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