Numerical Integration

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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:

\int_{\lambda_a}^{\lambda_b}\! f(\lambda)\, d\lambda.

where \lambda \, 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:

I=\int_{-1}^{+1} f(\xi) d\xi


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   \xi_i \,   and weights   w_i \,   for   i = 1, \cdots p \,.


\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)


The coordinates and related weights are:


Number of points, p Points, ±ξi Weights, wi
1\, 0.0 \, 2.0\,
2\, \pm\sqrt{1/3} 1.0\,
3\, 0.0 \, \frac{8}{9}
\pm\sqrt{3/5} \frac{5}{9}
4\, \pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7} \tfrac{18+\sqrt{30}}{36}
\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7} \tfrac{18-\sqrt{30}}{36}
5\, 0.0 \, \frac{128}{225}
\pm\tfrac13\sqrt{5-2\sqrt{10/7}} \tfrac{322+13\sqrt{70}}{900}
\pm\tfrac13\sqrt{5+2\sqrt{10/7}} \tfrac{322-13\sqrt{70}}{900}


or, using numerical values:


Number of points, p Points, ±ξi Weights, wi
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

References

  1. 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]
  2. Numerical Integration
  3. Gaussian Quadrature

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