ||
d | 1/2 | 1/2 | 0 | 1/15 | ||
e | 0 | 1 | 0 | 1/40 | ||
f | 0 | 1/2 | 1/2 | 1/15 | ||
g | 1/3 | 1/3 | 1/3 | 9/40 |
Note that the weight values has been normalised in order to sum 1/2 to maintain the exact value for the element area.
Therefore:
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