Description
Book SynopsisAn 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 cond
Table of Contents
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 Introduction—Piecewise 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 Integration—Gauss 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 Theorem—Divergence 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 Vectors—The 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
8 Finite Element Formulation for Two-Dimensional Elasticity 226
8.1 Piecewise Finite Element Approximation—Assembly Equations 226
8.2 Accounting for Restrained (Fixed) Displacements 231
8.3 Postprocessing 232
8.4 Continuity—Completeness Requirements 232
8.5 Finite Elements for Two-Dimensional Elasticity 232
Problems 251
9 Finite Element Formulation for Three-Dimensional Elasticity 257
9.1 Weak Form for Three-Dimensional Elasticity 257
9.2 Piecewise Finite Element Approximation—Assembly Equations 258
9.3 Isoparametric Finite Elements for Three-Dimensional Elasticity 264
Problems 287
Reference 288
10 Topics in Applied Finite Element Analysis 289
10.1 Concentrated Loads in Multidimensional Analysis 289
10.2 Effect of Autogenous (Self-Induced) Strains—The Special Case of Thermal Strains 291
10.3 The Patch Test for Verification of Finite Element Analysis Software 294
10.4 Subparametric and Superparametric Elements 295
10.5 Field-Dependent Natural Boundary Conditions: Emission Conditions and Compliant Supports 296
10.6 Treatment of Nodal Constraints 302
10.7 Treatment of Compliant (Spring) Connections Between Nodal Points 309
10.8 Symmetry in Analysis 311
10.9 Axisymmetric Problems and Finite Element Analysis 316
10.10 A Brief Discussion on Efficient Mesh Refinement 319
Problems 321
References 323
11 Convergence of Multidimensional Finite Element Analysis, Locking Phenomena in Multidimensional Solids and Reduced Integration 324
11.1 Convergence of Multidimensional Finite Elements 324
11.2 Effect of Element Shape in Multidimensional Analysis 327
11.3 Incompatible Modes for Quadrilateral Finite Elements 328
11.4 Volumetric Locking in Continuum Elements 333
11.5 Uniform Reduced Integration and Spurious Zero-Energy (Hourglass) Modes 337
11.6 Resolving the Problem of Hourglass Modes: Hourglass Stiffness 339
11.7 Selective-Reduced Integration 346
11.8 The B-bar Method for Resolving Locking 348
Problems 351
References 352
12 Multifield (Mixed) Finite Elements 353
12.1 Multifield Weak Forms for Elasticity 354
12.2 Mixed (Multifield) Finite Element Formulations 359
12.3 Two-Field (Stress-Displacement) Formulations and the Pian-Sumihara Quadrilateral Element 367
12.4 Displacement-Pressure (u-p) Formulations and Finite Element Approximations 370
12.5 Stability of Mixed u-p Formulations—the inf-sup Condition 374
12.6 Assumed (Enhanced)-Strain Methods and the B-bar Method as a Special Case 377
12.7 A Concluding Remark for Multifield Elements 381
References 381
13 Finite Element Analysis of Beams 383
13.1 Basic Definitions for Beams 383
13.2 Differential Equations and Boundary Conditions for 2D Beams 385
13.3 Euler-Bernoulli Beam Theory 388
13.4 Strong Form for Two-Dimensional Euler-Bernoulli Beams 392
13.5 Weak Form for Two-Dimensional Euler-Bernoulli Beams 394
13.6 Finite Element Formulation: Two-Node Euler-Bernoulli Beam Element 397
13.7 Coordinate Transformation Rules for Two-Dimensional Beam Elements 404
13.8 Timoshenko Beam Theory 408
13.9 Strong Form for Two-Dimensional Timoshenko Beam Theory 411
13.10 Weak Form for Two-Dimensional Timoshenko Beam Theory 411
13.11 Two-Node Timoshenko Beam Finite Element 415
13.12 Continuum-Based Beam Elements 418
13.13 Extension of Continuum-Based Beam Elements to General Curved Beams 424
13.14 Shear Locking and Selective-Reduced Integration for Thin Timoshenko Beam Elements 440
Problems 443
References 446
14 Finite Element Analysis of Shells 447
14.1 Introduction 447
14.2 Stress Resultants for Shells 451
14.3 Differential Equations of Equilibrium and Boundary Conditions for Flat Shells 452
14.4 Constitutive Law for Linear Elasticity in Terms of Stress Resultants and Generalized Strains 456
14.5 Weak Form of Shell Equations 464
14.6 Finite Element Formulation for Shell Structures 472
14.7 Four-Node Planar (Flat) Shell Finite Element 480
14.8 Coordinate Transformations for Shell Elements 485
14.9 A “Clever” Way to Approximately Satisfy C1 Continuity Requirements for Thin Shells—The Discrete Kirchhoff Formulation 500
14.10 Continuum-Based Formulation for Nonplanar (Curved) Shells 510
Problems 521
References 522
15 Finite Elements for Elastodynamics, Structural Dynamics, and Time-Dependent Scalar Field Problems 523
15.1 Introduction 523
15.2 Strong Form for One-Dimensional Elastodynamics 525
15.3 Strong Form in the Presence of Material Damping 527
15.4 Weak Form for One-Dimensional Elastodynamics 529
15.5 Finite Element Approximation and Semi-Discrete Equations of Motion 530
15.6 Three-Dimensional Elastodynamics 536
15.7 Semi-Discrete Equations of Motion for Three-Dimensional Elastodynamics 539
15.8 Structural Dynamics Problems 539
15.9 Diagonal (Lumped) Mass Matrices and Mass Lumping Techniques 546
15.10 Strong and Weak Form for Time-Dependent Scalar Field (Parabolic) Problems 549
15.11 Semi-Discrete Finite Element Equations for Scalar Field (Parabolic) Problems 555
15.12 Solid and Structural Dynamics as a “Parabolic” Problem: The State-Space Formulation 557
Problems 558
References 559
16 Analysis of Time-Dependent Scalar Field (Parabolic) Problems 560
16.1 Introduction 560
16.2 Single-Step Algorithms 562
16.3 Linear Multistep Algorithms 568
16.4 Predictor-Corrector Algorithms—Runge-Kutta (RK) Methods 569
16.5 Convergence of a Time-Stepping Algorithm 572
16.6 Modal Analysis and Its Use for Determining the Stability for Systems with Many Degrees of Freedom 583
Problems 587
References 587
17 Solution Procedures for Elastodynamics and Structural Dynamics 588
17.1 Introduction 588
17.2 Modal Analysis: What Will NOT Be Presented in Detail 589
17.3 Step-by-Step Algorithms for Direct Integration of Equations of Motion 594
17.4 Application of Step-By-Step Algorithms for Discrete Systems with More than One Degrees of Freedom 608
17.4 Problems 613
References 613
18 Verification and Validation for the Finite Element Method 615
18.1 Introduction 615
18.2 Code Verification 615
18.3 Solution Verification 622
18.4 Numerical Uncertainty 627
18.5 Sources and Types of Uncertainty 629
18.6 Validation Experiments 630
18.7 Validation Metrics 631
18.8 Extrapolation of Model Prediction Uncertainty 633
18.9 Predictive Capability 634
References 634
19 Numerical Solution of Linear Systems of Equations 637
19.1 Introduction 637
19.2 Direct Methods 638
19.3 Iterative Methods 640
19.4 Parallel Computing and the Finite Element Method 644
19.5 Parallel Conjugate Gradient Method 649
References 653
Appendix A: Concise Review of Vector and Matrix Algebra 654
A.1 Preliminary Definitions 654
A.2 Matrix Mathematical Operations 656
A.3 Eigenvalues and Eigenvectors of a Matrix 660
A.4 Rank of a Matrix 662
Appendix B: Review of Matrix Analysis for Discrete Systems 664
B.1 Truss Elements 664
B.2 One-Dimensional Truss Analysis 666
B.3 Solving the Global Stiffness Equations of a Discrete System and Postprocessing 671
B.4 The ID Array Concept (for Equation Assembly) 673
B.5 Fully Automated Assembly: The Connectivity (LM) Array Concept 680
B.6 Advanced Interlude—Programming of Assembly When the Restrained Degrees of Freedom Have Nonzero Values 682
B.7 Advanced Interlude 2: Algorithms for Postprocessing 683
B.8 Two-Dimensional Truss Analysis—Coordinate Transformation Equations 684
B.9 Extension to Three-Dimensional Truss Analysis 693
Problem 694
Appendix C: Minimum Potential Energy for Elasticity—Variational Principles 695
Appendix D: Calculation of Displacement and Force Transformations for Rigid-Body Connections 700
Index 706
