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.


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

A general integral expression form for two dimensional domains can be written in terms of the area coordinates and, therefore, computed by using the Gauss quadrature:


\int_0^1 \int_0^{1-L_3} f(L_1,L_2,L_3) dL_2 dL_3 = \sum_{p=1}^{n_p} f(L_{1_p},L_{2_p},L_{3_p}) W_p

with:

  • n_p \,   the number of integration points;
  • L_{1_p}, L_{2_p}, L_{3_p} \,   the value of the area coordinates;
  • W_p \,   the weight in the integration point p;

The following table and picture shows the integration points and weights for triangles obtained from the Gaussian quadrature (precision means the degree of polynomial for exact integration):


Number of points, n precision Points L1 L2 L3 Wi
1 Linear a 1/3 1/3 1/3 1/2
3 Quadratic a 1/2 1/2 0 1/6
b 0 1/2 1/2 1/6
c 1/2 0 1/2 1/6
4 Cubic a 1/3 1/3 1/3 -9/32
b 0.6 0.2 0.2 25/96
c 0.2 0.6 0.2 25/96
d 0.2 0.2 0.6 25/96
7 Quartic a 0 0 1 1/40
b 1/2 0 1/2 1/15
c 1 0 0 1/40
d 1/2 1/2 0 1/15
e 0 1 0 1/40
f 0 1/2 1/2 1/15
g 1/3 1/3 1/3 9/40


IntegrationPointsTriangularElement.jpg


Note that the weight values has been normalised in order to sum 1/2 to maintain the exact value for the element area.

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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