Trefftz Finite Element Method and Its Applications

[+] Author and Article Information
Qing-Hua Qin

Department of Engineering, Australian National University, Canberra ACT 0200, Australia

Appl. Mech. Rev 58(5), 316-337 (Sep 01, 2005) (22 pages) doi:10.1115/1.1995716 History:

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.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

Configuration of the T-element model

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Figure 2

Special element containing a singular corner

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Figure 3

Orthotropic configuration of potential problem

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Figure 4

FE version of approach: (a) subdivision into subdomains Ω1,Ω2,… with piecewise approximations u1,u2,…; and (b) corresponding FE mesh with nodes 1,2,…etc.

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Figure 5

Isolated concentrated loads in infinite plane

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Figure 6

Singular V-notched element

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Figure 7

A typical HT element with linear frame function

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Figure 8

Geometry and loading condition of the thin plate

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Figure 9

Illustration for β and ϕ

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Figure 10

Stretched skew crack plate (μ=0.3)

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Figure 11

Uniformly loaded simply supported 30 deg skew plate (L∕t=100)

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Figure 12

Configuration of meshes used in finite element analysis

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Figure 13

Three element meshes in Example 3



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