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 \, and weights wi \, for i = 1,...,n \,.


\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)

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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