Automation of Finite Element Methods (eBook)

Preface 6
Contents 8
Notation 14
Introduction 23
1 Basic Equations of Continuum Mechanics 27
1.1 Kinematics 27
1.1.1 Motion and Deformation Gradient 27
1.1.2 Strain Measures 31
1.1.3 Transformation of Vectors and Tensors 33
1.1.4 Time Derivatives 35
1.2 Balance Equations 38
1.2.1 Balance of Mass 38
1.2.2 Balance of Linear and Angular Momentum 39
1.2.3 First Law of Thermodynamics 40
1.2.4 Introduction of Different Stress Tensors and Stress Rates 41
1.2.5 Balance Equations with Respect to Initial Configuration 44
1.3 Weak Form of Equilibrium, Variational Principles 45
1.3.1 Weak Form of Linear Momentum in the Initial Configuration 46
1.3.2 Weak Form of Linear Momentum in the Current Configuration 48
1.3.3 Variational Functionals 49
1.3.4 Mathematical Formalism for Weak Forms 52
References 53
2 Automation of Research in Computational Modeling 55
2.1 Introduction 56
2.1.1 Abstract Symbolic Description of a Computational Model 57
2.2 Advanced Software Tools and Techniques 59
2.2.1 Symbolic and Algebraic Computational Systems 59
2.2.2 Automatic Differentiation Tools 60
2.2.3 Problem Solving Environments 60
2.2.4 Hybrid Approaches 61
2.3 Automatic Generation of Numerical Codes 62
2.4 Automatic Differentiation 64
2.4.1 Principles of Automatic Differentiation 64
2.4.2 Generalized Notation of Automatic Differentiation 66
2.4.3 Local Definition of AD Exceptions 69
2.4.4 Global Definition of AD Exceptions 70
2.4.5 Differentiation with Respect to Variables with an Index 70
2.5 Automatic Code Generation with AceGen 71
2.5.1 Hybrid Symbolic-Numerical System AceGen 71
2.5.2 Typical AceGen Automatic Code Generation Procedure 73
2.5.3 Simultaneous Simplification Procedure 75
2.5.4 Efficiency and Limitations of Automation of Computational Modeling 77
2.6 Automatic Differentiation and Finite Element Method 77
2.6.1 ADB Form of General Potential Form 79
2.6.2 ADB Form of General Weak Form 81
2.6.3 Representative Formulas for Residual and Tangent Matrix 84
2.7 Automatic Generation of FE User Subroutines 89
References 93
3 Automation of Primal Analysis 95
3.1 Classification of Nonlinear Computational Problems 95
3.1.1 Time-Independent Problems 98
3.1.2 Time-Dependent Problems 98
3.1.3 Time-Independent Coupled Problems 99
3.1.4 Time-Dependent Coupled Problems 101
3.2 Solution of Nonlinear Systems of Equations 102
3.2.1 Newton--Raphson Method 105
3.2.2 Automation of Solution of Time-Independent and Time-Dependent Problems 108
3.3 Solution of Coupled Nonlinear Systems of Equations 109
3.3.1 Solution of Locally Coupled Problems 111
3.3.2 Automation of the Solution of Locally Coupled Problems 113
3.3.3 Solution and Automation of Locally Coupled Problems Using Static Elimination of Local Unknowns 114
3.3.4 Solution of Gauss Point Coupled Problems 117
3.3.5 Automation of the Solution of Gauss Point Coupled Problems 119
References 120
4 Automation of Discretization Techniques 123
4.1 General Isoparametric Concept 124
4.1.1 One-Dimensional Interpolations 130
4.1.2 Two-Dimensional Interpolations 132
4.1.3 Three-Dimensional Interpolation 135
4.2 Discretization of Potentials and Weak Forms 140
4.2.1 Three-Dimensional Solid Element, Initial Configuration 141
4.2.2 Three-Dimensional Solid Element, Current Configuration 148
4.2.3 Comparison of Formulations 152
4.3 Alternative Implementations 154
4.3.1 Comparison of Implementations 159
References 161
5 Materials 162
5.1 Elastic Materials 162
5.1.1 General Formulation 162
5.1.2 Neo-Hooke Material 164
5.1.3 Split in Isochoric and Volumetric Parts 166
5.1.4 Anisotropic Strain Energy Functions 167
5.1.5 Automation of Formulation of Elastic Materials 168
5.2 Elasto-Plastic Materials, Small Deformations 168
5.2.1 General Formulation 169
5.2.2 Integration of Constitutive Equations for Small Inelastic Deformations 172
5.2.3 Integration of General Elasto-Plastic Materials 173
5.2.4 Automation of Formulation of Small Strain Elasto-Plasticity 176
5.2.5 Example: von Mises Plasticity 180
5.2.6 Formulation of a von Mises Small Strain Elasto-Plastic Element 183
5.3 Elasto-Plastic Materials, Finite Deformations 191
5.3.1 General Formulation 192
5.3.2 Integration of Constitutive Equations for Finite Deformation Problems 196
5.3.3 Automation of Finite Strain Plasticity 199
5.3.4 Finite Strain Plasticity Example 199
References 201
6 Continuum Elements 205
6.1 Requirements for Continuum Finite Elements 205
6.2 Two-Dimensional Elements 207
6.2.1 Hyperelastic Triangular Element 207
6.2.2 Axisymmetric Element 211
6.2.3 Deformation Dependent Loads 213
6.3 Three-Dimensional Elements 217
6.3.1 Hyperelastic Solid Elements 217
6.4 Mixed Elements for Incompressibility 222
6.4.1 Mixed T2-P1 Element 224
6.4.2 Mixed T2-P0 Element 227
6.4.3 Mixed Q1-P0 Element 228
6.5 Enhanced Strain Element 230
6.5.1 General Concept and Formulation 231
6.5.2 Discretization of the Enhanced Strain Element 232
6.6 Example 235
References 237
7 Structural Elements 240
7.1 Nonlinear Truss Element 241
7.1.1 Kinematics and Strains 241
7.1.2 Constitutive Equations for the Truss 242
7.1.3 Variational Formulation 244
7.1.4 Finite-Element-Model 244
7.2 Two-Dimensional Geometrically Exact Beam Elements 247
7.2.1 Two-Dimensional Beam Kinematics 248
7.2.2 Constitutive Equations 249
7.2.3 Strain Energy Function 250
7.2.4 Finite Element Formulation for the Two-Dimensional Beam 250
7.2.5 Two-Dimensional Beam Example 255
7.3 Three-Dimensional Geometrically Exact Beam Element 255
7.3.1 Beam Kinematics 255
7.3.2 Constitutive Equation and Variational Form 257
7.3.3 Finite Element Formulation of the 3d-Beam 257
7.4 General Shell Element 261
7.4.1 Introductory Remarks 262
7.4.2 Shell Kinematics 265
7.4.3 Parametrization of the Rotations 266
7.4.4 Strain Energy Function 268
7.4.5 Finite Element Formulation for a Shear Deformable Shell 269
7.4.6 Example 273
References 275
8 Automation of Sensitivity Analysis 278
8.1 Introduction to Sensitivity Analysis 278
8.1.1 Parametrization of the Continuum and the Discretized Problem 280
8.1.2 Formulation and Solution of a Simple Sensitivity Problem 282
8.1.3 Introduction of Design Velocity Fields 283
8.1.4 Design Velocity Matrix 284
8.2 Classification and Formulation of Problems for Sensitivity Analysis 286
8.2.1 Classification of Sensitivity Problems 286
8.2.2 Classification of Sensitivity Design Velocity Fields 288
8.2.3 General Design Velocity Fields 289
8.2.4 Boundary Conditions Related Sensitivity Analysis 290
8.3 Solution and Automation of Sensitivity Problems 292
8.3.1 Direct Differentiation Method for Time-Independent Problems 294
8.3.2 Efficient Solution of Global Sensitivity Problem 297
8.3.3 Direct Differentiation Method for Time-Dependent Problems 298
8.3.4 Direct Differentiation Method for Time-Independent Locally Coupled Problems 300
8.3.5 Direct Differentiation Method for Time-Independent Gauss Point Coupled Problems 304
8.3.6 Direct Differentiation Method for Time-Dependent Locally Coupled Problems 305
8.3.7 Natural Boundary Condition Sensitivity Analysis 309
8.3.8 Implicit Dependency and the Cases with Multiple Domains 310
8.4 Generation of Sensitivity Analysis Related Subroutines 312
8.4.1 Axisymmetric, Hyper-Elastic Element for Primal and Sensitivity Analysis 313
8.4.2 Three-Dimensional, Elasto-Plastic Element for Primal and Sensitivity Analysis 316
8.5 Sensitivity Analysis of a Double-Layered Axisymmetric Cone by AceFEM 322
8.5.1 Definition of Sensitivity Parameters 323
8.5.2 Design Velocity Matrix 325
8.5.3 AceFEM Input for Primal and Sensitivity Analysis 326
References 331
9 Erratum to: Automation of Finite Element Methods 333
Erratum to: J. Korelc and P. Wriggers, Automation of Finite Element Methods, DOI 10.1007/978-3-319-39005-5 333
Appendix A Mathematica and AceGen Syntax 334
A.1 Highlights of Mathematica Syntax 334
A.2 Highlights of AceGen Syntax 336
A.2.1 AceGen Session 336
A.2.2 Assignment Operators 338
A.2.3 Program Flow Control 339
A.2.4 Generalized Automatic Differentiation with AceGen 339
A.3 Finite Elements Template Constants 341
A.4 Finite Elements User Subroutines Interface 343
Appendix B Vectors and Tensors 345
B.1 Tensor Algebra 345
B.1.1 Definition of a Tensor 345
B.1.2 Vectors and Tensors in a Base System 346
B.1.3 Operations with Vectors and Tensors 348
B.1.4 Special Forms of Tensors 350
B.1.5 Eigenvalues and Invariants of Tensors 351
B.1.6 Tensors of Higher Order 353
B.2 Tensor Analysis 354
B.2.1 Differentiation with Respect to a Real Variable 354
B.2.2 Gradient of a Field 355
B.2.3 Divergence of a Field 355
B.2.4 Pull Back and Push Forward Operations 356
B.2.5 Lie-Derivative of Stress Tensors 357
Appendix C Tables for Gauss Integration 359
C.1 One-Dimensional Integration 359
C.2 Two-Dimensional Integration 360
C.2.1 Quadrilateral Elements 360
C.2.2 Triangular Elements 360
C.3 Three-Dimensional Integration 360
C.3.1 Hexahedral Elements 360
C.3.2 Tetrahedral Elements 361
Index 364