  Summary This work presents variational concepts associated with reduced
Trefftz type approaches and discusses the interrelationship between various
concepts of the displacement, hybrid and Trefftz methods. The basic concept of
the displacement version of the reduced Trefftz method operates on the natural
boundary conditions enforced in an integral form whereas the stress version of
the reduced Trefftz type approach operates on the essential boundary conditions
enforced in an integral sense. The application of the method proposed in the
framework of the finite element method is briefly outlined. The methods used by
the reduced Trefftz type approach for enforcing conformity and interelement
continuity between neighboured elements are also discussed. Comparisons with
other known methods for the same purpose are performed. General strategy for
developing finite elements of general geometric form such as quadrilateral
elements with invariance properties is presented. The basic idea of this
strategy consists in using the natural coordinate system only for defining the
element geometry and performing the element integration in the biunit interval.
For defining the approximation functions a local coordinate system defined from
the directions of the covariant base vectors and the perpendicular
contravariant base vectors computed in the geometric centre of the element is
used. This strategy can also be used to implement other versions of finite
elements and other forms of finite elements. Different numerical calculations
and comparisons in the linear statics and kinetics are performed in order to
assess the convergence and the numerical performance of finite elements
developed by applying the reduced Trefftz type approach.
