Numerical Integration

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(Gauss-Legendre Numerical Integration)
(Gauss-Legendre Numerical Integration)
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! Number of points, ''p'' !! Points, ''&plusmn;&xi;''<sub>''i'' !! Weights, ''w''<sub>''i''</sub>
 
! Number of points, ''p'' !! Points, ''&plusmn;&xi;''<sub>''i'' !! Weights, ''w''<sub>''i''</sub>
 
|- align="center"
 
|- align="center"
| 1 || <math>0.0 \,</math> || <math>2.0\,</math>
+
| <math>1\,</math> || <math>0.0 \,</math> || <math>2.0\,</math>
 
|- align="center"
 
|- align="center"
| 2 || <math>\pm\sqrt{1/3}</math> || <math>1.0\,</math>
+
| <math>2\,</math> || <math>\pm\sqrt{1/3}</math> || <math>1.0\,</math>
 
|- align="center"
 
|- align="center"
| rowspan="2" | 3 || <math>0.0 \,</math> || <math>\frac{8}{9}</math>
+
| rowspan="2" | <math>3\,</math> || <math>0.0 \,</math> || <math>\frac{8}{9}</math>
 
|- align="center"
 
|- align="center"
 
| <math>\pm\sqrt{3/5}</math> || <math>\frac{5}{9}</math>
 
| <math>\pm\sqrt{3/5}</math> || <math>\frac{5}{9}</math>
 
|- align="center"
 
|- align="center"
| rowspan="2" | 4 || <math>\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18+\sqrt{30}}{36}</math>
+
| rowspan="2" | <math>4\,</math> || <math>\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18+\sqrt{30}}{36}</math>
 
|- align="center"
 
|- align="center"
 
| <math>\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18-\sqrt{30}}{36}</math>
 
| <math>\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18-\sqrt{30}}{36}</math>
 
|- align="center"
 
|- align="center"
| rowspan="3" | 5 || <math>0.0 \,</math> || <sup>128</sup>⁄<sub>225</sub>\,</math>
+
| rowspan="3" | <math>5\,</math> || <math>0.0 \,</math> || <sup>128</sup>⁄<sub>225</sub>\,</math>
 
|- align="center"
 
|- align="center"
 
| <math>\pm\tfrac13\sqrt{5-2\sqrt{10/7}}</math> || <math>\tfrac{322+13\sqrt{70}}{900}</math>
 
| <math>\pm\tfrac13\sqrt{5-2\sqrt{10/7}}</math> || <math>\tfrac{322+13\sqrt{70}}{900}</math>

Revision as of 11:44, 3 November 2009

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 \, 128225\,</math>
\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

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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