In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz. It falls within the class of finite element methods; the hybrid Trefftz finite-element method has been advanced since its introduction about 30 years ago. The conventional method of finite element analysis involves converting the differential equation that governs the problem into a variational functional from which element nodal properties – known as field variables – can be found; this can be solved by substituting in approximate solutions to the differential equation and generating the finite element stiffness matrix, combined with all the elements in the continuum to obtain the global stiffness matrix. Application of the relevant boundary conditions to this global matrix, the subsequent solution of the field variables rounds off the mathematical process, following which numerical computations can be used to solve real life engineering problems.
An important aspect of solving the functional requires us to find solutions which satisfy the given boundary conditions and satisfy inter-element continuity since we define independently the properties over each element domain. The hybrid Trefftz method differs from the conventional finite element method in the assumed displacement fields and the formulation of the variational functional. In contrast to the conventional method the Trefftz method assumes the displacement field is composed of two independent components; the frame field here is the same as that used in the conventional finite element method but defined on the boundary of the element – hence the use of the term “hybrid” in the method’s nomenclature. The variational functional must thus include additional terms to account for boundary conditions, since the assumed solution field only satisfies the governing differential equation; the main advantages of the hybrid Trefftz method over the conventional method are: the formulation calls for integration along the element boundaries only which allows for curve-sided or polynomial shapes to be used for the element boundary, presents expansion bases for elements that do not satisfy inter-element continuity through the variational functional, this method allows for the development of crack singular or perforated elements through the use of localized solution functions as the trial functions.
Since its mainstream introduction some 30 years ago, this modified finite element method has become popular to applications such as elasticity, Kirchhoff plates, thick plates, general three-dimensional solid mechanics, antisymmetric solid mechanics, potential problems, elastodynamic problems, geometrically nonlinear plate bending, transient heat conduction analysis among various others. It is being applied to steady, non-turbulent, Newtonian fluid flow applications through ongoing research at the Faculty of Engineering and Information Technology at the Australian National University in Canberra, Australia; the hybrid Trefftz method is being applied to some fields, e.g. computational modeling of hydrated soft tissues or water-saturated porous media, through ongoing research project at the Technical University of Lisbon, Instituto Superior Técnico in Portugal. Qin, Q. H; the Trefftz Finite and Boundary Element Method, England: WIT Press, pp. 1–55 Connor, J. J.. A. Finite Element Techniques for Fluid Flow, England: Newnes-Butterworths Qin, Q.
H. "Formulation of hybrid Trefftz finite element method for elastoplasticity", Applied Mathematical Modelling, 29: 235–252, doi:10.1016/j.apm.2004.09.004 Hazewinkel, Michiel, ed. "Trefftz method", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Hybrid-Trefftz research project at Instituto Superior Técnico in Lisbon, Portugal
The Junior Kangaroos side represents Australia in the sport of rugby league. They are known as the Junior Kangaroos, after the native marsupial of that name. Since 2019, they are an under-23 side, with players selected from the NRL, they are administered by the Australian Rugby League Commission. Prior to 2010, the Junior Kangaroos was selected on an under-19 age basis; the best Australian players 19 years of age or younger, would represent the country against the likes of the Junior Kiwis, Papua New Guinea and Tonga, sometimes as curtain raisers to the senior Kangaroos games. Players such as Greg Bird, Matt Bowen, Justin Hodges, Luke Lewis, Jamie Lyon, Cameron Smith and Johnathan Thurston represented the Junior Kangaroos early in their careers and went on to play at the senior level for Australia; the last Junior Kangaroos side selected under the under 19 rules was in 2007. This would be the last Junior Kangaroos team until 2010. In 2010, with the advent of the National Youth Competition, the Junior Kangaroos was switched to an under-20 team and played in a two Test series against the Junior Kiwis.
From 2011 to 2018, the Junior Kangaroos have faced the Junior Kiwis at the end of each season in a one off Test match as a curtain raiser to a senior level Australia-New Zealand Test match. In 2019, the ARL announced that Junior Kangaroos would be moving to an under-23 format and would play France at WIN Stadium. Played as a curtain raiser to the New Zealand-England Four Nations match. Played as a curtain raiser to the New Zealand-Papua New Guinea Four Nations match. Played as a curtain raiser to the Australia-New Zealand pre-Four Nations tournament match. Both coaches agreed to allow an 18th man be used during the match. Played as a curtain raiser to the Australia-New Zealand end of year Test match. Played as a curtain raiser to the Papua New Guinea-Fiji Melanesian Cup match and the Samoa-Tonga Polynesian Cup match. Played as a curtain raiser to the Papua New Guinea-Fiji Melanesian Cup match and the Samoa-Tonga Polynesian Cup match. Played as a curtain raiser to the 2017 Anzac Test. Played as a curtain raiser to the Australia-New Zealand end of year Test match.
Played as a curtain raiser to the Australia-New Zealand Oceania Cup match. Ben Jones Dale Finucane Chad Townsend Alex McKinnon Mitchell Cornish Michael Lichaa Cameron McInnes Matthew Lodge Tevita Pangai Junior Brodie Croft Jake Clifford David Fifita David Hamilton Andrew Patmore Kelly Egan Justin Holbrook Neil Henry Australia national rugby league team Australian Aboriginal rugby league team