Finite Elements, An Introduction

Abstract

REVIEWED BY T. BELYTSCHKO 1 The finite element method has been the topic of approximately 30 books and monographs that have been published over the last 15 years. Nevertheless, many instructors still have difficulties in finding a text for advanced undergraduate or beginning graduate courses which will develop a sound, fundamental understanding of the method. This book presents a significant advance in that direction for those who wish a more rigorous, mathematical development. The book consists of six chapters. The first two are devoted to one-dimensional problems, emphasizing the development of a symmetric variational formulation for second-order, two-point boundary value problems and the smoothness required in the space of approximating functions. In the third chapter, a finite element program for one-dimensional problems is described, including the FORTRAN statements. Chapters 4 and 5 repeat the same material for twodimensional problems, including shape functions for triangles and quadrilaterals and numerical quadrature. Chapter 6 presents an introduction to three-dimensional problems, fourth-order problems, and time-dependent problems. A notable feature of this book is that it develops the weak, or variational form, from the partial differential equations, rather than simply presenting the variational form as given; the latter approach bothers many of the better students who usually wonder where the variational form comes from. The concepts in this book are all developed with rigor, clarity, and conciseness. Once a student has mastered this book, he will certainly have a broader understanding of the mathematics of the finite element method than would be obtained from more conventional treatments. In using this book in my class, I found two types of response. Engineering students with a modest mathematical background found the book a little difficult as an introduction; it requires simultaneously tackling the concepts of the weak form, finite element approximations, and notation and concepts to which they are unaccustomed. On the other hand, mathematically inclined students tend to find this book delightful. In addition to its value as a text, it is also recommended to finite element specialists who wish to familiarize themselves with the more recent developments in the mathematical aspects of the method. Even recently I have

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