Interpolation Functions for General Element Formulation - Fundamentals of Finite Element Analysis - page 220

Use Gaussian quadrature to obtain exact values for the following integrals in two
dimensions. Verify exactness by analytical integration.

6.35

1 2

a.

xy d x d y

0 0

1 1

( r 2 2 rs + s 2 )d r d s

b.

+

− 1 − 1

1 1

( r 3 1)( s 2

c.

− + s )d r d s

− 1 − 1

1 1

( r 5 2 r 3 )( s 3

d.

− + s )d r d s

− 1 − 1

1 1

(6 r 4 1)( s 2

e.

− + s + 1) d r d s

− 1 − 1

Use Gaussian quadrature to obtain exact values for the following three-
dimensional integrals. Verify exactness by analytical integration.

6.36

2 2 2

a.

xyz d x d y d z

0 0 0

1 1 1

1)( s 2

b.

t ( r − − 2) d r d s d t

− 1 − 1 − 1

1 1 1

t 3 r 3 s 2 d r d s d t

c.

− 1 − 1 − 1

1 1 1

t 2 ( r − 2) 4 ( s 2

d.

− 1) d r d s d t

− 1 − 1 − 1

1 1 1

( r 3 4)( t 2

e.

− r )( s + − 1) d r d s d t

− 1 − 1 − 1

Evaluate each of the following integrals using two-point Gaussian quadrature.
Compare each result with the corresponding analytical solution.

6.37

1

cos 2

a.

r d r

− 1