Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 448

Once this equation is solved for u , the approximate series solution

u ( x ) = N u ( x ) u (7.35b)

is formed. We must, of course, address such important computational considerations as
the ease of solving Eq. (7.35a) and the convergence characteristics of the resulting series

u

[Eq. (7.35b)].

Symmetry of the Equations for the Unknown Coefficients
Matrix equations of the form of Eq. (7.35a) can be solved with fewer computations if the
matrix k u is symmetric. The integral expressions for the various residual methods can be
identified readily in terms of the elements k ij of the matrix k u

. For example, for collocation,

k ij = LN uj ( x i ) p i = f ( x i ) (7.36a)

for the subdomain method,

k ij = LN uj dV p i = fdV (7.36b)

V i V i

for Galerkin's method,

k ij = N ui LN uj dV p i = N ui fdV (7.36c)

V V

and for the least squares method,

k ij = LN ui LN uj dV p i = LN ui fdV (7.36d)

V V

As can be seen from these relations, for the collocation (and minimax) and subdomain
procedures, the matrix k u is not in general symmetric i.e., k ij

= .

This is also the case for
Galerkin's method. The matrix k u is always symmetric for “quadratic” formulations such
as the least squares approach.

k ji

Stability and Convergence
The term accuracy refers to the closeness of a solution to the true or exact solution. Stabil-
ity refers to the growth of error as a computation proceeds. In an unstable computation,
truncation, roundoff, or other errors accumulate such that the progress toward the true
solution is overcome or swamped by the error. Convergence refers to achieving progressive
closeness to a particular solution as successive solutions are computed as a parameter is
changed. In such a calculation, typically, the number of terms in a trial solution is adjusted.
Convergence is also used in reference to an iterative computational procedure. In such iter-
ative techniques, the results for a particular computation become the starting point for the
next computation. As this procedure is repeated, it is said to be convergent if the difference
between successive results becomes smaller. Convergence studies for many trial function
methods are available in several references, e.g., Finlayson (1972).

7.3.10 Weak Formulation and Boundary Conditions
In the previous sections, weighted-residual formulations such as

W ( L u − f ) dV = 0 (7.37)

V