Numerical Integration

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(Gauss-Legendre Numerical Integration)
(Gauss-Legendre Numerical Integration)
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| 2 || <math>\pm\sqrt{1/3}</math> || 1
 
| 2 || <math>\pm\sqrt{1/3}</math> || 1
 
|- align="center"
 
|- align="center"
| rowspan="2" | 3 || 0 || <sup>8</sup>⁄<sub>9</sub>
+
| rowspan="2" | 3 || 0.0 || <sup>8</sup>⁄<sub>9</sub>
 
|- align="center"
 
|- align="center"
 
| <math>\pm\sqrt{3/5}</math> || <sup>5</sup>⁄<sub>9</sub>
 
| <math>\pm\sqrt{3/5}</math> || <sup>5</sup>⁄<sub>9</sub>
Line 42: Line 42:
 
| <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 || 0 || <sup>128</sup>⁄<sub>225</sub>
+
| rowspan="3" | 5 || 0.0 || <sup>128</sup>⁄<sub>225</sub>
 
|- 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>
Line 56: Line 56:
 
! 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 || 0 || 2
+
| 1 || 0.0 || 2.0
 
|- align="center"
 
|- align="center"
| 2 || <math>\pm\sqrt{1/3}</math> || 1
+
| 2 || 0.5773502692 || 1
 
|- align="center"
 
|- align="center"
| rowspan="2" | 3 || 0 || <sup>8</sup>⁄<sub>9</sub>
+
| rowspan="2" | 3 || 0.0 || 0.8888888889
 
|- align="center"
 
|- align="center"
| <math>\pm\sqrt{3/5}</math> || <sup>5</sup>⁄<sub>9</sub>
+
| 0.774596697 || 0.5555555556
 
|- 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" | 4 || 0.3399810436 || 0.6521451549
 
|- align="center"
 
|- align="center"
| <math>\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18-\sqrt{30}}{36}</math>
+
| 0.8611363116 || 0.3478548451
 
|- align="center"
 
|- align="center"
| rowspan="3" | 5 || 0 || <sup>128</sup>⁄<sub>225</sub>
+
| rowspan="3" | 5 || 0.0 || 0.5688888889
 
|- align="center"
 
|- align="center"
| <math>\pm\tfrac13\sqrt{5-2\sqrt{10/7}}</math> || <math>\tfrac{322+13\sqrt{70}}{900}</math>
+
| 0.5384693101 || 0.4786286705
 
|- align="center"
 
|- align="center"
| <math>\pm\tfrac13\sqrt{5+2\sqrt{10/7}}</math> || <math>\tfrac{322-13\sqrt{70}}{900}</math>
+
| 0.9061798459 || 0.2369268851
 
|}
 
|}
  

Revision as of 11:40, 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
3 0.0 89
\pm\sqrt{3/5} 59
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
\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
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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