3R2. Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform. Lecture Notes in Applied Mechanics Volume 5. - FME Duddeck (Lehrstuhl fur Baumechanik, Technische Univ, Arcisstr 21, Munchen, 80333, Germany). Springer-Verlag, Berlin. 2002. 181 pp. ISBN 3-540-43138-1. $79.95.
Reviewed by G Maier (Dept of Struct Eng, Tech Univ Politecnico, Piazza Leonardo Da Vinci 32, Milan, 20133, Italy).
The boundary element methods (BEMs), namely the traditional “collocation” method (CBEM) and the more recent “symmetric Galerkin” (SGBEM), are at present fairly well consolidated and understood in their potentialities and limitations. In particular, the latter BEM, after space modeling, leads to symmetric (through still densely populated) matrix operators, can be generated by variational approaches and in some application areas is endowed with useful theoretical results such as solution characterizations and convergence criteria.
This research book (and perhaps textbook) presents a remarkable and promising generalization of the SGBEM referred to here as “Fourier Boundary Element Method” (FBEM). The author uses spatial and temporal Fourier transform to reformulate the SGBEM. It is shown that no inverse transform is required, since all variables (including fundamental solutions, their derivatives and trial and test functions) can be dealt with in Fourier space only.
The main positive feature of the FBEM is represented by its direct applicability to several technically meaningful problems for which the fundamental solutions (such as Kelvin’s kernel in linear isotropic elasticity) are not known, but can be explicitly generated in Fourier space only. Non-isotropic linear elastic analysis is a typical problem of this kind, although various researchers with diverse approximations and numerical approaches challenged it.
After a brief survey of CBEM and SGBEM (Chapter 2) and of mathematical concepts and tools belonging to the theories of distributions and of Fourier integral transform (Chapter 3 and part of the appendices), the FBEM is developed for a fairly comprehensive set of problems, with emphasis on anisotropic situations, namely: heat conduction, elasticity, thin (Kirchhoff) and thick (Reissner) plates without and with Winkler foundations, transient and stationary waves in elasticity and poroelasticity, coupled and uncoupled thermoelasticity, Mises-Huber elastoplasticity and geometric no linearity in elasticity.
Chapter 11 devotes special attention to wavelets and wavelet transform, specifically to a multi-resolution algorithm apt to induce matrix sparsity and reduce the computational effort.
In this reviewer’s opinion, the contents of this book, which gathers in a systematic concise fashion several original results due to its author, represents a major contribution to the development of BEMs and to the literature on them. Perhaps the presentation is too concise in various parts; the numerical tests are rather academic and small size for computational validation; some earlier contributions to SGBEM would have been worth citation; among the pictures of pertinent great scientists, which enrich all chapters, many readers would like to see one of Carlo Somigliana.
However, most readers, are likely to appreciate the originality of Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform in content and style and to welcome some peculiar features of it, like the biographical section of the introduction (engineers can read, eg, that Fourier as Prefect in Grenoble was responsible for the construction of a highway from Grenoble to Turin, and that Galerkin was a consultant in planning and building many of the largest dams and hydroelectric power plants in Soviet Union).
Fabian Duddek’s book starts with a citation from Laurent Schwartz’s The´orie des Distributions (namely: engineers use Heaviside symbolic calculus with their own personal conception, with a more or less tranquil conscience). This reminds me of another citation from Schwartz: “Il n’y a pas de mathematiques sans larmes a` l’usage des physiciens et des inge´nieurs” (there is no mathematics used by physicists and engineers without tears). In this reviewer’s opinion, even that great mathematician would find no reason for tears in this book; on the contrary, he would probably welcome, as most readers will, an elegant and potentially fruitful combination of mathematics and computational mechanics.
