# The Finite Element Method

## 1st Edition

### Fundamentals and Applications

**Authors:**Douglas H. Norrie Gerard de Vries

**eBook ISBN:**9781483218915

**Imprint:**Academic Press

**Published Date:**28th January 1973

**Page Count:**336

## Description

The Finite Element Method: Fundamentals and Applications demonstrates the generality of the finite element method by providing a unified treatment of fundamentals and a broad coverage of applications. Topics covered include field problems and their approximate solutions; the variational method based on the Hilbert space; and the Ritz finite element method. Finite element applications in solid and structural mechanics are also discussed.

Comprised of 16 chapters, this book begins with an introduction to the formulation and classification of physical problems, followed by a review of field or continuum problems and their approximate solutions by the method of trial functions. It is shown that the finite element method is a subclass of the method of trial functions and that a finite element formulation can, in principle, be developed for most trial function procedures. Variational and residual trial function methods are considered in some detail and their convergence is examined. After discussing the calculus of variations, both in classical and Hilbert space form, the fundamentals of the finite element method are analyzed. The variational approach is illustrated by outlining the Ritz finite element method. The application of the finite element method to solid and structural mechanics is also considered.

This monograph will appeal to undergraduate and graduate students, engineers, scientists, and applied mathematicians.

## Table of Contents

Preface

Acknowledgments

Chapter 1 The Formulation of Physical Problems

1.1 Introduction

1.2 Classification of Physical Problems

1.3 Classification of the Equations of a System

References

Chapter 2 Field Problems and Their Approximate Solutions

2.1 Formulation of Field (Continuous) Problems

2.2 Classification of Field Problems

2.3 Equilibrium Field Problems

2.4 Eigenvalue Field Problems

2.5 Propagation Field Problems

2.6 Summary of Governing Equations

2.7 Approximate Solution of Field Problems

2.8 Trial Function Methods in Equilibrium Problems

2.9 Trial Function Methods in Eigenvalue Problems

2.10 Trial Function Methods in Propagation Problems

2.11 Accuracy, Stability, and Convergence

2.12 Approximate Solutions for Nonlinear Problems

2.13 The Extension to Vector Problems

2.14 The Finite Element Method

References

Chapter 3 The Variational Calculus and Its Application

3.1 Maxima and Minima of Functions

3.2 The Lagrange Multipliers

3.3 Maxima and Minima of Functionals

3.4 Variational Principles in Physical Phenomena

References

Chapter 4 The Variational Method Based on the Hilbert Space

4.1 The Hilbert Function Space

4.2 Equilibrium and Eigenvalue Problems

4.3 The Variational Solution of the Equilibrium Problem

4.4 Inhomogeneous Boundary Conditions

4.5 Natural Boundary Conditions

4.6 The Variational Solution of the Eigenvalue Problem

References

Chapter 5 Fundamentals of the Finite Element Approach

5.1 Classification of Finite Element Methods

5.2 The Finite Element Approximation

5.3 Elements and Their Shape Functions

5.4 Variational Finite Element Methods

5.5 Residual Finite Element Methods

5.6 The Direct Finite Element Method

5.7 Significant Features of a Finite Element Method

5.8 The Coefficient Finite Element Method

5.9 The Cell Finite Element Method

5.10 Convergence in the Finite Element Method

References

Chapter 6 The Ritz Finite Element Method (Classical)

6.1 Statement of the Problem

6.2 The Equivalent Variational Problem

6.3 The Subdivision of the Region

6.4 The Element Shape Function

6.5 The Subdivision of the Functional

6.6 The Minimization Condition

6.7 The Element Matrix Equation

6.8 The System Matrix Equation

6.9 Insertion of the Dirichlet Boundary Condition

6.10 The Finite Element Approximation

6.11 The Two-Dimensional Region

6.12 Structural Formulations of the Finite Element Method

References

Chapter 7 The Ritz Finite Element Method (Hilbert Space)

7.1 The Ritz Finite Element Method for the Equilibrium Problem

7.2 Rayleigh-Ritz Finite Element Solution for the Eigenvalue Problem

References

Chapter 8 Finite Element Applications in Solid and Structural Mechanics

8.1 The Solid Mechanics Formulation of the Finite Element Method

8.2 The Structural Formulation of the Finite Element Method

References

Chapter 9 The Laplace or Potential Field

9.1 The Laplace Equation

9.2 The Variational Formulation for the Laplace Field

9.3 The Ritz Finite Element Solution of the Laplace Field

9.4 Summary

References

Chapter 10 Laplace and Associated Boundary-Value Problems

10.1 The Potential Flow Field

10.2 The Electrostatic Field

10.3 The Thermal Conduction Field

10.4 Porous Media Flows

10.5 The Quasi-Harmonic Equation

10.6 The Poisson Equation

10.7 Unsteady Potential Fields (Moving Boundaries)

10.8 Lifting Bodies with Appreciable Boundary Displacement

References

Chapter 11 The Helmholtz and Wave Equations

11.1 Physical Phenomena and the Helmholtz Equation

11.2 Physical Phenomena and the Wave Equation

References

Chapter 12 The Diffusion Equation

12.1 Forms of the Diffusion Equation

12.2 The Finite Element Solution of the Diffusion Equation

References

Chapter 13 Finite Element Applications to Viscous Flow

13.1 Oden and Somogyi

13.2 Tong

13.3 Baker

13.4 Leonard

13.5 Atkinson et al.

13.6 Reddi

13.7 Argyris and Scharpf

13.8 Other Formulations

References

Chapter 14 Finite Element Applications to Compressible Flow

14.1 Leonard

14.2 Gelder

14.3 De Vries, Berard, and Norrie

14.4 Reddi and Chu

14.5 Other Formulations

References

Chapter 15 Finite Element Applications to More General Fluid Flows

15.1 Skiba

15.2 Oden—I

15.3 Oden—II

15.4 Oden—III

15.5 De Vries and Norrie

15.6 Baker

15.7 Bramlette and Mallett

15.8 Other Formulations

References

Chapter 16 Other Finite Element Applications

16.1 Solid-Fluid Coupled Vibrations

16.2 Further Finite Element Applications

16.3 Further Developments

References

Appendix A Matrix Algebra

A.1 Matrix Definitions

A.2 Matrix Algebra

A.3 Quadratic and Linear Forms

References

Appendix B The Differential and Integral Calculus of Matrices

B.1 Definition of Differentiation and Integration of Matrices

B.2 Differentiation of a Function of a Matrix with Respect to the Matrix

B.3 Partial Differentiation of Matrices

B.4 Differentiation of Functions of Several Variables

References

Appendix C The Transformation Matrix

C.1 The Functional Relationship Between Coordinate Systems

C.2 The Local Transformation Matrix

C.3 The Linear Transformation

C.4 The Translation Matrix

C.5 The Rotation Matrix

C.6 Successive Transformations

C.7 Transformation of Matrices

C.8 Principal Axes, Diagonalization, and Eigenvalues

References

Additional References

A. Mathematical Methods

B. Variational Principles and Formulations

C. Finite Element Analysis

D. Finite Element Solutions of Physical Problems

E. Finite Element Computational Procedures

F. Input/Output Data Procedures

G. Large-Scale Finite Element Systems

Author Index

Subject Index

## Details

- No. of pages:
- 336

- Language:
- English

- Copyright:
- © Academic Press 1973

- Published:
- 28th January 1973

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483218915