Numerical Integration
(→Gauss-Legendre Numerical Integration) |
(→Gauss-Legendre Numerical Integration) |
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:<math>\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)</math> | :<math>\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)</math> | ||
+ | |||
+ | |||
+ | The coordinates and related weights are: | ||
+ | |||
+ | |||
+ | {| class="wikitable" style="margin:auto; background:white;" | ||
+ | ! Number of points, ''p'' !! Points, ''xi''<sub>''i'' !! Weights, ''w''<sub>''i''</sub> | ||
+ | |- align="center" | ||
+ | | 1 || 0 || 2 | ||
+ | |- align="center" | ||
+ | | 2 || <math>\pm\sqrt{1/3}</math> || 1 | ||
+ | |- align="center" | ||
+ | | rowspan="2" | 3 || 0 || <sup>8</sup>⁄<sub>9</sub> | ||
+ | |- align="center" | ||
+ | | <math>\pm\sqrt{3/5}</math> || <sup>5</sup>⁄<sub>9</sub> | ||
+ | |- align="center" | ||
+ | | rowspan="2" | 4 || <math>\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18+\sqrt{30}}{36}</math> | ||
+ | |- align="center" | ||
+ | | <math>\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18-\sqrt{30}}{36}</math> | ||
+ | |- align="center" | ||
+ | | rowspan="3" | 5 || 0 || <sup>128</sup>⁄<sub>225</sub> | ||
+ | |- align="center" | ||
+ | | <math>\pm\tfrac13\sqrt{5-2\sqrt{10/7}}</math> || <math>\tfrac{322+13\sqrt{70}}{900}</math> | ||
+ | |- align="center" | ||
+ | | <math>\pm\tfrac13\sqrt{5+2\sqrt{10/7}}</math> || <math>\tfrac{322-13\sqrt{70}}{900}</math> | ||
+ | |} | ||
== References == | == References == |
Revision as of 11:27, 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:
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, xi_{i } | Weights, w_{i} |
---|---|---|
1 | 0 | 2 |
2 | 1 | |
3 | 0 | ^{8}⁄_{9} |
^{5}⁄_{9} | ||
4 | ||
5 | 0 | ^{128}⁄_{225} |
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