ANSYS Mechanical APDL Theory Reference

988 Pages·2013·20.43 MB·English

ANSYS Mechanical APDL Theory Reference
ANSYS, Inc. Release 15.0
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Table of Contents
1. Introduction............................................................................................................................................ 1
1.1. Purpose of the Theory Reference....................................................................................................... 1
1.2. Understanding Theory Reference Notation........................................................................................ 2
1.3. Applicable Products.......................................................................................................................... 3
2. Structures................................................................................................................................................ 5
2.1. Structural Fundamentals................................................................................................................... 5
2.1.1. Stress-Strain Relationships........................................................................................................ 5
2.1.2. Orthotropic Material Transformation for Axisymmetric Models.................................................. 9
2.1.3.Temperature-Dependent Coefficient of Thermal Expansion..................................................... 10
2.2. Derivation of Structural Matrices..................................................................................................... 12
2.3. Structural Strain and Stress Evaluations........................................................................................... 15
2.3.1. Integration Point Strains and Stresses..................................................................................... 15
2.3.2. Surface Stresses..................................................................................................................... 16
2.3.3. Shell Element Output............................................................................................................. 16
2.4. Combined Stresses and Strains........................................................................................................ 18
2.4.1. Combined Strains................................................................................................................... 18
2.4.2. Combined Stresses................................................................................................................. 19
2.4.3. Failure Criteria........................................................................................................................ 20
2.4.3.1. Maximum Strain Failure Criteria..................................................................................... 20
2.4.3.2. Maximum Stress Failure Criteria..................................................................................... 22
2.4.3.3.Tsai-Wu Failure Criteria................................................................................................... 23
2.4.3.4. Physical Failure Criteria.................................................................................................. 24
2.4.3.4.1. Hashin Fiber Failure Criterion................................................................................ 24
2.4.3.4.2. Hashin Matrix Failure Criterion.............................................................................. 24
2.4.3.4.3. Puck Fiber Failure Criterion ([401])......................................................................... 24
2.4.3.4.4. Puck Matrix Failure Criterion ([401])....................................................................... 24
2.4.3.4.5. LaRc03 Fiber Failure Criterion ([402])...................................................................... 25
2.4.3.4.6. LaRc03 Matrix Failure Criterion ([402]).................................................................... 26
2.4.3.4.7. LaRc04 Fiber Failure Criterion ([403])...................................................................... 26
2.4.3.4.8. LaRc04 Matrix Failure Criterion ([403]).................................................................... 27
3. Structures with Geometric Nonlinearities............................................................................................ 29
3.1. Understanding Geometric Nonlinearities......................................................................................... 29
3.2. Large Strain.................................................................................................................................... 29
3.2.1.Theory .................................................................................................................................. 30
3.2.2. Implementation..................................................................................................................... 31
3.2.3. Definition of Thermal Strains.................................................................................................. 33
3.2.4. Element Formulation.............................................................................................................. 34
3.2.5. Applicable Input .................................................................................................................... 35
3.2.6. Applicable Output.................................................................................................................. 35
3.3. Large Rotation................................................................................................................................ 35
3.3.1.Theory ................................................................................................................................... 35
3.3.2. Implementation..................................................................................................................... 36
3.3.3. Element Transformation......................................................................................................... 36
3.3.4. Deformational Displacements................................................................................................ 37
3.3.5. Updating Rotations................................................................................................................ 39
3.3.6. Applicable Input .................................................................................................................... 39
3.3.7. Applicable Output.................................................................................................................. 39
3.3.8. Consistent Tangent Stiffness Matrix and Finite Rotation........................................................... 39
3.4. Stress Stiffening.............................................................................................................................. 41
3.4.1. Overview and Usage.............................................................................................................. 41
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3.4.2.Theory ................................................................................................................................... 41
3.4.3. Implementation..................................................................................................................... 44
3.4.4. Pressure Load Stiffness........................................................................................................... 46
3.4.5. Applicable Input .................................................................................................................... 47
3.4.6. Applicable Output.................................................................................................................. 47
3.5. Spin Softening................................................................................................................................ 47
3.6. General Element Formulations........................................................................................................ 50
3.6.1. Fundamental Equations ......................................................................................................... 51
3.6.2. Classical Pure Displacement Formulation................................................................................ 52
3.6.3. Mixed u-P Formulations.......................................................................................................... 54
3.6.4. u-P Formulation I.................................................................................................................... 56
3.6.5. u-P Formulation II................................................................................................................... 58
3.6.6. u-P Formulation III.................................................................................................................. 59
3.6.7.Volumetric Constraint Equations in u-P Formulations.............................................................. 59
3.7. Constraints and Lagrange Multiplier Method................................................................................... 60
4. Structures with Material Nonlinearities................................................................................................ 63
4.1. Understanding Material Nonlinearities............................................................................................ 63
4.2. Rate-Independent Plasticity............................................................................................................ 64
4.2.1.Theory ................................................................................................................................... 65
4.2.2.Yield Criterion........................................................................................................................ 65
4.2.3. Flow Rule............................................................................................................................... 68
4.2.4. Hardening Rule...................................................................................................................... 68
4.2.5. Plastic Strain Increment.......................................................................................................... 70
4.2.6. Implementation..................................................................................................................... 72
4.2.7. Elastoplastic Stress-Strain Matrix............................................................................................. 73
4.2.8. Specialization for Hardening................................................................................................... 74
4.2.9. Specification for Nonlinear Isotropic Hardening...................................................................... 75
4.2.10. Specialization for Bilinear Kinematic Hardening.................................................................... 76
4.2.11. Specialization for Multilinear Kinematic Hardening............................................................... 78
4.2.12. Specialization for Nonlinear Kinematic Hardening................................................................. 80
4.2.13. Specialization for Anisotropic Plasticity................................................................................. 81
4.2.14. Hill Potential Theory............................................................................................................. 81
4.2.15. Generalized Hill Potential Theory.......................................................................................... 83
4.2.16. Specialization for Drucker-Prager.......................................................................................... 87
4.2.16.1. Classic Drucker-Prager Model....................................................................................... 87
4.2.16.2. Extended Drucker-Prager Model.................................................................................. 89
4.2.17. Extended Drucker-Prager Cap Model.................................................................................... 90
4.2.17.1. Shear Failure Envelope Function.................................................................................. 91
4.2.17.2. Compaction Cap Function........................................................................................... 92
4.2.17.3. Expansion Cap Function.............................................................................................. 92
4.2.17.4. Lode Angle Function.................................................................................................... 93
4.2.17.5. Hardening Functions................................................................................................... 94
4.2.18. Gurson's Model.................................................................................................................... 96
4.2.19. Gurson Plasticity with Isotropic/Chaboche Kinematic Hardening......................................... 100
4.2.20. Cast Iron Material Model..................................................................................................... 101
4.3. Rate-Dependent Plasticity (Including Creep and Viscoplasticity)..................................................... 105
4.3.1. Creep................................................................................................................................... 105
4.3.1.1. Definition and Limitations............................................................................................ 106
4.3.1.2. Calculation of Creep.................................................................................................... 106
4.3.1.3.Time Step Size............................................................................................................. 108
4.3.2. Rate-Dependent Plasticity ................................................................................................... 108
4.3.2.1. Perzyna Option............................................................................................................ 109
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4.3.2.2. Peirce Option............................................................................................................... 109
4.3.2.3. Exponential Visco-Hardening (EVH) Option.................................................................. 109
4.3.2.4. Anand Viscoplasticity Option....................................................................................... 109
4.3.3. Extended Drucker-Prager (EDP) Creep Model........................................................................ 112
4.3.3.1. EDP Inelastic Strain Rate Decomposition...................................................................... 113
4.3.3.2. EDP Yielding and Hardening Conditions....................................................................... 114
4.3.3.3. EDP Creep Measurements............................................................................................ 114
4.3.3.4. EDP Equivalent Creep Stress......................................................................................... 115
4.3.3.5. EDP Elastic Creep and Stress Projection........................................................................ 116
4.3.4. Cap Creep Model.................................................................................................................. 117
4.3.4.1. Assumptions and Restrictions...................................................................................... 118
4.3.4.2. Functions and Potentials.............................................................................................. 118
4.3.4.3. Cap Zones................................................................................................................... 118
4.3.4.4. Equivalent Creep Stress................................................................................................ 119
4.3.4.4.1. Equivalent Creep Stress in the Compaction Zone................................................. 120
4.3.4.5. Elastic Creep and Stress Projection............................................................................... 121
4.3.4.6. Commands Used for Cap Creep ................................................................................... 121
4.4. Gasket Material............................................................................................................................. 121
4.4.1. Stress and Deformation........................................................................................................ 122
4.4.2. Material Definition ............................................................................................................... 122
4.4.3.Thermal Deformation........................................................................................................... 123
4.5. Nonlinear Elasticity....................................................................................................................... 123
4.5.1. Overview and Guidelines for Use.......................................................................................... 123
4.6. Hyperelasticity.............................................................................................................................. 124
4.6.1. Finite Strain Elasticity............................................................................................................ 125
4.6.2. Deviatoric-Volumetric Multiplicative Split............................................................................. 127
4.6.3. Isotropic Hyperelasticity....................................................................................................... 127
4.6.3.1.Arruda-Boyce Model ..................................................................................................... 127
4.6.3.2.Blatz-Ko Model............................................................................................................. 128
4.6.3.3.Extended Tube Model.................................................................................................... 128
4.6.3.4.Gent Model.................................................................................................................. 128
4.6.3.5.Mooney-Rivlin.............................................................................................................. 129
4.6.3.6.Neo-Hookean............................................................................................................... 130
4.6.3.7.Ogden Compressible Foam Model.................................................................................. 130
4.6.3.8.Ogden Potential ........................................................................................................... 131
4.6.3.9.Polynomial Form.......................................................................................................... 131
4.6.3.10.Yeoh Model................................................................................................................ 132
4.6.4. Anisotropic Hyperelasticity................................................................................................... 132
4.6.5. USER Subroutine.................................................................................................................. 133
4.6.6. Output Quantities................................................................................................................ 134
4.6.7. Hyperelasticity Material Curve Fitting.................................................................................... 134
4.6.7.1. Uniaxial Tension (Equivalently, Equibiaxial Compression).............................................. 137
4.6.7.2. Equibiaxial Tension (Equivalently, Uniaxial Compression).............................................. 138
4.6.7.3. Pure Shear................................................................................................................... 138
4.6.7.4.Volumetric Deformation .............................................................................................. 139
4.6.7.5. Least Squares Fit Analysis............................................................................................. 140
4.6.8. Experimental Response Functions ........................................................................................ 140
4.6.9. Material Stability Check........................................................................................................ 142
4.7. Bergstrom-Boyce .......................................................................................................................... 142
4.8. Mullins Effect ............................................................................................................................... 144
4.8.1.The Pseudo-elastic Model..................................................................................................... 145
4.9.Viscoelasticity............................................................................................................................... 146
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4.9.1. Small Strain Viscoelasticity.................................................................................................... 146
4.9.2. Constitutive Equations......................................................................................................... 146
4.9.3. Numerical Integration .......................................................................................................... 148
4.9.4.Thermorheological Simplicity............................................................................................... 149
4.9.5. Large-Deformation Viscoelasticity......................................................................................... 150
4.9.6.Visco-Hypoelasticity............................................................................................................. 150
4.9.7. Large-Strain Visco-Hyperelasticity......................................................................................... 151
4.9.8. Large-Strain Visco-Anisotropic Hyperelasticity....................................................................... 153
4.9.9. Shift Functions..................................................................................................................... 153
4.9.9.1.Williams-Landel-Ferry Shift Function............................................................................ 153
4.9.9.2.Tool-Narayanaswamy Shift Function............................................................................. 153
4.9.9.3.Tool-Narayanaswamy Shift Function with Fictive Temperature...................................... 153
4.9.9.4. User-Defined Shift Function......................................................................................... 154
4.10. Concrete..................................................................................................................................... 155
4.10.1.The Domain (Compression - Compression - Compression)................................................... 156
4.10.2.The Domain (Tension - Compression - Compression)........................................................... 159
4.10.3.The Domain (Tension - Tension - Compression)................................................................... 160
4.10.4.The Domain (Tension - Tension - Tension)............................................................................ 160
4.11. Swelling...................................................................................................................................... 162
4.12. Cohesive Zone Material (CZM) Model.......................................................................................... 162
4.12.1. Interface Elements.............................................................................................................. 163
4.12.1.1. Material Model - Exponential Behavior....................................................................... 163
4.12.1.2. Material Model - Bilinear Behavior.............................................................................. 165
4.12.1.3.Viscous Regularization............................................................................................... 169
4.12.2. Contact Elements............................................................................................................... 169
4.12.2.1. Material Model - Bilinear Behavior.............................................................................. 169
4.13. Fluid Material Models.................................................................................................................. 174
4.13.1. Liquid ................................................................................................................................ 174
4.13.2. Gas .................................................................................................................................... 175
4.13.3. Pressure-Volume Data ........................................................................................................ 176
5. Electromagnetics ................................................................................................................................ 177
5.1. Electromagnetic Field Fundamentals............................................................................................. 177
5.1.1. Magnetic Scalar Potential..................................................................................................... 180
5.1.2. Solution Strategies............................................................................................................... 180
5.1.2.1. RSP Strategy................................................................................................................ 182
5.1.2.2. DSP Strategy................................................................................................................ 182
5.1.2.3. GSP Strategy................................................................................................................ 183
5.1.3. Magnetic Vector Potential..................................................................................................... 184
5.1.4. Limitation of the Node-Based Vector Potential...................................................................... 185
5.1.5. Edge-Based Magnetic Vector Potential ................................................................................. 187
5.1.6. Harmonic Analysis Using Complex Formalism....................................................................... 187
5.1.7. Nonlinear Time-Harmonic Magnetic Analysis........................................................................ 189
5.1.8. Electric Scalar Potential......................................................................................................... 190
5.1.8.1. Quasistatic Electric Analysis......................................................................................... 191
5.1.8.2. Electrostatic Analysis................................................................................................... 192
5.2. Derivation of Electromagnetic Matrices......................................................................................... 193
5.2.1. Magnetic Scalar Potential..................................................................................................... 193
5.2.1.1. Degrees of freedom..................................................................................................... 193
5.2.1.2. Coefficient Matrix........................................................................................................ 193
5.2.1.3. Applied Loads ............................................................................................................. 193
5.2.2. Magnetic Vector Potential..................................................................................................... 194
5.2.2.1. Degrees of Freedom.................................................................................................... 194
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5.2.2.2. Coefficient Matrices..................................................................................................... 194
5.2.2.3. Applied Loads ............................................................................................................. 195
5.2.3. Edge-Based Magnetic Vector Potential.................................................................................. 196
5.2.4. Electric Scalar Potential......................................................................................................... 199
5.2.4.1. Quasistatic Electric Analysis......................................................................................... 199
5.2.4.2. Electrostatic Analysis................................................................................................... 200
5.3. Electromagnetic Field Evaluations................................................................................................. 200
5.3.1. Magnetic Scalar Potential Results.......................................................................................... 200
5.3.2. Magnetic Vector Potential Results......................................................................................... 201
5.3.3. Edge-Based Magnetic Vector Potential.................................................................................. 202
5.3.4. Magnetic Forces................................................................................................................... 203
5.3.4.1. Lorentz forces.............................................................................................................. 203
5.3.4.2. Maxwell Forces............................................................................................................ 204
5.3.4.2.1. Surface Integral Method...................................................................................... 204
5.3.4.2.2.Volumetric Integral Method ................................................................................ 205
5.3.4.3.Virtual Work Forces...................................................................................................... 205
5.3.4.3.1. Element Shape Method....................................................................................... 206
5.3.4.3.2. Nodal Perturbation Method................................................................................ 206
5.3.5. Joule Heat in a Magnetic Analysis......................................................................................... 207
5.3.6. Electric Scalar Potential Results............................................................................................. 207
5.3.6.1. Quasistatic Electric Analysis......................................................................................... 207
5.3.6.2. Electrostatic Analysis ................................................................................................... 209
5.3.7. Electrostatic Forces............................................................................................................... 209
5.4. Stranded Coil Analyses.................................................................................................................. 210
5.4.1. Governing Equations............................................................................................................ 210
5.4.2. A-VOLT-EMF Formulation...................................................................................................... 211
5.4.3. A-CURR Formulation............................................................................................................. 213
5.5. Inductance, Flux and Energy Computation..................................................................................... 214
5.5.1. Differential Inductance Definition......................................................................................... 214
5.5.2. Review of Inductance Computation Methods........................................................................ 215
5.5.3. Inductance Computation Method Used................................................................................ 216
5.5.4.Transformer and Motion Induced Voltages............................................................................ 216
5.5.5. Absolute Flux Computation.................................................................................................. 217
5.5.6. Inductance Computations.................................................................................................... 217
5.5.7. Absolute Energy Computation.............................................................................................. 218
5.6. Electromagnetic Particle Tracing.................................................................................................... 219
5.7. Capacitance Computation............................................................................................................. 220
5.8. Conductance Computation........................................................................................................... 222
5.9. Hall Effect..................................................................................................................................... 224
6. Heat Flow............................................................................................................................................. 227
6.1. Heat Flow Fundamentals............................................................................................................... 227
6.1.1. Conduction and Convection................................................................................................. 227
6.1.2. Radiation ............................................................................................................................. 230
6.1.2.1.View Factors................................................................................................................ 231
6.1.2.2. Radiation Usage .......................................................................................................... 234
6.2. Derivation of Heat Flow Matrices................................................................................................... 235
6.3. Heat Flow Evaluations................................................................................................................... 236
6.3.1. Integration Point Output...................................................................................................... 236
6.3.2. Surface Output..................................................................................................................... 237
6.4. Radiation Matrix Method............................................................................................................... 237
6.4.1.View Factor Calculation (2-D): Radiation Matrix Method......................................................... 238
6.4.1.1. Non-Hidden Method ................................................................................................... 238
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6.4.1.2. Hidden Method........................................................................................................... 239
6.4.2.View Factors of Axisymmetric Bodies.................................................................................... 240
6.4.3. Space Node.......................................................................................................................... 241
6.5. Radiosity Solution Method............................................................................................................ 241
6.5.1.View Factor Calculation (3-D): Hemicube Method.................................................................. 243
7. Fluid Flow............................................................................................................................................ 245
7.1. Squeeze Film Theory..................................................................................................................... 245
7.1.1. Flow Between Flat Surfaces.................................................................................................. 245
7.1.2. Flow in Channels.................................................................................................................. 249
7.2. Slide Film Theory........................................................................................................................... 250
8. Acoustics ............................................................................................................................................. 253
8.1. Acoustic Fundamentals................................................................................................................. 253
8.1.1. Governing Equations............................................................................................................ 253
8.1.2. Finite Element Formulation of the Wave Equation................................................................. 255
8.2. Derivation of Acoustic Matrices..................................................................................................... 256
8.3. Propagation, Radiation, and Scattering of Acoustic Pressure Waves................................................. 257
8.3.1. Acoustic Boundary Conditions.............................................................................................. 257
8.3.2. Absorbing Boundary Condition (ABC)................................................................................... 259
8.3.3. Perfectly Matched Layers (PML)............................................................................................ 262
8.3.4. Acoustic Excitation Sources.................................................................................................. 264
8.3.4.1. Specified Pressure or Normal Velocity on the Domain Boundary................................... 264
8.3.4.2. Mass Source in the Wave Equation ............................................................................... 264
8.3.4.3. Analytic Wave Sources................................................................................................. 265
8.3.5. Sophisticated Acoustic Media............................................................................................... 268
8.3.5.1. Non-uniform Acoustic Media....................................................................................... 269
8.3.5.2. Equivalent Fluid of Perforated Materials....................................................................... 269
8.3.5.2.1. Johnson-Chapoux-Allard Model.......................................................................... 269
8.3.5.2.2. Delany-Bazley and Miki Models........................................................................... 271
8.3.5.2.3. Complex Media Properties.................................................................................. 272
8.3.5.3. Impedance Sheet Approximation................................................................................. 273
8.3.5.4.Viscous-Thermal Media................................................................................................ 273
8.3.5.4.1. Boundary Layer Impedance (BLI) Model............................................................... 273
8.3.5.4.2. Low Reduced Frequency (LRF) Model.................................................................. 274
8.4. Acoustic Fluid-Structural Interaction (FSI)...................................................................................... 276
8.4.1. Coupled Acoustic Fluid-Structural System with an Unsymmetric Matrix Equation................... 276
8.4.2. Coupled Acoustic Fluid-Structural System with Symmetric Matrix Equation for Full Harmonic
Analysis........................................................................................................................................ 278
8.4.3. Coupled Acoustic Fluid-Structural System with Symmetric Matrix Equation for Lossless Modal
Analysis........................................................................................................................................ 279
8.5. Pure Scattered Pressure Formulation............................................................................................. 280
8.6. Acoustic Output Quantities........................................................................................................... 282
8.7.Transfer Admittance Matrix........................................................................................................... 285
8.7.1.Transfer Admittance Matrix Connected to Acoustic Domains................................................. 286
8.7.2.Transfer Admittance Matrix Connected to the Structural and Acoustic Domain...................... 286
9. Diffusion.............................................................................................................................................. 289
9.1. Diffusion Fundamentals................................................................................................................ 289
9.2. Normalized Concentration Approach............................................................................................ 290
9.3. Derivation of Diffusion Matrices.................................................................................................... 290
9.4. Diffusion Analysis Results.............................................................................................................. 291
10. Coupling............................................................................................................................................ 293
10.1. Coupled Effects........................................................................................................................... 293
10.1.1. Elements............................................................................................................................ 293
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10.1.1.1. Advantages ............................................................................................................... 294
10.1.1.2. Disadvantages........................................................................................................... 294
10.1.2. Coupling Methods.............................................................................................................. 294
10.1.2.1.Thermal-Structural Analysis........................................................................................ 296
10.1.2.2. Magneto-Structural Analysis (Vector Potential)........................................................... 297
10.1.2.3. Magneto-Structural Analysis (Scalar Potential)............................................................ 297
10.1.2.4. Electromagnetic Analysis........................................................................................... 298
10.1.2.5. Stranded Coil Analysis................................................................................................ 298
10.1.2.6. Electro-Thermo-Structural Analysis ............................................................................ 299
10.1.2.7. Electro-Magneto-Thermo-Structural Analysis ............................................................. 299
10.1.2.8. Electro-Magneto-Thermal Analysis............................................................................. 300
10.1.2.9. Piezoelectric Analysis................................................................................................. 300
10.1.2.10. Electroelastic Analysis.............................................................................................. 301
10.1.2.11.Thermo-Piezoelectric Analysis.................................................................................. 301
10.1.2.12. Piezoresistive Analysis.............................................................................................. 302
10.1.2.13.Thermo-Pressure Analysis......................................................................................... 303
10.1.2.14. Acoustic-Structural Analysis..................................................................................... 303
10.1.2.15.Thermo-Electric Analysis.......................................................................................... 303
10.1.2.16. Magnetic-Thermal Analysis...................................................................................... 304
10.1.2.17. Circuit-Magnetic Analysis......................................................................................... 304
10.1.2.18. Structural-Diffusion Analysis .................................................................................... 304
10.1.2.19.Thermal-Diffusion Analysis....................................................................................... 305
10.1.2.20. Structural-Thermal-Diffusion Analysis....................................................................... 305
10.2.Thermoelasticity ......................................................................................................................... 309
10.3.Thermoplasticity......................................................................................................................... 312
10.4. Piezoelectrics.............................................................................................................................. 313
10.5. Electroelasticity........................................................................................................................... 317
10.6. Piezoresistivity............................................................................................................................ 318
10.7.Thermoelectrics .......................................................................................................................... 319
10.8. Review of Coupled Electromechanical Methods........................................................................... 321
10.9. Porous Media Flow...................................................................................................................... 322
10.10. Structural-Diffusion Coupling.................................................................................................... 323
11. Shape Functions................................................................................................................................ 327
11.1. Understanding Shape Function Labels......................................................................................... 327
11.2. 2-D Lines .................................................................................................................................... 328
11.2.1. 2-D Lines without RDOF...................................................................................................... 329
11.2.2. 2-D Lines with RDOF........................................................................................................... 329
11.3. 3-D Lines .................................................................................................................................... 329
11.3.1. 3-D 2-Node Lines (Not Combining Translations and Rotations)............................................. 329
11.3.2. 3-D 2-Node Lines (Combining Translations and Rotations)................................................... 330
11.3.3. 3-D 3-Node Lines................................................................................................................ 330
11.3.4. 3-D 4-Node Lines................................................................................................................ 331
11.4. Axisymmetric Shells.................................................................................................................... 331
11.4.1. Axisymmetric Shell without ESF.......................................................................................... 331
11.5. Axisymmetric Harmonic Shells and General Axisymmetric Surfaces.............................................. 332
11.5.1. Axisymmetric Harmonic Shells............................................................................................ 332
11.5.1.1. Axisymmetric Harmonic Shells without ESF................................................................ 332
11.5.1.2. Axisymmetric Harmonic Shells with ESF..................................................................... 333
11.5.2. General Axisymmetric Surfaces........................................................................................... 333
11.6. 3-D Shells.................................................................................................................................... 335
11.6.1. 3-D 3-Node Triangular Shells without RDOF (CST)................................................................ 336
11.6.2. 3-D 6-Node Triangular Shells without RDOF (LST)................................................................ 336
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11.6.3. 3-D 3-Node Triangular Shells with RDOF but without SD...................................................... 337
11.6.4. 3-D 4-Node Quadrilateral Shells without RDOF and without ESF (Q4)................................... 337
11.6.5. 3-D 4-Node Quadrilateral Shells without RDOF but with ESF (QM6)...................................... 338
11.6.6. 3-D 8-Node Quadrilateral Shells without RDOF.................................................................... 338
11.6.7. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and without ESF....................... 339
11.6.8. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and with ESF............................. 339
11.7. 2-D and Axisymmetric Solids....................................................................................................... 339
11.7.1. 2-D and Axisymmetric 3-Node Triangular Solids (CST)......................................................... 340
11.7.2. 2-D and Axisymmetric 6-Node Triangular Solids (LST).......................................................... 340
11.7.3. 2-D and Axisymmetric 4-node Quadrilateral Solid without ESF (Q4)..................................... 341
11.7.4. 2-D and Axisymmetric 4-node Quadrilateral Solids with ESF (QM6)...................................... 341
11.7.5. 2-D and Axisymmetric 8-Node Quadrilateral Solids (Q8)...................................................... 342
11.7.6. 2-D and Axisymmetric 4-Node Quadrilateral Infinite Solids.................................................. 343
11.7.6.1. Lagrangian Isoparametric Shape Functions................................................................ 343
11.7.6.2. Mapping Functions.................................................................................................... 343
11.7.7. 2-D and Axisymmetric 8-Node Quadrilateral Infinite Solids.................................................. 344
11.7.7.1. Lagrangian Isoparametric Shape Functions................................................................ 344
11.7.7.2. Mapping Functions.................................................................................................... 344
11.8. Axisymmetric Harmonic Solids.................................................................................................... 344
11.8.1. Axisymmetric Harmonic 3-Node Triangular Solids............................................................... 345
11.8.2. Axisymmetric Harmonic 6-Node Triangular Solids............................................................... 345
11.8.3. Axisymmetric Harmonic 4-Node Quadrilateral Solids without ESF........................................ 346
11.8.4. Axisymmetric Harmonic 4-Node Quadrilateral Solids with ESF............................................. 346
11.8.5. Axisymmetric Harmonic 8-Node Quadrilateral Solids........................................................... 346
11.9. 3-D Solids................................................................................................................................... 347
11.9.1. 4-Node Tetrahedra.............................................................................................................. 347
11.9.2. 10-Node Tetrahedra............................................................................................................ 348
11.9.3. 5-Node Pyramids................................................................................................................ 348
11.9.4. 13-Node Pyramids.............................................................................................................. 349
11.9.5. 6-Node Wedges without ESF............................................................................................... 350
11.9.6. 6-Node Wedges with ESF.................................................................................................... 351
11.9.7. 15-Node Wedges................................................................................................................ 351
11.9.8. 8-Node Bricks without ESF.................................................................................................. 352
11.9.9. 8-Node Bricks with ESF....................................................................................................... 353
11.9.10. 20-Node Bricks................................................................................................................. 354
11.9.11. 8-Node Infinite Bricks........................................................................................................ 355
11.9.11.1. Lagrangian Isoparametric Shape Functions.............................................................. 355
11.9.11.2. Mapping Functions.................................................................................................. 356
11.9.12. 3-D 20-Node Infinite Bricks............................................................................................... 357
11.9.12.1. Lagrangian Isoparametric Shape Functions.............................................................. 357
11.9.12.2. Mapping Functions.................................................................................................. 358
11.9.13. General Axisymmetric Solids............................................................................................. 358
11.9.13.1. General Axisymmetric Solid with 4 Base Nodes......................................................... 360
11.9.13.2. General Axisymmetric Solid with 3 Base Nodes......................................................... 360
11.9.13.3. General Axisymmetric Solid with 8 Base Nodes......................................................... 360
11.9.13.4. General Axisymmetric Solid with 6 Base Nodes......................................................... 361
11.10. Electromagnetic Tangential Vector Elements.............................................................................. 361
11.10.1.Tetrahedral Elements........................................................................................................ 362
11.10.2. Hexahedral Elements........................................................................................................ 363
12. Element Tools.................................................................................................................................... 365
12.1. Element Shape Testing................................................................................................................ 365
12.1.1. Overview ........................................................................................................................... 365
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