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