5R5. Variational Methods for Structural Optimization. Applied Mathematical Sciences, Vol 140. - A Cherkaev (Dept of Math, Univ of Utah, Salt Lake City UT 84112). Springer-Verlag, New York. 2000. 545 pp. ISBN 0-387-98462-3. \$79.95.

Reviewed by D Givoli (Dept of Aerospace Eng, Technion-Israel, Haifa, 32000, Israel).

This is an excellent book on material and microstructure optimization by one of the world’s main authorities on the subject. It is a comprehensive (545 pages) treatment and state-of-the-art review of this area, which is an active field in applied mathematics. The title of the book may be slightly misleading for readers from the engineering community, since the book does not deal with the methods of classical engineering structural optimization, on the macro scale. The latter has been the subject of many books, like the collection of papers Structural Optimization: Status and Promise, edited by MP Kamat (AIAA, 1993). Cherkaev’s unique book is concerned with the microstructure and with questions such as how to distribute the different materials constituting a two-phase or a multi-phase solid in an optimal way. The main practical application of the subject is to the design of composite materials on the micro scale. (The last chapter also touches on composite optimization on the macro scale.) Perhaps the words “Composite Optimization” instead of “Structural Optimization” in the title would have been more revealing.

The book is divided into five parts, each containing up to four chapters. Part I explains the key problems and ideas in a simple one-dimensional context. It is demonstrated that in some non-convex variational problems, the minimizer suffers from fine-scale oscillations. Then the notion of relaxation via the convex envelope is introduced as a remedy (Ch 1). Conductivity problems (eg, in heat transfer) in composites are discussed, along with the concept of homogenization (Ch 2). A way to find bounds for the effective properties of the composite is shown. The notion of G-closure is defined: this is the set of all the effective properties of all possible composites that can be constructed from the given materials (Ch 3). Part II goes deeper and discusses various approaches to attack some composite optimization problems in conductivity, using the techniques introduced in Part I. First a basic problem is dealt with thoroughly (Ch 4), and then more general problems are discussed (Ch 5). Part III generalizes the concept of relaxation to the multi-dimensional case, first by introducing the notion of the quasiconvex envelope (Ch 6), and then by discussing three different methods (Ch 7–9) for finding upper and lower bounds for this envelope.

Part IV concentrates on methods for finding the G-closure. First the theory is discussed (Ch 10), then it is applied to a few examples (Ch 11), and finally it is extended to problems with more than two materials (Ch 12) and to problems involving dissipation (Ch 13). Part V, clearly the highlight of both Cherkaev’s work and the book, discusses optimization of composites in linear elasticity. The theory of inhomogeneous elasticity is outlined (Ch 14), and various optimization problems are solved in a simple case (Ch 15), and in more general cases (Ch 16), by finding bounds for the effective properties. The last chapter (Ch 17) discusses some global optimization problems of elastic composites.

In general, the book is very well written. The exposition is mathematical in nature, but by no means dry. The book includes many illustrations and examples and is certainly accessible to mathematically-oriented readers of the applied mechanics and engineering communities. Not many mathematicians can write in a style which is both rigorous and readable at the same time, and in this, the author has done an important service to researchers in applied mechanics. Cherkaev wisely avoids the treatment of abstract subjects like Young measures. The book is full of up-to-date information on the problems and methods of composite optimization, with reference to a very large number of papers on the subject. Each chapter ends with a short summary and with a few well-designed problems for the reader.

The book could have seen better proofing. This reviewer was amazed to find out that more than half of the entries in the index have wrong page numbers. (There is a shift of 2 in all the page numbers starting from around page 200.) In addition, the page numbers appearing in the Contents are wrong from the middle of Chapter 15. There are also many errors in equation numbers appearing in the text (the first three errors are on pages 5, 6, and 12). This reviewer is sure these errors will be a source of frustration to many readers. It is really a pity that such a book, which is so thorough on the professional level, is less than perfect from the publishing aspect.

Despite this unnecessary deficiency, Cherkaev’s book is indeed a very welcome contribution to the field of composite optimization. Variational Methods for Structural Optimization is highly recommended to researchers and students interested in the field, and should certainly be added to every library collection of books in applied mechanics.