**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.

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