Finite Elements: An Introduction by Becker, Carey and Oden
Finite elements are a powerful mathematical technique for solving complex problems in engineering and science. They are widely used in fields such as structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Finite elements can handle problems with complicated geometries, material properties, boundary conditions, and loading scenarios.
In this book, Becker, Carey and Oden provide a comprehensive introduction to the finite element method (FEM) for beginners and advanced users alike. The book covers the basic theory and applications of FEM in one-dimensional, two-dimensional, and three-dimensional problems. The book also presents several numerical examples and exercises to illustrate the concepts and techniques of FEM.
The book is divided into two parts. Part I covers the fundamentals of FEM, including the variational formulation, the element interpolation functions, the assembly procedure, the solution methods, and the error analysis. Part II covers the applications of FEM in various types of problems, such as heat conduction, elasticity, fluid mechanics, diffusion-reaction, and electromagnetics.
The book is written in a clear and concise style, with an emphasis on physical understanding and practical implementation. The book is suitable for undergraduate and graduate students of engineering and applied mathematics, as well as researchers and practitioners who want to learn more about FEM.
One of the main features of the book is that it introduces the concept of isoparametric elements, which are elements that have the same shape and interpolation functions in both the physical and the parametric domains. Isoparametric elements allow for a great flexibility and simplicity in the formulation and implementation of FEM. The book shows how to use isoparametric elements for different types of problems and geometries.
Another feature of the book is that it discusses the use of FEM for nonlinear problems, which are problems that involve nonlinear material behavior, large deformations, or nonlinear boundary conditions. Nonlinear problems are more challenging and realistic than linear problems, but they require special techniques and considerations to solve. The book explains how to deal with nonlinear problems using FEM, such as the incremental-iterative method, the Newton-Raphson method, and the arc-length method.
The book also provides a comprehensive overview of the software aspects of FEM, such as the data structures, the input/output formats, the graphical user interfaces, and the debugging tools. The book includes several computer programs and subroutines that illustrate how to implement FEM in a general-purpose programming language such as Fortran or C. The book also gives some examples of commercial and open-source FEM software packages that are available for different applications.
In addition to the theoretical and practical aspects of FEM, the book also discusses some of the historical and philosophical aspects of FEM. The book traces the origins and development of FEM from the early works of Euler, Schellbach, Courant, and others, to the modern advances and applications of FEM in various fields. The book also reflects on the role and importance of FEM in the scientific and engineering community, as well as its limitations and challenges. a474f39169