The Finite Element Method for Boundary Value Problems Mathematics and Computations Applied and Computational Mechanics Series

Written by two well-respected experts in the field, The Finite Element Method for Boundary Value Problems: Mathematics and Computations bridges the gap between applied mathematics and application-oriented computational studies using FEM. Mathematically rigorous, the FEM is presented as a method of approximation for differential operators that are mathematically classified as self-adjoint, non-self-adjoint, and non-linear, thus addressing totality of all BVPs in various areas of engineering, applied mathematics, and physical sciences. These classes of operators are utilized in various methods of approximation: Galerkin method, Petrov-Galerkin Method, weighted residual method, Galerkin method with weak form, least squares method based on residual functional, etc. to establish unconditionally stable finite element computational processes using calculus of variations. Readers are able to grasp the mathematical foundation of finite element method as well as its versatility of applications. h-, p-, and k-versions of finite element method, hierarchical approximations, convergence, error estimation, error computation, and adaptivity are additional significant aspects of this book.

1 Introduction

1. General Comments and Basic Philosophy
2. Basic Concepts of the Finite Element Method
3. Summary

1. Concepts from Functional Analysis

1. General Comments
2. Sets, Spaces, Functions, Functions Spaces, and Operators
3. Elements of Calculus of Variations
4. Examples of Differential Operators and their Properties
5. 2.5 Summary
2. Classical Methods of Approximation

1. Introduction
2. Basic Steps in Classical Methods of Approximation based on Integral Forms
3. Integral forms using the Fundamental Lemma of the Calculus of Variations
4. Approximation Spaces for Various Methods of Approximation
5. Integral Formulations of BVPs using the Classical Methods of Approximations
6. Numerical Examples
7. Summary
3. The Finite Element Method

1. Introduction
2. Basic steps in the finite element method
3. Summary
4. Self-Adjoint Differential Operators

1. Introduction
2. One-dimensional BVPs in a single dependent variable

5.3 Two-dimensional boundary value problems

5.4 Three-dimensional boundary value problems

5.5 Summary

6 Non-Self-Adjoint Differential Operators

6.1 Introduction

6.2 1D convection-diffusion equation

6.3 2D convection-diffusion equation

6.4 Summary

7 Non-Linear Differential Operators

7.1 Introduction

7.2 One dimensional Burgers equation

7.3 Fully developed ow of Giesekus Fluid between parallel plates (polymer flow)

7.4 2D steady-state Navier-Stokes equations

7.5 2D compressible Newtonian fluid Flow

7.6 Summary

8 Basic Elements of Mapping and Interpolation Theory

8.1 Mapping in one dimension

8.2 Elements of interpolation theory over

8.4 Local approximation over : quadrilateral elements

8.5 2D p-version local approximations

8.6 2D approximations for quadrilateral elements

8.10 Serendipity family of interpolations

8.11 Interpolation functions for 3D elements

8.12 Summary

9 Linear Elasticity using the Principle of Minimum Total Potential Energy

9.1 Introduction

9.2 New notation

9.3 Approach

9.4 Element equations

9.5 Finite element formulation for 2D linear elasticity

9.6 Summary

10 Linear and Nonlinear Solid Mechanics using the Principle of Virtual Displacements

10.1 Introduction

10.2 Principle of virtual displacements

10.3 Virtual work statements

10.4 Solution method

10.5 Finite element formulation for 2D solid continua

10.6 Finite element formulation for 3D solid continua

10.7 Axisymmetric solid finite elements

10.8 Summary

11 Additional Topics in Linear Structural Mechanics

11.1 Introduction

11.2 1D axial spar or rod element in R1 (1D space)

11.3 1D axial spar or rod element in R2

11.4 1D axial spar or rod element in R3 (3D space)

11.5 The Euler-Bernoulli beam element

11.6 Euler-Bernoulli frame elements in R2

11.7 The Timoshenko beam elements

11.8 Finite element formulations in R2 and R3

11.9 Summary

12 Convergence, Error Estimation, and Adaptivity

12.1 Introduction

12.2 h-, p-, k-versions of FEM and their convergence

12.3 Convergence and convergence rate

12.4 Error estimation and error computation

12.5 A priori error estimation

12.6 Model problems

12.7 A posteriori error estimation and computation

12.8 Adaptive processes in finite element computations

12.9 Summary

Appendix A: Numerical Integration using Gauss Quadrature

A.1 Gauss quadrature in R1, R2 and R3

A.2 Gauss quadrature over triangular domains

Karan S. Surana attended undergraduate school at Birla Institute of Technology and Science (BITS), Pilani, India and received a B.E. in mechanical engineering in 1965. He then attended the University of Wisconsin, Madison where he obtained M.S. and Ph.D. in mechanical engineering in 1967 and 1970. He joined The University of Kansas, Department of Mechanical Engineering faculty where he is currently serving as Deane E. Ackers University Distinguished Professor of Mechanical Engineering. His areas of interest and expertise are computational mathematics, computational mechanics, and continuum mechanics. He is the author of over 350 research reports, conference papers, and journal papers.

J. N. Reddy is a Distinguished Professor, Regents’ Professor, and inaugural holder of the Oscar S. Wyatt Endowed Chair in Mechanical Engineering at Texas A&M University, College Station, Texas. Dr. Reddy earned a Ph.D. in Engineering Mechanics in 1974 from University of Alabama in Huntsville. He worked as a Post-Doctoral Fellow in Texas Institute for Computational Mechanics (TICOM) at the University of Texas at Austin, Research Scientist for Lockheed Missiles and Space Company, Huntsville, during l974-75, and taught at the University of Oklahoma from 1975 to 1980, Virginia Polytechnic Institute & State University from 1980 to 1992, and at Texas A&M University from 1992. Professor Reddy also played active roles in professional societies as the President of USACM, founding member of the General Council of IACM, Secretary of Fellows of AAM, member of the Board of Governors of SES, Chair of the Engineering Mechanics Executive Committee, among several others.