An introductory textbook covering the fundamentals of linear finite element analysis (FEA) This book constitutes the first volume in a two-volume set that introduces readers to the theoretical foundations and the implementation of the finite element method (FEM). The first volume focuses on the use of the method for linear problems. A general procedure is presented for the finite element analysis (FEA) of a physical problem, where the goal is to specify the values of a field function. First, the strong form of the problem (governing differential equations and boundary conditions) is formulated. Subsequently, a weak form of the governing equations is established. Finally, a finite element approximation is introduced, transforming the weak form into a system of equations where the only unknowns are nodal values of the field function. The procedure is applied to one-dimensional elasticity and heat conduction, multi-dimensional steady-state scalar field problems (heat conduction, chemical diffusion, flow in porous media), multi-dimensional elasticity and structural mechanics (beams/shells), as well as time-dependent (dynamic) scalar field problems, elastodynamics and structural dynamics. Important concepts for finite element computations, such as isoparametric elements for multi-dimensional analysis and Gaussian quadrature for numerical evaluation of integrals, are presented and explained. Practical aspects of FEA and advanced topics, such as reduced integration procedures, mixed finite elements and verification and validation of the FEM are also discussed. * Provides detailed derivations of finite element equations for a variety of problems. * Incorporates quantitative examples on one-dimensional and multi-dimensional FEA. * Provides an overview of multi-dimensional linear elasticity (definition of stress and strain tensors, coordinate transformation rules, stress-strain relation and material symmetry) before presenting the pertinent FEA procedures. * Discusses practical and advanced aspects of FEA, such as treatment of constraints, locking, reduced integration, hourglass control, and multi-field (mixed) formulations. * Includes chapters on transient (step-by-step) solution schemes for time-dependent scalar field problems and elastodynamics/structural dynamics. * Contains a chapter dedicated to verification and validation for the FEM and another chapter dedicated to solution of linear systems of equations and to introductory notions of parallel computing. * Includes appendices with a review of matrix algebra and overview of matrix analysis of discrete systems. * Accompanied by a website hosting an open-source finite element program for linear elasticity and heat conduction, together with a user tutorial. Fundamentals of Finite Element Analysis: Linear Finite Element Analysis is an ideal text for undergraduate and graduate students in civil, aerospace and mechanical engineering, finite element software vendors, as well as practicing engineers and anybody with an interest in linear finite element analysis.
IOANNIS KOUTROMANOS, PHD, is an Assistant Professor in the Department of Civil and Environmental Engineering at the Virginia Polytechnic Institute and State University. His research primarily focuses on the analytical simulation of structural components and systems under extreme events, with an emphasis on reinforced concrete, masonry and steel structures under earthquake loading. He has authored and co-authored research papers and reports on finite element analysis (element formulations, constitutive models, verification and validation of modeling schemes). He is a voting member of the joint ACI/ASCE Committee 447, Finite Element Analysis of Reinforced Concrete Structures.
Klappentext
An Introductory Textbook Covering the Fundamentals of Linear Finite Element Analysis (FEA)
This book constitutes the first volume in a two-volume set that introduces readers to the theoretical foundations and the implementation of the finite element method (FEM). The first volume focuses on the use of the method for linear problems. A general procedure is presented for the finite element analysis (FEA) of a physical problem, where the goal is to specify the values of a field function. First, the strong form of the problem (governing differential equations and boundary conditions) is formulated. Subsequently, a weak form of the governing equations is established. Finally, a finite element approximation is introduced, transforming the weak form into a system of equations where the only unknowns are nodal values of the field function. The procedure is applied to one-dimensional elasticity and heat conduction, multi-dimensional steady-state scalar field problems (heat conduction, chemical diffusion, flow in porous media), multi-dimensional elasticity and structural mechanics (beams/shells), as well as time-dependent (dynamic) scalar field problems, elastodynamics and structural dynamics. Important concepts for finite element computations, such as isoparametric elements for multi-dimensional analysis and Gaussian quadrature for numerical evaluation of integrals, are presented and explained. Practical aspects of FEA and advanced topics, such as reduced integration procedures, mixed finite elements and verification and validation of the FEM are also discussed.
Fundamentals of Finite Element Analysis: Linear Finite Element Analysis is an ideal text for undergraduate and graduate students in civil, aerospace and mechanical engineering, finite element software vendors, as well as practicing engineers and anybody with an interest in linear finite element analysis.
Inhalt
Preface xiv
About the Companion Website xviii
1 Introduction 1
1.1 Physical Processes and Mathematical Models 1
1.2 Approximation, Error, and Convergence 3
1.3 Finite Element Method for Differential Equations 5
1.4 Brief History of the Finite Element Method 6
1.5 Finite Element Software 8
1.6 Significance of Finite Element Analysis for Engineering 8
1.7 Typical Process for Obtaining a Finite Element Solution for a Physical Problem 12
1.8 A Note on Linearity and the Principle of Superposition 14
References 16
2 Strong and Weak Form for One-Dimensional Problems 17
2.1 Strong Form for One-Dimensional Elasticity Problems 17
2.2 General Expressions for Essential and Natural B.C. in One-Dimensional Elasticity Problems 23
2.3 Weak Form for One-Dimensional Elasticity Problems 24
2.4 Equivalence of Weak Form and Strong Form 28
2.5 Strong Form for One-Dimensional Heat Conduction 32
2.6 Weak Form for One-Dimensional Heat Conduction 37
Problems 44
References 46
3 Finite Element Formulation for One-Dimensional Problems 47
3.1 IntroductionPiecewise Approximation 47
3.2 Shape (Interpolation) Functions 51
3.3 Discrete Equations for Piecewise Finite Element Approximation 59
3.4 Finite Element Equations for Heat Conduction 66
3.5 Accounting for Nodes with Prescribed Solution Value (Fixed Nodes) 67
3.6 Examples on One-Dimensional Finite Element Analysis 68
3.7 Numerical IntegrationGauss Quadrature 91
3.8 Convergence of One-Dimensional Finite Element Method 100
3.9 Effect of Concentrated Forces in One-Dimensional Finite Element Analysis 106
Problems 108
References 111
4 Multidimensional Problems: Mathematical Preliminaries 112
4.1 Introduction 112
4.2 Basic Definitions 113
4.3 Green's TheoremDivergence Theorem and Green's Formula 118
4.4 Procedure for Multidimensional Problems 121
Problems 122
References 122
5 Two-Dimensional Heat Conduction and Other Scalar Field Problems 123
5.1 Strong Form for Two-Dimensional Heat Conduction 123
5.2 Weak Form for Two-Dimensional Heat Conduction 129
5.3 Equivalence of Strong Form and Weak Form 131
5.4 Other Scalar Field Problems 133
Problems 139
6 Finite Element Formulation for Two-Dimensional Scalar Field Problems 141
6.1 Finite Element Discretization and Piecewise Approximation 141
6.2 Three-Node Triangular Finite Element 148
6.3 Four-Node Rectangular Element 153
6.4 Isoparametric Finite Elements and the Four-Node Quadrilateral (4Q) Element 158
6.5 Numerical Integration for Isoparametric Quadrilateral Elements 165
6.6 Higher-Order Isoparametric Quadrilateral Elements 176
6.7 Isoparametric Triangular Elements 178
6.8 Continuity and Completeness of Isoparametric Elements 181
6.9 Concluding Remarks: Finite Element Analysis for Other Scalar Field Problems 183
Problems 183
References 188
7 Multidimensional Elasticity 189
7.1 Introduction 189
7.2 Definition of Strain Tensor 189
7.3 Definition of Stress Tensor 191
7.4 Representing Stress and Strain as Column VectorsThe Voigt Notation 193
7.5 Constitutive Law (Stress-Strain Relation) for Multidimensional Linear Elasticity 194
7.6 Coordinate Transformation Rules for Stress, Strain, and Material Stiffness Matrix 199
7.7 Stress, Strain, and Constitutive Models for Two-Dimensional (Planar) Elasticity 202
7.8 Strong Form for Two-Dimensional Elasticity 208
7.9 Weak Form for Two-Dimensional Elasticity 212
7.10 Equivalence between the Strong Form and the Weak Form 215
7.11 Strong Form for Three-Dimensional Elasticity 218
7.12 Using Polar (Cylindrical) Coordinates 220
References 225<...