ANSYS, Inc. Theory ReferenceANSYS Release 9.0

002114 November 2004

ANSYS, Inc. is a UL registered ISO 9001: 2000 Company.

ANSYS, Inc. Theory ReferenceANSYS Release 9.0

ANSYS, Inc. Southpointe 275 Technology Drive Canonsburg, PA 15317 ansysinfo@ansys.com http://www.ansys.com (T) 724-746-3304 (F) 724-514-9494

Copyright and Trademark InformationCopyright 2004 SAS IP, Inc. All rights reserved. Unauthorized use, distribution or duplication is prohibited. ANSYS, DesignSpace, CFX, DesignModeler, DesignXplorer, ANSYS Workbench environment, AI*Environment, CADOE and any and all ANSYS, Inc. product names referenced on any media, manual or the like, are registered trademarks or trademarks of subsidiaries of ANSYS, Inc. located in the United States or other countries. ICEM CFD is a trademark licensed by ANSYS, Inc. All other trademarks and registered trademarks are property of their respective owners. ANSYS, Inc. is a UL registered ISO 9001: 2000 Company. ANSYS Inc. products may contain U.S. Patent No. 6,055,541. Microsoft, Windows, Windows 2000 and Windows XP are registered trademarks of Microsoft Corporation. Inventor and Mechanical Desktop are registered trademarks of Autodesk, Inc. SolidWorks is a registered trademark of SolidWorks Corporation. Pro/ENGINEER is a registered trademark of Parametric Technology Corporation. Unigraphics, Solid Edge and Parasolid are registered trademarks of Electronic Data Systems Corporation (EDS). ACIS and ACIS Geometric Modeler are registered trademarks of Spatial Technology, Inc. FLEXlm License Manager is a trademark of Macrovision Corporation. This ANSYS, Inc. software product and program documentation is ANSYS Confidential Information and are furnished by ANSYS, Inc. under an ANSYS software license agreement that contains provisions concerning non-disclosure, copying, length and nature of use, warranties, disclaimers and remedies, and other provisions. The Program and Documentation may be used or copied only in accordance with the terms of that license agreement. See the ANSYS, Inc. online documentation or the ANSYS, Inc. documentation CD for the complete Legal Notice. If this is a copy of a document published by and reproduced with the permission of ANSYS, Inc., it might not reflect the organization or physical appearance of the original. ANSYS, Inc. is not liable for any errors or omissions introduced by the copying process. Such errors are the responsibility of the party providing the copy. Edited by: Peter Kohnke, Ph.D.

Table of Contents1. Introduction ........................................................................................................................................ 11 1.1. Purpose of the ANSYS Theory Reference ........................................................................................ 11 1.2. Notation ....................................................................................................................................... 11 1.3. Applicable Products ...................................................................................................................... 13 1.3.1. ANSYS Products ................................................................................................................... 13 1.3.2. ANSYS Workbench Products ................................................................................................. 14 1.4. Using the ANSYS, Inc. Theory Reference for the ANSYS Workbench Product ................................... 14 1.4.1. Elements Used by the ANSYS Workbench Product ................................................................ 14 1.4.2. Solvers Used by the ANSYS Workbench Product ................................................................... 14 1.4.3. Other Features ..................................................................................................................... 15 2. Structures ............................................................................................................................................ 21 2.1. Structural Fundamentals ............................................................................................................... 21 2.1.1. Stress-Strain Relationships ................................................................................................... 21 2.1.2. Orthotropic Material Transformation for Axisymmetric Models .............................................. 25 2.1.3. Temperature-Dependent Coefficient of Thermal Expansion .................................................. 26 2.2. Derivation of Structural Matrices ................................................................................................... 28 2.3. Structural Strain and Stress Evaluations ....................................................................................... 212 2.3.1. Integration Point Strains and Stresses ................................................................................. 212 2.3.2. Surface Stresses ................................................................................................................. 212 2.3.3. Shell Element Output ......................................................................................................... 213 2.4. Combined Stresses and Strains .................................................................................................... 215 2.4.1. Combined Strains ............................................................................................................... 215 2.4.2. Combined Stresses ............................................................................................................. 216 2.4.3. Failure Criteria .................................................................................................................... 217 2.4.4. Maximum Strain Failure Criteria .......................................................................................... 217 2.4.5. Maximum Stress Failure Criteria .......................................................................................... 218 2.4.6. Tsai-Wu Failure Criteria ....................................................................................................... 218 2.4.7. Safety Tools in the ANSYS Workbench Product .................................................................... 219 3. Structures with Geometric Nonlinearities .......................................................................................... 31 3.1. Large Strain .................................................................................................................................. 31 3.1.1. Theory ................................................................................................................................ 31 3.1.2. Implementation ................................................................................................................... 33 3.1.3. Definition of Thermal Strains ................................................................................................ 35 3.1.4. Element Formulation ........................................................................................................... 36 3.1.5. Applicable Input .................................................................................................................. 37 3.1.6. Applicable Output ............................................................................................................... 37 3.2. Large Rotation .............................................................................................................................. 37 3.2.1. Theory ................................................................................................................................. 38 3.2.2. Implementation ................................................................................................................... 38 3.2.3. Element Transformation ....................................................................................................... 39 3.2.4. Deformational Displacements ............................................................................................ 310 3.2.5. Updating Rotations ............................................................................................................ 311 3.2.6. Applicable Input ................................................................................................................ 311 3.2.7. Applicable Output .............................................................................................................. 311 3.2.8. Consistent Tangent Stiffness Matrix and Finite Rotation ...................................................... 311 3.3. Stress Stiffening .......................................................................................................................... 313 3.3.1. Overview and Usage .......................................................................................................... 313 3.3.2. Theory ............................................................................................................................... 313 3.3.3. Implementation ................................................................................................................. 316 3.3.4. Pressure Load Stiffness ....................................................................................................... 318ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . SAS IP, Inc.

ANSYS, Inc. Theory Reference 3.3.5. Applicable Input ................................................................................................................ 319 3.3.6. Applicable Output .............................................................................................................. 319 3.4. Spin Softening ............................................................................................................................ 319 3.5. General Element Formulations .................................................................................................... 322 3.5.1. Fundamental Equations ..................................................................................................... 323 3.5.2. Classical Pure Displacement Formulation ............................................................................ 324 3.5.3. Mixed u-P Formulations ..................................................................................................... 326 3.5.4. u-P Formulation I ............................................................................................................... 327 3.5.5. u-P Formulation II ............................................................................................................... 329 3.5.6. u-P Formulation III .............................................................................................................. 330 3.5.7. Volumetric Constraint Equations in u-P Formulations .......................................................... 330 3.6. Constraints and Lagrange Multiplier Method ............................................................................... 331 4. Structures with Material Nonlinearities .............................................................................................. 41 4.1. Rate-Independent Plasticity .......................................................................................................... 42 4.1.1. Theory ................................................................................................................................. 43 4.1.2. Yield Criterion ...................................................................................................................... 43 4.1.3. Flow Rule ............................................................................................................................. 46 4.1.4. Hardening Rule .................................................................................................................... 46 4.1.5. Plastic Strain Increment ........................................................................................................ 48 4.1.6. Implementation ................................................................................................................... 49 4.1.7. Elastoplastic Stress-Strain Matrix ......................................................................................... 411 4.1.8. Specialization for Hardening ............................................................................................... 411 4.1.9. Specification for Nonlinear Isotropic Hardening .................................................................. 412 4.1.10. Specialization for Bilinear Kinematic Hardening ................................................................ 413 4.1.11. Specialization for Multilinear Kinematic Hardening ........................................................... 415 4.1.12. Specialization for Nonlinear Kinematic Hardening ............................................................. 417 4.1.13. Specialization for Anisotropic Plasticity ............................................................................. 419 4.1.14. Hill Potential Theory ......................................................................................................... 419 4.1.15. Generalized Hill Potential Theory ...................................................................................... 421 4.1.16. Specialization for Drucker-Prager ...................................................................................... 425 4.1.17. Cast Iron Material Model ................................................................................................... 427 4.2. Rate-Dependent Plasticity ........................................................................................................... 431 4.2.1. Creep Option ..................................................................................................................... 431 4.2.1.1. Definition and Limitations ......................................................................................... 431 4.2.1.2. Calculation of Creep .................................................................................................. 432 4.2.1.3. Time Step Size ........................................................................................................... 434 4.2.2. Rate-Dependent Plasticity ................................................................................................. 434 4.2.2.1. Perzyna Option .......................................................................................................... 434 4.2.2.2. Peirce Option ............................................................................................................ 434 4.2.3. Anand Viscoplasticity ......................................................................................................... 435 4.2.3.1. Overview ................................................................................................................... 435 4.2.3.2. Theory ...................................................................................................................... 435 4.2.3.3. Implementation ........................................................................................................ 436 4.3. Gasket Material ........................................................................................................................... 437 4.3.1. Stress and Deformation ...................................................................................................... 438 4.3.2. Material Definition ............................................................................................................. 438 4.3.3. Thermal Deformation ......................................................................................................... 439 4.4. Nonlinear Elasticity ..................................................................................................................... 439 4.4.1. Overview and Guidelines for Use ........................................................................................ 439 4.5. Shape Memory Alloy Material Model ........................................................................................... 440 4.5.1. Background ....................................................................................................................... 440 4.5.2. The Continuum Mechanics Model ...................................................................................... 441

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ANSYS, Inc. Theory Reference 4.6. Hyperelasticity ............................................................................................................................ 444 4.6.1. Introduction ....................................................................................................................... 444 4.6.2. Finite Strain Elasticity ......................................................................................................... 444 4.6.3. Deviatoric-Volumetric Multiplicative Split ........................................................................... 445 4.6.4. Strain Energy Potentials ..................................................................................................... 446 4.6.4.1. Neo-Hookean ............................................................................................................ 446 4.6.4.2. Mooney-Rivlin ........................................................................................................... 447 4.6.4.3. Polynomial Form ....................................................................................................... 448 4.6.4.4. Ogden Potential ........................................................................................................ 448 4.6.4.5. Arruda-Boyce Model .................................................................................................. 449 4.6.4.6. Gent Model ............................................................................................................... 449 4.6.4.7. Yeoh Model ............................................................................................................... 450 4.6.4.8. Ogden Compressible Foam Model ............................................................................. 450 4.6.4.9. Blatz-Ko Model .......................................................................................................... 451 4.6.5. USER Subroutine ................................................................................................................ 451 4.6.6. Mooney-Rivlin (Using TB,MOONEY Command) .................................................................... 451 4.6.7. Output Quantities .............................................................................................................. 453 4.6.8. Determining Mooney-Rivlin Material Constants .................................................................. 453 4.6.9. Uniaxial Tension (Equivalently, Equibiaxial Compression) .................................................... 456 4.6.10. Equibiaxial Tension (Equivalently, Uniaxial Compression) .................................................. 457 4.6.11. Pure Shear ....................................................................................................................... 457 4.6.12. Least Squares Fit Analysis ................................................................................................. 458 4.6.13. Material Stability Check .................................................................................................... 459 4.7. Viscoelasticity ............................................................................................................................. 460 4.7.1. Small Strain Viscoelasticity .................................................................................................. 460 4.7.2. Constitutive Equations ....................................................................................................... 460 4.7.3. Numerical Integration ........................................................................................................ 462 4.7.4. Thermorheological Simplicity ............................................................................................. 463 4.7.5. Large Deformation Viscoelasticity ....................................................................................... 464 4.7.6. Visco-Hypoelasticity ........................................................................................................... 464 4.7.7. Large Strain Viscoelasticity ................................................................................................. 465 4.7.8. Shift Functions ................................................................................................................... 467 4.8. Concrete ..................................................................................................................................... 469 4.8.1. The Domain (Compression - Compression - Compression) ................................................... 471 4.8.2. The Domain (Tension - Compression - Compression) ........................................................... 474 4.8.3. The Domain (Tension - Tension - Compression) ................................................................... 474 4.8.4. The Domain (Tension - Tension - Tension) ........................................................................... 475 4.9. Swelling ..................................................................................................................................... 476 5. Electromagnetics ................................................................................................................................. 51 5.1. Electromagnetic Field Fundamentals ............................................................................................ 51 5.1.1. Magnetic Scalar Potential ..................................................................................................... 54 5.1.2. Solution Strategies ............................................................................................................... 54 5.1.2.1. RSP Strategy ................................................................................................................ 55 5.1.2.2. DSP Strategy ............................................................................................................... 56 5.1.2.3. GSP Strategy ............................................................................................................... 57 5.1.3. Magnetic Vector Potential .................................................................................................... 58 5.1.4. Edge Flux Degrees of Freedom ............................................................................................. 59 5.1.5. Limitation of the Nodal Vector Potential ............................................................................. 510 5.1.6. Harmonic Analysis Using Complex Formalism ..................................................................... 511 5.1.7. Nonlinear Time-Harmonic Magnetic Analysis ...................................................................... 513 5.1.8. Electric Scalar Potential ...................................................................................................... 514 5.1.8.1. Quasistatic Electric Analysis ....................................................................................... 515

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ANSYS, Inc. Theory Reference 5.1.8.2. Electrostatic Analysis ................................................................................................. 516 5.2. Derivation of Electromagnetic Matrices ....................................................................................... 517 5.2.1. Magnetic Scalar Potential ................................................................................................... 517 5.2.1.1. Degrees of freedom ................................................................................................... 517 5.2.1.2. Coefficient Matrix ...................................................................................................... 517 5.2.1.3. Applied Loads ........................................................................................................... 517 5.2.2. Magnetic Vector Potential .................................................................................................. 518 5.2.2.1. Degrees of Freedom .................................................................................................. 518 5.2.2.2. Coefficient Matrices ................................................................................................... 518 5.2.2.3. Applied Loads ........................................................................................................... 519 5.2.3. Electric Scalar Potential ...................................................................................................... 521 5.2.3.1. Quasistatic Electric Analysis ....................................................................................... 521 5.2.3.2. Electrostatic Analysis ................................................................................................. 522 5.3. Electromagnetic Field Evaluations ............................................................................................... 522 5.3.1. Magnetic Scalar Potential Results ....................................................................................... 522 5.3.2. Magnetic Vector Potential Results ....................................................................................... 523 5.3.3. Magnetic Forces ................................................................................................................. 524 5.3.3.1. Lorentz forces ............................................................................................................ 524 5.3.3.2. Maxwell Forces .......................................................................................................... 525 5.3.3.3. Virtual Work Forces .................................................................................................... 526 5.3.3.3.1. Element Shape Method ..................................................................................... 526 5.3.3.3.2. Nodal Perturbation Method .............................................................................. 527 5.3.4. Joule Heat in a Magnetic Analysis ....................................................................................... 527 5.3.5. Electric Scalar Potential Results ........................................................................................... 528 5.3.5.1. Quasistatic Electric Analysis ....................................................................................... 528 5.3.5.2. Electrostatic Analysis ................................................................................................ 529 5.3.6. Electrostatic Forces ............................................................................................................ 529 5.3.7. Electric Constitutive Error ................................................................................................... 530 5.4. Voltage Forced and Circuit-Coupled Magnetic Field ..................................................................... 531 5.4.1. Voltage Forced Magnetic Field ........................................................................................... 531 5.4.2. Circuit-Coupled Magnetic Field .......................................................................................... 532 5.5. High-Frequency Electromagnetic Field Simulation ....................................................................... 533 5.5.1. High-Frequency Electromagnetic Field FEA Principle .......................................................... 533 5.5.2. Boundary Conditions and Perfectly Matched Layers (PML) .................................................. 538 5.5.2.1. PEC Boundary Condition ............................................................................................ 538 5.5.2.2. PMC Boundary Condition ........................................................................................... 538 5.5.2.3. Impedance Boundary Condition ................................................................................ 539 5.5.2.4. Perfectly Matched Layers ........................................................................................... 540 5.5.2.5. Periodic Boundary Condition ..................................................................................... 542 5.5.3. Excitation Sources .............................................................................................................. 542 5.5.3.1. Waveguide Modal Sources ......................................................................................... 542 5.5.3.2. Current Excitation Source .......................................................................................... 542 5.5.3.3. Plane Wave Source .................................................................................................... 543 5.5.3.4. Surface Magnetic Field Source ................................................................................... 543 5.5.3.5. Electric Field Source ................................................................................................... 544 5.5.4. High-Frequency Parameters Evaluations ............................................................................. 544 5.5.4.1. Electric Field .............................................................................................................. 544 5.5.4.2. Magnetic Field ........................................................................................................... 544 5.5.4.3. Poynting Vector ......................................................................................................... 544 5.5.4.4. Power Flow ............................................................................................................... 545 5.5.4.5. Stored Energy ............................................................................................................ 545 5.5.4.6. Dielectric Loss ........................................................................................................... 545

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ANSYS, Inc. Theory Reference 5.5.4.7. Surface Loss .............................................................................................................. 545 5.5.4.8. Quality Factor ............................................................................................................ 545 5.5.4.9. Voltage ..................................................................................................................... 546 5.5.4.10. Current .................................................................................................................... 546 5.5.4.11. Characteristic Impedance ........................................................................................ 546 5.5.4.12. Scattering Matrix (S-Parameter) ................................................................................ 546 5.5.4.13. Surface Equivalence Principle ................................................................................... 548 5.5.4.14. Radar Cross Section (RCS) ......................................................................................... 549 5.5.4.15. Antenna Pattern ...................................................................................................... 550 5.5.4.16. Antenna Radiation Power ........................................................................................ 551 5.5.4.17. Antenna Directive Gain ............................................................................................ 551 5.5.4.18. Antenna Power Gain ................................................................................................ 551 5.5.4.19. Antenna Radiation Efficiency ................................................................................... 552 5.5.4.20. Electromagnetic Field of Phased Array Antenna ....................................................... 552 5.5.4.21. Specific Absorption Rate (SAR) ................................................................................. 552 5.5.4.22. Power Reflection and Transmission Coefficient ......................................................... 552 5.5.4.23. Reflection and Transmission Coefficient in Periodic Structure ................................... 553 5.5.4.24. The Smith Chart ....................................................................................................... 554 5.5.4.25. Conversion Among Scattering Matrix (S-parameter), Admittance Matrix (Y-parameter), and Impedance Matrix (Z-parameter) .................................................................................... 554 5.6. Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros .............................. 555 5.6.1. Differential Inductance Definition ....................................................................................... 556 5.6.2. Review of Inductance Computation Methods ...................................................................... 557 5.6.3. Inductance Computation Method Used .............................................................................. 557 5.6.4. Transformer and Motion Induced Voltages ......................................................................... 557 5.6.5. Absolute Flux Computation ................................................................................................ 558 5.6.6. Inductance Computations .................................................................................................. 559 5.6.7. Absolute Energy Computation ........................................................................................... 559 5.7. Electromagnetic Particle Tracing ................................................................................................. 560 5.8. Maxwell Stress Tensor ................................................................................................................. 561 5.8.1. Notation ............................................................................................................................ 561 5.8.2. Fundamental Relations ....................................................................................................... 562 5.8.3. Derived Relations ............................................................................................................... 563 5.8.4. Maxwell Tensor From Maxwell's Equations .......................................................................... 564 5.9. Electromechanical Transducers ................................................................................................... 565 5.10. Capacitance Computation ......................................................................................................... 565 5.11. Open Boundary Analysis with a Trefftz Domain .......................................................................... 568 5.12. Circuit Analysis, Reduced Order Modeling ................................................................................. 569 5.12.1. Mechanical Circuit Elements ............................................................................................. 569 5.12.2. Electrical Circuit Elements ................................................................................................. 570 5.12.3. Coupled Field Circuit Elements ......................................................................................... 570 5.13. Conductance Computation ....................................................................................................... 570 6. Heat Flow ............................................................................................................................................. 61 6.1. Heat Flow Fundamentals .............................................................................................................. 61 6.1.1. Conduction and Convection ................................................................................................. 61 6.1.2. Radiation ............................................................................................................................. 63 6.1.2.1. View Factors ................................................................................................................ 64 6.1.2.2. Radiation Usage .......................................................................................................... 64 6.2. Derivation of Heat Flow Matrices ................................................................................................... 65 6.3. Heat Flow Evaluations ................................................................................................................... 67 6.3.1. Integration Point Output ...................................................................................................... 67 6.3.2. Surface Output .................................................................................................................... 67

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ANSYS, Inc. Theory Reference 6.4. Radiation Matrix Method .............................................................................................................. 68 6.4.1. Non-Hidden Method ............................................................................................................ 69 6.4.2. Hidden Method .................................................................................................................... 69 6.4.3. View Factors of Axisymmetric Bodies .................................................................................. 610 6.4.4. Space Node ........................................................................................................................ 612 6.5. Radiosity Solution Method .......................................................................................................... 612 6.5.1. View Factor Calculation - Hemicube Method ....................................................................... 613 7. Fluid Flow ............................................................................................................................................ 71 7.1. Fluid Flow Fundamentals .............................................................................................................. 71 7.1.1. Continuity Equation ............................................................................................................. 71 7.1.2. Momentum Equation ........................................................................................................... 72 7.1.3. Compressible Energy Equation ............................................................................................. 73 7.1.4. Incompressible Energy Equation .......................................................................................... 74 7.1.5. Turbulence .......................................................................................................................... 75 7.1.5.1. Standard k- Model ..................................................................................................... 77 7.1.5.2. RNG Turbulence Model ................................................................................................ 79 7.1.5.3. NKE Turbulence Model .............................................................................................. 710 7.1.5.4. GIR Turbulence Model ............................................................................................... 711 7.1.5.5. SZL Turbulence Model ............................................................................................... 712 7.1.5.6. Standard k- Model ................................................................................................... 712 7.1.5.7. SST Turbulence Model ............................................................................................... 713 7.1.5.8. Near-Wall Treatment ................................................................................................. 714 7.1.6. Pressure ............................................................................................................................. 716 7.1.7. Multiple Species Transport ................................................................................................. 717 7.1.8. Arbitrary Lagrangian-Eulerian (ALE) Formulation ................................................................. 718 7.2. Derivation of Fluid Flow Matrices ................................................................................................ 719 7.2.1. Discretization of Equations ................................................................................................. 719 7.2.2. Transient Term ................................................................................................................... 720 7.2.3. Advection Term ................................................................................................................. 722 7.2.4. Monotone Streamline Upwind Approach (MSU) .................................................................. 722 7.2.5. Streamline Upwind/Petro-Galerkin Approach (SUPG) .......................................................... 723 7.2.6. Collocated Galerkin Approach (COLG) ................................................................................ 724 7.2.7. Diffusion Terms .................................................................................................................. 724 7.2.8. Source Terms ..................................................................................................................... 725 7.2.9. Segregated Solution Algorithm .......................................................................................... 725 7.3. Volume of Fluid Method for Free Surface Flows ........................................................................... 731 7.3.1. Overview ........................................................................................................................... 731 7.3.2. CLEAR-VOF Advection ........................................................................................................ 731 7.3.3. CLEAR-VOF Reconstruction ................................................................................................ 733 7.3.4. Treatment of Finite Element Equations ............................................................................... 734 7.3.5. Treatment of Volume Fraction Field .................................................................................... 735 7.3.6. Treatment of Surface Tension Field ..................................................................................... 737 7.4. Fluid Solvers ............................................................................................................................... 738 7.5. Overall Convergence and Stability ............................................................................................... 739 7.5.1. Convergence ..................................................................................................................... 739 7.5.2. Stability ............................................................................................................................. 740 7.5.2.1. Relaxation ................................................................................................................. 740 7.5.2.2. Inertial Relaxation ...................................................................................................... 740 7.5.2.3. Artificial Viscosity ....................................................................................................... 740 7.5.3. Residual File ....................................................................................................................... 741 7.5.4. Modified Inertial Relaxation ................................................................................................ 741 7.6. Fluid Properties .......................................................................................................................... 742

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ANSYS, Inc. Theory Reference 7.6.1. Density .............................................................................................................................. 742 7.6.2. Viscosity ............................................................................................................................. 743 7.6.3. Thermal Conductivity ......................................................................................................... 746 7.6.4. Specific Heat ...................................................................................................................... 747 7.6.5. Surface Tension Coefficient ................................................................................................ 747 7.6.6. Wall Static Contact Angle ................................................................................................... 748 7.6.7. Multiple Species Property Options ...................................................................................... 748 7.7. Derived Quantities ...................................................................................................................... 749 7.7.1. Mach Number .................................................................................................................... 749 7.7.2. Total Pressure .................................................................................................................... 749 7.7.3. Y-Plus and Wall Shear Stress ............................................................................................... 750 7.7.4. Stream Function ................................................................................................................. 751 7.7.4.1. Cartesian Geometry ................................................................................................... 751 7.7.4.2. Axisymmetric Geometry (about x) .............................................................................. 751 7.7.4.3. Axisymmetric Geometry (about y) .............................................................................. 751 7.7.4.4. Polar Coordinates ...................................................................................................... 751 7.7.5. Heat Transfer Film Coefficient ............................................................................................. 752 7.7.5.1. Matrix Procedure ....................................................................................................... 752 7.7.5.2. Thermal Gradient Procedure ...................................................................................... 752 7.7.5.3. Film Coefficient Evaluation ........................................................................................ 752 7.8. Squeeze Film Theory ................................................................................................................... 753 7.8.1. Flow Between Flat Surfaces ................................................................................................ 753 7.8.2. Flow in Channels ................................................................................................................ 754 7.9. Slide Film Theory ........................................................................................................................ 755 8. Acoustics ............................................................................................................................................. 81 8.1. Acoustic Fluid Fundamentals ........................................................................................................ 81 8.1.1. Governing Equations ........................................................................................................... 81 8.1.2. Discretization of the Lossless Wave Equation ........................................................................ 82 8.2. Derivation of Acoustics Fluid Matrices ........................................................................................... 83 8.3. Absorption of Acoustical Pressure Wave ........................................................................................ 84 8.3.1. Addition of Dissipation due to Damping at the Boundary ...................................................... 84 8.4. Acoustics Fluid-Structure Coupling ............................................................................................... 86 8.5. Acoustics Output Quantities ......................................................................................................... 87 9. This chapter intentionally omitted. .................................................................................................... 91 10. This chapter intentionally omitted. ................................................................................................ 101 11. Coupling .......................................................................................................................................... 111 11.1. Coupled Effects ......................................................................................................................... 111 11.1.1. Introduction ..................................................................................................................... 111 11.1.1.1. Advantages ............................................................................................................. 112 11.1.1.2. Disadvantages ......................................................................................................... 112 11.1.2. Coupling .......................................................................................................................... 112 11.1.2.1. Thermal-Structural Analysis ..................................................................................... 114 11.1.2.2. Magneto-Structural Analysis (Vector Potential) ......................................................... 114 11.1.2.3. Magneto-Structural Analysis (Scalar Potential) .......................................................... 114 11.1.2.4. Electromagnetic Analysis ......................................................................................... 115 11.1.2.5. Electro-Magneto-Thermo-Structural Analysis ........................................................... 115 11.1.2.6. Electro-Magneto-Thermal Analysis ........................................................................... 116 11.1.2.7. Piezoelectric Analysis ............................................................................................... 116 11.1.2.8. Piezoresistive Analysis ............................................................................................. 116 11.1.2.9. Thermo-Pressure Analysis ........................................................................................ 117 11.1.2.10. Velocity-Thermo-Pressure Analysis ......................................................................... 117 11.1.2.11. Pressure-Structural (Acoustic) Analysis ................................................................... 118ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . SAS IP, Inc.

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ANSYS, Inc. Theory Reference 11.1.2.12. Thermo-Electric Analysis ........................................................................................ 119 11.1.2.13. Magnetic-Thermal Analysis .................................................................................... 119 11.1.2.14. Circuit-Magnetic Analysis ..................................................................................... 1110 11.2. Piezoelectrics .......................................................................................................................... 1113 11.2.1. Structural Mass ............................................................................................................... 1116 11.2.2. Structural Damping ........................................................................................................ 1117 11.2.3. Structural Stiffness ......................................................................................................... 1117 11.2.4. Dielectric Conductivity ................................................................................................... 1117 11.2.5. Piezoelectric Coupling Matrix ......................................................................................... 1117 11.2.6. Structural Load Vector .................................................................................................... 1117 11.2.7. Electrical Load Vector ..................................................................................................... 1117 11.2.8. Elastic Energy ................................................................................................................. 1117 11.2.9. Dielectric Energy ............................................................................................................ 1117 11.2.10. Electromechanical Coupling Energy .............................................................................. 1117 11.3. Piezoresistivity ........................................................................................................................ 1118 11.4. Thermoelectrics ...................................................................................................................... 1119 11.5. Review of Coupled Electromechanical Methods ....................................................................... 1121 12. Shape Functions .............................................................................................................................. 121 12.1. 2-D Lines .................................................................................................................................. 122 12.1.1. 2-D Lines without RDOF ................................................................................................... 122 12.1.2. 2-D Lines with RDOF ......................................................................................................... 122 12.2. 3-D Lines .................................................................................................................................. 123 12.2.1. 3-D 2 Node Lines without RDOF ........................................................................................ 123 12.2.2. 3-D 2 Node Lines with RDOF ............................................................................................. 124 12.2.3. 3-D 3 Node Lines .............................................................................................................. 124 12.3. Axisymmetric Shells .................................................................................................................. 125 12.3.1. Axisymmetric Shell without ESF ........................................................................................ 125 12.3.2. Axisymmetric Shell with ESF ............................................................................................. 125 12.4. Axisymmetric Harmonic Shells .................................................................................................. 126 12.4.1. Axisymmetric Harmonic Shells without ESF ....................................................................... 126 12.4.2. Axisymmetric Harmonic Shells with ESF ............................................................................ 126 12.5. 3-D Shells ................................................................................................................................. 127 12.5.1. 3-D 3-Node Triangular Shells without RDOF (CST) ............................................................. 128 12.5.2. 3-D 6-Node Triangular Shells without RDOF (LST) .............................................................. 128 12.5.3. 3-D 3-Node Triangular Shells with RDOF but without SD ................................................... 128 12.5.4. 3-D 3-Node Triangular Shells with RDOF and with SD ........................................................ 129 12.5.5. 3-D 6-Node Triangular Shells with RDOF and with SD ...................................................... 1210 12.5.6. 3-D 4-Node Quadrilateral Shells without RDOF and without ESF (Q4) ............................... 1211 12.5.7. 3-D 4-Node Quadrilateral Shells without RDOF but with ESF (QM6) .................................. 1211 12.5.8. 3-D 8-Node Quadrilateral Shells without RDOF ................................................................ 1212 12.5.9. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and without ESF ................... 1212 12.5.10. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and with ESF ....................... 1213 12.5.11. 3-D 4-Node Quadrilateral Shells with RDOF and with SD ................................................ 1213 12.5.12. 3-D 8-Node Quadrilateral Shells with RDOF and with SD ................................................ 1215 12.6. 2-D and Axisymmetric Solids ................................................................................................... 1215 12.6.1. 2-D and Axisymmetric 3 Node Triangular Solids (CST) ...................................................... 1216 12.6.2. 2-D and Axisymmetric 6 Node Triangular Solids (LST) ...................................................... 1217 12.6.3. 2-D and Axisymmetric 4 Node Quadrilateral Solid without ESF (Q4) ................................. 1217 12.6.4. 2-D and Axisymmetric 4 Node Quadrilateral Solids with ESF (QM6) .................................. 1218 12.6.5. 2-D and Axisymmetric 8 Node Quadrilateral Solids (Q8) ................................................... 1218 12.6.6. 2-D and Axisymmetric 4 Node Quadrilateral Infinite Solids .............................................. 1219 12.6.6.1. Lagrangian Isoparametric Shape Functions ............................................................ 1219

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ANSYS, Inc. Theory Reference 12.6.6.2. Mapping Functions ................................................................................................ 1220 12.6.7. 2-D and Axisymmetric 8 Node Quadrilateral Infinite Solids .............................................. 1220 12.6.7.1. Lagrangian Isoparametric Shape Functions ............................................................ 1220 12.6.7.2. Mapping Functions ................................................................................................ 1221 12.7. Axisymmetric Harmonic Solids ................................................................................................ 1221 12.7.1. Axisymmetric Harmonic 3 Node Triangular Solids ........................................................... 1221 12.7.2. Axisymmetric Harmonic 6 Node Triangular Solids ........................................................... 1222 12.7.3. Axisymmetric Harmonic 4 Node Quadrilateral Solids without ESF .................................... 1222 12.7.4. Axisymmetric Harmonic 4 Node Quadrilateral Solids with ESF ......................................... 1222 12.7.5. Axisymmetric Harmonic 8 Node Quadrilateral Solids ....................................................... 1223 12.8. 3-D Solids ............................................................................................................................... 1223 12.8.1. 4 Node Tetrahedra ......................................................................................................... 1224 12.8.2. 10 Node Tetrahedra ........................................................................................................ 1224 12.8.3. 5 Node Pyramids ............................................................................................................ 1225 12.8.4. 13 Node Pyramids .......................................................................................................... 1226 12.8.5. 6 Node Wedges without ESF ........................................................................................... 1227 12.8.6. 6 Node Wedges with ESF ................................................................................................ 1228 12.8.7. 15 Node Wedges as a Condensation of 20 Node Brick ...................................................... 1228 12.8.8. 15 Node Wedges Based on Wedge Shape Functions ....................................................... 1229 12.8.9. 8 Node Bricks without ESF .............................................................................................. 1230 12.8.10. 8 Node Bricks with ESF .................................................................................................. 1231 12.8.11. 20 Node Bricks .............................................................................................................. 1232 12.8.12. 8 Node Infinite Bricks .................................................................................................... 1233 12.8.12.1. Lagrangian Isoparametric Shape Functions .......................................................... 1233 12.8.12.2. Mapping Functions .............................................................................................. 1234 12.8.13. 3-D 20 Node Infinite Bricks ............................................................................................ 1234 12.8.13.1. Lagrangian Isoparametric Shape Functions .......................................................... 1235 12.8.13.2. Mapping Functions .............................................................................................. 1236 12.9. Electromagnetic Edge Elements .............................................................................................. 1236 12.9.1. 2-D 8 Node Quad Geometry and DOFs ............................................................................ 1237 12.9.2. 3-D 20 Node Brick Geometry and DOFs ........................................................................... 1239 12.10. High Frequency Electromagnetic Tangential Vector Elements ................................................ 1241 12.10.1. Tetrahedral Elements (HF119) ....................................................................................... 1241 12.10.2. Hexahedral Elements (HF120) ....................................................................................... 1243 12.10.3. Triangular Elements (HF118) ......................................................................................... 1245 12.10.4. Quadrilateral Elements (HF118) ..................................................................................... 1247 13. Element Tools .................................................................................................................................. 131 13.1. Integration Point Locations ....................................................................................................... 131 13.1.1. Lines (1, 2, or 3 Points) ...................................................................................................... 131 13.1.2. Quadrilaterals (2 x 2 or 3 x 3 Points) ................................................................................... 131 13.1.3. Bricks and Pyramids (2 x 2 x 2 Points) ................................................................................ 132 13.1.4. Triangles (1, 3, or 6 Points) ................................................................................................ 133 13.1.5. Tetrahedra (1, 4, 5, or 11 Points) ........................................................................................ 133 13.1.6. Triangles and Tetrahedra (2 x 2 or 2 x 2 x 2 Points) ............................................................. 135 13.1.7. Wedges (3 x 2 or 3 x 3 Points) ............................................................................................ 135 13.1.8. Wedges (2 x 2 x 2 Points) .................................................................................................. 136 13.1.9. Bricks (14 Points) .............................................................................................................. 136 13.1.10. Nonlinear Bending (5 Points) .......................................................................................... 137 13.2. Lumped Matrices ...................................................................................................................... 138 13.2.1. Diagonalization Procedure ............................................................................................... 138 13.2.2. Limitations of Lumped Mass Matrices ............................................................................... 139 13.3. Reuse of Matrices ...................................................................................................................... 139

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ANSYS, Inc. Theory Reference 13.3.1. Element Matrices .............................................................................................................. 139 13.3.2. Structure Matrices .......................................................................................................... 1310 13.3.3. Override Option ............................................................................................................. 1310 13.4. Temperature-Dependent Material Properties ........................................................................... 1310 13.5. Positive Definite Matrices ........................................................................................................ 1311 13.5.1. Matrices Representing the Complete Structure ............................................................... 1311 13.5.2. Element Matrices ............................................................................................................ 1312 13.6. Nodal and Centroidal Data Evaluation ..................................................................................... 1312 13.7. Element Shape Testing ............................................................................................................ 1313 13.7.1. Overview ........................................................................................................................ 1313 13.7.2. 3-D Solid Element Faces and Cross-Sections .................................................................... 1313 13.7.3. Aspect Ratio ................................................................................................................... 1316 13.7.4. Aspect Ratio Calculation for Triangles ............................................................................. 1316 13.7.5. Aspect Ratio Calculation for Quadrilaterals ...................................................................... 1317 13.7.6. Angle Deviation ............................................................................................................. 1318 13.7.7. Angle Deviation Calculation ........................................................................................... 1318 13.7.8. Parallel Deviation ........................................................................................................... 1319 13.7.9. Parallel Deviation Calculation ......................................................................................... 1319 13.7.10. Maximum Corner Angle ................................................................................................ 1320 13.7.11. Maximum Corner Angle Calculation .............................................................................. 1320 13.7.12. Jacobian Ratio .............................................................................................................. 1322 13.7.12.1. Jacobian Ratio Calculation ................................................................................... 1322 13.7.13. Warping Factor ............................................................................................................. 1324 13.7.13.1. Warping Factor Calculation for Quadrilateral Shell Elements ................................. 1324 13.7.13.2. Warping Factor Calculation for 3-D Solid Elements ................................................ 1326 14. Element Library ............................................................................................................................... 141 14.1. LINK1 - 2-D Spar (or Truss) ......................................................................................................... 141 14.1.1. Assumptions and Restrictions ........................................................................................... 141 14.1.2. Other Applicable Sections ................................................................................................ 141 14.2. PLANE2 - 2-D 6-Node Triangular Structural Solid ........................................................................ 142 14.2.1. Other Applicable Sections ................................................................................................ 142 14.3. BEAM3 - 2-D Elastic Beam .......................................................................................................... 142 14.3.1. Element Matrices and Load Vectors .................................................................................. 143 14.3.2. Stress Calculation ............................................................................................................. 145 14.4. BEAM4 - 3-D Elastic Beam .......................................................................................................... 146 14.4.1. Stiffness and Mass Matrices .............................................................................................. 146 14.4.2. Gyroscopic Damping Matrix ........................................................................................... 1410 14.4.3. Pressure and Temperature Load Vector ........................................................................... 1410 14.4.4. Local to Global Conversion ............................................................................................. 1410 14.4.5. Stress Calculations .......................................................................................................... 1412 14.5. SOLID5 - 3-D Coupled-Field Solid ............................................................................................. 1413 14.5.1. Other Applicable Sections .............................................................................................. 1414 14.6. Not Documented .................................................................................................................... 1414 14.7. COMBIN7 - Revolute Joint ....................................................................................................... 1414 14.7.1. Element Description ....................................................................................................... 1414 14.7.2. Element Matrices ............................................................................................................ 1416 14.7.3. Modification of Real Constants ........................................................................................ 1418 14.8. LINK8 - 3-D Spar (or Truss) ....................................................................................................... 1419 14.8.1. Assumptions and Restrictions ......................................................................................... 1419 14.8.2. Element Matrices and Load Vector .................................................................................. 1419 14.8.3. Force and Stress ............................................................................................................. 1422 14.9. INFIN9 - 2-D Infinite Boundary ................................................................................................. 1423

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ANSYS, Inc. Theory Reference 14.9.1. Introduction ................................................................................................................... 1423 14.9.2. Theory ........................................................................................................................... 1423 14.10. LINK10 - Tension-only or Compression-only Spar ................................................................... 1426 14.10.1. Assumptions and Restrictions ....................................................................................... 1426 14.10.2. Element Matrices and Load Vector ................................................................................ 1426 14.11. LINK11 - Linear Actuator ........................................................................................................ 1428 14.11.1. Assumptions and Restrictions ....................................................................................... 1429 14.11.2. Element Matrices and Load Vector ................................................................................ 1429 14.11.3. Force, Stroke, and Length ............................................................................................. 1430 14.12. CONTAC12 - 2-D Point-to-Point Contact ................................................................................ 1431 14.12.1. Element Matrices .......................................................................................................... 1431 14.12.2. Orientation of the Element ........................................................................................... 1433 14.12.3. Rigid Coulomb Friction ................................................................................................. 1433 14.13. PLANE13 - 2-D Coupled-Field Solid ........................................................................................ 1434 14.13.1. Other Applicable Sections ............................................................................................. 1435 14.14. COMBIN14 - Spring-Damper .................................................................................................. 1435 14.14.1. Types of Input .............................................................................................................. 1436 14.14.2. Stiffness Pass ................................................................................................................ 1436 14.14.3. Output Quantities ........................................................................................................ 1437 14.15. Not Documented .................................................................................................................. 1438 14.16. PIPE16 - Elastic Straight Pipe .................................................................................................. 1438 14.16.1. Other Applicable Sections ............................................................................................. 1439 14.16.2. Assumptions and Restrictions ....................................................................................... 1439 14.16.3. Stiffness Matrix ............................................................................................................. 1439 14.16.4. Mass Matrix .................................................................................................................. 1440 14.16.5. Gyroscopic Damping Matrix .......................................................................................... 1440 14.16.6. Stress Stiffness Matrix ................................................................................................... 1441 14.16.7. Load Vector .................................................................................................................. 1441 14.16.8. Stress Calculation ......................................................................................................... 1443 14.17. PIPE17 - Elastic Pipe Tee ........................................................................................................ 1449 14.17.1. Other Applicable Sections ............................................................................................. 1449 14.18. PIPE18 - Elastic Curved Pipe (Elbow) ....................................................................................... 1449 14.18.1. Other Applicable Sections ............................................................................................. 1450 14.18.2. Stiffness Matrix ............................................................................................................. 1450 14.18.3. Mass Matrix .................................................................................................................. 1453 14.18.4. Load Vector .................................................................................................................. 1453 14.18.5. Stress Calculations ........................................................................................................ 1454 14.19. Not Documented .................................................................................................................. 1454 14.20. PIPE20 - Plastic Straight Pipe .................................................................................................. 1454 14.20.1. Assumptions and Restrictions ....................................................................................... 1455 14.20.2. Other Applicable Sections ............................................................................................. 1455 14.20.3. Stress and Strain Calculation ......................................................................................... 1455 14.21. MASS21 - Structural Mass ...................................................................................................... 1459 14.22. Not Documented .................................................................................................................. 1460 14.23. BEAM23 - 2-D Plastic Beam .................................................................................................... 1460 14.23.1. Other Applicable Sections ............................................................................................. 1461 14.23.2. Integration Points ......................................................................................................... 1461 14.23.3. Tangent Stiffness Matrix for Plasticity ............................................................................ 1465 14.23.4. Newton-Raphson Load Vector ...................................................................................... 1467 14.23.5. Stress and Strain Calculation ......................................................................................... 1470 14.24. BEAM24 - 3-D Thin-walled Beam ............................................................................................ 1471 14.24.1. Assumptions and Restrictions ...................................................................................... 1472

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ANSYS, Inc. Theory Reference 14.24.2. Other Applicable Sections ............................................................................................. 1472 14.24.3. Temperature Distribution Across Cross-Section ............................................................. 1472 14.24.4. Calculation of Cross-Section Section Properties ............................................................. 1473 14.24.5. Offset Transformation ................................................................................................... 1478 14.25. PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid ................................................... 1481 14.25.1. Other Applicable Sections ............................................................................................. 1482 14.25.2. Assumptions and Restrictions ....................................................................................... 1482 14.25.3. Use of Temperature ...................................................................................................... 1482 14.26. CONTAC26 - 2-D Point-to-Ground Contact ............................................................................. 1482 14.27. MATRIX27 - Stiffness, Damping, or Mass Matrix ...................................................................... 1482 14.27.1. Assumptions and Restrictions ....................................................................................... 1482 14.28. SHELL28 - Shear/Twist Panel .................................................................................................. 1483 14.28.1. Assumptions and Restrictions ....................................................................................... 1483 14.28.2. Commentary ................................................................................................................ 1483 14.28.3. Output Terms ............................................................................................................... 1484 14.29. FLUID29 - 2-D Acoustic Fluid ................................................................................................. 1485 14.29.1. Other Applicable Sections ............................................................................................. 1485 14.30. FLUID30 - 3-D Acoustic Fluid ................................................................................................. 1486 14.30.1. Other Applicable Sections ............................................................................................. 1486 14.31. LINK31 - Radiation Link .......................................................................................................... 1486 14.31.1. Standard Radiation (KEYOPT(3) = 0) .............................................................................. 1486 14.31.2. Empirical Radiation (KEYOPT(3) = 1) .............................................................................. 1487 14.31.3. Solution ....................................................................................................................... 1487 14.32. LINK32 - 2-D Conduction Bar ................................................................................................. 1488 14.32.1. Other Applicable Sections ............................................................................................. 1488 14.32.2. Matrices and Load Vectors ............................................................................................ 1488 14.33. LINK33 - 3-D Conduction Bar ................................................................................................. 1489 14.33.1. Other Applicable Sections ............................................................................................. 1489 14.33.2. Matrices and Load Vectors ............................................................................................ 1489 14.33.3. Output ......................................................................................................................... 1490 14.34. LINK34 - Convection Link ....................................................................................................... 1490 14.34.1. Conductivity Matrix ...................................................................................................... 1490 14.34.2. Output ......................................................................................................................... 1491 14.35. PLANE35 - 2-D 6-Node Triangular Thermal Solid ..................................................................... 1492 14.35.1. Other Applicable Sections ............................................................................................. 1492 14.36. SOURC36 - Current Source ..................................................................................................... 1493 14.36.1. Description .................................................................................................................. 1493 14.37. COMBIN37 - Control .............................................................................................................. 1493 14.37.1. Element Characteristics ................................................................................................ 1494 14.37.2. Element Matrices .......................................................................................................... 1495 14.37.3. Adjustment of Real Constants ....................................................................................... 1495 14.37.4. Evaluation of Control Parameter ................................................................................... 1496 14.38. FLUID38 - Dynamic Fluid Coupling ........................................................................................ 1497 14.38.1. Description .................................................................................................................. 1497 14.38.2. Assumptions and Restrictions ....................................................................................... 1498 14.38.3. Mass Matrix Formulation .............................................................................................. 1498 14.38.4. Damping Matrix Formulation ........................................................................................ 1499 14.39. COMBIN39 - Nonlinear Spring .............................................................................................. 14100 14.39.1. Input .......................................................................................................................... 14100 14.39.2. Element Stiffness Matrix and Load Vector .................................................................... 14101 14.39.3. Choices for Element Behavior ...................................................................................... 14102 14.40. COMBIN40 - Combination .................................................................................................... 14105

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ANSYS, Inc. Theory Reference 14.40.1. Characteristics of the Element ..................................................................................... 14105 14.40.2. Element Matrices for Structural Applications ............................................................... 14106 14.40.3. Determination of F1 and F2 for Structural Applications ................................................ 14107 14.40.4. Thermal Analysis ......................................................................................................... 14108 14.41. SHELL41 - Membrane Shell .................................................................................................. 14108 14.41.1. Assumptions and Restrictions ..................................................................................... 14109 14.41.2. Wrinkle Option ........................................................................................................... 14109 14.42. PLANE42 - 2-D Structural Solid ............................................................................................. 14110 14.42.1. Other Applicable Sections ........................................................................................... 14110 14.43. SHELL43 - 4-Node Plastic Large Strain Shell .......................................................................... 14111 14.43.1. Other Applicable Sections ........................................................................................... 14112 14.43.2. Assumptions and Restrictions ..................................................................................... 14112 14.43.3. Assumed Displacement Shape Functions .................................................................... 14112 14.43.4. Stress-Strain Relationships .......................................................................................... 14112 14.43.5. In-Plane Rotational DOF .............................................................................................. 14113 14.43.6. Spurious Mode Control with Allman Rotation .............................................................. 14113 14.43.7. Natural Space Extra Shape Functions with Allman Rotation ......................................... 14115 14.43.8. Warping ..........................................