Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 438

7.3 Residual Methods

In the residual method, the constants (functions) u i are chosen such that an error term or a

residual is zero at selected points, zero in an average sense, or minimized in some fashion.
For Eqs. (7.1) and (7.2), the residuals for the interior and boundary, respectively, can be
expressed as

R V = L u − f

or R = L u − f (7.10)

R S = B u − g

where the p 1

vectors. As shown, often the subscripts are dropped. For solids and beams, Eq. (7.10) would
be

× 1 vector u has been replaced by the approximate u , and R V and R S are p ×

D T ED

w ) −

= + = (

R V u p V R V EI p z (7.11)

w − w
θ − θ

as V

− V

R S = u − u as p − p R S = (7.12)

M

−

M

As indicated in Section 7.2, the trial solutions are usually chosen to satisfy either the bound-
ary conditions or the governing equations. In either case, one of the residuals would be zero.
For example, with the interior method, if N u is chosen to satisfy all of the boundary condi-
tions, then R S =

.
There are many techniques for selecting

0

u i to minimize or make the residuals zero.
Several procedures are described in the following sections. Frequently, the selection of
the u i is based on a scalar R expressed in the form

Interior Method

W j h 1 ( R V ) dV = 0 j = 1 , 2 , ... ,m (7.13a)

V

Boundary Method

W j h 2 ( R S ) dV = 0 j = 1 , 2 , ... ,m (7.13b)

S

where h 1 ,h 2 are prescribed functions of R V ,R S ; the W j are weights or independent weighting
functions , and m is equal to the number of unknown coefficients u j in Eq. (7.9a).

In most of the subsequent discussion, it suffices to adopt the interior residual expressions

( ) = = ...

W j h R dV 0 j 1 , 2 , ,m (7.14a)

V

where the subscripts associated with h and R have been dropped.
Equation (7.14) is a set of m simultaneous equations to be solved for the coefficients

u i ,i .

For a general p -dimensional problem, the residual R is a p

= 1 , 2 , ... ,m of the trial solution u ( x )

× 1 vector. If the mapping h is

chosen such that h satisfies m conditions

similar to Eq. (7.14a). The weighted residual method in this case may be expressed as

( R ) is also a p × 1 vector, then each element of h ( R )

W ( k ) h

( R ) dV = 0 k = 1 , 2 , ... ,m (7.14b)

V

is a diagonal matrix with p non-zero entries W ( k )
1

,W ( k )
2

where W ( k ) W ( k )

p

, ... . These are mp

≤ ≤

equations to be solved for mp coefficients u i , 1 i mp .