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, ±ξ_i	Weights, w_i
$1\,$	$0.0 \,$	$2.0\,$
$2\,$	$\pm\sqrt{1/3}$	$1.0\,$
$3\,$	$0.0 \,$	$\frac{8}{9}$
$3\,$	$\pm\sqrt{3/5}$	$\frac{5}{9}$
$4\,$	$\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}$	$\tfrac{18+\sqrt{30}}{36}$
$4\,$	$\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}$	$\tfrac{18-\sqrt{30}}{36}$
$5\,$	$0.0 \,$	$\frac{128}{225}$
	$\pm\tfrac13\sqrt{5-2\sqrt{10/7}}$	$\tfrac{322+13\sqrt{70}}{900}$
	$\pm\tfrac13\sqrt{5+2\sqrt{10/7}}$	$\tfrac{322-13\sqrt{70}}{900}$

or, using numerical values:

Number of points, p	Points, ±ξ_i	Weights, w_i
1	0.0	2.0
2	0.5773502692	1.0
3	0.0	0.8888888889
3	0.774596697	0.5555555556
4	0.3399810436	0.6521451549
4	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

Example of a one dimensional integration

For the function: $f(x)=1+x+x^2+x^3+x^4 \,$ , the exact integration in [-1,+1] is:

$I=\int_{-1}^{+1} f(x) dx = \left . \left ( x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} \right ) \right |_{-1}^{+1} = 2 + 2 \frac{1}{3} + 2 \frac{1}{5} = 3.0666$

Numerically:

First order Gauss-Legendre Quadrature:

$p=1, x_1=0, W_1=2; \qquad I=W_1 f(x_1)=2$

Second order Gauss-Legendre Quadrature:

$p=2 \begin{cases} x_1 = - 0.57735, & W_1 = 1 \\ x_2 = + 0.57735, & W_2 = 1 \end{cases} \qquad I=W_1 f(x_1) + W_2 f(x_2) = 0.67464 + 2.21424 = 2.8888$

Third order Gauss-Legendre Quadrature:

$p=3 \begin{cases} x_1 = - 0.77459, & W_1 = 0.5555 \\ x_2 = - 0.00000, & W_2 = 0.8888 \\ x_3 = + 0.77459, & W_3 = 0.5555 \end{cases}$

$I=W_1 f(x_1) + W_2 f(x_2) + W_3 f(x_3) = 0.7204·0.5555 + 1.0·0.8888 + 3.19931·0.5555 = 3.0666 \,$

That is the exact value, because for any polynomial function of p^th order it is enough to use p-1 integration points.

Two Dimensional Numerical Integration

By using isoparametric formulation we can use natural coordinates to compute any integration. In addition, we can still use the Gauss-Legendre quadrature.