Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 474

FIGURE 7.7
Configurations for which the boundary methods have
been found to be useful.

In this section, a somewhat simpler and less generally applicable boundary method
will be presented. This is sometimes referred to as the Trefftz 13 method [Trefftz, 1926].
This method utilizes trial functions which satisfy the governing differential equations and,
through a work expression, lead to an approximation of the boundary conditions. In the
Ritz method, a variational principle, using trial functions which satisfy the displacement
boundary conditions and the kinematic conditions, provides an approximation to the con-
ditions of equilibrium. However, with the Trefftz method, by employing trial functions
which satisfy the differential equations of equilibrium as well as the kinematic conditions,
all of the boundary conditions are approximated with the aid of a variational expression.
The global form of the boundary conditions will constitute the variational expression.
For the Trefftz method, the trial solution must satisfy

D T σ + 0

E − 1 σ = =

p V =

in V (7.79)

Du

that is, all of the governing differential equations in V are to be satisfied. A variational
expression in terms of boundary integrals of the boundary conditions u = u on S u and

13 Erich Trefftz (1888-1937), son of a Leipzig, Germany merchant, studied in Aachen, G ottingen, and Strassburg.
He received his doctor's degree at Strassburg in 1913 with his research based on a suggestion by R. von Mises. In
1919, he became a professor of applied mathematics in Aachen. In 1922, he accepted a professorship in applied
mechanics in Dresden, a post that he held until his death. His work on applied mechanics dealt chiefly with
hydrodynamics, the theory of vibrations, and elasticity.