Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 473

0
0
0
6
24

0
0
2
0
0

0
0
0
6
0

EI
L 3

R u + R p = [11111] + [01000] − [10000]

0
1
2
3
4

00000
0026 2
0

EI
L 3

+ [002612] = 2 4 12 24

0 6 12 24 42

24 24 32 48 72

This leads to the same system of equations as obtained for case C of Example 7.9.
For the third and final trial function, consider that of case D of Example 7.9. As might be
expected, this leads to the same results found for Ritz's method in Example 7.9.

It is of interest that the extended Ritz and Galerkin methods gave the same results for the
same trial solutions for the beam in Examples 7.9 and 7.10. This is not surprising since both
of these variational methods are set up such that the same restrictions, i.e., no boundary
conditions need to be satisfied, are imposed on their trial solutions and both involve the
global form of the equilibrium equations. Also, for a beam, it is readily shown that

R T [ k u + ( R + ) ] Ritz = [ k u + ( R p + R u ) ] Galerkin (7.77)

To do so, integrate k u Ritz twice by parts. Thus

EI L
0

N T
u

N u dx

k u Ritz =

EI L
0

EI N u N u L

N i u N u −

N u N u

= dx + (7.78)

0

R T , with R from Eq. (7.70), gives k u

Adding this to R + + R p + R u , where k u , R p , and R u

are given by Eq. (7.75).
In general, the nonextended Ritz and Galerkin methods can be expected to give different
results.

7.4.5 Trefftz's Method: A Boundary Method

The classical trial function methods and the finite element method, which can be treated as
an extension of the classical trial function methods wherein trial solutions apply to elements
into which the system has been divided, sometimes encounter difficulties when the domain
(volume) is extremely large or when singularities occur in some of the variables (Fig. 7.7).
Frequently, exact solutions exist for the differential equations in the volume and, sometimes,
there are solutions that take the singularity into account. In such cases, it may be useful to
use a boundary rather than an interior method.
Often for the boundary method, the selection of trial functions is not straightforward. In
general, singular functions such as Green's functions can be employed. These lead to an
approximation in the form of a set of integral equations, which are the basis of an important
computational technique, the boundary element method (BEM) or the boundary finite element
method, which is considered in Chapter 9.