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Introduction to the Finite Element Method, Third Edition
1 online resource (766 pages) : illustrations.
J.N. Reddy's, An Introduction to the Finite Element Method, third edition is an update of one of the most popular FEM textbooks available. The book retains its strong conceptual approach, clearly examining the mathematical underpinnings of FEM, and providing a general approach of engineering application areas. Known for its detailed, carefully selected example problems and extensive selection of homework problems, the author has comprehensively covered a wide range of engineering areas making the book approriate for all engineering majors, and underscores the wide range of use FEM has in the professional world. A supplementary text Web site located at http://www.mhhe.com/reddy3e contains password-protected solutions to end-of-chapter problems, general textbook information, supplementary chapters on the FEM1D and FEM2D computer programs, and more!.
Preface -- 1 Introduction -- 1.1 General Comments -- 1.2 Mathematical Models -- 1.3 Numerical Simulations -- 1.4 The Finite Element Method -- 1.4.1 The Basic Idea -- 1.4.2 The Basic Features -- 1.4.3 Some Remarks -- 1.4.4 A Brief Review of History and Recent Developments -- 1.5 The Present Study -- 1.6 Summary -- Problems -- References for Additional Reading -- 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods -- 2.1 General Introduction -- 2.1.1 Variational Principles and Methods -- 2.1.2 Variational Formulations -- 2.1.3 Need for Weighted-Integral Statements -- 2.2 Some Mathematical Concepts and Formulae -- 2.2.1 Coordinate Systems and the Del Operator -- 2.2.2 Boundary Value, Initial Value, and Eigenvalue Problems -- 2.2.3 Integral Identities -- 2.2.4 Linear and Bilinear Functionals -- 2.3 Elements of Calculus of Variations -- 2.3.1 Introduction -- 2.3.2 Variational Operator and First Variation -- 2.3.3 Fundamental Lemma of Variational Calculus -- 2.3.4 The Euler Equations -- 2.3.5 Natural and Essential Boundary Conditions -- 2.3.6 Hamilton?s Principle -- 2.4 Integral Formulations -- 2.4.1 Introduction -- 2.4.2 Weighted-Integral and Weak Formulations -- 2.4.3 Linear and Bilinear Forms and Quadratic Functionals -- 2.4.4 Examples -- 2.5 Variational Methods -- 2.5.1 Introduction -- 2.5.2 The Ritz Method -- 2.5.3 Approximation Functions -- 2.5.4 Examples -- 2.5.5 The Method of Weighted Residuals -- 2.6 Summary -- Problems -- References for Additional Reading -- 3 Second-Order Differential Equations in One Dimension: Finite Element Models -- 3.1 Background -- 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 Postcomputation of the Solution. -- 3.3 Some Remarks -- 3.4 Axisymmetric Problems -- 3.4.1 Model Equation -- 3.4.2 Weak Form -- 3.4.3 Finite Element Model -- 3.5 Summary -- Problems -- References for Additional Reading -- 4 Second-Order Differential Equations in One Dimension: Applications -- 4.1 Preliminary Comments -- 4.2 Discrete Systems -- 4.2.1 Linear Elastic Spring -- 4.2.2 Torsion of Circular Shafts -- 4.2.3 Electrical Resistor Circuits -- 4.2.4 Fluid Flow through Pipes -- 4.3 Heat Transfer -- 4.3.1 Governing Equations -- 4.3.2 Finite Element Models -- 4.3.3 Numerical Examples -- 4.4 Fluid Mechanics -- 4.4.1 Governing Equations -- 4.4.2 Finite Element Model -- 4.5 Solid and Structural Mechanics -- 4.5.1 Preliminary Comments -- 4.5.2 Finite Element Model of Bars and Cables -- 4.5.3 Numerical Examples -- 4.6 Plane Trusses -- 4.6.1 Introduction -- 4.6.2 Basic Truss Element -- 4.6.3 General Truss Element -- 4.6.4 Constraint Equations: Penalty Approach -- 4.6.5 Constraint Equations: A Direct Approach -- 4.7 Summary -- Problems -- References for Additional Reading -- 5 Beams and Frames -- 5.1 Introduction -- 5.2 Euler?Bernoulli Beam Element -- 5.2.1 Governing Equation -- 5.2.2 Discretization of the Domain -- 5.2.3 Derivation of Element Equations -- 5.2.4 Assembly of Element Equations -- 5.2.5 Imposition of Boundary Conditions -- 5.2.6 Postprocessing of the Solution -- 5.2.7 Numerical Examples -- 5.3 Timoshenko Beam Elements -- 5.3.1 Governing Equations -- 5.3.2 Weak Form -- 5.3.3 General Finite Element Model -- 5.3.4 Consistent Interpolation Elements -- 5.3.5 Reduced Integration Element -- 5.3.6 Numerical Examples -- 5.4 Plane Frame Elements -- 5.4.1 Introductory Comments -- 5.4.2 Frame Element -- 5.5 Summary -- Problems -- References for Additional Reading -- 6 Eigenvalue and Time-Dependent Problems -- 6.1 Eigenvalue Problems -- 6.1.1 Introduction.
6.1.2 Formulation of Eigenvalue Problems -- 6.1.3 Finite Element Formulation -- 6.2 Time-Dependent Problems -- 6.2.1 Introduction -- 6.2.2 Semidiscrete Finite Element Models -- 6.2.3 Parabolic Equations -- 6.2.4 Hyperbolic Equations -- 6.2.5 Mass Lumping -- 6.2.6 Applications -- 6.3 Summary -- Problems -- References for Additional Reading -- 7 Computer Implementation -- 7.1 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 Solving Equations and Postprocessing -- 7.3 Applications of Program FEM1D -- 7.3.1 General Comments -- 7.3.2 Illustrative Examples -- 7.4 Summary -- Problems -- References for Additional Reading -- 8 Single-Variable Problems in Two Dimensions -- 8.1 Introduction -- 8.2 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 Derivation of Interpolation Functions -- 8.2.6 Evaluation of Element Matrices and Vectors -- 8.2.7 Assembly of Element Equations -- 8.2.8 Postcomputations -- 8.2.9 Axisymmetric Problems -- 8.3 A Numerical Example -- 8.4 Some Comments on Mesh Generation and Imposition of Boundary Conditions -- 8.4.1 Discretization of a Domain -- 8.4.2 Generation of Finite Element Data -- 8.4.3 Imposition of Boundary Conditions -- 8.5 Applications -- 8.5.1 Conduction and Convection Heat Transfer -- 8.5.2 Fluid Mechanics -- 8.5.3 Solid Mechanics -- 8.6 Eigenvalue and Time-Dependent Problems -- 8.6.1 Introduction -- 8.6.2 Parabolic Equations -- 8.6.3 Hyperbolic Equations -- 8.7 Summary -- Problems -- References for Additional Reading -- 9 Interpolation Functions, Numerical Integration, and Modeling Considerations -- 9.1 Introduction -- 9.2 Element Library -- 9.2.1 Triangular Elements -- 9.2.2 Rectangular Elements -- 9.2.3 The Serendipity Elements -- 9.2.4 Hermite Cubic Interpolation Functions -- 9.3 Numerical Integration -- 9.3.1 Preliminary Comments -- 9.3.2 Coordinate Transformations -- 9.3.3 Integration over a Master Rectangular Element -- 9.3.4 Integration over a Master Triangular Element -- 9.4 Modeling Considerations -- 9.4.1 Preliminary Comments -- 9.4.2 Element Geometries -- 9.4.3 Mesh Generation -- 9.4.4 Load Representation -- 9.5 Summary -- Problems -- References for Additional Reading -- 10 Flows of Viscous Incompressible Fluids -- 10.1 Preliminary Comments -- 10.2 Governing Equations -- 10.3 Velocity-Pressure Formulation -- 10.3.1 Weak Formulation -- 10.3.2 Finite Element Model -- 10.4 Penalty Function Formulation -- 10.4.1 Preliminary Comments -- 10.4.2 Formulation of the Flow Problem as a Constrained Problem -- 10.4.3 Lagrange Multiplier Model -- 10.4.4 Penalty Model -- 10.4.5 Time Approximation -- 10.5 Computational Aspects -- 10.5.1 Properties of the Matrix Equations -- 10.5.2 Choice of Elements -- 10.5.3 Evaluation of Element Matrices in the Penalty Model -- 10.5.4 Postcomputation of Stresses -- 10.6 Numerical Examples -- 10.7 Summary -- Problems -- References for Additional Reading -- 11 Plane Elasticity -- 11.1 Introduction -- 11.2 Governing Equations -- 11.2.1 Plane Strain -- 11.2.2 Plane Stress -- 11.2.3 Summary of Equations -- 11.3 Weak Formulations -- 11.3.1 Preliminary Comments -- 11.3.2 Principle of Virtual Displacements in Vector Form.
11.3.3 Weak Form of the Governing Differential Equations -- 11.4 Finite Element Model -- 11.4.1 General Model -- 11.4.2 Eigenvalue and Transient Problems -- 11.5 Evaluation of Integrals -- 11.6 Assembly of Finite Element Equations -- 11.7 Examples -- 11.8 Summary -- Problems -- References for Additional Reading -- 12 Bending of Elastic Plates -- 12.1 Introduction -- 12.2 Classical Plate Theory -- 12.2.1 Displacement Field -- 12.2.2 Virtual Work Statement -- 12.2.3 Finite Element Model -- 12.2.4 Plate Bending Elements -- 12.3 Shear Deformation Plate Theory -- 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 -- 12.4 Eigenvalue and Time-Dependent Problems -- 12.5 Examples -- 12.6 Summary -- Problems -- References for Additional Reading -- 13 Computer Implementation of Two-Dimensional Problems -- 13.1 Introduction -- 13.2 Preprocessor -- 13.3 Element Computations (Processor) -- 13.4 Applications of the Computer Program FEM2D -- 13.4.1 Introduction -- 13.4.2 Description of Mesh Generators -- 13.4.3 Applications (Illustrative Examples) -- 13.5 Summary -- Problems -- References for Additional Reading -- 14 Prelude to Advanced Topics -- 14.1 Introduction -- 14.2 Alternative Finite Element Models -- 14.2.1 Introductory Comments -- 14.2.2 Weighted Residual Finite Element Models -- 14.2.3 Mixed Formulations -- 14.3 Three-Dimensional Problems -- 14.3.1 Heat Transfer -- 14.3.2 Flows of Viscous Incompressible Fluids -- 14.3.3 Elasticity -- 14.3.4 Three-Dimensional Finite Elements -- 14.3.5 A Numerical Example -- 14.4 Nonlinear Problems -- 14.4.1 General Comments -- 14.4.2 Bending of Euler?Bernoulli Beams -- 14.4.3 The Navier?Stokes Equations in Two Dimensions -- 14.4.4 Solution Methods for Nonlinear Algebraic Equations -- 14.4.5 Numerical Examples -- 14.5 Errors in Finite Element Analysis -- 14.5.1 Types of Errors -- 14.5.2 Measures of Errors -- 14.5.3 Convergence and Accuracy of Solutions -- 14.6 Summary -- Problems -- References for Additional Reading -- Index.
Electronic reproduction. New York, N.Y. : McGraw Hill, 2006. Mode of access: World Wide Web. System requirements: Web browser. Access may be restricted to users at subscribing institutions.
Mode of access: Internet via World Wide Web.
Includes bibliographical references and index.