Numerical Integration

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(Example of a one dimensional integration)
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:That is the exact value, because for any polynomial function of '''p<sup>th</sup>''' order it is enough to use '''p-1''' integration points.
 
:That is the exact value, because for any polynomial function of '''p<sup>th</sup>''' order it is enough to use '''p-1''' integration points.
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== Two Dimensional Numerical Integration ==
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By using isoparametric formulation we can use natural coordinates to compute any integration. In addition, we can still use the Gauss-Legendre quadrature.
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=== Numerical Integration for Isoparametric Triangular Domains ===
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:[[Image:IntegrationPointsTriangularElement.jpg]]
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== References ==
 
== References ==

Revision as of 18:05, 5 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.


Contents

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 \, \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, ±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
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 pth 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.

Numerical Integration for Isoparametric Triangular Domains

IntegrationPointsTriangularElement.jpg


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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