Numerical Integration

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 Revision as of 11:33, 3 November 2009 (view source)JMora (Talk | contribs) (→Gauss-Legendre Numerical Integration)← Older edit Revision as of 11:36, 3 November 2009 (view source)JMora (Talk | contribs) (→Gauss-Legendre Numerical Integration)Newer edit → Line 27: Line 27: − {| border="1" cellpadding="1" cellspacing="0" class="wikitable" style="margin:auto; background:white;" + {| border="1" cellpadding="5" cellspacing="0" class="wikitable" style="margin:auto; background:white;" − ! Number of points, ''p'' !! Points, ''ξ''''i'' !! Weights, ''w''''i'' + ! Number of points, ''p'' !! Points, ''±ξ''''i'' !! Weights, ''w''''i'' + |- align="center" + | 1 || 0.0 || 2.0 + |- align="center" + | 2 || $\pm\sqrt{1/3}$ || 1 + |- align="center" + | rowspan="2" | 3 || 0 || 89 + |- align="center" + | $\pm\sqrt{3/5}$ || 59 + |- align="center" + | rowspan="2" | 4 || $\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}$ || $\tfrac{18+\sqrt{30}}{36}$ + |- align="center" + | $\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}$ || $\tfrac{18-\sqrt{30}}{36}$ + |- align="center" + | rowspan="3" | 5 || 0 || 128225 + |- align="center" + | $\pm\tfrac13\sqrt{5-2\sqrt{10/7}}$ || $\tfrac{322+13\sqrt{70}}{900}$ + |- align="center" + | $\pm\tfrac13\sqrt{5+2\sqrt{10/7}}$ || $\tfrac{322-13\sqrt{70}}{900}$ + |} + + + or, using numerical values: + + + {| border="1" cellpadding="5" cellspacing="0" class="wikitable" style="margin:auto; background:white;" + ! Number of points, ''p'' !! Points, ''±ξ''''i'' !! Weights, ''w''''i'' |- align="center" |- align="center" | 1 || 0 || 2 | 1 || 0 || 2

Revision as of 11:36, 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 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 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 2
2 $\pm\sqrt{1/3}$ 1
3 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 128225
$\pm\tfrac13\sqrt{5-2\sqrt{10/7}}$ $\tfrac{322+13\sqrt{70}}{900}$
$\pm\tfrac13\sqrt{5+2\sqrt{10/7}}$ $\tfrac{322-13\sqrt{70}}{900}$

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