Numerical Integration - Numerical Methods in Engineering with MATLAB - page 236

specifying the limits of integration on x and y ,quadrature is not apractical meansof
evaluating integrals over irregular regions. However, an irregular region A can always
be approximatedas an assembly of triangular or quadrilateralsubregions A 1 , ,

called finite elements , as illustrated inFig. 6.6. The integralover A can thenbeevaluated
by summing the integrals over the finite elements:

A 2 ,...

f ( x , y ) dx dy ≈ f ( x , y ) dx dy

A A i

i

Volume integrals can computedinasimilar manner, using tetrahedra orrectangular
prismsfor the finite elements.

Boundary of region A

A I

Figure 6.6. Finite element model of an irregular
region.

Gauss-Legendre Quadrature over a Quadrilateral Element

η

η = 1

3

1

4

ξ = − 1

ξ = 1

0

ξ

y

1

1

2

1 x

η= −1

1 0

(a)
Figure 6.7. Mappinga quadrilateral into the standard rectangle.

(b)

Consider the double integral

1 1

I = f ( ξ,η ) d ξ d η

− −

1 1

over the rectangular element shown in Fig. 6.7(a).Evaluating each integral in
turn by Gauss-Legendrequadrature using n nodes in each coordinate direction,