AN INTRODUCTION TO THE FINITE ELEMENT METHOD Second Edition
J. N. Reddv Oscar S. Wyatt Chair in Mechanical Engineering Texas A&M University College S...
AN INTRODUCTION TO THE FINITE ELEMENT METHOD Second Edition
J. N. Reddv Oscar S. Wyatt Chair in Mechanical Engineering Texas A&M University College Station, Texas 77843
McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Singapore Sydney Tokyo Toronto
CONTENTS
Preface to the Second Edition Preface to the First Edition
Part 1 Preliminaries Introduction
3
1.1 1.2 1.3
General Comments Historical Background The Basic Concept of the Finite Element Method 1.3.1 General Comments 1.3.2 Approximation of the Circumference of a Circle 1.3.3 Approximate Determination of the Center of Mass 1.3.4 Solution of Differential Equation 1.3.5 Some Remarks 1.4 The Present Study 1.5 Summary References for Additional Reading
3 5 5 5 6 8 10 13 15 15 16
Integral Formulations and Variational Methods
18
2.1 2.2
18 20 20 22 26 27 28 28 28 33 35
2.3
Need for Weighted-Integral Forms Some Mathematical Concepts and Formulae 2.2.1 Boundary, Initial, and Eigenvalue Problems 2.2.2 Integral Relations 2.2.3 Functionals 2.2.4 The Variational Symbol Weak Formulation of Boundary Value Problems 2.3.1 Introduction 2.3.2 Weighted-Integral and Weak Formulations 2.3.3 Linear and Bilinear Forms and Quadratic Functionals 2.3.4 Examples
ix
X
CONTENTS
2.4 Variational Methods of Approximation 2.4.1 Introduction 2.4.2 The Rayleigh-Ritz Method 2.4.3 The Method of Weighted Residuais 2.5 Summary Problems References for Additional Reading
40 40 40 51 57 59 63
Part 2 Finite Element Analysis of One-Dimensional Problems 3
Second-Order Boundary Value Problems
67
3.1 Introduction 3.2 Basic Steps of Finite Element Analysis 3.2.1 Model Boundary Value Problem 3.2.2 Discretization of the Domain 3.2.3 Derivation of Element Equations 3.2.4 Connectivity of Elements 3.2.5 Imposition of Boundary Conditions 3.2.6 Solution of Equations 3.2.7 Postprocessing of the Solution 3.2.8 Radially Symmetrie Problems 3.3 Applications 3.3.1 keat Transfer 3.3.2 Fluid Mechanics 3.3.3 Solid Mechanics 3.4 Summary Problems References for Additional Reading
Introduction The Euler-Bernoulli Beam Element 4.2.1 Governing Equation 4.2.2 Discretization of the Domain 4.2.3 Derivation of Element Equations 4.2.4 Assembly of Element Equations 4.2.5 Imposition of Boundary Conditions 4.2.6 Solution 4.2.7 Postprocessing of the Solution 4.2.8 Examples 4.3 Plane Truss and Euler-Bernoulli Frame Elements 4.4 The Timoshenko Beam and Frame Elements 4.4.1 Governing Equations 4.4.2 Weak Form 4.4.3 Finite Element Model 4.5 Inclusion of Constraint Equations 4.6 Summary
CONTENTS
5
6
7
xi
Problems References for Additional Reading
192 198
Finite Element Error Analysis
199
5.1 Approximation Errors 5.2 Various Measures of Errors 5.3 Convergence of Solution 5.4 Accuracy of the Solution 5.5 Summary Problems References for Additional Reading
199 200 201 202 207 207 208
Eigenvalue and Time-Dependent Problems
209
6.1
Eigenvalue Problems 6.1.1 Introduction 6.1.2 Formulation of Eigenvalue Problems 6.1.3 Finite Element Models 6.1.4 Applications 6.2 Time-Dependent Problems 6.2.1 Introduction 6.2.2 Semidiscrete Finite Element Models 6.2.3 Time Approximations 6.2.4 Mass Lumping 6.2.5 Applications 6.3 Summary Problems References for Additional Reading
Isoparametric Formulations and Numerical Integration 7.1.1 Background 7.1.2 Natural Coordinates 7.1.3 Approximation of Geometry 7.1.4 Isoparametric Formulations 7.1.5 Numerical Integration 7.2 Computer Implementation 7.2.1 Introductory Comments 7.2.2 General Outline 7.2.3 Preprocessor 7.2.4 Calculation of Element Matrices (Processor) 7.2.5 Assembly of Element Equations (Processor) 7.2.6 Imposition of Boundary Conditions (Processor) 7.2.7 Solution of Equations and Postprocessing 7.3 Applications of the Computer Program FEM1DV2 7.3.1 General Comments 7.3.2 Illustrative Examples 7.4 Summary Problems References for Additional Reading
Part 3 Finite Element Analysis of Two-Dimensional Problems 8
9
Single-Variable Problems
295
8.1 8.2
295 297 297 298 299 301 303 311 318 322 323 324
Introduction Boundary Value Problems 8.2.1 The Model Equation 8.2.2 Finite Element Discretization 8.2.3 Weak Form 8.2.4 Finite Element Model 8.2.5 Interpolation Functions 8.2.6 Evaluation of Element Matrices and Vectors 8.2.7 Assembly of Element Equations 8.2.8 Postprocessing 8.2.9 Axisymmetric Problems 8.2.10 AnExample 8.3 Some Comments on Mesh Generation and Imposition of Boundary Conditions 8.3.1 Discretization of a Domain 8.3.2 Generation of Finite Element Data 8.3.3 Imposition of Boundary Conditions 8.4 Applications 8.4.1 Heat Transfer 8.4.2 Fluid Mechanics 8.4.3 Solid Mechanics 8.5 Eigenvalue and Time-Dependent Problems 8.5.1 Introduction 8.5.2 Parabolic Equations 8.5.3 Hyperbolic Equations 8.6 Summary Problems References for Additional Reading
Interpolation Functions, Numerical Integration, and Modeling Considerations
404
9.1
Library of Elements and Interpolation Functions 9.1.1 Introduction 9.1.2 Triangulär Elements 9.1.3 Rectangular Elements 9.1.4 The Serendipity Elements 9.2 Numerical Integration 9.2.1 Preliminary Comments 9.2.2 Coordinate Transformations 9.2.3 Integration over a Master Rectangular Element 9.2.4 Integration over a Master Triangulär Element 9.3 Modeling Considerations 9.3.1 Preliminary Comments 9.3.2 Element Geometries 9.3.3 Mesh Generation 9.3.4 Load Representation 9.4 Summary
10 Plane Elasticity 10.1 Introduction 10.2 Governing Equations 10.2.1 Assumptions of Plane Elasticity 10.2.2 Basic Equations 10.3 Weak Formulations 10.3.1 Preliminary Comments 10.3.2 Principle of Virtual Displacements in Matrix Form 10.3.3 Weak Form of the Governing Differential Equations 10.4 Finite Element Model 10.4.1 Matrix Form of the Model 10.4.2 Weak Form Model 10.4.3 Eigenvalue and Transient Problems 10.5 Evaluation of Integrals 10.6 Assembly and Boundary and Initial Conditions 10.7 Examples 10.8 Summary Problems References for Additional Reading
11 Flows of Viscous Incompressible Fluids 11.1 11.2 11.3 11.4
Preliminary Comments Governing Equations Velocity-Pressure Finite Element Model Penalty-Finite Element Model 11.4.1 Penalty Function Method 11.4.2 Formulation of the Flow Problem as a Constrained Problem . 11.4.3 Lagrange Multiplier Formulation 11.4.4 Penalty Function Formulation 11.4.5 Computational Aspects 11.5 Examples 11.6 Summary Problems References for Additional Reading
12 Bending of Elastic Plates 12.1 12.2
12.3
12.4
Introduction Classical Plate Model 12.2.1 Displacement Field 12.2.2 Virtual Work Statement 12.2.3 Finite Element Model Shear Deformable Plate Model 12.3.1 Displacement Field 12.3.2 Virtual Work Statement 12.3.3 Finite Element Model 12.3.4 Shear Locking and Reduced Integration Eigenvalue and Time-Dependent Problems
12.5 Examples 12.6 Summary Problems References for Additional Reading
522 529 529 531
C o m p u t e r Implementation 13.1 Introduction 13.2 Preprocessor 13.3 Element Computations: Processor 13.4 Applications of the Computer Program FEM2DV2 13.4.1 Introduction 13.4.2 Description of Mesh Genrators 13.4.3 Applications (Illustrative Examples) 13.5 Summary Problems References for Additional Reading
533 533 534 535 540 540 548 551 563 570 575
Part 4 Advanced Topics 14
Weighted-Residual Finite E l e m e n t Models, and Finite E l e m e n t Models of Nonlinear and Three-Dimensional Problems 14.1 Introduction 14.2 Alternative Formulations 14.2.1 Introductory Comments 14.2.2 Weighted-Residual Finite Element Models 14.2.3 Mixed Formulations 14.3 Nonlinear Problems 14.3.1 General Comments 14.3.2 Large-Deflection Bending of (Euler-Bernoulli) Beams 14.3.3 Solution Methods for Nonlinear Algebraic Equations 14.3.4 The 2-D Navier-Stokes Equations 14.4 Three-Dimensional Problems 14.5 Summary Problems References for Additional Reading
