An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics

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 Classication of Nonlinearities . . . . . . . . . . . . . . . 8
1.6 Review of Vectors and Tensors . . . . . . . . . . . . . . . . 12
1.6.1 Preliminary Comments . . . . . . . . . . . . . . . . . 12
1.6.2 Denition 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
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 Functionals, 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 Congurations 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
2.4 Strain Measures . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Deformation Tensors . . . . . . . . . . . . . . . . . . 56
2.4.2 The Green{Lagrange Strain Tensor . . . . . . . . . . . . 57
2.4.3 The Cauchy and Euler Strain Tensors . . . . . . . . . . . 58
2.4.4 Innitesimal Strain Tensor and Rotation Tensor . . . . . . 59
2.4.4.1 Innitesimal strain tensor . . . . . . . . . . . . . . 59
2.4.4.2 Innitesimal 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{Kirchho Stress Tensors . . . . . . . . . . . . . . 65
2.5.3.1 First Piola{Kirchho stress tensor . . . . . . . . . . . 65
2.5.3.2 Second Piola{Kirchho stress tensor . . . . . . . . . 67
2.6 Material Frame Indierence . . . . . . . . . . . . . . . . . 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{Kirchho stress tensor . . . . . . . . . . 70
2.6.3.3 Second Piola{Kirchho 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
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 G^ateaux 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 Dierential 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 Dierential 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
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 Dierential 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 coecients . . . . . . . . . . . 194
4.5.3.2 Assembly of element coecients . . . . . . . . . . . 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 coecients . . . . . . . . . . . . . . . . . 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 Stiness Matrix . . . . . . . . . . . . . . . . . 249
5.3.6 Numerical Examples . . . . . . . . . . . . . . . . . . 251
5.3.7 Functionally Graded Material Beams . . . . . . . . . . . 256
5.3.7.1 Material variation and stiness coecients . . . . . . . 256
5.3.7.2 Equations of equilibrium . . . . . . . . . . . . . . 257
5.3.7.3 Finite element model . . . . . . . . . . . . . . . . 258
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 261
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 (-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
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 Kirchho free-edge condition . . . . . . . . . . . 320
7.3.4.2 Typical edge conditions . . . . . . . . . . . . . . . 321
7.3.5 Stress Resultant{De
ection Relations . . . . . . . . . . . 322
7.4 Finite Element Models of the CPT . . . . . . . . . . . . . . 324
7.4.1 General Formulation . . . . . . . . . . . . . . . . . . 324
7.4.2 Tangent Stiness Coecients . . . . . . . . . . . . . . . 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
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 Stiness Coecients . . . . . . . . . . . . . . . 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
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 Denitions 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
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 (0/90/90/0) 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 Rened 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/hp 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
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
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 
ow past a circular cylinder . . . 595
11.2.2.2 Steady 
ow over a backward facing step . . . . . . . . 600
11.2.2.3 Lid-driven cavity 
ow . . . . . . . . . . . . . . . . 604
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 rst-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 modied L2-norm based least-squares model with
improved element-level mass conservation . . . . . . . 611
11.3.3 Numerical Examples: Verication Problems . . . . . . . . 613
11.3.3.1 Steady Kovasznay 
ow . . . . . . . . . . . . . . . 613
11.3.3.2 Steady 
ow in a 1  2 rectangular cavity . . . . . . . 616
11.3.3.3 Steady 
ow past a large cylinder in a narrow channel . . 619
11.3.3.4 Unsteady 
ow past a circular cylinder . . . . . . . . . 621
11.3.3.5 Unsteady 
ow 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
A1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 637
A1.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . 639
A1.2.1 Preliminary Comments . . . . . . . . . . . . . . . . . 639
A1.2.2 Symmetric Solver . . . . . . . . . . . . . . . . . . . . 640
A1.2.3 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 Modied Riks Methods . . . . . . . . . . . . . 654
References . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679