Introduction to Hybrid-Trefftz Finite Element Method

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.