An Introduction to the Finite Element Method

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1 An Introduction to the Finite Element Method Third Edition J. N. REDDY Department 01 Mechanical Engineering Texas A&M University College Station, Texas, USA Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco SI. Louis Bangkok Bogote. Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

2 CONTENTS Preface xiv 1 Introduction 1.1 General Comments 1.2 Mathematical Models 1.3 Numerical Simulations 1.4 The Finite Element Method The Basic Idea The Basic Features Some Remarks A Brief Review of History and Recent Developments 1.5 The Present Study 1.6 Summary 1 I Mathematical Preliminaries, Integral Formulations, and Variational Methods 2.1 General Introduction Variational Principles and Methods Variational Formulations Need for Weighted-Integral Statements 2.2 Some Mathematical Concepts and Formulae Coordinate Systems and the Dei Operator Boundary Value, Initial Value, and Eigenvalue Integral Identities Linear and Bilinear Functionals 2.3 Elements of Calculus of Variations Introduction Variational Operator and First Variation Fundamental Lemma of Variational Calculus The Euler Equations Natural and Essential Boundary Conditions Hamilton's Principle vii

3 viii CONTENTS 2.4 Integral Formulations Introduction Weighted-Integral and Weak Formulations Linear and Bilinear Forms and Quadratic Functionals Examples 2.5 Variational Methods Introduction The Ritz Method Approximation Functions Examples The Method of Weighted Residuals 2.6 Summary Second-Order Differential Equations in One Dimension: Finite Element Models 3.1 Background 3.2 Basic Steps of Finite Element Analysis Model Boundary Value Problem Discretization of the Domain Derivation of Element Equations Connectivity of Elements Imposition of Boundary Conditions Solution of Equations Postcomputation of the Solution 3.3 Some Remarks 3.4 Axisymmetric Model Equation Weak Form Finite Element Model 3.5 Summary Second-Order Differential Equations in One Dimension: Applications 4.1 Preliminary Comments 4.2 Discrete Systems Linear Elastic Spring Torsion of Circular Shafts Electrical Resistor Circuits Fluid Flow through Pipes 4.3 Heat Transfer Governing Equations Finite Element Models Numerical Examples

4 CONTENTS ix 4.4 Fluid Mechanics Governing Equations [ Finite Element Model Solid and Structural Mechanics Preliminary Comments Finite Element Model of Bars and Cables Numerical Examples Plane Trosses Introduction Basic Tross Element General Tross Element Constraint Equations: Penalty Approach Constraint Equations: A Direct Approach Summary S References far Additional Reading Beams and Frames Introduction Euler-Bernoulli Beam Element Governing Equation Discretization of the Domain Derivation of Element Equations Assembly of Element Equations Imposition of Boundary Conditions Postprocessing of the Solution Numerical Examples Timoshenko Beam Elements Governing Equations WeakForm General Finite Element Model Consistent Interpolation Elements Reduced Integration Element Numerical Examples 271 S.4 Plane Frame Elements Introductory Comments Frame Element Summary Eigenvalue and Time-Dependent Eigenvalue Introduction Farmulation of Eigenvalue Finite Element Farmulation Time-Dependent Introduction Semidiscrete Finite Element Models 316

5 X CONTENTS Parabolic Equations Hyperbolic Equations Mass Lumping Applications 6.3 Summary Computer Implementation 7.1 Numerical Integration Background Natural Coordinates Approximation of Geometry Isoparametric Formulations Numerical Integration 7.2 Computer Implementation Introductory Comments General Outline Preprocessor Calculation of Element Matrices (Processor) Assembly of Element Equations (Processor) Imposition of Boundary Conditions (Processor) Solving Equations and Postprocessing 7.3 Applications of Program FEMID General Comments Illustrative Examples 7.4 Summary Single-Variable in Two Dimensions Introduction Boundary Value The Model Equation Finite Element Discretization Weak Form Finite Element Model Derivation of Interpolation Functions Evaluation of Element Matrices and Yectors Assembly of Element Equations Postcomputations Axisymmetric A Numerical Example Some Comments on Mesh Generation and Imposition of Boundary Conditions Discretization of a Domain Generation of Finite Element Data Imposition of Boundary Conditions 456

6 CONTENTS xi 8.5 Applications Conduction and Convection Heat Transfer Fluid Mechanics Solid Mechanics Eigenvalue and Time-Dependent Introduction Parabolic Equations Hyperbolic Equations Summary References far Additional Reading Interpolation Functions, Numericallntegration, and Modeling Considerations 9.1 Introduction 9.2 Element Library Triangular Elements Rectangular Elements The Serendipity Elements Hermite Cubic Interpolation Functions 9.3 Numerical Integration Preliminary Comments Coardinate Transformations Integration over a Master Rectangular Element Integration over a Master Triangular Element 9.4 Modeling Considerations Preliminary Comments Element Geometries Mesh Generation Load Representation 9.5 Summary References far Additional Reading Flows of Viscous Incompressible Fluids 10.1 Preliminary Comments 10.2 Governing Equations 10.3 Ye1ocity-Pressure Farmulation Weak Farmulation Finite Element Model 10.4 Penalty Function Farmulation Preliminary Comments ]0.4.2 Farmulation of the Flow Problem as a Constrained Problem Lagrange Multiplier Model Penalty Model Time Approximation

7 xii CONTENTS 10.5 Computational Aspects Properties ofthe Matrix Equations Choice of Elements Evaluation of Element Matrices in the Penalty Model Postcomputation of Stresses Numerical Examples Summary Plane Elasticity Introduction Governing Equations Plane Strain Plane Stress Summary of Equations Weak Formulations Preliminary Comments Principle of Virtua1 Displacements in Vector Form Weak Form of the Governing Differential Equations Finite Element Model General Model Eigenva1ue and Transient Evaluation of Integrals Assembly of Finite Element Equations Examp1es Summary Bending of Elastic Plates Introduction Classical PIate Theory Displacement Field Virtual Work Statement Finite Element Model Plate Bending Elements Shear Deformation Plate Theory Displacement Fie1d Virtua1 Work Statement Finite Element Model Shear Locking and Reduced Integration 652 J2.4 Eigenvalue and Time-Dependent Examples Summary

8 CONTENTS xiii 13 Computer Implementation of Two-Dimensional 13.1 Introduction 13.2 Preprocessor 13.3 Element Computations (Processor) 13.4 Applications of the Computer Program FEM2D Introduction Description of Mesh Generators Applications (Illustrative Examples) 13.5 Summary 14 Prelude to Advanced Topics 14.1 Introduction 14.2 Alternative Finite Element Models Introductory Comments Weighted Residual Finite Element Models Mixed Formulations 14.3 Three-Dimensional Heat Transfer Flows of Viscous Incompressible Fluids Elasticity Three-Dimensional Finite Elements A Numerical Example 14.4 Nonlinear General Comments Bending of Euler-Bernoulli Beams The Navier-Stokes Equations in Two Dimensions Solution Methods for Nonlinear Algebraic Equations Numerical Examples 14.5 Errors in Finite Element Analysis Types of Errors Measures of Errors Convergence and Accuracy of Solutions 14.6 Summary Index

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