11R3. Trefftz Finite and Boundary Element Method. - Qing-Hua Qin (Dept of Mech and Mechatronic Eng, Univ of Sydney, Sydney, Australia). WIT Press, Southampton, UK. 2000. 282 pp. ISBN 1-85312-855-4. $183.00.
Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).
This textbook provides a good up-to-date account of some modern methods in numerical analysis. The book should be of interest to researchers in finite and boundary element methods and be accessible to graduate students interested in these topics as well.
Chapter 1 is introductory. It surveys, among other things, the finite element technique, various modified variational principles, and the basic concept of Trefftz-complete solution. Many of these ideas are applied in the next chapter for the treatment of several potential problems such as seepage, heat conduction, electrostatics, and many others which can be written as a function of a potential and whose governing equation is the classical Laplace, Poisson, or Helmholtz equation.
Chapter 3 deals with the hybrid Trefftz finite element theory in linear elasticity. Chapters 4 and 5 apply Trefftz’s method to thin and thick plates. The approach for these two cases is different in the following sense. In contrast to Kirchoff’s thin plate theory, the transverse shear deformation effect of the plate is taken into account in thick plate theory, so that the governing equation is a sixth-order boundary value problem. As a consequence, three boundary conditions should be considered for each boundary. In contrast, only two boundary conditions are considered in thin plate theory.
Chapter 6 develops a hybrid Trefftz element formulation for numerical solutions of two-dimensional transient heat conduction problems.
In Chapter 7, a family of hybrid Trefftz elements is presented for the nonlinear analysis of plate bending problems. Examples include post-buckling problems of thin plates and large deflection of thick plates with or without elastic foundations.
Chapter 8 expounds the approach proposed by Freitas and Wang for the application of Trefftz stress elements to elastoplasticity. The next chapter deals with applications of Trefftz elements for solving dynamic problems of plate bending. Based on the time-step method, the dynamic plate equation is discretized with respect to time, and then the resulting set of elliptic equations is solved by the corresponding time-independent element approach.
The last chapter expounds the main techniques arising in the Trefftz boundary element method. The two possible alternative techniques (direct and indirect formulations) are described briefly in order to provide a primary introduction to this method.
Presentation, style, and layout in the book under review are all very good. This reviewer can warmly recommend Trefftz Finite and Boundary Element Method to anyone looking for a clear introduction to the subject, especially to a reader who enjoys the study of concrete examples.