This chapter presents a review on the hybrid-Trefftz (HT) finite element
method (FEM) and its applications in applied mechanics and mechanical engineering.
First, fundamental issues on the hybrid Trefftz FE approach are described, including
elements associated with special-purposed functions, generalized variational functionals,
T-complete functions, internal fields and boundary fields in an element. Then, the Trefftz
FEM of contact problems, two-dimensional elastic problems, elastoplasticity, and
piezoelectricity are described. Mathematical expressions for the cases mentioned are
derived by means of T-complete solutions and a modified variational functional. In the
case of plane elasticity and elastoplastic problems, exact solutions derived in a wellestablished
way from complex variable equations are used for the intra-element fields,
and an iterative form of the fundamental equations is employed in the case of
elastoplasticity. The creation of force-displacement equations from the variational
functional is also addressed. Finally, some conclusions are presented and some open
questions in this area are discussed.
Keywords: Contact problem, Elastoplasticity, FEM, Piezoelectricity, Plane
elasticity, Trefftz function, Variational principle.
