Me´thode des E´le´ments Finis en Me´canique des Structures

Abstract

7R3. Me´thode des E´le´ments Finis en Me´canique des Structures. Finite Element Methods in the Mechanics of Structures. (French). - T Gmur. Presses Polytech et Univ Romandes, Lausanne. 2000. 252 pp. ISBN 2-88074-461-X. Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).This textbook deals with the study of some elementary basic problems raised by Applied Mechanics. The book is intended for a large audience: students in Applied Sciences, researchers, engineers, etc. Let Ω be a smooth bounded domain in RN. The problems studied in the present work are of the following (weak) form: find a vector field u such that ∫Ω vT[Du-f]dΩ=0 ∀v,where D is a certain linear differential operator and f is a given vector field. Throughout the work, the study is limited to the case of linear elliptic partial differential equations of second order. In many cases, the exact solution of this problem (if it exists!) cannot be found explicitly. That is why several methods have been developed in Numerical Analysis in order to approximate the solutions of wide classes of differential equations. The Finite Element Method was introduced in the 50s, and it can be viewed as an extension of the classical Galerkin Method. The Finite Element Method consists in the decomposition of Ω into a finite number of subdomains, called finite elements. The solution of the initial problem is then found as an assemblage of functions with compact support. After a short introduction, the author introduces in Chapter 2 the strong and the weak mathematical formulation in the one-dimensional case. This elementary frame-work enables the author to explain carefully how numerical methods can be applied for approximating the solution. The Finite Element Method is based in this simple case by an integral approach. In the next chapter, the weak formulation of the approached problem is developed, and one-dimensional linear finite problems of second degree are considered. Chapter 4 is devoted to the presentation of the weak formulation to linear bidimensional problems of the second degree. The study is developed on the model of the heat-transfer equation. In this case, the approach is based on planar finite elements. The author develops the mathematical treatment of the problem, which includes convergence and numerical integration. A general situation is treated in Chapter 5, which includes applications in Linear Elasticity for two or three dimensions. The last chapter is devoted to some applications of the Finite Element Method. Here, the numerical solutions are confronted to experimental results. In this chapter, but also throughout the book, the exposition simplifies many results that have so far only been accessible in several journal articles. In short, there is an enormous wealth of content provided by the author, much of it cannot be found in any single source. Certainly, many techniques related to the Finite Element Method are discussed, so the title is unquestionably appropriate. The book and its extensive bibliography (100 titles strictly related to the subject) should serve as a tremendous reference for all researchers in the field and certainly belongs on the bookshelf next to other references on the Finite Element Method. One of the reasons why the author is able to cover such vast material in the book is that he tends to develop the main ideas exactly to the right degree of generality that he needs. Moreover, the treatment is always done by progressing from the special to the general case, essentially avoiding pedantic verifications. Sometimes the proof is even given through the right picture. In this case, the author has really saved the reader from a lot of unnecessary heavy notation. Moreover, the notation is not only simple, but also everywhere consistent. One basic question this reviewer would like to answer is whether this book is meant for students. My feeling is that, first of all, it is very nice for students to see so many concrete examples and pictures. The reader here is well motivated by simple, but illuminating examples before being faced with general notions. This is due quite systematically in the book, where examples and pictures always precede the proofs and allow those to be easily stated and easily understood. This reviewer concludes that Me´thode des E´le´ments Finis en Me´canique des Structures is an attractive book, full of concrete information, which gives a clear and lucid view of the current knowledge on the resolution of differential equations with the Finite Element Method.