J. N. REDDY
Department of Mechanical EngineeringTexas A&M University
College Station, Texas, 77843-3123, USA
An Introduction to Nonlinear
Finite Element Analysis
with applications to heat transfer, fluid mechanics,and solid mechanics
Second Edition
OXPORDUNIVERSITY PRESS
Contents
Preface to the Second Edition v
Preface to the First Edition ix
About the Author xxv
List of Symbols xxvii
1 General Introduction and Mathematical Preliminaries...
1
1.1 General Comments 1
1.2 Mathematical Models 2
1.3 Numerical Simulations 4
1.4 The Finite Element Method 6
1.5 Nonlinear Analysis 8
1.5.1 Introduction 8
1.5.2 Classification of Nonlinearities 8
1.6 Review of Vectors and Tensors 12
1.6.1 Preliminary Comments 12
1.6.2 Definition of a Physical Vector 13
1.6.2.1 Vector addition 13
1.6.2.2 Multiplication of a vector by a scalar 13
1.6.3 Scalar and Vector Products 14
1.6.3.1 Scalar product (or "dot" product) 14
1.6.3.2 Vector product 14
1.6.3.3 Plane area as a vector 15
1.6.3.4 Linear independence of vectors 16
1.6.3.5 Components of a vector 16
1.6.4 Summation Convention and Kronecker Delta and
Permutation Symbol 17
1.6.4.1 Summation convention 17
1.6.4.2 Kronecker delta symbol 17
1.6.4.3 The permutation symbol 18
1.6.5 Tensors and their Matrix Representation 19
1.6.5.1 Concept of a second-order tensor 19
1.6.5.2 Transformation laws for vectors and tensors 20
xii CONTENTS
1.6.6 Calculus of Vectors and Tensors 22
1.7 Concepts from Functional Analysis 27
1.7.1 Introduction 27
1.7.2 Linear Vector Spaces 28
1.7.2.1 Vector addition 28
1.7.2.2 Scalar multiplication 29
1.7.2.3 Linear subspaces 29
1.7.2.4 Linear dependence and independence of vectors....
29
1.7.3 Normed Vector Spaces 30
1.7.3.1 Holder inequality 30
1.7.3.2 Minkowski inequality 30
1.7.4 Inner Product Spaces 32
1.7.4.1 Orthogonality of vectors 33
1.7.4.2 Cauchy-Schwartz inequality 33
1.7.4.3 Hilbert spaces 35
1.7.5 Linear Transformations 36
1.7.6 Linear Functional, Bilinear Forms, and Quadratic Forms . . 37
1.7.6.1 Linear functional 38
1.7.6.2 Bilinear forms 38
1.7.6.3 Quadratic forms 38
1.8 The Big Picture 40
1.9 Summary 42
Problems 42
2 Elements of Nonlinear Continuum Mechanics 47
2.1 Introduction 47
2.2 Description of Motion 48
2.2.1 Configurations of a Continuous Medium 48
2.2.2 Material and Spatial Descriptions 49
2.2.3 Displacement Field 53
2.3 Analysis of Deformation 54
2.3.1 Deformation Gradient 54
2.3.2 Volume and Surface Elements in the Material and
Spatial Descriptions 55
CONTENTS xiii
2.4 Strain Measures 56
2.4.1 Deformation Tensors 56
2.4.2 The Green-Lagrange Strain Tensor 57
2.4.3 The Cauehy and Euler Strain Tensors 58
2.4.4 Infinitesimal Strain Tensor and Rotation Tensor 59
2.4.4.1 Infinitesimal strain tensor 59
2.4.4.2 Infinitesimal rotation tensor 60
2.4.5 Time Derivatives of the Deformation Tensors 60
2.5 Measures of Stress 62
2.5.1 Stress Vector 62
2.5.2 Cauchy's Formula and Stress Tensor 63
2.5.3 Piola-Kirchhoff Stress Tensors 65
2.5.3.1 First Piola-Kirchhoff stress tensor 65
2.5.3.2 Second Piola-Kirchhoff stress tensor 67
2.6 Material Frame Indifference 67
2.6.1 The Basic Idea 67
2.6.2 Objectivity of Strains and Strain Rates 69
2.6.3 Objectivity of Stress Tensors 69
2.6.3.1 Cauchy stress tensor 69
2.6.3.2 First Piola-Kirchhoff stress tensor 70
2.6.3.3 Second Piola-Kirchhoff stress tensor 70
2.7 Equations of Continuum Mechanics 70
2.7.1 Introduction 70
2.7.2 Conservation of Mass 71
2.7.2.1 Spatial form of the continuity equation 71
2.7.2.2 Material form of the continuity equation 72
2.7.3 Reynolds Transport Theorem 72
2.7.4 Balance of Linear Momentum 73
2.7.4.1 Spatial form of the equations of motion 73
2.7.4.2 Material form of the equations of motion 73
2.7.5 Balance of Angular Momentum 74
2.7.6 Thermodynamic Principles 74
2.7.6.1 Energy equation in the spatial description 75
xiv CONTENTS
2.7.6.2 Energy equation in the material description 76
2.7.6.3 Entropy inequality 77
2.8 Constitutive Equations for Elastic Solids 78
2.8.1 Introduction 78
2.8.2 Restrictions Placed by the Entropy Inequality 79
2.8.3 Elastic Materials and the Generalized Hooke's Law 80
2.9 Energy Principles of Solid Mechanics 83
2.9.1 Virtual Displacements and Virtual Work 83
2.9.2 First Variation or Gateaux Derivative 83
2.9.3 The Principle of Virtual Displacements 84
2.10 Summary 88
Problems 91
3 The Finite Element Method: A Review 97
3.1 Introduction 97
3.2 One-Dimensional Problems 98
3.2.1 Governing Differential Equation 98
3.2.2 Finite Element Approximation 98
3.2.3 Derivation of the Weak Form 101
3.2.4 Approximation Functions 104
3.2.5 Finite Element Model 107
3.2.6 Natural Coordinates 114
3.3 Two-Dimensional Problems 116
3.3.1 Governing Differential Equation 116
3.3.2 Finite Element Approximation 118
3.3.3 Weak Formulation 118
3.3.4 Finite Element Model 121
3.3.5 Approximation Functions: Element Library 122
3.3.5.1 Linear triangular element 122
3.3.5.2 Linear rectangular element 125
3.3.5.3 Higher-order triangular elements 126
3.3.5.4 Higher-order rectangular elements 128
3.3.6 Assembly of Elements 130
CONTENTS XV
3.4 Axisymmetric Problems 136
3.4.1 Introduction 136
3.4.2 One-Dimensional Problems 137
3.4.3 Two-Dimensional Problems 138
3.5 The Least-Squares Method 139
3.5.1 Background 139
3.5.2 The Basic Idea 141
3.6 Numerical Integration 143
3.6.1 Preliminary Comments 143
3.6.2 Coordinate Transformations 143
3.6.3 Integration Over a Master Rectangular Element 147
3.6.4 Integration Over a Master Triangular Element 148
3.7 Computer Implementation 149
3.7.1 General Comments 149
3.7.2 One-Dimensional Problems 152
3.7.3 Two-Dimensional Problems 157
3.8 Summary 163
Problems 164
4 One-Dimensional Problems Involving a Single Variable 175
4.1 Model Differential Equation 175
4.2 Weak Formulation 177
4.3 Finite Element Model 177
4.4 Solution of Nonlinear Algebraic Equations 180
4.4.1 General Comments 180
4.4.2 Direct Iteration Procedure 180
4.4.3 Newton's Iteration Procedure 185
4.5 Computer Implementation 192
4.5.1 Introduction 192
4.5.2 Preprocessor Unit 193
4.5.3 Processor Unit 194
4.5.3.1 Calculation of element coefficients 194
xvi CONTENTS
4.5.3.2 Assembly of element coefficients 198
4.5.3.3 Imposition of boundary conditions 200
4.6 Summary 208
Problems 209
5 Nonlinear Bending of Straight Beams 213
5.1 Introduction 213
5.2 The Euler-Bernoulli Beam Theory 214
5.2.1 Basic Assumptions 214
5.2.2 Displacement and Strain Fields 214
5.2.3 The Principle of Virtual Displacements: Weak Form 216
5.2.4 Finite Element Model 222
5.2.5 Iterative Solution Strategies 224
5.2.5.1 Direct iteration procedure 225
5.2.5.2 Newton's iteration procedure 225
5.2.6 Load Increments 228
5.2.7 Membrane Locking 228
5.2.8 Computer Implementation 230
5.2.8.1 Rearrangement of equations and computation of
element coefficients 230
5.2.8.2 Computation of strains and stresses 235
5.2.9 Numerical Examples 237
5.3 The Timoshenko Beam Theory 242
5.3.1 Displacement and Strain Fields 242
5.3.2 Weak Forms 243
5.3.3 General Finite Element Model 245
5.3.4 Shear and Membrane Locking 247
5.3.5 Tangent Stiffness Matrix 249
5.3.6 Numerical Examples 251
5.3.7 Functionally Graded Material Beams 256
5.3.7.1 Material variation and stiffness coefficients 256
5.3.7.2 Equations of equilibrium 257
5.3.7.3 Finite element model 258
5.4 Summary 261
CONTENTS xvii
Problems 261
6 Two-Dimensional Problems Involving a Single Variable 265
6.1 Model Equation 265
6.2 Weak Form 266
6.3 Finite Element Model 268
6.4 Solution of Nonlinear Equations 269
6.4.1 Direct Iteration Scheme 269
6.4.2 Newton's Iteration Scheme 269
6.5 Axisymmetric Problems 271
6.5.1 Introduction 271
6.5.2 Governing Equation and the Finite Element Model 272
6.6 Computer Implementation 273
6.6.1 Introduction 273
6.6.2 Numerical Integration 273
6.6.3 Element Calculations 275
6.7 Time-Dependent Problems 282
6.7.1 Introduction 282
6.7.2 Semidiscretization 283
6.7.3 Full Discretization of Parabolic Equations 284
6.7.3.1 Eigenvalue problem 284
6.7.3.2 Time (a-family of) approximations 284
6.7.3.3 Fully discretized equations 286
6.7.3.4 Direct iteration scheme 286
6.7.3.5 Newton's iteration scheme 287
6.7.3.6 Explicit and implicit formulations and mass lumping . .287
6.7.4 Full Discretization of Hyperbolic Equations 289
6.7.4.1 Newmark's scheme 289
6.7.4.2 Fully discretized equations 289
6.7.5 Stability and Accuracy 292
6.7.5.1 Preliminary comments 292
6.7.5.2 Stability criteria 293
6.7.6 Computer Implementation 294
6.7.7 Numerical Examples 299
xviii CONTENTS
6.8 Summary 306
Problems 306
7 Nonlinear Bending of Elastic Plates 311
7.1 Introduction 311
7.2 The Classical Plate Theory 312
7.2.1 Assumptions of the Kinematics 312
7.2.2 Displacement and Strain Fields 312
7.3 Weak Formulation of the CPT 315
7.3.1 Virtual Work Statement 315
7.3.2 Weak Forms 318
7.3.3 Equilibrium Equations 318
7.3.4 Boundary Conditions 319
7.3.4.1 The Kirchhoff free-edge condition 320
7.3.4.2 Typical edge conditions 321
7.3.5 Stress Resultant-Deflection Relations 322
7.4 Finite Element Models of the CPT 324
7.4.1 General Formulation 324
7.4.2 Tangent Stiffness Coefficients 327
7.4.3 Non-Conforming and Conforming Plate Elements 331
7.5 Computer Implementation of the CPT Elements 333
7.5.1 General Remarks 333
7.5.2 Programming Aspects 335
7.5.3 Post-Computation of Stresses 339
7.6 Numerical Examples using the CPT Elements 340
7.6.1 Preliminary Comments 340
7.6.2 Results of Linear Analysis 340
7.6.3 Results of Nonlinear Analysis 344
7.7 The First-Order Shear Deformation Plate Theory 348
7.7.1 Introduction 348
7.7.2 Displacement Field 348
7.7.3 Weak Forms using the Principle of Virtual Displacements . .349
CONTENTS xix
7.7.4 Governing Equations 350
7.8 Finite Element Models of the FSDT 352
7.8.1 Weak Forms 352
7.8.2 The Finite Element Model 354
7.8.3 Tangent Stiffness Coefficients 356
7.8.4 Shear and Membrane Locking 358
7.9 Computer Implementation and Numerical Results of
the FSDT Elements 359
7.9.1 Computer Implementation 359
7.9.2 Results of Linear Analysis 359
7.9.3 Results of Nonlinear Analysis 363
7.10 Transient Analysis of the FSDT 370
7.10.1 Equations of Motion 370
7.10.2 The Finite Element Model 371
7.10.3 Time Approximation 374
7.10.4 Numerical Examples 375
7.11 Summary 378
Problems 379
8 Nonlinear Bending of Elastic Shells 385
8.1 Introduction 385
8.2 Governing Equations 387
8.2.1 Geometric Description 387
8.2.2 General Strain-Displacement Relations 392
8.2.3 Stress Resultants 394
8.2.4 Displacement and Strain Fields 395
8.2.5 Equations of Equilibrium 397
8.2.6 Shell Constitutive Relations 399
8.3 Finite Element Formulation 399
8.3.1 Weak Forms 399
8.3.2 Finite Element Model 400
8.3.3 Linear Analysis 402
8.3.4 Nonlinear Analysis 410
xx CONTENTS
8.4 Summary 413
Problems 415
9 Finite Element Formulations of Solid Continua 417
9.1 Introduction 417
9.1.1 Background 417
9.1.2 Summary of Definitions and Concepts from
Continuum Mechanics 418
9.1.3 Energetically-Conjugate Stresses and Strains 419
9.2 Various Strain and Stress Measures 421
9.2.1 Introduction 421
9.2.2 Notation 422
9.2.3 Conservation of Mass 423
9.2.4 Green-Lagrange Strain Tensors 423
9.2.4.1 Green-Lagrange strain increment tensor 424
9.2.4.2 Updated Green-Lagrange strain tensor 424
9.2.5 Euler-Almansi Strain Tensor 425
9.2.6 Relationships Between Various Stress Tensors 426
9.2.7 Constitutive Equations 426
9.3 Total Lagrangian and Updated Lagrangian Formulations 429
9.3.1 Principle of Virtual Displacements 429
9.3.2 Total Lagrangian Formulation 430
9.3.2.1 Weak form 430
9.3.2.2 Incremental decompositions 431
9.3.2.3 Linearization 432
9.3.3 Updated Lagrangian Formulation 433
9.3.3.1 Weak form 433
9.3.3.2 Incremental decompositions 435
9.3.3.3 Linearization 436
9.3.4 Some Remarks on the Formulations 437
9.4 Finite Element Models of 2-D Continua 439
9.4.1 Introduction 439
9.4.2 Total Lagrangian Formulation 439
CONTENTS xxi
9.4.3 Updated Lagrangian Formulation 444
9.4.4 Computer Implementation 445
9.4.5 A Numerical Example 450
9.5 Conventional Continuum Shell Finite Element 458
9.5.1 Introduction 458
9.5.2 Incremental Equations of Motion 459
9.5.3 Finite Element Model of a Continuum 460
9.5.4 Shell Finite Element 462
9.5.5 Numerical Examples 468
9.5.5.1 Simply-supported orthotropic plate under uniform load . 468
9.5.5.2 Four-layer (0o/9079070°) clamped plate under
uniform load 469
9.5.5.3 Cylindrical shell roof under self-weight 470
9.5.5.4 Simply-supported spherical shell under point load....
471
9.5.5.5 Shallow cylindrical shell under point load 472
9.6 A Refined Continuum Shell Finite Element 473
9.6.1 Backgound 473
9.6.2 Representation of Shell Mid-Surface 474
9.6.3 Displacement and Strain Fields 478
9.6.4 Constitutive Relations 480
9.6.4.1 Isotropic and functionally graded shells 481
9.6.4.2 Laminated composite shells 483
9.6.5 The Principle of Virtual Displacements and its Discretization.
486
9.6.6 The Spectral//ip Basis Functions 488
9.6.7 Finite Element Model and Solution of Nonlinear Equations . .
491
9.6.7.1 The Newton procedure 491
9.6.7.2 The cylindrical arc-length procedure 493
9.6.7.3 Element-level static condensation and
assembly of elements 495
9.6.8 Numerical Examples 497
9.6.8.1 A cantilevered plate strip under an end transverse load.
498
9.6.8.2 Post-buckling of a plate strip under axial compressive load 500
9.6.8.3 An annular plate with a slit under an end transverse load 501
9.6.8.4 A cylindrical panel subjected to a point load 504
9.6.8.5 Pull-out of an open-ended cylindrical shell 509
xxii CONTENTS
9.6.8.6 A pinched half-cylindrical shell 512
9.6.8.7 A pinched cylinder with rigid diaphragms 513
9.6.8.8 A pinched hemisphere with an 18° hole 515
9.6.8.9 A pinched composite hyperboloidal shell 517
9.7 Summary 521
Problems 522
10 Weak-Form Finite Element Models of Flows of
Viscous Incompressible Fluids 523
10.1 Introduction 523
10.2 Governing Equations 524
10.2.1 Introduction 524
10.2.2 Equation of Mass Continuity 525
10.2.3 Equations of Motion 525
10.2.4 Energy Equation 526
10.2.5 Constitutive Equations 526
10.2.6 Boundary Conditions 527
10.3 Summary of Governing Equations 529
10.3.1 Vector Form 529
10.3.2 Cartesian Component Form 529
10.4 Velocity-Pressure Finite Element Model 530
10.4.1 Weak Forms 530
10.4.2 Semidiscrete Finite Element Model 532
10.4.3 Fully Discretized Finite Element Model 534
10.5 Penalty Finite Element Models 535
10.5.1 Introduction 535
10.5.2 Penalty Function Method 536
10.5.3 Reduced Integration Penalty Model 538
10.5.4 Consistent Penalty Model 539
10.6 Computational Aspects 540
10.6.1 Properties of the Finite Element Equations 540
10.6.2 Choice of Elements 541
10.6.3 Evaluation of Element Matrices in Penalty Models 543
CONTENTS xxiii
10.6.4 Post-Computation of Pressure and Stresses 544
10.7 Computer Implementation 545
10.7.1 Mixed Model 545
10.7.2 Penalty Model 549
10.7.3 Transient Analysis 552
10.8 Numerical Examples 552
10.8.1 Preliminary Comments 552
10.8.2 Linear Problems 552
10.8.3 Nonlinear Problems 561
10.8.4 Transient Analysis 567
10.9 Non-Newtonian Fluids 570
10.9.1 Introduction 570
10.9.2 Governing Equations in Cylindrical Coordinates 570
10.9.3 Power-Law Fluids 572
10.9.4 White-Metzner Fluids 574
10.9.5 Numerical Examples 578
10.10 Coupled Fluid Flow and Heat Transfer 582
10.10.1 Finite Element Models 582
10.10.2 Numerical Examples 583
10.10.2.1 Heated cavity 583
10.10.2.2 Solar receiver 584
10.11 Summary 587
Problems 587
11 Least-Squares Finite Element Models of Flows of
Viscous Incompressible Fluids 589
11.1 Introduction 589
11.2 Least-Squares Finite Element Formulation 593
11.2.1 The Navier-Stokes Equations of Incompressible Fluids . . . .593
11.2.2 Numerical Examples 595
11.2.2.1 Low Reynolds number flow past a circular cylinder . . .
595
11.2.2.2 Steady flow over a backward facing step 600
11.2.2.3 Lid-driven cavity flow 604
xxiv CONTENTS
11.3 A Least-Squares Finite Element Model with Enhanced
Element-Level Mass Conservation 607
11.3.1 Introduction 607
11.3.2 Unsteady Flows 608
11.3.2.1 The velocity-pressure-vorticity first-order system ....609
11.3.2.2 Temporal discretization 609
11.3.2.3 The standard L2-norm based least-squares model....
610
11.3.2.4 A modified L2-norm based least-squares model with
improved element-level mass conservation 611
11.3.3 Numerical Examples: Verification Problems 613
11.3.3.1 Steady Kovasznay flow 613
11.3.3.2 Steady flow in a 1 x 2 rectangular cavity 616
11.3.3.3 Steady flow past a large cylinder in a narrow channel. .
619
11.3.3.4 Unsteady flow past a circular cylinder 621
11.3.3.5 Unsteady flow past a large cylinder in a narrow channel.
627
11.4 Summary and Future Direction 632
Problems 635
Appendix 1: Solution Procedures for Linear Equations ....637
A 1.1 Introduction 637
A1.2 Direct Methods 639
Al.2.1 Preliminary Comments 639
A1.2.2 Symmetric Solver 640
Al.2.3 Unsymmetric Solver 642
A1.3 Iterative Methods 644
Appendix 2: Solution Procedures for Nonlinear Equations . .
645
A2.1 Introduction 645
A2.2 The Picard Iteration Method 646
A2.3 The Newton Iteration Method 650
A2.4 The Riks and Modified Riks Methods 654
References 663
Index 679
