3R4. Underlying Principles of the Boundary Element Method. - D Cartwright (Col of Eng, Bucknell Univ PA). WIT Press, Southampton, UK. 2001. 276 pp. ISBN 1-85312-839-2. \$149.00.

Reviewed by DE Beskos (Dept of Civil Eng, Univ of Patras, Patras, GR-26500, Greece).

This is a very well written introductory textbook on the foundations of the direct Boundary Element Method (BEM). It is very useful to both teachers and their undergraduate students in applied mathematics and engineering, as well as those interested in learning the basics of the method.

The emphasis is on the principles and the mathematical derivations of the BEM and not on its numerical implementation. In that sense, the book is unique since most of the existing books emphasize the numerical implementation of the method. Detailed mathematical derivations are provided and solved problems are presented in detail in each chapter to help the student understand the subject matter of the book. Applications are described for one-, two- and three-dimensional problems of potential theory and elastostatics in a unified manner. Only constant elements are considered here for which the computation of singular integrals can be done analytically (in closed form).

There are two aspects of the book this reviewer considers very important and worth mentioning: $i$) The concepts of the Green’s function and the fundamental solution are both discussed in detail. It is further shown how one can obtain the latter as a combination of Green’s functions defined for different boundary conditions. In many other books these two concepts are used one for the other and this creates confusion. $ii$) The boundary integral equation for internal points is derived here through the method of weighted residuals, which is a very powerful and general method for formulating boundary element and finite element methods.

The whole book consists of seven chapters, one appendix, a bibliography, and a subject index. More specifically, Chapter 1 deals with a discussion on the derivation of the basic field equations (governing equations) of the scalar or vector type and of the second or fourth order in one or more dimensions. Chapter 2 discusses Green’s functions, their properties, derivation, and use in solving boundary value problems. The concept of the fundamental solution, its relation to Green’s functions, and its derivation are taken up in Chapter 3. The method of weighted residuals is described in Chapter 4, and its use in the derivation of the direct boundary integral equation for one-dimensional, potential, and elastostatic problems are described in Chapters 5, 6, and 7, respectively. In those last three chapters, related problems are solved to illustrate the method and demonstrate its advantages. Finally, the last chapter contains various appendices dealing with details of various mathematical derivations, which were presented in Chapters 3 and 5–7. The book concludes with a list of 20 reference books on the BEM and a short subject index.

The author has certainly succeeded in fulfilling his stated aim of providing an introductory textbook emphasizing the underlying principles of the BEM. Underlying Principles of the Boundary Element Method should be purchased by teachers, undergraduate and graduate students, researchers who would like to start working in the field, and certainly by libraries.