This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lamé-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article.

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