The Finite Element Method

This textbook introduces the widely used numerical technique FEM in various engineering disciplines for the analysis of structures, heat transfer, fluid dynamics, and other physical phenomena.
Appropriate for interested senior undergraduate engineering students and beginner graduate students in a one-semester introductory course, this book provides a clear understanding of the main issues in FEM.
Looking at the FEM as a variational approximation method that uses localized piecewise polynomial basis functions for the solution of boundary-value problems (BVP) and initial boundary-value problems (IBVP), the book uses examples to apply this technique to various problems of physical interest, e.g., elasticity, heat conduction, advection-diffusion, etc. One-dimensional (1D) problems are presented first to make it easier to grasp the fundamental concepts associated with the formulation and application of the FEM; then, the methodology is extended to more challenging 2D and 3D problems that involve somewhat greater mathematical complexity. For simplicity, the book deals with problems that are specified in terms of a single set of state variables, such as displacements or temperature. Finally, due to the introductory nature of this text, only linear problems are considered.

-Jacobo Bielak is Hamerschlag University Professor Emeritus in the Department of Civil and Environmental Engineering at Carnegie Mellon University. His research interests include earthquake engineering, engineering seismology, computational seismology, computational mechanics, and structural health monitoring. Bielak was a member of the original Applied Technology Council (ATC) committee that drafted the first seismic provisions for soil-structure interaction in the US, based mainly on his work. These provisions are now, in modified form, part of the National Earthquake Hazards Reduction Program (NEHRP) seismic provisions.He has received the Gordon Bell Prize for Special Accomplishment Based on Innovation. He is a Member of the National Academy of Engineering, a Distinguished Member of the American Society of Civil Engineers, an Honorary Member of the Earthquake Engineering Research Institute, and a Fellow of the U.S. Association for Computational Mechanics. He isalso a Corresponding Member of the Mexican Academy of Sciences and of the Mexican Academy of Engineering. Bielak received his Civil Engineer's degree from the National University of Mexico (UNAM), MS from Rice University, and PhD from Caltech.

Part I Fundamental Concepts, Implementation, and Applications.- Chapter 1 Prototype One-Dimensional Boundary-Value Problem (BVP) - Governing Differential Equation and Variational Formulations with Examples.- Chapter 2 Finite Elements for the Prototype One-Dimensional BVP.- Chapter 3 Prototype Two-Dimensional (2D) Problem - Variational Formulations.-Chapter 4 Finite Elements for the Model Two-Dimensional BVP.- Part II Extensions.- Chapter 5 Problems With Singularities.- Chapter 6 Three-Dimensional Boundary-Value Problems.- Chapter 7 Initial Boundary-Value Problems.- Chapter 8 Spectral Elements.- Chapter 9 One-Dimensional Fourth-Order Boundary Value Problems.