Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 435

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Direct Variational and Weighted Residual Methods:
Classical Tr ial Function Methods

In most of this chapter, we will consider trial function methods for solving the governing
equations in their differential (local) form. This contrasts with the previous chapter where
a trial function technique—the finite element method—was employed to solve the integral
(global) form of the governing equations. It will be shown that some of the trial function
methods of this chapter can also be applied to problems formulated in global form.
The key to the successful solution of the governing equations lies in the ability to develop
a reliable and robust approximate method. The trial function methods represent one class
of techniques which appear to possess these characteristics for solving structural mechanics
problems. This approach is of particular interest because, as seen in the previous chapter,
it can be extended to large-scale systems for which the trial function is applied to each of
the elements composing the system.
In the trial function methods, the unknown solution is approximated by a set of basis
functions containing constants or functions. These constants or functions are chosen by
a variety of criteria to provide the best approximation of the trial function family to the
correct solution. Finlayson (1972) contains an in-depth study of trial function methods.

7.1 Governing Differential Equations
The fundamental differential equations for linear elastic solids are provided in C h apter 1.
Differential equations are frequently expressed in operator form, e.g., D T EDu

=

p . Differ-
ential operators, such as D T ED , can be classified as being elliptic, parabolic, and hyperbolic
in form [Norrie and de Vries, 1973]. Generally, the static or stability problems of the me-
chanics of solids are so-called boundary value problems, which are elliptic in type, while
initial value problems correspond to parabolic and hyperbolic operators.
We wish to employ general expressions for the governing differential equations describ-
ing the behavior of solids. Suppose u is the dependent variable or u the vector of dependent
variables that are to be computed by solving the governing differential equations. These
governing equations can be of the form

Lu = f in V (7.1)

subject to boundary conditions

=

Bu g on S (7.2)

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