3R1. Boundary Element Method, Volume 1: Applications in Thermo-Fluids and Acoustics. - LC Wrobel (Brunel Univ, UK). Wiley, W Sussex, UK. 2002. 451 pp. ISBN 0-471-72039-9. $160.00.

Reviewed by AJ Kassab (Dept of Mech, Mat, and Aerospace Eng, Col of Eng, Univ of Central Florida, Orlando, FL 32816-2450).

This is Volume 1 of a two-volume set offering a comprehensive treatment of the fundamentals and applications of the boundary element method (BEM) in thermo-fluids, acoustics, and solid mechanics. Volume 1 is authored by Prof Luis Wrobel of Brunel University, and Volume 2, by Prof Ferri Aliabadi, of the College of London, both of whom are recognized authorities on and long-standing contributors to the development of the BEM. The BEM is a numerical method for the solution of boundary integral equations that can be derived for a variety of engineering field problems. This is possible if a fundamental solution is available for the differential operator governing the field problem of interest, and in such cases, the BEM only requires a discretization of the bounding surface, a distinct advantage over volume meshing techniques such as the finite volume and finite element method. This advantage is even more pronounced when dealing with moving surface and inverse problems as well as optimization.

This 11-chapter book begins with a historical review of integral equations and the BEM over the last 40 years. Basic mathematical notations and theory of integral equations methods useful in development of the BEM in potential theory are covered in the ensuing chapters. A brief treatment of hypersingular equations, dual and multiple reciprocity, axi-symmetry, Galerkin BEM formulation, and fast solvers is also provided. The next two chapters address heat transfer applications of the BEM in steady and unsteady regimes. Linear as well as nonlinear heat transfer are both covered with applications to nonlinear materials, radiation, phase change, and materials processing. Convection-diffusion, bio-heat transfer, hyperbolic heat conduction, as well as coupled heat and mass transfer are treated in some detail.

The next chapter develops the BEM for acoustics modeling in the frequency domain. Attention is given to the ensuing hyper-singular integrals and to regularization of the integral equations as well as to the treatment of the well-known non-uniqueness arising in exterior acoustics problems. Much detail is provided in the formulation and implementation of the BEM in acoustics as well as a wide range of applications ranging from muffler simulations to acoustic barrier modeling. The dual BEM is developed for the case of thin barriers. Finally, BEM in transient acoustics is presented using the time dependent fundamental solution along with time marching schemes. The chapter closes with a brief review of BEM in acoustics including FEM/BEM coupling, bio-engineering applications, and modeling of musical instruments.

The following chapter treats the application of the BEM in electrochemistry, which begins with a derivation of the potential equation governing the problem of electrochemistry as well as the definition of the polarization curve that serves as a boundary condition for these problems. Several examples are presented ranging from 2D to 3D applications in circular corrosion cells, buried tanks, seawater pump analysis, and cathodic protection of offshore platforms as well electro-deposition.

The next three chapters address fluid mechanics problems in ideal flows, slow viscous flows, and general viscous flows governed by the Navier-Stokes equations. The theory and applications of ideal flow to saltwater intrusion in confined and unconfined aquifers, flow in heterogeneous porous media, viscous fingering, evolution of capillary fountains, and propagation of nonlinear surface waves are all detailed. Integral equation theory for slow viscous flows governed by the Stokes equation is considered in detail, and the complete double layer potential boundary integral equation method is addressed. This is a class of viscous flow problems where the BEM excels as explicit fundamental solutions exist and are readily available to formulate the required boundary integral equation that is the theoretical foundation of the BEM. Moreover, the problem of suspended particles in Stokes flow is considered along with a review of the literature on fast solvers for such problems. BEM application to Oseen and non-Newtonian flows is briefly treated. The chapter on general viscous flows considers several alternatives to address the lack of fundamental solutions for the general Navier-Stokes equations. The alternatives range from cell integration to the dual reciprocity BEM. Velocity-pressure, velocity-vorticity, and penalty formulations are presented.

Following the series of chapters on fluid mechanics is an extensive chapter on the subject of BEM applications to inverse problems in all of the fields discussed in the prior chapters of the book. The mathematical theory of inverse problems is first reviewed, and the concept of regularization is introduced. Applications of the BEM to inverse heat transfer, acoustics, fluid mechanics, and electrochemical problems are provided.

The final chapter is dedicated to the important subject of numerical integration in 2D and 3D and to the treatment of special integrals appearing in the BEM. In particular, methods are discussed for the numerical evaluation of nearly singular, singular, and hyper-singular integrals.

The book is well organized, and written in a pleasant and approachable style. It provides an excellent up-to-date source of information on the BEM. The author is meticulous in his presentation and offers comprehensive references at the end of each chapter. The chapters are self-contained, and all rely on fundamentals of integral equations and the BEM detailed in Chapter 2. Each chapter contains numerous examples and high-quality illustrations. There are no exercises at the end of each chapter. The book has a table of contents and a subject index. Boundary Element Method, Volume 1: Applications in Thermo-Fluids and Acoustics is highly recommended as a textbook for graduate courses of the BEM, and it is also strongly recommended for acquisition by university and research center libraries as an important and modern contribution by a leading authority to the growing literature on the BEM.