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Unified theory of Trefftz methods and numerical implications

Herrera I.

Computer Assisted Mechanics and Engineering Sciences

2003

Vol. 10, No. 4

495-514

In the 1st International Workshop, devoted to Trefftz Method, the author presented an indirect approach to Trefftz Method (Trefftz-Herrera Method), while in the Second one some of the basic ideas of how to integrate different approaches to Trefftz method were introduced. The present Plenary Lecture, corresponding to the 3rd International Workshop of this series, is devoted to show that Trefftz Method, when formulated in an suitable framework, is a very broad concept capable of incorporating and unifying many numerical methods for partial differential equations. In this manner, the unified theory of Trefftz Method that was announced in the second publication of this series, has been developed. It includes Direct Trefftz Methods (Trefftz-Jirousek) and Indirect Trefftz Methods (Trefftz-Herrera). At present, the unified theory is fully developed and an overview is given here, as well as a brief description of its numerical implications.

On Jirousek method and its generalizations

Herrera I.

Computer Assisted Mechanics and Engineering Sciences

2001

Vol. 8, No. 2-3

325-342

Professor Jirousek has been a very important driving force in the modern development of Trefftz method, contributing to its application in many different fields such as elasticity, shells and plates theory, Poisson equation and transient heat analysis. This article is dedicated to him. The focus of the paper is to incorporate Jirousek method into a very general framework of Trefftz method which has been introduced by Herrera. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces of fully discontinuous functions - i.e., spaces in which the functions together with their derivatives may have jump discontinuities - and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz-Jirousek) and indirect (Trefftz-Herrera) methods. In turn, each one of them can be divided into overlapping and non-overlapping.