Advanced Nonlinear Tmg

NX Nastran Advanced Nonlinear Theory and Modeling

Guide

Proprietary & Restricted Rights Notice

© 2007 UGS Corp. All Rights Reserved. This software and related documentation are proprietary to UGS Corp. NASTRAN is a registered trademark of the National Aeronautics and Space Administration. NX Nastran is an enhanced proprietary version developed and maintained by UGS Corp. MSC is a registered trademark of MSC.Software Corporation. MSC.Nastran and MSC.Patran are trademarks of MSC.Software Corporation. All other trademarks are the property of their respective owners.

Table of contents

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Table of contents 1. Introduction ......................................................................................................... 1

1.1 Objective of this manual.................................................................................. 1 1.2 Overview of Advanced Nonlinear Solution .................................................... 2

1.2.1 Choosing Between Solutions 601 and 701............................................... 3 1.3 Structure of Advanced Nonlinear Solution ..................................................... 6

1.3.1 Executive Control..................................................................................... 6 1.3.2 Case Control ............................................................................................. 7 1.3.3 Bulk Data.................................................................................................. 9 1.3.4 Terminology used in Advanced Nonlinear Solution .............................. 11

2. Elements ............................................................................................................. 12 2.1 Rod elements ................................................................................................. 18

2.1.1 General considerations ........................................................................... 18 2.1.2 Material models and formulations.......................................................... 19 2.1.3 Numerical integration............................................................................. 19 2.1.4 Mass matrices ......................................................................................... 20

2.2 Beam elements .............................................................................................. 20 2.2.1 Elastic beam element.............................................................................. 22 2.2.2 Elasto-plastic beam element ................................................................... 23 2.2.3 Mass matrices ......................................................................................... 30

2.3 Shell elements................................................................................................ 30 2.3.1 Basic assumptions in element formulation............................................. 32 2.3.2 Material models and formulations.......................................................... 38 2.3.3 Shell nodal point degrees of freedom..................................................... 39 2.3.4 Composite shell elements (Solution 601 only)....................................... 44 2.3.5 Numerical integration............................................................................. 47 2.3.6 Mass matrices ......................................................................................... 48 2.3.7 Selection of elements for analysis of thin and thick shells..................... 48

2.4 Surface elements � 2-D solids (Solution 601 only)....................................... 49 2.4.1 General considerations ........................................................................... 49 2.4.2 Material models and formulations.......................................................... 54 2.4.3 Numerical integration............................................................................. 56 2.4.4 Mass matrices ......................................................................................... 56 2.4.5 Recommendations on use of elements.................................................... 56

2.5 Solid elements � 3-D ..................................................................................... 57 2.5.1 General considerations ........................................................................... 57 2.5.2 Material models and nonlinear formulations.......................................... 64 2.5.3 Numerical integration............................................................................. 65 2.5.4 Mass matrices ......................................................................................... 65

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iv Advanced Nonlinear Solution ⎯ Theory and Modeling Guide

2.5.5 Recommendations on use of elements.................................................... 66 2.6 Scalar elements � Springs, masses and dampers ........................................... 67 2.7 R-type elements............................................................................................. 69

2.7.1 Rigid elements........................................................................................ 70 2.7.2 RBE3 element......................................................................................... 73

2.8 Other element types....................................................................................... 73 2.8.1 Gap element............................................................................................ 73 2.8.2 Concentrated mass element .................................................................... 74 2.8.3 Bushing element ..................................................................................... 75

3. Material models and formulations................................................................... 76 3.1 Stress and strain measures ............................................................................. 78

3.1.1 Kinematic formulations .......................................................................... 78 3.1.2 Strain measures....................................................................................... 80 3.1.3 Stress measures....................................................................................... 81 3.1.4 Some theory on large strain plasticity .................................................... 82 3.1.5 Thermal Strains ...................................................................................... 87

3.2 Linear elastic material models....................................................................... 89 3.2.1 Elastic-isotropic material model............................................................. 91 3.2.2 Elastic-orthotropic material model ......................................................... 91

3.3 Nonlinear elastic material model................................................................... 92 3.4 Elasto-plastic material model ........................................................................ 95 3.5 Temperature-dependent elastic material models ......................................... 100 3.6 Thermal elasto-plastic and elastic-creep material models ........................... 101

3.6.1 Evaluation of thermal strains................................................................ 104 3.6.2 Evaluation of plastic strains ................................................................. 105 3.6.3 Evaluation of creep strains ................................................................... 107 3.6.4 Computational procedures.................................................................... 110

3.7 Hyperelastic material models ...................................................................... 112 3.7.1 Mooney-Rivlin material model ............................................................ 114 3.7.2 Ogden material model .......................................................................... 117 3.7.3 Arruda-Boyce material model .............................................................. 118 3.7.4 Hyperfoam material model................................................................... 120 3.7.5 Thermal strain effect ............................................................................ 122

3.8 Gasket material model (Solution 601 only)................................................. 124 4. Contact conditions........................................................................................... 128

4.1 Overview ..................................................................................................... 130 4.2 Contact algorithms for Solution 601 ........................................................... 137

4.2.1 Constraint-function method.................................................................. 138 4.2.2 Lagrange multiplier (segment) method ................................................ 140

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4.2.3 Rigid target method .............................................................................. 140 4.2.4 Selection of contact algorithm.............................................................. 140

4.3 Contact algorithms for Solution 701 ............................................................ 141 4.3.1 Kinematic constraint method................................................................ 141 4.3.2 Penalty method..................................................................................... 143 4.3.3 Rigid target method .............................................................................. 144 4.3.4 Selection of contact algorithm.............................................................. 144

4.4 Contact set properties .................................................................................. 144 4.5 Friction ........................................................................................................ 152

4.5.1 Basic friction model ............................................................................. 152 4.5.2 Pre-defined friction models (Solution 601 only).................................. 152

4.6 Contact features........................................................................................... 154 4.6.1 Dynamic contact/impact....................................................................... 154 4.6.2 Contact detection.................................................................................. 156 4.6.3 Suppression of contact oscillations (Solution 601) .............................. 157 4.6.4 Restart with contact .............................................................................. 158 4.6.5 Contact damping................................................................................... 159

4.7 Modeling considerations ............................................................................. 160 4.7.1 Contactor and target selection .............................................................. 160 4.7.2 General modeling hints ........................................................................ 161 4.7.3 Modeling hints specific to Solution 601............................................... 164 4.7.4 Modeling hints specific to Solution 701............................................... 166 4.7.5 Convergence considerations (Solution 601 only) ................................ 168 4.7.6 Handling improperly supported bodies ................................................ 170

4.8 Rigid target contact algorithm..................................................................... 172 4.8.1 Introduction .......................................................................................... 172 4.8.2 Basic concepts ...................................................................................... 174 4.8.3 Modeling considerations ...................................................................... 186 4.8.4 Rigid target contact reports for Solution 601 ....................................... 196 4.8.5 Rigid target contact report for Solution 701......................................... 198 4.8.6 Modeling hints and recommendations.................................................. 199 4.8.7 Conversion of models set up using the NXN4 rigid target algorithm.. 201

5. Loads, boundary conditions and constraint equations ................................ 204 5.1 Introduction ................................................................................................. 204 5.2 Concentrated loads ...................................................................................... 211 5.3 Pressure and distributed loading.................................................................. 213 5.4 Inertia loads ─ centrifugal and mass proportional loading.......................... 216 5.5 Enforced motion .......................................................................................... 219 5.6 Applied temperatures .................................................................................. 220 5.7 Bolt preload ................................................................................................. 221

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vi Advanced Nonlinear Solution ⎯ Theory and Modeling Guide

5.8 Constraint equations .................................................................................... 221 5.9 Mesh glueing ............................................................................................... 222

6. Static and implicit dynamic analysis ............................................................. 226 6.1 Linear static analysis ................................................................................... 226 6.2 Nonlinear static analysis.............................................................................. 227

6.2.1 Solution of incremental nonlinear static equations .............................. 228 6.2.2 Line Search........................................................................................... 229 6.2.3 Low speed dynamics feature ................................................................ 230 6.2.4 Automatic-Time-Stepping (ATS) method............................................ 231 6.2.5 Total Load Application (TLA) method and Stabilized TLA (TLA-S) method............................................................................................................ 233 6.2.6 Load-Displacement-Control (LDC) method ........................................ 234 6.2.7 Convergence criteria for equilibrium iterations ................................... 238 6.2.8 Selection of incremental solution method ............................................ 242 6.2.9 Example................................................................................................ 246

6.3 Linear dynamic analysis.............................................................................. 252 6.3.1 Mass matrix .......................................................................................... 254 6.3.2 Damping ............................................................................................... 255

6.4 Nonlinear dynamic analysis ........................................................................ 256 6.5 Solvers......................................................................................................... 258

6.5.1 Direct sparse solver .............................................................................. 258 6.5.2 Iterative multi-grid solver..................................................................... 260 6.5.3 3D-iterative solver................................................................................ 262

7. Explicit dynamic analysis ............................................................................... 265 7.1 Formulation ................................................................................................. 265

7.1.1 Mass matrix .......................................................................................... 267 7.1.2 Damping ............................................................................................... 267

7.2 Stability ....................................................................................................... 268 7.3 Time step management................................................................................ 271

8. Additional capabilities .................................................................................... 273 8.1 Initial conditions.......................................................................................... 273

8.1.1 Initial displacements and velocities...................................................... 273 8.1.2 Initial temperatures............................................................................... 273

8.2 Restart.......................................................................................................... 273 8.3 Element birth and death feature................................................................... 275 8.4 Reactions calculation................................................................................... 280 8.5 Element death due to rupture....................................................................... 281 8.6 Stiffness stabilization (Solution 601 only) .................................................. 281

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8.7 Bolt feature .................................................................................................. 283 8.8 Parallel processing....................................................................................... 284 8.9 Usage of memory and disk storage ............................................................. 284

Additional reading ................................................................................................. 286 Index ....................................................................................................................... 291

1.1: Objective of this manual

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1. Introduction

1.1 Objective of this manual

This Theory and Modeling Guide serves two purposes:

! To provide a concise summary of the theoretical basis of Advanced Nonlinear Solution as it applies to Solution 601 and Solution 701. This includes the finite element procedures used, the elements and the material models. The depth of coverage of these theoretical issues is such that the user can effectively use Solutions 601 and 701. A number of references are provided throughout the manual which give more details on the theory and procedure used in the program. These references should be consulted for further details. Much reference is made however to the book Finite Element Procedures (ref. KJB).

ref. K.J. Bathe, Finite Element Procedures, Prentice Hall,

Englewood Cliffs, NJ, 1996.

! To provide guidelines for practical and efficient modeling using Advanced Nonlinear Solution. These modeling guidelines are based on the theoretical foundation mentioned above, and the capabilities and limitations of the different procedures, elements, material models and algorithms available in the program. NX Nastran commands and parameter settings needed to activate different analysis features are frequently mentioned.

It is assumed that the user is familiar with NX Nastran fundamentals pertaining to linear analysis. This includes general knowledge of the NX Nastran structure, commands, elements, materials, and loads.

We intend to update this report as we continue our work on

Advanced Nonlinear Solution. If you have any suggestions regarding the material discussed in this manual, we would be glad to hear from you.

Chapter 1: Introduction

2 Advanced Nonlinear Solution ⎯ Theory and Modeling Guide

1.2 Overview of Advanced Nonlinear Solution

• Advanced Nonlinear Solution is a Nastran solution option focused on nonlinear problems. It is capable of treating geometric and material nonlinearities as well as nonlinearities resulting from contact conditions. State-of-the-art formulations and solution algorithms are used which have proven to be reliable and efficient. • Advanced Nonlinear Solution supports static and implicit dynamic nonlinear problems via Solution 601, and explicit dynamic problems via Solution 701. • Advanced Nonlinear Solution supports many of the standard Nastran commands and several commands specific to Advanced Nonlinear Solution that deal with nonlinear features such as contact. The NX Nastran Quick Reference Guide provides more details on the Nastran commands and entries that are supported in Advanced Nonlinear Solution. • Advanced Nonlinear Solution supports many of the commonly used features of linear Nastran analysis. This includes most of the elements, materials, boundary conditions, and loads. Some of these features are modified to be more suitable for nonlinear analysis, and many other new features are added that are needed for nonlinear analysis. • The elements available in Advanced Nonlinear Solution can be broadly classified into rods, beams, 2-D solids, 3-D solids, shells, scalar elements and rigid elements. The formulations used for these elements have proven to be reliable and efficient in linear, large displacement, and large strain analyses. Chapter 2 provides more details on the elements. • The material models available in Advanced Nonlinear Solution are elastic isotropic, elastic orthotropic, hyperelastic, plastic isotropic and gasket. Thermal and creep effects can be added to some of these materials. Chapter 3 provides more details on these material models.

1.3: Structure of Solution 601

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• Advanced Nonlinear Solution has very powerful features for 3-D contact analysis. These include several contact algorithms and different contact types such as single-sided contact, double-sided contact, self-contact, and tied contact. Chapter 4 provides more details on contact. • Loads, boundary conditions and constraints are addressed in Chapter 5. Time varying loads and boundary conditions are common to nonlinear analysis and their input in Advanced Nonlinear Solution is slightly different from other Nastran solutions, as discussed in Chapter 5. • Solution 601 of Advanced Nonlinear Solution currently supports two nonlinear analysis types: static and implicit transient dynamic. Details on the formulations used are provided in Chapter 6. Other features of nonlinear analysis, such as time stepping, load displacement control (arc length method), line search, and available solvers are also discussed in Chapter 6. • Solution 701 of Advanced Nonlinear Solution is dedicated to explicit transient dynamic analysis. Details on the formulations used are provided in Chapter 7. Other features of explicit analysis, such as stability and time step estimation are also discussed in Chapter 7. • Additional capabilities present in Advanced Nonlinear Solution such as restarts, stiffness stabilization, initial conditions, and parallel processing are discussed in Chapter 8. • Most of the global settings controlling the solution in Advanced Nonlinear Solution are provided in the NXSTRAT bulk data entry. This includes parameters that control the solver selection, time integration values, convergence tolerances, contact settings, etc. An explanation of these parameters is found in the NX Nastran Quick Reference Guide.

1.2.1 Choosing Between Solutions 601 and 701 • The main criterion governing the selection of the implicit (Solution 601) or explicit (Solution 701) formulations is the time scale of the solution.



ref. KJBSection 9.2

• The implicit method can use much larger time steps since it is unconditionally stable. However, it involves the assembly and solution of a system of equations, and it is iterative. Therefore, the computational time per load step is relatively high. The explicit method uses much smaller time steps since it is conditionally stable, meaning that the time step for the solution has to be less than a certain critical time step, which depends on the smallest element size and the material properties. However, it involves no matrix solution and is non-iterative. Therefore, the computational time per load step is relatively low. • For both linear and nonlinear static problems, the implicit method is the only option. • For slow-speed dynamic problems, the time solution time spans a period of time considerably longer than the time it takes the wave to propagate through an element. The solution in this case is dominated by the lower frequencies of the structure. This class of problems covers most structural dynamics problems, certain metal forming problems, crush analysis, earthquake response and biomedical problems. When the explicit method is used for such problems the resulting number of time steps will be excessive, unless mass-scaling is applied, or the loads are artificially applied over a shorter time frame. No such modifications are needed in the implicit method. Hence, the implicit method is the optimal choice. • For high-speed dynamic problems, the solution time is comparable to the time required for the wave to propagate through the structure. This class of problems covers most wave propagation problems, explosives problems, and high-speed impact problems. For these problems, the number of steps required with the explicit method is not excessive. If the implicit method uses a similar time step it will be much slower and if it uses a much larger time step it will introduce other solution errors since it will not be capturing the pertinent features of the solution (but it will remain stable). Hence, the explicit method is the optimal choice. • A large number of dynamics problems cannot be fully classified as either slow-speed or high-speed dynamic. This includes many crash problems, drop tests and metal forming problems. For these problems both solution methods are comparable. However,


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whenever possible (when the time step is relatively large and there are no convergence difficulties) we recommend the use of the implicit solution method. • Note that the explicit solution provided in Solution 701 does not use reduced integration with hour-glassing. This technique reduces the computational time per load step. However, it can have detrimental effect on the accuracy and reliability of the solution. • Since the explicit time step size depends on the length of the smallest element, one excessively small element will reduce the stable time step for the whole model. Mass-scaling can be applied to these small elements to increase their stable time step. The implicit method is not sensitive to such small elements. • Since the explicit time step size depends on the material properties, a nearly incompressible material will also significantly reduce the stable time step. The compressibility of the material can be increased in explicit analysis to achieve a more acceptable solution time. The implicit method is not as sensitive to highly incompressible materials (provided that a mixed formulation is used). • Higher order elements such as the 10-node tetrahederal, 20 and 27 node brick elements are only available in implicit analysis. They are not used in explicit analysis because no suitable mass-lumping technique is available for these elements. • Model nonlinearity is another criterion influencing the choice between implicit and explicit solutions. As the level of nonlinearity increases, the implicit method requires more time steps, and each time step may require more iterations to converge. In some cases, no convergence is reached. The explicit method however, is less sensitive to the level of nonlinearity. Note that when the implicit method fails it is usually due to non-convergence within a time step, while when the explicit method fails it is usually due to a diverging solution. • The memory requirements is another factor. For the same mesh, the explicit method requires less memory since it does not store a



stiffness matrix and does not require a solver. This can be significant for very large problems. • Since Advanced Nonlinear Solution handles both Solution 601 and Solution 701 with very similar inputs, the user can in many cases restart from one analysis type to the other. This capability can be used, for example, to perform implicit springback analysis following an explicit metal forming simulation, or to perform an explicit analysis following the implicit application of a gravity load. It can also be used to overcome certain convergence difficulties in implicit analyses. A restart from the last converged implicit solution to explicit can be performed, then, once that stage is passed, another restart from explicit to implicit can be performed to proceed with the rest of the solution.

1.3 Structure of Advanced Nonlinear Solution

• The input data for Advanced Nonlinear Solution follows the standard Nastran format consisting of the following 5 sections: 1. Nastran Statement (optional) 2. File Management Statements (optional) 3. Executive control Statements 4. Case Control Statements 5. Bulk Data Entries The first two sections do not involve any special treatment in Advanced Nonlinear Solution. The remaining three sections involve some features specific to Advanced Nonlinear Solution, as described below.

1.3.1 Executive Control • Solution 601 is invoked by selecting solution sequence 601 in the SOL Executive Control Statement. This statement has the following form: SOL 601,N


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where N determines the specific analysis type selected by Solution 601. Currently, only static and direct time-integration implicit dynamic analyses are available as shown in Table 1.3-1:

N Solution 601 Analysis Type

106 Static 129 Transient dynamic

Table 1.3-1: Solution 601 Analysis Types • Solution 701 is invoked by selecting solution sequence 701 in the SOL Executive Control Statement. This statement has the following form: SOL 701 and is used for explicit dynamic analyses. • In many aspects Solution 601,106 is similar to Solution 106 for nonlinear static analysis. However, it uses the advanced nonlinear features of Solution 601. Likewise, Solution 601,129 is similar to Solution 129 for nonlinear transient response analysis. Solution 701 provides an alternative to the implicit nonlinear dynamic analysis of Solution 601,129.

1.3.2 Case Control • The Case Control Section supports several commands that control the solution, commands that select the input loads, temperatures and boundary conditions, commands that select the output data and commands that select contact sets. Table 1.3-2 lists the supported Case Control Commands.



Case Control Command Description

Solution control

SUBCASE1 Subcase delimiter

TSTEP2 Time step set selection

Loads and boundary conditions

LOAD Static load set selection

DLOAD3 Dynamic load set selection

SPC Single-point constraint set selection

MPC Multipoint constraint set selection

TEMPERATURE4

TEMPERATURE(LOAD) TEMPERATURE(INITIAL)

Temperature set selection Temperature load Initial temperature

IC Transient initial condition set selection

BGSET Glue contact set selection

BOLTLD Bold preload set selection

Output related

SET Set definition

DISPLACEMENT Displacement output request

VELOCITY Velocity output request

ACCELERATION Acceleration output request

STRESS Element stress/strain output request

SPCFORCES Reaction force output request

GKRESULTS Gasket results output request

TITLE Output title

SHELLTHK Shell thickness output request

Contact related

BCSET Contact set selection

BCRESULTS Contact results output request

Element related

EBDSET Element birth/death selection

Table 1.3-2: Case Control Commands


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Notes:

1. Only one subcase is allowed in Advanced Nonlinear Solution.

2. TSTEP is used for all analysis types in Advanced Nonlinear Solution. In explicit analysis with automatic time stepping it is used for determining the frequency of output of results.

3. DLOAD is used for time-varying loads for both static and transient dynamic analyses.

4. TEMPERATURE, TEMPERATURE(BOTH) and TEMPERATURE(MAT) are not allowed for Advanced Nonlinear Solution. Use TEMPERATURE(INIT) and TEMPERATURE(LOAD) instead.

1.3.3 Bulk Data

• The Bulk Data section contains all the details of the model. Advanced Nonlinear Solution supports most of the commonly used Bulk Data entries. In many cases, restrictions are imposed on some of the parameters in a Bulk Data entry, and in some other cases, different interpretation is applied to some of the parameters to make them more suitable for nonlinear analysis. Several Bulk Data entries are also specific to Advanced Nonlinear Solution. • Table 1.3-3 lists the supported Bulk Data entries.



Element Connectivity

CBAR CBEAM CBUSH1D CDAMP1 CDAMP2 CELAS1

CELAS2 CGAP CHEXA CMASS1 CMASS2 CONM1

CONM2 CONROD CPENTA CQUAD4 CQUAD8 CQUADR

CQUADX CROD CTRIA3 CTRIA6 CTRIAR CTRIAX

CTETRA RBAR RBE2 RBE3

Element Properties

EBDSET EBDADD PBAR PBARL

PBCOMP PBEAM PBEAML PBUSH1D

PCOMP PDAMP PELAS PGAP

PLPLANE PLSOLID PMASS

PROD PSHELL PSOLID

Material Properties

CREEP MAT1 MAT2

MAT8 MAT9 MATG

MATHE MATHP MATT1

MATT2 MATT9 MATS1

TABLEM1 TABLES1 TABLEST

Loads, Boundary Conditions and Constraints

BGSET BOLT BOLTFOR DLOAD FORCE FORCE1

FORCE2 GRAV LOAD MOMENT MOMENT1 MOMENT2

MPC MPCADD PLOAD PLOAD1 PLOAD2 PLOAD4

PLOADX1 RFORCE SPC SPC1 SPCADD SPCD

TABLED1 TABLED2 TEMP TEMPD TIC TLOAD1

Contact

BCPROP BCRPARA

BCTADD BCTPARA

BCTSET BSURF BSURFS

Other Commands

CORD1C CORD1R

CORD1S CORD2C

CORD2R CORD2S

GRID NXSTRAT1

PARAM2

TSTEP3

Table 1.3-3: Bulk Data entries supported by Advanced Nonlinear Solution


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Notes:

1. NXSTRAT is the main entry defining the solution settings for Advanced Nonlinear Solution.

2. Only a few PARAM variables are supported. Most are replaced by NXSTRAT variables.

3. TSTEP is used for both static and dynamic analyses. 1.3.4 Terminology used in Advanced Nonlinear Solution

The terminology used in Advanced Nonlinear Solution is for the most part the same as that used in other Nastran documents.

Chapter 2: Elements


2. Elements

• Advanced Nonlinear Solution supports most of the commonly used elements in linear Nastran analyses. Some of these elements are modified to be more suitable for nonlinear analysis. • The Advanced Nonlinear Solution elements are generally classified as line, surface, solid, scalar, or rigid elements. ! Line elements are divided into 2 main categories � rod elements and beam elements. Rod elements only possess axial stiffness, while beam elements also possess bending, shear and torsional stiffness.

! Surface elements are also divided into 2 main categories � 2-D solids and shell elements.

! 3-D solid elements are the only solid elements in Advanced Nonlinear Solution.

! The scalar elements are spring, mass and damper elements.

! R-type elements impose constraints between nodes, such as rigid elements. ! Other element types available in Advanced Nonlinear Solution are the gap element, concentrated mass element, and the bushing element.

• This chapter outlines the theory behind the different element classes, and also provides details on how to use the elements in modeling. This includes the materials that can be used with each element type, their applicability to large displacement and large strain problems, their numerical integration, etc. • More detailed descriptions of element input and output are provided in several other manuals, including:

- NX Nastran Reference Manual - NX Nastran Quick Reference Guide - NX Nastran DMAP Programmer�s Guide

2. Elements

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Table 2-1 shows the different elements available in Advanced Nonlinear Solution, and how they can be obtained from Nastran element connectivity and property ID entries. Restrictions related to Solution 701 are noted.

Element Connectivity Entry

Property ID Entry Advanced Nonlinear Solution Element

Rod Elements

CROD PROD 2-node rod element

CONROD None 2-node rod element

Beam Elements

CBAR PBAR, PBARL 2-node beam element

CBEAM PBEAM, PBEAML, PBCOMP

2-node beam element

Shell Elements3

CQUAD4 PSHELL1, PCOMP2 4-node quadrilateral shell element

CQUAD8 PSHELL1, PCOMP2 4-node to 8-node quadrilateral shell element

CQUADR PSHELL, PCOMP2 4-node quadrilateral shell element

CTRIA3 PSHELL1, PCOMP2 3-node triangular shell element

CTRIA6 PSHELL1, PCOMP2 3-node to 6-node triangular shell element

CTRIAR PSHELL, PCOMP2 3-node triangular shell element

Chapter 2: Elements


2D Solid Elements4

CQUAD PLPLANE 4-node to 9-node quadrilateral 2D plane strain element with hyperelastic material

CQUAD4 PLPLANE, PSHELL1 4-node quadrilateral 2D plane strain element

CQUAD8 PLPLANE, PSHELL1 4-node to 8-node 2D plane strain element

CTRIA3 PLPLANE, PSHELL1 3-node triangular 2D plane strain element

CTRIA6 PLPLANE, PSHELL1 3-node to 6-node triangular 2D plane strain element

CQUADX PLPLANE 4-node to 9-node quadrilateral 2D axisymmetric element with hyperelastic material

CTRIAX PLPLANE 3-node to 6-node triangular 2D axisymmetric element with hyperelastic material

3D Solid Elements5

CHEXA PSOLID, PLSOLID 8-node to 20-node brick 3D solid element

CPENTA PSOLID, PLSOLID 6-node to 15-node wedge 3D solid element

CTETRA PSOLID, PLSOLID 4-node to 10-node tetrahedral 3D solid element

Scalar Elements

CELAS1; CELAS2 PELAS; None Spring element

CDAMP1; CDAMP2 PDAMP; None Damper element

CMASS1; CMASS2 PMASS; None Mass element

2. Elements

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R-Type Elements

RBAR None Single rigid element

RBE2 None Multiple rigid elements

RBE3 None Interpolation constraint element

Other Elements

CGAP PGAP 2-node gap element

CONM1, CONM2 None Concentrated mass element

CBUSH1D PBUSH1D Bushing element

Table 2-1: Elements available in Advanced Nonlinear Solution

Notes: 1. CQUAD4, CQUAD8, CTRIA3, and CTRIA6 with a PSHELL property ID are

treated as either 2D plane strain elements or shell elements depending on the MID2 parameter.

2. Elements with a PCOMP property ID entry are treated as multi-layered shell elements. These elements are not supported in Solution 701.

3. Only 3-node and 4-node single layer shells are supported in Solution 701. 4. 2-D solid elements are not supported in Solution 701. 5. Only 4-node tetrahedral, 6-node wedge and 8-node brick 3-D solid elements are

supported in Solution 701.

Chapter 2: Elements


Table 2-2 lists the acceptable combination of elements and materials for Solution 601. Thermal effects in this table imply temperature dependent material properties. Thermal strains are usually accounted for in isothermal material models.

Rod Beam Shell 2D Solid 3D Solid

Elastic isotropic ! ! ! ! !

...Thermal ! ! ! !

...Creep ! ! ! !

Elastic orthotropic ! !

...Thermal ! !

Plastic isotropic !1 !1 ! ! !

...Thermal ! ! ! !

Hyperelastic ! !

Gasket !

Nonlinear elastic isotropic !

Table 2-2: Element and material property combinations in Solution 601

Note: 1. No thermal strains in these plastic isotropic material models.

2. Elements

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Table 2-3 lists the acceptable combination of elements and materials for Solution 701. Thermal effects in this table imply temperature dependent material properties. Thermal strains are usually accounted for in isothermal material models. Note that interpolation of temperature dependent material properties is only performed at the start of the analysis in Solution 701.

Rod Beam Shell 2D Solid 3D Solid

Elastic isotropic ! ! ! !

...Thermal ! ! !

...Creep

Elastic orthotropic ! !

...Thermal ! !

Plastic isotropic !1 !1 ! !

...Thermal ! ! !

Hyperelastic !

Gasket

Nonlinear elastic isotropic !

Table 2-3: Element and material property combinations in Solution 701

Note: 1. No thermal strains in these plastic isotropic material models.

Chapter 2: Elements


2.1 Rod elements

2.1.1 General considerations

• Rod elements are generated using the CONROD and CROD entries. These line elements only possess axial stiffness. Fig. 2.1-1 shows the nodes and degrees of freedom of a rod element.

Z

Y

X

�

w2

v2

u1

�

G2

(u, v, w) are nodal translationaldegrees of freedom

w1

v1

u2

G1

Figure 2.1-1: Rod element

• Note that the only force transmitted by the rod element is the longitudinal force as illustrated in Fig. 2.1-2. This force is constant throughout the element.

Z

YX

�

P

P

area A

Stress constant over

cross-sectional area

ref. KJBSections 5.3.1,

6.3.3

Figure 2.1-2: Stresses and forces in rod elements

2.3: Shell elements

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2.1.2 Mate

e material models that are com

• he rod elements can be used with small displacement/small

ssumed

ments and rotations can be very large. In all cases, the ross-sectional area of the element is assumed to remain

th.

All of the compatible material models listed in Table 2-2 can be mall and large displacement formulations.

2.1.3 Num egration

• The rod elements use one or two point Gauss integration.

• When the material is temperature-independent, the rod elements require only 1-point Gauss numerical integration for an exact evaluation of the stiffness matrix, because the force is constant in

. However, 2-point integration may be appropriate when a temperature-dependent material is used because, due to a varying

rial models and formulations

• See Table 2-2 for a list of thpatible with rod elements.

Tstrain or large displacement/small strain kinematics. In the small displacement case, the displacements and strains are ainfinitesimally small. In the large displacement case, the displacecunchanged, and the strain is equal to the longitudinal displacementdivided by the original leng

used with both the s

• Thermal strains are not available in the isotropic plasticity material model for rod elements. They can be obtained by switching to a temperature dependent isotropic plasticity material model. • Note that the nonlinear elastic isotropic material model is only available for rod elements.

erical int

the element

temperature, the material properties can vary along the length of an element.

Chapter 2: Elements


2.1.4 Mas

tes the mass distribution more accurately.

each node i

s matrices

• The consistent mass matrix is calculated using Eq. (4.25) in ref. KJB, p. 165 and calcula • The lumped mass matrix for the rod element is formed by dividing the element�s mass M among its nodes. The mass assigned

to s iML

⎛ ⎞⋅⎜ ⎟⎝ ⎠!

, in which L = total element length, =

total element length associated with element node i

.e., for the 2-node rod element,

i!

fraction of the

(i 1 2L

=! and 2 2has no rotational mass. • The same lumped mass matrix is used for both Solution 601 andSolution 701.

L=! ). The element

2.2 Beam

AR

aterial models that are compatible with the beam elem • itian beams with a constant cro e own in Fig r is along the line connecting the nodes GA and GB. The s-direction is bas A entry. odeled by the beam element are (see Fig. 2.2-2):

! Cubic transverse displacem nts

elements • Beam elements are generated using the CBAR and CBEAM entries. The properties for a CBAR entry are defined using PBor PBARL entries while the properties for CBEAM are defined using the PBEAM, PBEAML or PBCOMP entries. See Table 2-2 for a list of the m

ent.

he beam elements are 2-node HermTss-section and 6 degrees of freedom at ach node as sh. 2.2-1. The r-direction in the local coo dinate system

ed on the v vector defined in the CB R or CBEAMThe displacements m

v and we (s- and t-direction displacements) ! Linear longitudinal displacements u (r-direction

2.2: Beam elements

UGS Corp. 21

displacement) ! Linear torsional displacements rθ and warping displacements

�

�

node GA

s (y )elem

t (z )elem

r (x )elem

node GB

v

Plane 1

End A

End B

Figure 2.2-1: Beam element

Z

YX

�

�

1

2 w_

v_

u_

_�s

_�

r

_�t

s

r

t

Figure 2.2-2: Conventions used for 2-node Hermitian beaelement

m

Nonlinear Solution: ROD, TUBE, I, CHAN, T, BOX, BAR, H, T1, I1, CHAN1, CHAN2, and T2.

• The element is formulated based on the Bernoulli-Euler beam theory. • The following PBARL and PBEAML cross-sections are supported by Advanced

Chapter 2: Elements


• Axial forces applied to the beam are assumed to be acting along the beam�s centroid and hence cause no bending. Also shear forces pplied to the beam are assumed to be acting through the beam�s

In order to the bending due to an off-centroidal axial force or a shear force applied away from the shear center, the

can be applied directly or the forces can be applied at an offset location using rigid elements.

Stress and strain output is not supported in beam elements.

Off-centered beam elements can be modeled using rigid lation used

epends on the selected material (see Table 2-2).

rmulations exist in Advanced Nonlinear Solution, ne for elastic materials and one for plastic materials.

ic beam elements can be used to simulate bolts. See ection 8.6 for details.

ashear center and hence cause no twisting.

model

resulting moments

•

• elements (see Fig. 2.2-3). The beam element formud • Two basic foo • ElastS

I-beamI-beam

Rigid panel

Hollow square section

Beam element

Rigid elements

Beam elements

Physical problem: Finite element model:

Figure 2.2-3: Use of rigid elements for modeling off-centered

beams

Elastic beam

2.2.1 Elastic beam element

• elements can be used with !small displacement/small strain kinematics, or

2.2: Beam elements

UGS Corp. 23

n) formulation is used in the case of large displacements.

The elastic beam element only supports the isotropic material o

olution 701) are evaluated in closed form for both small and large

ollowing reference:

68.

E = Young's modulus

!large displacement/small strain kinematics. • A TL (Total Lagrangia

•m del. • The element�s force vector and stiffness matrix (except in Sdisplacement formulations. The stiffness matrix used is discussed in detail in the f

ref. J.S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hill Book Co., 19

Note that the element stiffness matrix is defined by the following quantities:

ν = Poisson's ratio L = length of the beam

, ,r s t

I I I = moments of inertia about the local principal axes r,

s, and t A = cross-sectional area

,sh shs tA A = effective shear areas in s and t directions

This stiffness matrix is transformed from the local degrees of freedom (in the r, s, t axes) to the global coordinate system and is then assembled into the stiffness matrix of the complete structure.

2.2.2 Elasto-plastic beam element

• Elasto-plastic beam elements can be used with the !small displacement/small strain kinematics, or !large displacement formulation/small strain kinematics.

Chapter 2: Elements


• An updated Lagrangian formulation is used in the case of large displacements. • Only isotropic bilinear plasticity is supported. Thermal strains are not admissible for the elasto-plastic beam. • The nonlinear elasto-plastic beam element can only be employed for circular (ROD, TUBE) and rectangular (BAR) cross-sections.

• The beam element matrices are formulated using the Hermitian displacement functions, which give the displacement interpolation matrix summarized in Table 2.2-2.

Table 2.2-2: Beam interpolation functions not including shear effects

in which

1 1 2 2 1

4 5 6

4 5

1 3 3 1 1

1 7

1 6 6 0 6

0 0 1 0 0

0 0 1 0 0 0

6 0 (1 6 ) (1 6 )

r

s

t

u

u

u

r s t r st sL L L L L

r t LL

r s LL

t t s t sL

r

Ψ Ψ Ψ Ψ Ψ

Ψ Ψ Ψ

Ψ Ψ

Ψ Ψ Ψ Ψ Ψ

− − −⎢⎢⎡ ⎤⎢⎢ ⎥ ⎛ ⎞= − −⎢ ⎜ ⎟⎢ ⎥ ⎝ ⎠⎢⎢ ⎥⎣ ⎦ ⎢ ⎛ ⎞− −⎢ ⎜ ⎟⎢ ⎝ ⎠⎣

⎤− − − − − ⎥⎥⎥

⎥⎦

hhh

⎡

1 7

6

0 0 0 ( )

0 ( ) 0

t r r LL

r s r r LL

Ψ Ψ

Ψ Ψ Ψ

− − − ⎥⎥⎥−

2.2: Beam elements

2 2 2

1

2

4

2

6 7

;

1 3

3 2 ; 3 2

r rL L

r rL

rL L L L L

2 3

3 2 3

5

3 2 3

1 4 3 ; 2 3

2 ; 2

r r r rL L L L

r r rL L L L

r r r r

Ψ Ψ Ψ

Ψ Ψ

Ψ

⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= − +⎜ ⎟⎝ ⎠

⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= − + = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞Ψ

UGS Corp. 25

Note: displacements correspond to the forces/mom Si in Fig. 2.2-4 after

condensand t-directions.

ents ation of the last two columns containing the shear effects in the s-

�

�

Z

YX

S4 S1

1

S5

S2

S3S6

s

r

t

S11

S8

2

S7

S10

S12

S9

Neutral axis

S = r-direction force at node 1 (axial force, positive in compression)1

S = r-direction force at node 2 (axial force, positive in tension)7

S = s-direction force at node 1 (shear force)2

S = s-direction force at node 2 (shear force)8

S = t-direction force at node 1 (shear force)3

S = t-direction force at node 2 (shear force)9

S = r-direction moment at node 1 (torsion)4

S = r-direction moment at node 2 (torsion)10

S = t-direction moment at node 1 (bending moment)6

S = t-direction moment at node 2 (bending moment)12

S = s-direction moment at node 1 (bending moment)5

S = s-direction moment at node 2 (bending moment)11

Figure 2.2-4: Elasto-plastic beam element

Chapter 2: Elements


ent�s stiffness matrix and load vector are then transformed from the local coordinate system used above to the

isplacement coordinate system.

• Shear deformatio incl non-zero K1 or K2 in the PBAR or PBEAM entries. Constant shear distortions

• The elem

d

ns can be uded in beams by selecting a

rsγ and rtγ along the length of the beam are assumed, as depicted in Fig. 2.2-5. In this case the displacement interpolation matri r the ditional displacements corresponding to these shear deformations.

x of e 2.2 Tabl -2 is modified fo ad

�rs �rt

r r

s t

Figure 2.2-5: Assumptions of shear deformations through element thickness for nonlinear elasto-plastic beam element

• The interpolation functions in Table 2.2-2 do not account for warping in torsional deformations. The circular section does not warp, but for the rectangular section the displacement function for longitudinal displacements is corrected for warping as described in the following reference:

ref. K.J. Bathe and A. Chaudhary, "On the Displacement

Formulation of Torsion of Shafts with Rectangular Cross-Sections", Int. Num. Meth. in Eng., Vol. 18, pp. 1565-1568, 1982.

• The derivation of the beam element matrices employed in the large displacement formulation is given in detail in the following paper:

ref. K.J. Bathe and S. Bolourchi, "Large Displacement

Analysis of Three-Dimensional Beam Structures," Int. J. Num. Meth. in Eng., Vol. 14, pp. 961-986, 1979.

2.2: Beam elements

UGS Corp. 27

The derivations in the above reference demonstrated that the updated Lagrangian formulation is more effective than the total Lagrangian formulation, and hence the updated Lagrangian formulation is employed in Solution 601.

• All element matrices in elasto-plastic analysis are calculated using numerical integ ders are given in Table 2.2-3. The locations of the integration points are given in Fig. 2.2-6.

te Section Integration scheme

Integration order

ration. The integration or

Coordina

R Any Newton-Cotes 5

Rectangular 7 S

Pipe Newton-Cotes

3

Rectangular Newton-Cotes 7 t or θ

Com osite Pipe

ptrapezoidal

rule 8

Integration orders in elasto-plastic beam analysis

ref. KJBSection 6.6.3

Table 2.2-3:

Chapter 2: Elements


s

sHEIGHT

�

Pipe section

WIDTH

t

Rectangular section

r = 0r = -1 r = 1

r

s

t

a) Integration point locations in r-direction

b) Integration point locations in s-direction

�

�

�

� � �

� ��

� � � �

igure 2.2-6: Integration point locaF tions in elasto-plastic beam

analysis

2.2: Beam elements

UGS Corp. 29

s

�

Pipe section (3-D action)

Rectangular section

c) Integration point locations in t-direction

��

s

t

��

��

��

��

��

Figure 2.2-6: (continued)

• The elasto-plastic stress-strain relation is based on the classiflow theory with the von Mises

cal yield condition and is derived from

the three-dimensional stress-strain law using the following assumptions:

- the stresses ssτ and ttτ are zero

e stra- th in stγ is zero

stic-plastic st

la re al stress Hence, the e ss-strain matrix for the normrrτ and the two she tressesar s rsτ and rtτ is obtained using static

condensation.

Chapter 2: Elements


ass m s

or a consistent s uses a lumped

ated in .

ped mass for im icit analysis (Solution 601) is

2.2.3 M atrice

• The beam element can be used with a lumped ass matrix, except for Solution 701 which alwaym

mass. • The consistent mass matrix of the beam element is evaluclosed form, and does not include the effect of shear deformations • The lumped mass for translational degrees of freedom is Mwhere M is the total mass of the element.

/ 2

The rotational lum pl

23 2rr

rrM IMA

= ⋅ ⋅ , in which Irr = polar moment of inertia of the

beam cross-section and A = beam cross-sectional area. This mass is applied to all rotational degrees of freedom.

lumped

Th )

is

e rotational lumped mass for explicit analysis (Solution 701

32

mrr

IMA

M = ⋅ ⋅ where Im is the maximum bending moment of

inertia of the beam ( )max ,m ss ttI I I= . This lumped mass islied to all rotational degrees of freed

app om. Note that this scaling of rotational masses ensures that the rotational degrees of freedom do

2.3 Shell e

• when a PS es one of the following Nastran shell entries: CQUAD4, CTRIA3, CQUAD8, CTRIA6, CQUADR, or CTRIAR. • The PSHELL entry results in a single-layered shell, while PCOMP produces a composite shell.

not affect the critical stable time step of the element.

lements

Shell elements in Advanced Nonlinear Solution are generated HELL or PCOMP property ID entry referenc

2.3: Shell elements

UGS Corp. 31

• Shell elements are classified based on the number of nodes in the element. Table 2.3-1 shows the correspondence between the different shell elements and the NX element connectivity entries. • Solution 701 only supports 3-node and 4-node single-layered shell elements. • The extra middle node in the 9-node shell element is automatically added by the program when ELCV is set to 1 in the NXSTRAT entry. This extra node improves the performance of the shell element. The boundary conditions at the added node are predicted from the neighboring nodes. Shell element NX element connectivity entry

3-node CTRIA3 4-node

1

D8 9-node1 CQUAD82

CQUAD4 6-node CTRIA6 8-node1 CQUA

Ta spondence be X

No1. 2. TRAT entry

• bb she ent degrees of freedom are introduced which are not associated with nodes; therefore the

elements odes

ses the flexibility of the element, especially in bending situations. For theoretical considerations, see reference KJB, Section 4.4.1. Note that these incompatible-mode elements are formulated to pass the patch test. Also note that element distortions deteriorate the element performance when

be used with 4-node

ble 2.3-1: Corre tween shell elements and Nelement connectivity

tes: Only for Solution 601

ith ELCV = 1 in NXSW

Incompatible modes (bu le functions) can be used with 4-nodell elements. Additional displacem

condition of displacement compatibility between adjacentis not satisfied in general. The addition of the incompatible m(bubble functions) increa

incompatible modes are used. The incompatible modes feature can only

Chapter 2: Elements


ULJ forailable to the shell

Shell element di

s

tion

single layer shell elements. The feature is available in linear and nonlinear analysis, for all formulations except the mulation.

able 2.3-2 lists the features and capabilities avTelement types mentioned above.

Large splacement/ mall strain

Large strain ULJ

formulation

Large strain ULH formulation

Bubble functions

Solu701

3-node ! ! !

4-node ! ! ! ! !

6-node ! 8-node 9-node

! ! !

Table 2.3-2: Features available to shell elements

2.3.1 Basic assumptions in element formulation

• The basic equations used in the formulation of the shell elements in Advanced Nonlinear Solution are given in ref. KJB. These elements are based on the Mixed Interpolation of Tensorial Components (MITC). Tying points are used to interpolate the transverse shear strain and the membrane strains if necessary. These elements show excellent performance.

�

�

�

��

�

��

(d) 8-node element

�

�

�

�

(b) 4-node element

�

��

��

�

(c) 6-node element

(a) 3-node element

�

�

�

Figure 2.3-1: Shell elements in Advanced Nonlinear Solution

2.3: Shell elements

UGS Corp. 33

• The shdimensional continuum with the following two assumptions:

ss n a

l to the zero.

ell analysis, these assumptions orrespond to a very general shell theory. See the reference below

r nt A lysis , 2003.

ell element formulation treats the shell as a three-

A umption 1: Material particles that originally lie ostraight line "normal" to the midsurface of the structure remainon that straight line during deformation. Assumption 2: The stress in the direction normamidsurface of the structure is

For the Timoshenko beam theory, the structure is the beam, and for the Reissner/Mindlin plate theory, the structure is the plate under consideration. In shcfor more details:

ef. D. Chapelle and K.J. Bathe, The Finite Eleme na

of Shells � Fundamentals, Springer

r

s

t

G3

G7

G4

G8

G1

G5

G2

G6

Midsurface nodes Figure 2.3-2: Some conventions for the shell element; local node

numbering; local element coordinate system

• In the calculations of the shell element matrices the following geometric quantities are used:

Chapter 2: Elements


t midsurface at (see Fig. 2.3-2); (the left superscript denotes the configuration at time t)

to the shell midsurface ! The shell thickness, , at the nodal points measured in the

direction of the director vectors (see Fig. 2.3-3).

• Based on these quantities the geometry of the shell is interpolated as follows:

! The coordinates of the node k that lies on the shell elemen, 1,2,3t k

ix i =

! The director vectors t k pointing in the direction "normal" nV

kat k

nV

1 1( 1, 2,3)

2

q qtt t k k

i k i k k nik k

tx h x a h V i= =

= + =∑ ∑

where q is the number of element nodes, are the

components of the shell director vector are the

• At the ele pendent degrees of ts about the displacement the displacement of the shell

idsurface and 2 rotations resulting from

1, 2,3( )t kniV i =

t knV and ( , )kh r s

2-D interpolation functions given in ref. KJB.

ment level the shell has 5 indefreedom per node: 3 displacemencoordinate system resulting fromm the motion of the shell direction vector k

nV :

( )0

1 12t t k

i k i k kk k

u h u a h= =

= +∑ ∑q q

t k kni ni

t V V−

d using 2 re 2 axes

erpendicular to the shell director V as shown in Fig. 2.3-3.

ref. KJBFig. 5.33page 437

The motion of the director vector at node k is describerotational degrees of freedom about 1V and 2V which a

k

k k

p n

2.3: Shell elements

UGS Corp. 35

1

2

kn×Y V

k k k

kk n×

=Y VV

2 1n= ×V V V

For the special case when the knV vector is parallel to the Y axis,

the program uses the following conventions:

1 2 when k k kn≡ ≡ ≡ +V Z V X V Y

and

1 2 when k k kn≡ − ≡ ≡ −V Z V X V Y

The two rotational degrees of freedom named kα and kβ are about

axes kV and kV respectively. 1 2

�

�

�

�

vk

wk

uk �k

Vkn

�k

No stiffness

for rotation

about Vkn

Z

YX

Vk1

Vk2

formulation, the definitions of e = 0 (in the initial conf e updated

Figure 2.3-3: Shell degrees of freedom at node k

When using the large displacement

1k and 2

kV are only used at timViguration) after which the vectors t k

nV and 1t kV ar

Chapter 2: Elements


usin points, and is calc No

behavior of the shell enters the r

director vectors nV ) m

necessarily exactly normal to the shell midsurface. Fig. 2.3-4(a)

the shell midsurface, after deformations have taken place this vector will in general not be exactly perpendicular to the midsurface because of shear deformations (see Fig. 2.3-4(b))

solution because the stress in the t-direction (i.e., in the direction of ) is imposed to be zero. This is achieved by using the stress-

strain relationship in the

2t kVg incremental rotations at the nodal

ulated by the cross-product 2 1t k t k t k

n= ×V V V . te that a shell node may however be assigned 3 rotational

degrees of freedom. In this case, the element�s two rotational degrees of freedom are transformed to the displacement coordinate system before assembly. • Assumption 1 on the kinematic finite element solution in that the particles along the director vectot

nV (interpolated from the nodal point t k re ain on a straight line during deformation.

Note that in the finite element solution, the vector t V is not n

demonstrates this observation for a very simple case, considering the shell initial configuration. Furthermore, even if t

nV is originally normal to

.

• Assumption 2 on the stress state enters the finite element

tnV

, ,r s t coordinate system, shown in Fig. 2.3-5, with the condition that the stress in the direction t is zero.

• The transverse shear deformations are assumed to be constant across the shell thickness.

• In large displacement analysis, the midsurface nodal point coordinates are updated by adding the translational displacements of the nodes, and the director vectors are updated using the rotations at the nodes by applying the large rotation update transformation described in p. 580 of ref. KJB (Exercise 6.56).


page 440

ref. KJBpp. 399, 440

2.3: Shell elements

UGS Corp. 37

��

�

�

Angle 90� �

Angle = 90�

Initial configuration

Final configurationZ

XY

��

�

( )L2

L1 + L2�

k k + 1

L1 L2

0 knV

0 k+1nV

0nV

0nV

tVn

a) Due to initial geometry

b) Due to displacements and deformations (with shear)

Figure 2.3-4: Examples of director vectors not normal to the shell midsurface.

Chapter 2: Elements


r

r

s

s

s

t

t

r

rs t

=�

s t�s

t r=

�

t r�2 2

Figure 2.3-5: Definition of the local Cartesian system ( )r, s , t at an

integration point in the shell

2.3.2 Materia

• com

• with atics, or !

I isplacements

and rotations are assumed to be infinitesimally small. Using a linea g a nonl rial results in a materially-nonlinear-only formulation.

mall.

can be Jaumann) formulation or a ULH

ref. KJBSection 6.

l models and formulations

See Table 2-2 for a list of the material models that are patible with shell elements.

The shell element can be used

! small displacement/small strain kinematics,

! large displacement/small strain kinem

large displacement/large strain kinematics.

n the small displacement/small strain case, the d

r material results in a linear element formulation, and usininear mate

In the large displacement/small strain case, the displacements and rotations can be large, but the strains are assumed to be sIn this case, a TL formulation is used.

The large displacement/large strain formulation for shells either a ULJ (updated Lagrangian

6

2.3: Shell elements

UGS Corp. 39

formulation (updated Lagrangian Hencky) depending on the setting

, the total strains can be large, but the incremental strain r each time step should be small (< 1%). The ULH formulation

n

ed

rted shell elements.

2.3.3 Shell nodal point degrees of freedom

• Shell nodes can have either 2 or 3 rotational degrees of freedom which results in nodes having either 5 or 6 degrees of freedom. • The criterion for determining whether a shell node is assigned 5 or 6 degrees of freedom is as follows. 5 degrees of freedom are initially assigned to all shell midsurface nodes. The following cases change the node to 6 degrees of freedom:

! Geometry. Shell elements at that node intersecting at an angle greater than a specified tolerance (SDOFANG parameter in the NXSTRAT entry

ents with nal

ving constrained rotations

• source for

ULFORM parameter in the NXSTRAT entry. In the ULJ formulationforequires more computations, however, it has no such restriction othe size of the incremental strains. Another significant difference is that incompatible modes cannot be used with the ULJ formulation.

The large displacement/large strain kinematics can be only uswith single layer shell elements with an isotropic plastic material. See Table 2.3-2 for a list of the suppo

).

! Other elements. If the node also has other elemrotational degrees of freedom, i.e., beam elements, rotatiosprings, rotational masses or rotational dampers.

! Rotational loads, constraints or boundary conditions. This includes the following cases:

- applied moment at the node

- rotational fixed boundary condition at the node

- rigid link connected to the node

- constraint equation involconnected to the node

Shell nodes with 6 degrees of freedom may be a potential singularity. In this case, very weak springs are automatically

Chapter 2: Elements


added to prevent the singularity. The cases in which this happens

rity.

are discussed later in this section. • Enforced rotations can only be applied to 6 degree of freedom hell nodes with potential singulas

• Fig. 2.3-6 shows examples of 5 and 6 degree of freedom shell nodes.

• Note that for both 5 and 6 degree of freedom shell nodes, the translations uk, vk, wk are referred to the chosen displacement coordinate system.

� � �

��

��

�

�

�

�

�

�

��

X

YZ

8 7 65

4 39

2 1

No X,Y,Z translations

No Y, Z rotations

beam elementsRigid element (Node 3

as independent node)

Concentrated

moments

Concentrated

forces

MY

FX

MZ

FZ

4-node shell elements

NodeNumberof DOFs

1

2

3

54

6

7

89

6

5

6

6

5

6

6

6

5

Potentialsingularity

Yes

No

No

Yes

No

No

No

Yes

No

Figure 2.3-6: Examples of shell nodes with 5 or 6 degrees of freedom

2.3: Shell elements

UGS Corp. 41

0 knV

0 knV

element 1

element 1

element 2

element 2

k

k k

�

�

�

� �

�

�

0 knV is the average of all element director vectors

0 knV

Figure 2.3-7: 5 degree of freedom shell with unique vector at node k

5 degrees of freedom node: A node "k" that is assigned 5 degrees of freedom incorporates the following assumptions:

• Only one director vector (denotedssociated with the node. The program calculates the director ec (one normal

vec de k) at the node. This is illustrated in Fig. 2.3-7. If two (or more) elements attached to the node have oppositely directed normals, the program reverses the oppositely directed

at time = 0 as 0 knV ) is

av tor by taking the average of all normal vectors

tor is generated per shell element attached to no

normals, so that all normals attached to the node have (nearly) the direction. same

Chapter 2: Elements


6 d es

• The program generates as many normal vectors at node k as

ached to the node. Hence each individual shell element establishes at node k a vector normal to its

egrees of freedom node: A node "k" that is assigned 6 degreof freedom incorporates the following assumptions:

there are shell elements att

midsurface. This is illustrated in Fig. 2.3-8. The components of the shell element matrices corresponding to the rotational degreesof freedom at this node are first formulated in the local midsurface system defined by the normal vector and then rotated to the displacement coordinate system.

0 knV

element 1

0 knV

element 2

element 1

element 1

element 2

element 2

k

k k

�

�

�

� �

�

�

director for element 2director for element 1

igure 2.3-8: 6 degree of freedom shell with separate director or

the respective element).

The three rotational degrees of freedom at node k referred to the

6 degree of freedom shell nodes

ne of the rotational degrees of freedom. In this case, a weak rotational

ffness

Fvectors at node k (each vector is used as a director f

•displacement coordinate system can be free or constrained.

Singularity at • When a shell node is forced to have 6 degrees of freedom due to the reasons explained above, there may be a singularity at o

spring is added to the 3 rotational degrees of freedom. The sti

2.3: Shell elements

UGS Corp. 43

y Advanced Nonlinear Solution and

oes not require user intervention.

• Not all the cases that lead to a shell node possessing 6 degrees of freedom t e a

!

! . Beamsingularity at n

All

fixed

connects two shell structures as shown in Fig. 2.3-9, and it is perpendicular to both shells, then the beam is free to rotate about this perpendicular direction (the

of the spring is set to be a small fraction of the average rotational stiffnesses at the shell node.

• This is done automatically bd

(listed at he beginning of this section) may introducsingularity at the node.

Geometry. No potential singularity exists in this case.

Other elements -stiffened shells do not cause any the shell odes. A potential singularity exists for

beams perpendicular to the shell surface and for rotational springs, masses and dampers. ! Rotational loads, constraints or boundary conditions. the items listed earlier for this feature result in a potential singularity except for the following cases: - All rotational degrees of freedom at the node are - The node is a dependent node of a rigid link - The node is a dependent node of a rotational constraint

ion equat

• If multiple factors lead to the presence of 6 degrees of freedomat a shell node, no singularity is present if any of the factors eliminates the singularity. For example, if a shell node has an applied moment and is attached to non-perpendicular beams elements there is no singularity.

• Fig. 2.3-6 shows examples where shell singularity may or may not occur. • The singularity that may result from beams attached to shells requires some clarification. If a beam

Chapter 2: Elements


ls at an

z-direction in this example). If the beam intersects the shelangle, this singularity is not present.

Shell elements

Beams

Perpendicularto both shells

Not perpendicularto shells

x

z

y

2.3.4 Com osite shell elements (Solution 601 only)

lements are generated when a PCOMP property

! An arbitrary number of layers can be used to make up the total thickness of the shell, and each laydifferent th

• The conventions for defining the director vectors, the local axes V1 and V2, and the 5/6 degree of freedom selection are all the same as those for the single layer shell. • In order to take into account the change of material properties from one layer to another, numerical integration of the mass and stiffness matrices is performed layer by layer using reduced natural

Figure 2.3-9: Beams intersecting shell elements p

• Composite shell e ID references one of the following Nastran shell

connectivity entries: CQUAD4, CTRIA3, CQUAD8, CTRIA6. • The composite shell elements are kinematically formulated in the same way as the single layer shell elements, but

er can be assigned a ickness.

! Each layer can be assigned one of the different material models available. The element is nonlinear if any of the material models is nonlinear, or if the large displacement formulation is used.

2.3: Shell elements

UGS Corp. 45

g. 2.3-10 and 2.3-11). The relation between the element natural coordinate t and the reduced natural coordinate tn of layer n is:

coordinates through the thickness of the element (see Fi

( )1

11 2 1n

i n n

it t

a =

⎡ ⎤⎛ ⎞= − + − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∑! ! (2.3-1)

with

t = element natural coordinate through the thickness tn = layer n natural coordinate through the thickness = thickness of layer i a = total element thickness

a and are functions of r and s.

i!

i!

�

�

�

�

�

�

�

��

3

V3n

V31

V32

�3

�3

4

1

2a2

t

r

s

Z

Y

X

7

6

5

8

Figure 2.3-10: 8-node composite shell element

Chapter 2: Elements


ln

aLayer n

Layer 1

E Gnxx

nxz

nxy

l

geom

Figure 2.3-11: Multilayered shel

The etry of layer n is given by:

1 1 2tn kk

k i k k nik

nN N

k

ti

t k nx h x m t h V=

+ +⎢ ⎥⎣ ⎦

∑ (2=

⎤∑ !.3-2)

⎡=

with

tix = coordinate of a point inside layer n in direction i

functions N = number of nodes

h = interpolation kt k

ix = Cartesian coordinates of node k

ormal vector at node k ess at node k

= thickness of layer j at node k

= distance between element midsurface and midsurface of layer n at node k

In the above formula, is given by

ni = component ot kV f n t knV

ka = total element thicknjk! nkm

nkm

12 2

nnn jk kk k

j

am=

= − + −∑ !! (2.3-3)

Accordingly, the displacements in layer n are:

2.3: Shell elements

UGS Corp. 47

( )0 02 1

1 1 2

n nN Nk n k kk

i k i k i k i kk k

th u m V Vu α β= =

⎛ ⎞= + + − +⎜ ⎟

⎝ ⎠∑ ∑ !

with (2.3-4)

u = components of nodal displacements at node k

ki

,k kα β = rotations of knV about 1

kV and 2kV (see Fig. 2.3-3)

Using the expressions of the coordinates and displacements

be evaluated.

2.3.5 Num

e 8-node element, 3×3 point integration is used. The 3-node -

ode triangular shell element uses 7-point Gauss integration.

- For elastic materials, 2-point Gaussian integration is always

plastic materials, 5-point Newton-Cotes integration is the default. Although using 5-point integration is computationally more expensive, it gives much more accurate results for elasto-plastic shells.

- For composite shells with elasto-plastic materials, 3-point ation is the default.

The order of through-thickness integration can be modified via the TINT parameter in the NXSTRAT entry. If TINT is specified, it will be applied to both single-layered and composite elasto-plastic shells.

e integration order is used for both Solution 601 and


defined in Eqs. 2.3-2 and 2.3-4, the contribution of each layer to the element stiffness and mass matrices can

erical integration

• Gauss numerical integration is used in the in-plane directions of the shell. For the 4-node shell element, 2×2 integration is used. For thtriangular shell element uses 4-point Gauss integration in the inplane directions, and the 6-n

• Numerical integration through the shell thickness is as follows:

used. - For elasto-

Newton-Cotes integr

• The sam701.

Chapter 2: Elements


ployed with a lumped ass is allowed in

Solution 701. • The consistent mass matrix is calculated using the isoparametric formulation with the shell element interpolation functions.

f

idsurface nodes is

2.3.6 Mass matrices • In Solution 601 shell element can be emor a consistent mass matrix. Only a lumped m

• The lumped mass for translational degrees of freedom o/M n where M is the total element mass and m

n is the number of nodes. No special distributory concepts are employed to distinguish between corner and midside nodes, or to account for element distortion. The rotational lumped mass for implicit analysis (Solution 601) is

( )2av

112

M tn

⋅ , where avt is the average shell thickness. The same

rotational mass matrix is assumed for 5- and 6-degree of freedom nodes, and is applied to all rotational degrees of freedom. Th rotational lumped mass for explicit analysis (Solution 701) is e

( )2 2av

112

M t An

⋅ + , where avt is the average shell thickness and A i

the cross-sectional area. The rotational masses are scaled up to ensures that the rotational degrees of freedom will not reduce thecritical time step for shell elements. The same rotational mass matrix is assumed for 5- and 6-degree of freedom nodes and is ap

s

p to

2.3.7 Selection of ele

The mo of general shells is sually the 4-node element. This element does not lock and has a igh predictive capability and hence can be used for effective

analysis of thin • The phenomenon of an elem t being oo stiff is, in the literature, referred cking. I ssence, the phenomenon a polation functions used for an element are not �abundantly� able to represent zero (or very small)

ref. KJBpp. 403-40

lied all rotational degrees of freedom.

ments for analysis of thin and thick shells

st effective element for analysis •uh

and thick shells.

en to as element lo

much tn e

rises because the inter8

2.3: Shell elements

UGS Corp. 49

shearing or memb he element cannot represent zero shearing strains, but the phyvery small) shearing strains, then the element becomes very stiff as its thickness over length ratio decreases. The MITC elements are implemented to g prob . More details on

embrane terms are

2.4 Surface elements � 2-D solids (Solution 601 only)

X, c

ity entries CQUAD4, CQUAD8, leads to a plane strain 2-D

in Solution 701.

2.4.1 Gene

show some typical 2-D s.

rane strains. If tsical situation corresponds to zero (or

overcome the lockin lemthe interpolations used for the transverse and mprovided in ref. KJB, pp. 403 � 406.

• 2-D solid elements are obtained in the following cases: ! PLPLANE property ID entry that references the shell connectivity entries CQUAD, CQUAD4, CQUAD8, CQUADCTRIA3, CTRIA6, and CTRIAX. This leads to a hyperelastiplane strain or axisymmetric 2-D element. ! PSHELL property ID entry with MID2 = -1 that also references the shell connectivCTRIA3, and CTRIA6. Thiselement. ! 2-D elements are not supported

ral considerations

• The following kinematic assumptions are available for two-dimensional elements in Solution 601: plane strain and axisymmetric. Fig. 2.4-1 and Fig. 2.4-2elements and the assumptions used in the formulation

Chapter 2: Elements


�

�

�

��

�

��

(a) 8- & 9-node quadrilateral elements

�

�

�

�

(b) 3-node triangular element

(d) 6- & 7-node triangular elements

�

��

��

�

�

� ��

��

�

�

�

��

�

�

�

��

�

��

(c) 4-node quadrilateral element

Figure 2.4-1: 2-D solid elements

he .4-1

• 2-D solid elements in Solution 601 are classified based on tnumber of nodes in the element and the element shape. Table 2shows the correspondence between the different 2-D solid elements and the NX element connectivity entries.

a) Plane strain element

��

�

�

�

�xz = 0

�yz = 0

zz = 0

x

y

z

b)Axisymmetric element

��

�

�

�

�xz = 0

�yz = 0

zz = u/x

(linear analysis)

x

y

z

Figure 2.4-2: Basic assumptions in 2-D analysis

2.4: Surface elements � 2-D solids

UGS Corp. 51

2-D solid m

ele ent NX element connectivity entry

3-node triangle CTRIA3 CTRIAX1 (with 3 input nodes)

4-node quadrilateral CQUAD4 CQUAD2, CQUADX1 (with 4 input nodes)

6-node tr

7-node tr CTRIAX (with 6 input nodes)

8-node qua

9-node qua ADX (with 8 input nodes) CQUAD , CQUADX1 (with 9 input nodes)

iangle CTRIA6 CTRIAX1 (with 6 input nodes)

angle CTRIA6i3

1,3

drilateral CQUAD8 CQUAD2, CQUADX1 (with 8 input nodes)

drilateral CQUAD83 CQUAD2,3, CQU 1,3

2

Table 2.4-1: Correspondence between 2-D solid elem ent

Notes: 1. Axisym2. Plane strain hyperelastic only 3. With ELCV = 1 in NXSTRAT entry

ents and NX elemconnectivity entries

metric hyperelastic only

Chapter 2: Elements


t nodes improve the

erformance of the 2-D elements as explained later in this section.

e +X half plane.

axisymmetric element provides for the stiffness of one

dian of the structure. Hence, when this element is combined with

ples 5.9 and 5.10, p. 56.

• The basic 2-D elements used in Solution 601 are isoparametric displacement-based elements, and their formulation is described in detail in ref. KJB, Section 5.3.

• The basic finite element assumptions for the coordinates are (see Fig. 2.4-3):

• Note that the extra middle node in the 7-node and 9-node 2-D elements is automatically added by the program when ELCV is seto 1 in the NXSTRAT entry. These extrapThe boundary conditions at the added node are predicted from the neighboring nodes.

• The plane strain and axisymmetric 2-D elements must be defined in the XY plane. The axisymmetric element must, in addition, lie in th • 2-D solid elements can be combined with any other elements available in Advanced Nonlinear Solution.

• The raother elements, or when concentrated loads are defined, these must also refer to one radian, see ref. KJB, Exam3• The plane strain element provides for the stiffness of a unit thickness of the structure.

1

q

i ii

x h x=

= ∑ ; y1

q

i ii

y h=

= ∑

and for the displacements:

u1

q

i ii

u h=

= ∑ ; 1

v vq

i ii

h=

= ∑

where

hi(r,s) = interpolation function corresponding to node i

ref. KJBSections 5.3.1

and 5.3.2


UGS Corp. 53

q = number of element nodes xi, yi = nodal point coordinates ui, vi = nodal point displacements

(r,s) = isoparametric coordinates

��

�

�

�

�

� �

�

v2

u2

y2

x2

Displacement degrees

of freedomr

s

X

Y

34

1 2

7

8

5

9 6

Z

Figure 2.4-3: Conventions used for the 2-D solid element • In addition to the displacement-based elements, special mixed-interpolated elements are also available, in which the displacements and pressure are interpolated separately. These elements are effective and should be preferred in the analysis of incompressible media and inelastic materials (ela aterials with Poisson's ratio close to d elasto-plastic materials). Table 2.4-2 shows the number of pressure degrees of freedom used for each 2-D element type. For more details on the number of degrees of freedom ideal for each element, see the ref. KJB, Section 4.4.3 and Table 4.6, pp. 292-295. • The mixed interpolation is the default setting for hyperelastic materials. It can be activated for other materials, such as elasto-plastic, creep, and elastic with high Poisson�s ratio, via the UPFORM flag in the NXSTRAT entry. The 4-node element (1 pressure degree of freedom) and 9-node element (3 pressure degrees of freedom) are recommended with the mixed formulation.

stic m0.5, rubber-like materials, creep an

Chapter 2: Elements


ref. T. Sussman and K.J. Bathe, "A Finite Element lation fo and

. 1 87.

2-D pressure DOFs

Formu r Nonlinear Incompressible Elastic, Vol. Inelastic Analy

26, Nosis," J. Computers and Structures

/2, pp. 357-409, 19

solid element Number of

3- odn e triangle - 4-no 1

6- 1 7- od

8-node quadrilateral 1

de quadrilateral node triangle n e triangle 3

9-node quadrilateral 3 Table

2.4-2: Mixed formulation settings for 2-D solid elements

ions used for 2-D solid elements are ef

recting

2.4.2 Mate a

• See Table 2-2 for a list of the material models that are compatible with 2-D solid elements.

ended for elastic-plastic materials and also elastic materials with a Poisson ratio close to 0.5. For these


• The interpolation functd ined in ref. KJB, Fig. 5.4, p. 344.

• The 6-node spatially isotropic triangle is obtained by corthe interpolation functions of the collapsed 8-node element. It then uses the same interpolation functions for each of the 3 corner nodes and for each of the midside nodes.

The 3-node triangular element is obtained by collapsing one side of the 4-node element. This element exhibits the constant strain conditions (except that the hoop strain in axisymmetric analysis varies over the element).

ri l models and formulations

• Advanced Nonlinear Solution automatically uses the mixed r hyperelastic materials. The mixed interpolation formulation fo

formulation is also recomm


UGS Corp. 55

materials, the u/p mixed formulation can be activated by setting UPFORM = 1 in the NXSTRAT entry.

ensional elements can be used with

tics, or

s. ! The small displacement/sm strain and large

materia l

p

materials. A ULH (updated Lagran lation mpatible material models except the hyper For the hypere L (total Lagr gian) formulation is used

• The basic formulations of 2-D solid elements are described in ref. KJB, pp. 497-537, and the finite element discretization is given in ref. KJB pp. 538-542, 549-555.

• Note that all ns can be mixed in the same finite element model. are initially co atible, then they will remain com sis.

ref. KJBSections 6.2 an

• The two-dim - small displacement/small strain kinematics, - large displacement/small strain kinema - large displacement/large strain kinematic

alldisplacement/small strain kinematics can be used with any of the compatible material models, except for the hyperelastic

l. The use of a linear material with smaldisplacement/small strain kinematics corresponds to a linearformulation, and the use of a nonlinear material with the small displacement/small strain kinematics corresponds to a materially-nonlinear-only formulation.

! The program uses the TL (total Lagrangian) formulation

when the large displacement/small strain formulation is selected.

! The large displacement/large strain kinematics can be used

with plastic materials including those with thermal and creeeffects, as well as hyperelastic

gian Hencky) formu is used for all coelastic material.

lastic material, the T an.

continuum mechanics

these formulatioIf the elements mppatible throughout the analy

d 6.3.4


Chapter 2: Elements


2.4.3 Numerical integration

The 4-node quadrilateral element uses 2×2 Gauss integration for

ular elements are spatially otropic with respect to integration point locations and

ode 4-point

on in the axisymmetric case. The 6-node and 7-ode triangular elements use 7-point Gauss integration.

2.4.4 Mass matrices

•the calculation of element matrices. The 8-node and 9-node elements use 3×3 Gauss integration. • The 3-node, 6-node and 7-node triangisinterpolation functions (see Section 5.3.2, ref. KJB). The 3-nelement uses a single point integration in plane strain and Gauss integratin • Note that in geometrically nonlinear analysis, the spatial positions of the Gauss integration points change continuously as the element undergoes deformations, but throughout the response the same material particles are at the integration points.

• The consistent mass matrix is always calculated using either 3×3 Gauss integration for rectangular elements or 7-point Gauss integration for triangular elements. • The lumped mass matrix of an element is formed by dividing the element�s mass M equally among its n nodes. Hence, the mass assigned to each node is /M n . No special distributory conceare employed to distinguish between corner and midside nodeto account for element distortion. mmendations on use of elements

• The 9-node element is usually the most effective.

• he linear elements (3-node

pts s, or

2.4.5 Reco

and 4-node) should only be used in


5.5.4 and 5.5.5

Tanalyses when bending effects are not dominant.


UGS Corp. 57

r r

s

s

a) Rectangular elements

1-point

s

r

4-point

b) Triangular elements

7-point

Figure 2.4-4: Integration point positions for 2-D solid elements

the d.

2.5 Solid elements � 3-D

2.5.1 Gene

• 3-D solid elements are generated using the CHEXA, CPENTA -

• In near incompressible elastic, elasto-plastic and creep analysis,

use of the u/p mixed formulation elements is recommende

ral considerations

and CTETRA element connectivity entries. They generate 6-, 5and 4-sided 3-D elements, respectively.

Chapter 2: Elements


orted

linear Solution are

lassified based on the number of nodes in the element, and the

ifferent 3- solid elements and the NX element connectivity entries. Note

ir number of odes.

• So6-node • Adquadra have 8

• The PSOLID property ID entry is used for all of the suppmaterials, except hyperelastic which uses PLSOLID.

• 3-D solid elements in Advanced Noncelement shape. • Table 2.5-1 shows the correspondence between the dDthat the elements are frequently referred to just by then

lution 701 only supports linear elements (4-node tetrahedron, wedge and 8-node brick elements).

vanced Nonlinear Solution does not support incomplete tic 3-D elements. For example, a CHEXA entry can only nodes or 20 nodes. Anything in between is not supported.

2.5: Solid elements � 3-D

UGS Corp. 59

�

�

�

��

��

��

�

�

�

�

�

�

�

�

�

�

�

�

��

�

��

��

��

�

�

��

�

�

�

��

��

�

�

�

�

�

� �

�

�

��

(a) 8-, 20- & 27-node brick elements (CHEXA)

(b) 4-, 10- and 11-node tetrahedral elements (CTETRA)

�

�

�

�

�

�

��

��

�

�

�

�

�

�

� �

�

�

�

�

��

�

�

��

�

�

�

��

� �

�

�

�

�

�

�

� ��

� �

� ��

� �

� �

� �

� �

� �

� �

� �

� �

� �

(c) 6-, 15-, and 21- node wedge elements (CPENTA)

�

Figure 2.5-1: 3-D solid elements

Chapter 2: Elements


3-D solid element NX element connectivity entry

4-node tetrahedron CTETRA 10-node tetrahedron1 CTETRA 11-

CPENTA 12

227-node brick CHEXA and ELCV = 1 in NXSTRAT

node tetrahedron1 CTETRA and ELCV = 1 in NXSTRAT 6-node wedge 5-node wedge1 CPENTA 1-node wedge1 CPENTA and ELCV = 1 in NXSTRAT 8-node brick CHEXA 0-node brick1 CHEXA

1

Table 2.5-1: Correspondence between 3-D solid elements and NX

element connectivity entries Note: 1. Only for Solution 601 • Note that the mid-volume and midsurface nodes in the 27-node21-node and 11-node elements are automatically added by Advanced N

,

onlinear Solution when ELCV is set to 1 in the XSTRAT entry. The boundary conditions at the added nodes are

•

ent assumptions for the coordinates are

ee Fi

q

Npredicted from the neighboring nodes.

The elements used in Advanced Nonlinear Solution are

isoparametric displacement-based elements, and their formulation is described in ref. KJB, Section 5.3.

• The basic finite elem(s g. 2.5-2, for the brick element):

q q

x1 1 1

i i i i i ii i i

h x y h y z h z= = =

= = =∑ ∑ ∑

nd for the displacements:

ref. KJBSection 5.3

a


UGS Corp. 61

q

u h h w= =

node i r coordinates

1 1 1i i i= = =

where h

v vq q

u h w =∑ ∑ ∑i i i i i i

i (r, s, t) = interpolation function corresponding to , s, t = isoparametric

q = number of element nodes xi, yi, zi = nodal point coordinates ui, vi, wi = nodal point displacements

�

�

�

�

�

��

�

�

�

�

��

�

�

�

�

�

�

�

�

�

�

�

�

��

Z

X

Y

w8

u8

v88

4

t

s

rz4

y4

x4

2

9

112

11

310

13

175 14

6

187

16

19

1520

Figure 2.5-2: Conventions used for the nodal coordinates and displacements of the 3-D solid element

• In addition to the displacement-based elements, special mixed-interpolated elements are also available, in which the displacements and pressure are interpolated separately. These elements are effective and should be preferred in the analysis of incompressible media and inelastic materials (specifically for materials in which Poisson's ratio is close to 0.5, for rubber-like materials and for elasto-plastic materials). Table 2.5-2 shows the number of pressure degrees of freedom for each 3-D element type. For more details on

e mixed interpolation of pressure and displacement degrees of freedo .6 and 4.7, pp. 292 - 295 in ref. KJB.

thm for 3-D solids, see Section 4.4.3, p. 276, and Tables 4

Chapter 2: Elements


ent Number of pressure DOFs 3-D solid elem

4-node tetrahedron - 10-node tetrahedron 4 11-node tetrahedron 4

6-node wedge 1 15-node wedge 4

1

4

21-node wedge 4 8-node brick

20-node brick 4 27-node brick

xed u/p formulations available for 3-D solid elements

r

trahedron, the 15-node wedge and the 20-node brick

idside nodes if

d

m

Table 2.5-2: Mi • The mixed formulation is the default setting for hyperelastic materials, and it can be activated for other materials, such as elastic-plastic, creep, and elastic with high Poisson�s ratio, via the UPFORM flag in the NXSTRAT entry. The use of the 8-node (one pressure DOF) or 27-node (4 •

pressure DOFs) element is recommended with the mixed mulation. fo

• Note that 4 pressure degrees of freedom are used for the 10-

ode tenelement. Even though this setting does not satisfy the inf-sup test, the elements generally perform better than with a single pressure

gree of freedom. Still, it is better to add the mdepossible. This is done by setting ELCV = 1 in the NXSTRAT entry. • In addition to the displacement-based and mixed-interpolateelements, Advanced Nonlinear Solution also includes the possibility of including incompatible modes (bubble functions) in the formulation of the 6-node wedge and the 8-node brick element. Within this element, additional displacement degrees of freedoare introduced. These additional displacement degrees of freedom


UGS Corp. 63

re not associated with nodes; therefore the condition of isplacement compatibility between adjacent elements is not

satisfied in general. The addition of the incompatible modes ubble functions) increases the flexibility of the element,

ee reference KJB, Section 4.4.1. Note that these incompatible-mode lements are formulated to pass the patch test. Also note that

modes

he 6- node wedge and the 8-node brick elements. In articular, it is not available for the 4-node tetrahedral element.

ad

(bespecially in bending situations. For theoretical considerations, s

eelement distortions deteriorate the element performance when incompatible modes are used.

• Table 2.5-3 shows which elements support incompatible(bubble functions). The incompatible modes feature is only available for tp

3-D solid element Support for

incompatible modes

4-node tetrahedron No 10-node tetrahedron No 11-node tetrahedron No

6-node wedge Yes

20-node brick No No

15-node wedge No 21-node wedge No

8-node brick Yes

27-node brick

ble 2.5-3: Incompatible modes (bubble functions) available for 3-D lid elements

Taso

The incompatible modes feature cannot be used in conjunction

with the mixed-interpolation formulation.

• The interpolation functions used for 3-D solid elements for q ≤ 20 are shown in Fig. 5.5, ref. KJB, p. 345.

•

Chapter 2: Elements


The 10-node tetrahedron (see Fig. 2.5-1(c)) is obtained by co

otropic 10-node and 11-node tetrahedra are used in Solution 601.

lement exhibits constant strain conditions.

2.5.2 Material models and nonlinear formulations

re ompatible with 3-D solid elements.

Advanced Nonlinear Solution automatically uses the mixed

he mixed rmulation is also recommended for elastic-plastic materials and

ntry.

• The 3-D elements can be used with - displaceme /small strain kinematics,

- large displacement/large strain kinematics. ! all displac all strain and large

lacement/sm ain kinematics can be used with any of

small displacement/small strain kinematics corresponds to a linear

small strain kinematics corresponds to a materially-nonlinear-only formulation.

rogram uses the TL (total Lagrangian) form large d /small strain kinematics is selected.

! large displa t/large str atics can be used

with plastic materials including thermal and creep effects, as d Lagrangian

•llapsing nodes and sides of rectangular elements. Spatially

isThe 4-node tetrahedron (see Fig. 2.5-1(c)) is obtained by

collapsing nodes and sides of the 8-node rectangular element. This e

• See Table 2-2 for a list of the material models that ac

•interpolation formulation for hyperelastic materials. Tfoelastic materials with a Poisson ratio close to 0.5. It can be activated by setting UPFORM = 1 in the NXSTRAT e

small nt - large displacement/small strain kinematics, or

The sm ement/smdisp all strthe compatible material models, except for the hyperelastic material. The use of a linear material with

formulation, and the use of a nonlinear material with the small displacement/

! The p ulation when isplacement

The cemen ain kinem

well as hyperelastic materials. A ULH (update


UGS Corp. 65

Hencky) formulation is used for all compatible material ic

al models, a TL (total Lagrangian) formulation is used.

Note that all these formulations can be used in the same finite

ration

e s

tegration point locations and interpolation functions. For the 4-ent, 1-point Gauss integration is used. For the lement, 17-point Gauss integration is used,

and 17-point Gauss integration is also used for the 11-node l element.

he response

d for both Solution 601 and 701.

2.5.4 Mass matrices

The consistent mass matrix is always calculated using 3×3×3

point Gauss integration.

ent is formed by dividing the element�s mass M equally among each of its n nodal points.

ref. KJBSections 6.2

and 6.3.

models except the hyperelastic material. For the hyperelastmateri

• The basic continuum mechanics formulations are described in ref. KJB, pp. 497-568. The finite element discretization is summarized in Table 6.6, p. 555, ref. KJB. •element model. If the elements are initially compatible, they will remain compatible throughout the analysis.

2.5.3 Numerical integ

5



5.5.4 and 5.5.5

• The 8-node brick element uses 2×2×2 Gauss integration for thcalculation of element matrices. The 20-node and 27-node elementuse 3×3×3 Gauss integration.

• Tetrahedral elements are spatially isotropic with respect to innode tetrahedral elem10-node tetrahedral e

tetrahedra

• Note that in geometrically nonlinear analysis, the spatial positions of the Gauss integration points change continuously as the element undergoes deformations, but throughout tthe same material particles are at the integration points. • The same integration order is use

•Gauss integration except for the tetrahedral 4-node, 10-node and 11-node elements which use a 17- • The lumped mass matrix of an elem

Chapter 2: Elements


Hence the mass assigned to each node is /M n . No special distributory concepts are employed to distinguish between corner nd midside nodes, or to account for element distortion.

r both Solution 601 and olution 701.

2.5.5 Recommendations on use of elements

• The linear elements (4-node, 6-node and 8-node) usually perform better in contact problems. If bending effects are insignificant, it is usually best to not use incompatible modes.

• The linear elements (6-node and 8-node brick elements, without

compatible modes) should only be used in analyses when

rain element, many lements, that is fine meshes, must usually be used in analyses.

• In near incompressible elastic, elasto-plastic and creep analyses, the use of the u/p mixed formulation elements is recommended. • When the structure to be modeled has a dimension which is extremely small compared with the others, e.g., thin plates and shells, the use of the 3-D solid element usually results in too stiff a model and a poor conditioning of the stiffness matrix. In this case, the use of shell elements, particularly the 4-node shell element (see

nd 2.6), is more effective.

all available

.

f

e, in this case, the corresponding segment tractions will be accurate.

ref. KJBPage 383

a

• The same lumped matrix is used foS

inbending effects are not dominant. Since the 4-node tetrahedron is a constant st•

e

Sections 2.7 a

Recommendations specific to Solution 601

• The 27-node element is the most accurate among lements. However, the use of this element can be costlye

• The 20-node element is usually the most effective, especially ithe element is rectangular (undistorted). However, it is not recommended to use the 20-node element in contact analysis becausin

2.6: Scalar elements � Springs, masses and dampers

UGS Corp. 67

2.6 Scalar elements � Springs, masses and dampers

• Scalar elements in Advanced Nonlinear Solution either connect 2 degrees of freedom together or just a single degree of freedom to the ground. Ther ts: springs, masses, and dampers. ! Spring elements are defined using the CELAS1 and CELAS2 element connectivity ntries. ! Mass elements are defined using the CMASS1 and CMASS2 element connectivity entries.

! Damper elements are defined using the CDAMP1 and CDAMP2 element connectivity entries.

• Fig. 2.6-1 shows the spring, mass and damper single degree of freedom elements available in Advanced Nonlinear Solution. They correspond to a grounded spring, a concentrated mass, and a grounded damper, respectively. • Fig. 2.6-2 shows the available scalar elements connecting two degrees of freedom. Only the translational version of the spring and damper are shown in the figure, but they can connect rotational degrees of freedom as well.

e are three forms of scalar elemen

e

Chapter 2: Elements


� �

��

k u

Single

translational

DOF spring

�

k

Single

rotational

DOF spring

ü

m

Single

translational

DOF mass

Single

rotational

DOF mass

Mass = [m]M

c c

Single

translational

DOF damper

Single

rotational

DOF damper

Damping = [c]C

Stiffness = [k]K

.u

(b) mass element(a) spring element

(c) damper element

Figure 2.6-1: Single degree of freedom scalar elements

2.6: Scalar elements � Springs, masses and dampers

UGS Corp. 69

K =

C =

k

c

c

-k

-c

-c

-k k

a) spring element

b) damper element

1 2

U 1 U2

1 2

U1 U2

k

c

c) mass element

m

U1 U2

M =lumped

m/2 0

0

M =consistent

m/2

m/3

m/3

m/6

m/6

Figure 2.6-2: Two-degrees-of-freedom scalar elements

2.7 R-type elements

• R-type elements impose multipoint constraints on one or more nodes. The constraints are created automatically by the program based on the element�s input. The following R-type elements are supported in Advanced Nonlinear Solution: RBAR, RBE2 and RBE3. • Rigid elements are a subgroup of R-type elements that include RBAR and RBE2. • RBE3 is an interpolation constraint element which also produces constraint equations.

Chapter 2: Elements


• Solution 601 provides several options for modeling the Rigid

e how the Rigid elements are treated.

2

nnect

id option is selected, Rigid elements are

internally represented either as standard multipoint constraints, or tant constraint

coefficients and therefore do not give accurate results in large

nts that are updated based on the deformation of the structure. This is

re used.

2.7.1 Rigid elements

elements. They can be modeled as perfectly rigid elements using constraint equations or as flexible (but stiff) elements. TheEQRBAR or EQRBE2 parameters in the NXSTRAT entry determin

• Solution 701 does not support the flexible option. • The RBAR entry generates a single Rigid element betweennodes. • The RBE2 entry generates multiple Rigid elements. They coone independent node to several nodes.

• If the perfectly rig

as rigid links. Multipoint constraints have cons

displacements (unless the 2 nodes are coincident). Rigid links also create multipoint constraints but with variable coefficie

illustrated in Fig. 2.7-1. Therefore, whenever possible, large-displacement rigid links a

2.7: R-type elements

UGS Corp. 71

�

using MPC — small rotations

3 RBAR elements with

one independent node

�

��

��

� ��

��

�

�

�

��

�

��

�rotation

using large displacement

rigd links Figure 2.7-1: Difference between sm

large displacement all displacement MPC and

rigid links • 60 Rig rameters and the distance between the nodes (RBLCRIT paramtransla in NXST • Th lus and cross-sectional area of the internal beams can be automatically determBE a • The rigconstraint. en using the fnonlinear • bec his allows multiple constraints to be defined at a node, but msince freedo

If the flexible option is selected for Rigid elements, Solution 1 internally generates beam or spring elements depending on theid element pa

eter in NXSTRAT), or a spring element tion can be always requested (in the EQRBAR parameter RAT).

e stiffness of the internal springs and the Young�s modu

ined by Solution 601 or set by the user (see SPRINGK, AME nd BEAMA parameters in NXSTRAT entry).

id option results in more accurate enforcement of the However, the compliance introduced in the model whlexible option can lead to easier convergence in problems.

The flexible option results in none of the degrees of freedom oming dependent. T

ore importantly it is beneficial for contact contactor points cannot have all dependent degrees ofm.

Chapter 2: Elements


Classi • The internal representation of an RBAR rigid element depends on the options present in CNA, CNB, CMA and CMB, as shown in Fig. 2.7-2.

fication of Rigid elements

Independent: CNA

Dependent: CMA

�

�

GB

GAIndependent: CNB

Dependent: CMB

Figure 2.7-2: Relevant parameters in the RBAR rigid element • Cu f RBAR n.

one point are dependent on rds,

CNA

CNA Class egrees of freedom active but not all dependent

edom belong to 1 point. For example,

rrently, Advanced Nonlinear Solution identifies 3 classes o settings. Each class gets a different internal representatio

Class 1: All 6 degrees of freedom ofhose of the other point. In other wot

= 123456, CNB = 0, CMA = 0, CMB = 123456

r o

= 0, CNB = 123456, CMA = 123456, CMB = 0

2: All 6 ddegrees of fre

CNA CNB CMA CMB123 456 0 0

12346 5 5 12346 Class 3: Not all the 6 degrees of freedom are active in the con ,

CNA CNB CMA CMB

straint. For example

123456 0 0 123 123 456 4 3

2.7: R-type elements

UGS Corp. 73

Note that there are some other valid settings for RBAR that are not supported in Advanced Nonlinear Solution. • he internal representation of Rigid elements for each class is described in Table 2.7-1.

Rigid option Flexible option

T

Class 1

L < Lcrit L > Lcrit L < Lcrit L > Lcrit

MPC Rigid link Springs Beam Class 2 Springs Beam Class 3

MPC MPC MPC MPC Springs Springs

T

t

• RBE2 is interpolated in the sa manner as RBAR except that it produces m n only belong to Class 1 or 3, and their internal representation is dictated by the EQRBE2 parameter in NXSTRAT.

t that in large displacement analysis.

2.7.2 RBE e

R-type element defines the motion of a reference

2.8 Other element types

2.8.1 Gap element

• t is used in Advanced No inear Solution to connect two nodes as shown in Fig. 2.8-1. Gap elements are defined using the CGAP element connectivity entry.

able 2.7-1: Internal representation of Rigid elements

• See Section 5.7 for more details on rigid links and multipoinconstraints.

meultiple Rigid elements. These elements ca

• Note that only Class 1 represents an �ideal� Rigid elemen

accurateis lement 3

The RBE3•

node as a weighted average of the motion of a set of other nodes. This element is a useful tool for distributing applied load and mass in a model. It is internally represented in Advanced Nonlinear Solution with multipoint constraints.

The gap elemen nl

Chapter 2: Elements


• 0. When the gap is closed the ele hould be stiff), d when it is open the stiffness is KB (should be soft).

The initial gap opening is Ument has a stiffness of KA (s an

r

K - KA B

GA

KB

GB

s

t

U0

Figure 2.8-1: CGAP element coordinate system • The tangential behavior of the gap element represented by KT,

2.8.2 Conc

, only the diagonal mass terms are supported, and :

0 0 0 0 0M

MUI and MUZ is not supported. entrated mass element

• Advanced Nonlinear Solution supports the CONM1 and CONM2 entries for defining concentrated masses. • For CONM1the resulting mass matrix is given by

11

22

33

0 0 0 0 00 0 0 0 00 0 0 0 0

MM

M

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢M44

55

66

0 0 0 0 00 0 0 0 0

MM

⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

matrix is

• For CONM2, the off-diagonal mass moments of inertia termsare neglected, and the resulting mass

2.8: Other element types

UGS Corp. 75

0 0 0 0 00 0 0 0 0M

M

11

22

33

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

MI

I

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥

I⎢ ⎥⎢ ⎥⎣ ⎦

M

2.8.3 Bushing element

n

• The stiffness and damping act alonwhich is the line connecting its two nodes. In large displacement

s thbe specified via the CID parameter in the CBUSH

ntry.

• The one-dimensional bushing element CBUSH1D is used iAdvanced Nonlinear Solution to provide an axial stiffness and damping between two nodes as shown in Fig. 2.8-2.

g the axis of the element,

analysi e element axis is updated with deformation. A fixed element axis can e The element can have a constant or a nonlinear stiffness defined via a lookup table.

GA GB

Figure 2.8-2: BUSH1D e

lement

Chapter 3: Material models and formulations


3. Material m u

The objective l use of the ma il nced Nonlinear Solution. The stress and strain measures used by different materials and formulations are first summari ecThe table belo ial models available in Advanced Nonlinear Solution, and how they can be obtained from the material entry cards.

Material En i

odels and form lations of this chapter is to summarize the theoretical basis and practica

rial models and formulations ava able in Advate

zed in S tion 3.1. w lists the mater

tr es Advanced Nonlinear Solution material Sol 701availability

MAT1 Elastic isotropic !

MAT2 Elastic orthotropic (surface elemen

MAT8 !

MAT9 Elastic orthotropic (solid elements) !

MATHE

MATHP !

MATG, MA

MATT1, MA

MATT2, MA

MATT9, MA

MATS1, MA

MATS1, MA

MATS1, MATT1, MAT12

Thermal elasto-plastic !5

MATS1, MAT13 Thermal elasto-plastic !5

CREEP, MAT1 Elastic-creep

ts) !

Elastic orthotropic (surface elements)

Hyperelastic (Mooney-Rivlin, Ogden, Arruda-Boyce and Hyperfoam)

!6

Hyperelastic (Mooney-Rivlin only)

T1 Gasket

T1 Thermal elastic isotropic !5

T2 Thermal elastic orthotropic (surface elements)

!5

T9 Thermal elastic orthotropic (solid elements)

!5

T11 Elastic isotropic nonlinear4 !

T12 Elasto-plastic !

3: Material models and formulations

UGS Corp. 77

Notes: 1. With TYPE=NELAST in MATS1 2. With TYPE=PLASTIC in MATS1 3. With TYPE=PLASTIC and TID pointing to a TABLEST entry in MATS1 4. Only valid w5. Temperature6. Only Moone

ith CROD rod element interpolation at the start of the analysis only in Solution 701 y-Rivlin and Ogden materials are available in Solution 701



3.1 Stress

important to recognize which stress and strain measures are mployed in each material model: this is necessary in the

ho hey are uar feature More deta s on

tress/strain measures are provided in ref. KJB, Section 6.2.

the acceptable combinations of lements and material properties for Solution 601.

3.1.1 Kine

mall displacement/small strain kinematics

put of material parameters: All elements and material models use

ses

od m

all

.

re small, Green-Lagrange strains are practically e as engineering strains in the element coordinate system.

and strain measures

• It is epreparation of the input data and the interpretation of the analysis results.

• This section summarizes the stress and strain measures in Advanced Nonlinear Solution and w t sed with the different element types and nonline s. ils • Note that Table 2-2 listed e

matic formulations

S

Inthe engineering stress-engineering strain relationship.

Output: All elements and material models output Cauchy stresand engineering strains. • Using a linear material m el with small displacement/s all strain kinematics results in a linear finite element formulation. • Using a nonlinear material model with small displacement/smstrain kinematics results in a materially-nonlinear only (MNO) formulation.

Large displacement/small strain kinematics

Input of material parameters: Cauchy (true) stresses and logarithmic (true) strains should be less than 2%

Output: The output depends on the element type. Note that as

ng as the strains alothe sam

3.1: Stress and strain measures

UGS Corp. 79

Similarly, Second Piola-Kirchhoff stresse stresses in the element coordinate sy

mooutput Cauchy stresses and Green-Lagrange strains.

d

(3) Rods: all supported material models output Cauchyand engineering strains in the element c

Large displacement/large strain kinematics

This kind of formulation can only be used with 2-D and 3-D solid

d elements

s are practically the same stem. as Cauchy

(1) 2-D, 3-D solid elements: all supported material dels

(2) Shell elements: all supported material models output SeconPiola-Kirchhoff stresses and Green-Lagrange strains.

stresses

oordinate system.

elements and with shell elements.

For 2-D and 3-D soli

Input of material parameters: Cauchy (true) stresses and logarithmic (true) strains

Ou in the ele

ge Strain Elasto-Plastic Constitutive Formulation with Combined Isotropic-Kinematic Hardening Using the Logarithmic Stress and Strain Measures", Int. J. Numerical Methods in Engineering, Vol. 30, pp. 1099-

(2) For hyperelastic materials a total Lagrangian formulation is used. In this case,

Input of material parameters: Hyperelastic material constants.

(1) The updated Lagrangian Hencky formulation is used with elastic-plastic materials (including thermal and creep effects). In this case,

.

tput: Cauchy stresses and logarithmic strainsment coordinate system.

ref. A.L. Eterovic and K.J. Bathe, "A Hyperelastic-Based Lar

1114, 1990.



trains in th

ati linear Incompressible Elastic and Inelastic Analysis," J. Computers and Structures, Vol. 26, 1987.

Output: Cauchy stresses and Green-Lagrange s e element coordinate system.

ref. T. Sussman and K.J. Bathe, "A Finite ElementFormul on for Non

For shell elements Tw ain

ents J)

lati

hen the ULJ formulation is used:

ic strains.

tresses and logarithmic

3.1.2 Strain m

The strain m

o formulations are available for large displacement/large strshell elem . The updated Lagrangian Jaumann (ULformulation and the updated Lagrangian Hencky (ULH) formulation. For more details on how these formu ons apply to shell elements, see Section 2.3.

Input of material parameters: Cauchy (true) stresses and logarithmic (true) strains.

W

Output: Cauchy stresses and logarithm

When the ULH formulation is used:

Output: Kirchhoff stresses and left Hencky strains (practically equivalent to Cauchy sstrains).

easures

easures used in Advanced Nonlinear Solution are illustrated here in the simplified case of a rod under uniaxial tension (see Fig. 3.1-1).

Engineering strain: 00

0

e −=! !!

Green-Lagrange strain: 2 2

01202

−ε =

! ! !


UGS Corp. 81

Logarithmic and Hencky strain: 00

ln de⎛ ⎞

= =⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦

∫!

! !! !

⎡ ⎤!

Stretch: 0

λ =!!

• Note that for the small strains assumption to be valid, the strains should be less than about 2%. • Green-Lagrange strains are used in the large displacemestrain formulations. This is because this strain measure is iwith respect to

nt/small nvariant

rigid-body rotations. Therefore, for small strains, Green-Lagrange strains and the rotated engineering strains are equivalent.

�

�

�

0

F

F

A

A0

A = initial cross-sectional area0

S = final cross-sectionalF = applied force

= +0 �

Figure 3.1-1: Rod under uniaxial tension

terature.

3.1.3 Stress measures The stress measures used in Advanced Nonlinear Solution include

ref. KJBSec. 6.2.2

• Engineering strains are also called nominal strains in the li • Logarithmic strains are also known as true strains.



sures ial

engineering stresses, Second Piola-Kirchhoff stresses, Kirchhoff stresses, and Cauchy stresses (see ref. KJB). These stress meaare illustrated here in the simplified case of a rod under uniaxtension (see Fig. 3.1-1)

Engineering stress: 0

FA

σ =

Cauchy stress: 0AFA A

στ = =

0 0

0

FSA

σ= =! !! !

2nd Piola-Kirchhoff stress:

Kirchhoff stress: 0 0 0

FJA

στ = =! ! ! !

.

pressible,

• Cauchy stresses are also called true stresses in the literature

• For the case in which the material is incom

J0

στ τ= = can be used to compute the Cauchy stress and the

the engineering stress.

e Second Piola-Kirchhoff stresses are nearly equal to the Cauchy stresses from which the rigid body

When the volume change of the material is small, the Kirchhoff

n

, the differences between Kirchhoff and Cauchy stresses

3.1.4 Some theory on large strain plasticity

stic

!!

Kirchhoff stress from • When the strains are small, th

rotations of the material have been removed.

•stresses are nearly equal to the Cauchy stresses.

• Since Kirchhoff stresses are input/output only for large straianalysis with materials that are nearly incompressible, practically

gspeakinare negligible.

• This section provides a quick summary of large strain inela


UGS Corp. 83

sis using the ULH (updated Lagrangian Hencky) formulation. or further information, see ref KJB, Section 6.6.4 and also the

f. F. J. Montáns and K.J. Bathe, "Computational issues in large

plastic spin", Int. J. Numer. Meth. Engng, 2005; 63;159-196. ref. M. Kojić and K.J. Bathe, Inelastic Analysis of Solids and Structures, Springer-Verlag, 2003. • The following measures can be used to determine the state of a solid or a structure undergoing large displacements and large strains: stretches, logarithmic strains, Green-Lagrange strains.

Let be the total deformation gradient tensor at time t with respect to an initial configuration taken at time 0 (see ref. KJB,

subscripts are removed for better

analyFfollowing references: restrain elasto-plasticity: an algorithm for mixed hardening and

X

p.503). The left superscripts and readability. This total deformation gradient tensor X can be decomposed into a material rigid-body rotation R and a symmetricpositive-definite right stretch tensor U (polar decomposition):

=X R U (3.1-1)

The right stretch tensor can be represented in its principal directions by a diagonal tensor , such that

(3.1-2)

here is a rotation tensor with respect to the fixed global axes

ote that the rotation does not correspond to a material rigid-body rotation, but to a rotation of the coordinate system: and are two representations of the same deformed state, respectivelythe basic coordinate system and in the principal directions coordinate system. • The right Hencky strain tensor (computed in the right basis) is given by

UΛ

TL L=U R Λ R

LRw(see Fig. 3.1-2).

LRU Λ

in U

N



ln lnR TL= =E U R LΛ R (3.1-3)

mmetric n):

The superscript �R� symbolizes the right basis. • The total deformation gradient tensor X can also be

ecomposed into a material rigid-body rotation R and a sydpositive-definite (left) stretch tensor V (polar decompositio

=X V R (3.1-4)

in •

R (3.1-4) is the same as R in (3.1-1).

The left stretch tensor V can be represented in its principal directions by the diagonal tensor Λ , such that

TE E

where

=V R Λ R (3.1-5)

ER is a rotation tensor with respect to the fixed global axes and is given by E L=R R R (3.1-6)

omputed in the left basis) is given by

• The left Hencky strain tensor (c

ln lnL TE E= =E V R Λ R (3.1

The superscript �L� sy

-7)

mbolizes the left basis. • Comparison of left and right Hencky strain tensors:

ence both of s can be considered to be e

The principal values of the left and right Hencky strain tensors are identical, and equal to the logarithms of the principal stretches.

these strain tensorHlogarithmic strain tensors. However, the principal directions of thleft and right Hencky strain tensors are different. The principal directions of the right Hencky strain tensor do not contain the rigidbody rotations of the material, but the principal directions of the left Hencky strain tensor contain the rigid body rotations of the material.


UGS Corp. 85

Therefore, for a material undergoing rigid body rotations, the principal directions of the right Hencky strain tensor do not rotate, however the principal directions of the left Hencky strain tensor rotate with the material. Hence, the left Hencky strain tensor is preferred for output and visualization of the strain state.

L(t)

(0)x

y

R

(0)

(t)

L

Initial configuration at time 0

Configuration at time t not including rigid body rotation

Configuration at time t

Directions of maximum/minimum total stretchesand strains

Material rigid-body rotation between time 0 and time t

Directions of initial configuration fibers withmaximum/minimum total stretches and strains

R

R

R

Fig es and

ure 3.1-2: Directions of maximum/minimum total stretchstrains

Inelastic deformations

In inelastic analysis, the following multiplicative decom

position of stic deformation gradient the total deformation gradient into an ela

EX and an inelastic deformation gradient is assumed:

PX

E P=X X X (3.1-8)



To aterial under a given stress state with deformation gradient . If this regsubjec te, the deformation gradient is still

. Now if the stress state is removed, (3.1-8) implies that the s

the strains associated with the elastic deform

understand Eq. (3.1-8), consider a small region of mX

ion of material is separated from the rest of the model and ted to the same stress sta

Xdeformation gradient of the unloaded material is PX . The stresseare due entirely to

EX ation gradient

-8) is equivalent the additive decomposition of the displacements into elastic

It can be shown (see Montáns and Bathe), that (3.1todisplacements and plastic displacements. For the materials considered here, det 1P =X .

Polar decomposition of elastic deformation gradient: The elastic deformation gradient can be decomposed into an elastic rotation tensor ER a EU , EV : nd elastic right and left stretch tensors

E E E E E= =X R U V R (3.1-9a,b)

Elastic Hencky strain tensors: The elastic Hencky strain tensors in the right and left bases are given by

lnER E=E U , lnEL E=E V (3.1-10a,b)

Stress-strain relationships: The stresses are computed from the elastic Hencky strain tensors using the usual stress-strain law of isotropic elasticity. However, the stress measures used depend upon the strain measures used. When the right Hencky strain measure is used, the stress measure used is the rotated Kirchhoff stress ( )TE EJτ=τ R R (3.1-11)

stress measure and when the left Hencky strain measure is used, theis the (unrotated) Kirchhoff stress Jτ . detJ = X is the volume change of the material, and, using det 1P =X , det EJ = X .


UGS Corp. 87

stresses and

rain

3.1.5 Ther

• c

With theses choices of stress and strain measures, thestrains are work-conjugate. The choice of right Hencky strain and rotated Kirchhoff stresses gives the same numerical results as the choice of left Hencky stand (unrotated) Kirchhoff stresses). mal Strains

Calculation of thermal strains is needed for temperature-dependent material models (thermo-elastic isotropic, thermo-elastiorthotropic, thermo-plastic), as well as temperature-invariant material models with non-zero thermal expansion coefficients. • The current temperature tθ and the initial temperature 0θ aboth needed for the calculation of thermal strains. The current temperature field is set via the TEMPERATURE(LOAD) cacontrol entry, while the initial temperature is

re

se set via the

TEMPERATURE(INITIAL) case control entry. See Section 5.6 for more details. • The te ed on the nodal temperatures and the element shape functions, and then

• If the thermal expansion coefficient is constant, the following expression is used.

mperature at an integration point is evaluated bas

used to calculate the thermal strains.

( )0t TH tij ije α θ θ δ= − (3.1-12)

where ijδ is the = 1 for i = j and = 0 for Kronecker delta ( ijδ ijδi j≠ ). • If, however, the thermal expansion coefficient is temperature dependent, the thermal strains are calculated as follows: ( )0t TH t t

ij ije α θ θ δ= − (3.1-13)

where



( ) ( )( ) ( )( )( )1 θ−

(3.1-14) and

0 0t t tα α θ θ θ α θ θ= − −0 REF REFtθ θ−

REFθ is the reference temperature at which the thermal strain in the material is zero. • ns (3.1-13) and (3.1-14) are derived as follows: Suppose tha ental data, the dependence of the length of a bar as a function of temperature is obtained, as shown in Fig. 3.1-3.

Equatiot, from experim

Temperature, ��REF �

Len

gth

,L Secant to curve

L

LREF

ay be cal ted

Figure 3.1-3: Length of bar vs. temperature The thermal strain with respect to the reference length m

cula as

TH REFL LεREFL

−=

Then we define the mean coefficient of thermal expansion for a given temperature as follows:

( ) ( )TH

REF

ε θα θ

θ θ=

−

With this definition, the secant slope in Fig. 3.1-3 is ( )REFL α θ .

Now, in Solution 601, we assume that the thermal strains are initially zero. To do this, we subtract the thermal strain


UGS Corp. 89

corresponding to 0θ to obtain

( )( ) ( )( )0 0t TH t tREF REFε α θ θ θ α θ θ θ= − − −

which leads to Equations (3.1-13) and (3.1-14).

constant, REF

Notice that if the mean coefficient of thermal expansion is θ no longer enters into the definition of tα and the

equations reduce to Eq. 3.1-12. In general, when the mean coefficient of thermal expansion is not constant, REFθ must be chosen based on knowledge of the experime ine nt used to determ

( )α θ since the same material curve, different choices of REFθ

yield different values of ( )α θ .

r elastic material models

• The following material models are discussed in this section:

3.2 Linea

Elastic-isotropic: isotropic linear elastic non-thermal dependent

endent material model obtained with MAT2 and MAT8 for surface elements and MAT9 for 3-D solid elements

• These models can be employed using small

s

tic-isotropic and elastic-orthotropic materials are e

material model obtained with MAT1

Elastic-orthotropic: orthotropic linear elastic non-thermal dep

di placement/small strain or large displacement/small strain kinematics. The strains are always assumed to be small. • Thermal strains are supported for elastic isotropic materials, andthey are not supported for elastic-orthotropic materials. Thermal strains for orthotropic materials can be obtained by using a temperature-dependent elastic orthotropic material model (see Section 3.5). When the elas•

us d with the small displacement formulation, the formulation is linear.



mployed in a large displacement nalysis, the total or the updated Lagrangian formulation is

more effective. 2-D, 3-D solid elements nd shell elements use a total Lagrangian (TL) formulation, while

formulation, the stress-strain

rel

• If the material models are eaautomatically selected by the program depending on which formulation is numericallyarods and beams use an updated Lagrangian (UL) formulation.

• In the small displacement ationship is

0 0t tσ = C e

n which ti 0σ = engineering stresses and t e = engineering strain0 s.

lation, the

tress-strain relationship is

• In the large displacement total Lagrangian formus

0 0t t= εS C

in which 0t S = second Piola-Kirchhoff stresses and 0

t ε = Green-Lagrange strains. • In the large displacement formulation, used by rod and beam lements, the stress-strain relationship is e

*t t

tτ = εC

in which t τ = Cauchy stresses and *tt ε = rotated engineering strain.

• In the presence of thermal strains the following stress-strainrelationship is used instead in small displacement analysis:

( )0 0 0t t t THσ = −C e e

term

dded for the TL and UL formulations, respectively. The culation of thermal strain is detailed in Section 3.1.5.


where 0t THe are the thermal strains. A similar 0

t THε and *0t THε

is acal

3.2: Linear elastic material models

UGS Corp. 91

• s lonneg

• However, if the strains are large, the difference in the response

large, it is recommended that these linear elastic material models not be used.

3.2.1 Elastic-isotropic material model

• his material model is available for the rod, 2-D solid, 3-D solid • he two material constants used to define the constitutive rel

on's ratio

• The thermal expansion coefficient

The same matrix C is employed in all of these formulations. Ag as the strains remain small, the difference in the responses is ligible.

predictions is very significant (see ref. KJB, pp 589-590). If the strains are

T, beam, and shell type elements.

Tation (the matrix C) are

E = Young's modulus, v = Poiss

α is also u

strains are present.

3.2.2 Elas otr

-D

Material constants are defined along material axes (1,2,3). The

-D

G4

ref. KJBTable 4.3,

p. 194

sed if thermal

tic-orth opic material model

• The elastic-orthotropic material model is available for the 3solid and shell elements. •local constitutive matrix Cmat is then transformed to obtain the stress-strain matrix C corresponding to the global coordinate system. 3 solid elements: The orthotropic 3-D material is defined using the MAT9 command with the following assumptions: G14 = G15 = G16 = G24 = G25 =G26 = G34 = G35 = G36 = 0 and

5 G46 = G56 = 0. =



This leads to the following stress-strain relationship for 3-D solids:

33 33

12 44 12

66 31

0 0 00 0

G eG

G

σ11 11 12 13 11

22 22 23

33

0 0 00 0 0

G G G eG G e

⎡

22

⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥σ⎢ ⎥ ⎢⎢ ⎥ ⎥⎢ ⎥ ⎢

=

23 55 23

31

symmetric 0G

⎢ ⎥⎥σ⎢ ⎥ ⎢ ⎥⎢ ⎥τ γ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥τ γ⎢ ⎥ ⎢ ⎥⎢ ⎥τ γ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0 0 0

Shell elements: The orthotropic shell material is preferably definedhich leads to the following inverse stress-using the MAT8 entry, w

strain relationship defined in the shell material coordinate system (1,2,3):

11 111 12 11/ / 0 0 0/ 1/

e E Ee E E22 2212 1 2

12 1212

13 1

0 0 1/ 0 00 0 0 1/ 0z

GG

σ−ν⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ σ−ν⎢

13

23 2320 0 0 0 1/ zG

⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥=γ σ⎢ ⎥ ⎢ ⎥⎢ ⎥γ σ⎢ ⎥ ⎢

⎢ ⎥⎥⎢ ⎥ ⎢ ⎥⎢ ⎥γ σ⎣ ⎦⎣ ⎦ ⎣ ⎦

The MAT2 entry can also be used to define a shell material with

nly in-plane orthotropy:

o

11 11 12 11

22 12 22 22

12 33 12

0 0 00 0 0

0 0 0 00 0 0

G G eG G e

G

σ

13 33 13

23 33 23

00 0 0 0

GG

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥σ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥=τ γ⎢ ⎥ ⎢ ⎥⎢ ⎥τ

γ⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥τ γ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

3.3 Nonlin a

vanced Nonlinear Solution only supports nonlinear elastic materials with the rod elements. The nonlinear effect is obtained

e r elastic material model

• Ad

3.3: Nonlinear elastic material model

UGS Corp. 93

wi A • The stress-strain relationship in this case is defined as a piecewise linear function, as shown in Fig. 3.3-1.

th a M TS1 entry which has TYPE = �NELAST�.

�

�

�

�

�

�

Stress

�6

�5

�4

�3

�2

�1

e1 e2 e3

e4 e5 e6 Strain

Figure 3.3-1: Nonlinear elastic material for rod elements

A sufficient range (in terms of the strain) must be used in the in

Fig.

Note that the stress is uniquely defined as a function of the

strain only; hence for a specific strain te, reached in loading or unloading, a unique stress is obtained from the curve in Fig. 3.3-1.

•definition of the stress-strain relation so that the element straevaluated in the solution lies within that range; i.e., referring to3.3-1, we must have 1 6

te e e≤ ≤ for all t. • The stress-strain curve does not necessarily have to pass through the origin.

• A typical example of the nonlinear elastic model for rod elements is shown in Fig. 3.3-2. This example corresponds to a able-like beh

avior in which the rod supports tensile but no

ccompressive loading.



Stress

Strain

�

� �

Point 1 Point 2

Point 3

Figure 3.3-2: Nonlinear elastic material model corresponding to a tension-only cable

. Note that to use this element to simulate a contact gap, it is necessary to know which node of one body will come into contact with which node of the ther body, and connect these two nodes with a rod element.

• The rod element with this nonlinear elastic material model is particularly useful in modeling gaps between structures. This modeling feature is illustrated in Fig. 3.3-3

o

�

�m

n

Gap �Gap element

(between nodes and )m n

�_L

Strain

Stress

L

Figure 3.3-3: Modeling of gaps

3.4: Elasto-plastic material model

UGS C

3.4 Elasto

ed using the MATS1 material ntry with TYPE = �PLASTIC�. This section describes the

l:

596-604, ref. KJB.


-plastic material model

• Elasto-plastic materials are definefollowing material mode

Plastic: Elasto-plastic isotropic non-thermal dependent material.

• All elasto-plasticity models use the flow theory to describe theelastic-plastic response; the basic formulations for the von Mises models are summarized on pp.

• These material models are based on

orp. 95

(see p. 597, ref. KJB)

! An associated flow rule using the von Mises yield function An isotropic or kinematic, or mixed hardening

Figs. 3.4-1 to 3.4-3 summarize some important features of these ma rial models.

• , 2-D solid, 3-D solid, beam (plastic-bilinear only), and shell elements. • astic and plastic material constants are thermally invariant. However, thermal strains can be present when there is a tem erature load and a coefficient of thermal expansion.

! The von Mises yield condition

!

! Bilinear or multilinear stress-strain curves (based on H and

TID fields in MATS1)

te

hese models can be used with the rodT

All el

p



0y�

Stress

Strain

Bilinear stress-strain curve

Multilinear stress-strain curve

Figure 3.4-1: von Mises model

�3

s3

�

s ��s�

a) Principal stress space

�23

0y�

�23

0y�

Elastic region

b) Deviatoric stress space

(1,1,1)

Figure 3.4-2: von Mises yield surface

• These models can be used with small displacement/small strain, large displacement/small strain and large displacement/large strain kinematics. When used with sm l disma linear-only formulation is employed. When used with the displacement/small strain kinematics, either a TL or a UL formulation is emplo When used with large displacem atics, a ULH formulation (solid elements) or a ULJ formulation (shell ele l ment/large strain kinematics

al placement/small strain kinematics, a terially-non

yed. ent/large strain kinem

ment) is employed. Large disp ace


UGS Corp. 97

can only be used with the 2-D solid, 3-D solid and shell elements (inele

• For multilinear plasticity in rod elements, up to 7 stress-strain points on the stress-strain curve can be entered. For other elements, there is no restriction on the number of stress-strain points in the stress-strain curve. • 2-D plane strain or 3-D solid elements that reference these material models should preferably employ the mixed displacement-pressure (u/p) element formulation. This is done by setting UPFORM = 1 in the NXSTRAT comm nd.

that case, the shell elements must be 4-node single layer shell ments).

a

b) Bilinear kinematic hardening

E

E

E

E

E

T

T

0y�

a) Bilinear isotropic hardening

Stress

Straint

y�

ty�

E

E

E

E

E

T

T

0y�

Stress

Strain

20y�

0y� 0

y�t

y�

ty�

0y�

Strain Strain

Stress Stress

c) Multilinear isotropic hardening

2

d) Multilinear kinematic hardening

Figure 3.4-3: Isotropic and kinematic hardening



ises model with isotropic hardening, the following ield surface equation is used:

• In the von My

( ) 21 1 02 3

t t t ty yf σ= ⋅ − =s s

twhere s is the deviatoric stress tensor and 0 2

yσ the updated yield

stress at time t.

In the von Mises model with kinematic hardening, the following yield surface equation is used:

( ) ( ) 0 21 1 02 3

t t t t ty yf σ= − ⋅ − − =s α s α

where is the shift of the center of the yield surface (back stress tαtensor) and 0 2

yσ is the virgin, or initial, yield stress. In the vo odel with mixed hardening, the following d surface equation is used:

n Mises myiel

( ) ( ) 21 1 02 3

t t t t t ty yf σ= − ⋅ − − =s α s α

p

The back stress is evolved by

where 0t

y y pMEσ σ= + e

tα ( )1 p

pd C M d= −α e Cp is Prager�s hardening parameter, related to the p

modulus Ep and M is the factor used in general mixed hardening

Mixed hardening is only available for bilinear plasticity. The

lastic

(0 < M < 1) which is currently restricted to 0.5.


UGS Corp. 99

formulation for the von Mises model with mixed hardening is given in the following reference:

ref K.J. Bathe and F.J. Montáns, �On Modeling Mixed

Hardening in Computational Plasticity�, Computers and Structures, Vol. 82, No. 6, pp. 535�539, 2004.

The yield stress is a function of the effective plastic strain,

aterial. The effective plastic str

which defines the hardening of the main is defined as

0

23

tPt p pe d d= ⋅∫ e e

in which is the tensor of differential plastic strain increments.

In finite element analysis, we approximate

pdePt e as the sum of all of

the plastic strain increments up to the current solution time:

Pt p

all solution steps

e = ∆∑ e

where

23

P p pe d d∆ = ⋅e e and p∆e is the tensor of plastic strain

inc

solution st

rements in a solution step. Because of the summation over the

eps, we refer to the calculated value of Pt e as the

ed effective plastic strain. accumulat

the structure, the thermal strains terial characteristics are

considered to be temperature independent.

r in exists for the bilinear

case.

• If a thermal load is applied toare taken into account but the ma

Modeling of rupture: Rupture conditions can also be modeled. For the multilinear stress-strain curve, the rupture plastic strain corresponds to the effective plastic strain at the last point input fothe stress-strain curve. No rupture stra



When rupture is reached at a given element integration point, n

3.5 Temp a

d thermal orthotropic material models re discussed in this section. The thermal isotropic material is

ments with e MAT2 and MATT2 material entries and for solid elements with

era c nt in ese materials.

f

odels can be used with small displacement/small strain nd large displacement/small strain kinematics. The strains are lways assumed to be small.

When used with small displacement/small strain kinematics, a lly

ployed. 2-D, 3-D solids and hells use the TL formulations, and rods use a UL formulation.

In the data input for the analysis, the nodal point temperatures ust be defined for all time steps. See Section 5.6.

oduli, the shear moduli, the ts of thermal expansion defined

Section 3.2 e input as p ewise ear functions of the mperature, as illustrated in Fig. 3.5-1. Linear interpolation is

the corresponding element is removed from the model (see Sectio8.4).

er ture-dependent elastic material models

• The thermal isotropic anaobtained with the MAT1 and MATT1 material entries, and the thermal orthotropic material is obtained for surface eleththe MAT9 and MATT9 material entries. These commands allow the different elastic material constants to vary with temp ture. Thermal strains are taken into ac outh • The thermal isotropic model is available for the rod, 2-D solid, 3-D solid and shell elements. • The thermal orthotropic model is also available or the 3-D solid shell and elements. • Both maa

materia -nonlinear-only formulation is employed. When used with large displacement/small strain kinematics,

either the TL or UL formulation is ems •m For these models, the elastic m•

Poisson's ratios and the coefficienin ar iec linteused to calculate the material properties between input points.

3.5: Temperature-dependent elastic material models

UGS Corp. 101

� �

�

� �

�

� �

�

� �

�

� �

�

E

�

tEt

t�

t t

t

4 4

4

1 1

1

2 2

2

3 3

3

� �

�

� �

�

� �

�

� �

�

Temperature Temperature

Temperature Figure 3.5-1: Variation of material properties for thermo-elastic

model

independent, and the material is isotropic or orthotropic, then thermal strains could alternatively have been modeled using the elastic isotropic or orthotropic material (non-thermal) of Section

3.6 Thermal elasto-plastic and elastic-creep material models

er thermal elasto-plastic materials and ce a unified general solution can be

ure is

effects of


• The calculation of thermal strains is described in Section 3.1.5. • Note that if the material constants are all temperature

3.2.

• This section groups togethelastic-creep materials, sinapplied to these material types. The computational procedbased on the effective stress function algorithm, detailed inSection 3.6.4.

• The thermal elasto-plastic and elastic-creep models include the



! Isotropic elastic strains, via the MAT1 entry

! Thermal strains, , via the MATT1 or the MAT1 entries.

Time-independent plastic strains, e , via the MATS1 entry

e

t THrse

! rs

t p

! Time-dependent creep strains, s , via the CREEP entry

• The constitutive relation used is

t Cr

( )t t E t t P t C t THij ijrs rs rs rs rsC e e e eσ = − − − (3.6-

where t

1)

ijσ is the stress tensor at time t and is the elasticity

r of You

temperature-dependent.

ial models are optional. If, however, the omitted strain components result in one of the material models detailed in one of the previous

aterial model.

general, and can describe any material combining elastic, plastic, thermal and creep strains. However, currently only two material combinations are introduced in this section, These are:

EP

! thermal elasto-plastic - perature dependent elastic-plastic isotropic material,

obtained with MATS1, MATT1 and MAT1 entries, or only g to a

t EijrsC

t EijrsC tensor at the temperature corresponding to time t. The tenso

can be expressed in terms ng's modulus tE and Poisson's ratio tv both of which may be • Note that the thermal, plastic and creep parts of these mater

sections, then the program will select that m • The formulations provided in this section are very

! elastic-creep - Elastic isotropic material with creep, obtained with CRE

and MAT1 entries, and

Tem

MATS1, MAT1 with TID in MATS1 pointinTABLEST entry.

3.6: Thermo-elasto-plasticity and creep material models

UGS Corp. 103

• These material models can be used with the rod, 2-D solid, 3-D solid, and shell elements. • These models can be used with small displacement/small strain, large displacement/small strain and large displacement/ large strain kinematics.

When used with small displacement/small strain kinematics, a

ed. This

odels

ed u/p element formulation. This is done by setting UPFORM =1 in the NXSTRAT entry. • Note thcreep straiinteraction

h

• int, n

8.4).

ref. M.D. Snyder and K.J. BaThermo-Elastic-Plastic and Creep ProblemEng. and Design, Vol. 64, pp. 49-80, 1981.

ref. M. Kojić and K.J. Bathe, "The Effective-Stress-Function

lasticity and Creep," Int. J. Numer. Meth. Engng., Vol. 24, No. 8, pp. 1509-1532,

materially-nonlinear-only formulation is employed. When used with large displacement/small strain kinematics,

either a TL or a UL formulation is employed (TL for 2-D and 3-D solids and shells, and UL for rods).

hen used with large displacement/large strain kinematics, the WULH (updated Lagrangian Hencky) formulation is employis only supported for 2-D solid and 3-D solid elements.

• 2-D or 3-D solid elements that reference these material mshould preferably employ the mix

at the constitutive relations for the thermal, plastic and ns are independent of each other; hence the only between the strains comes from the fact that all strains

affect the stresses according to Eq. 3.6-1. Fig. 3.6-1 summarizes the constitutive description for a one-dimensional stress situationand a bilinear stress-strain curve.

Since there is no direct coupling in the evaluation of the different strain components, we can discuss the calculation of eacstrain component independently.

When rupture is reached at a given element integration po

the corresponding element is removed from the model (see Sectio

the, "A Solution Procedure for

s," J. Nuclear

Algorithm for Thermo-Elasto-P

1987.



(b) Strains considered in the model

� �

tP

Area A

(a) Model problem of rod element under constant load

Stress

t�

tT

tE ( )�

t tE( )�

tyv�

t Pe (time independent)

t Ee (time independent)

Strain

Creep

strain

t Time

te (time dependent)C

Temperature t�

( )t�

Figure 3.6-1: Thermo-elasto-plasticity and creep constitutive

description in one-dimensional analysis

3.6.1 Evaluation of thermal strains

ns are calculated as described in Section 3.1.5. he

• The thermal straiT y are used in the thermal elasto-plastic material model.


UGS Corp. 105

3.6.2 Evaluation of plastic strains

• Plasticity effects are included in the thermal elasto-plastic material model and is based on the von Mises yield criterion, an associated flow rule, isotropic or kinematic hardening (no mixed hardening), and bilinear or multilinear stress-strain curves (based

The multilinear plasticity case is more general. In this case, the tempera g

TABLES1 entry. The elastic material parameters can also be e yield curves are interpolated as shown

• The plastic strains are calculated using the von Mises plasticity model (see Section 3.4) with temperature-dependent material parameters (Young's modulus, Poisson's ratio, stress-strain curves).

on the H and TID fields in MATS1). • In the case of bilinear plasticity only the elastic materialparameters can be temperature dependent (Young�s modulus, Poisson�s ratio and coefficient of thermal expansion). •stress-strain curves can be made ture dependent by settinTID in MATS1 to point to a TABLEST entry instead of a

temperature dependent. Thin Fig. 3.6-2.

• The yield function is in isotropic hardening

2v

12 3

t t t ty yf 1 σ= ⋅ −s s

and in kinematic hardening

( ) ( ) 2v

1 12

t t t t t ty yf

3σ= − ⋅ − −s α s α

where ts is the deviatoric stress tensor, vy

tσ is the virgin yield

str e g.

ess corresponding to temperature tθ and tα is the shift of thstress tensor due to kinematic hardenin



a) Stress-strain curves input data

b) Yield curves

Stress

Stress

1

1

2

2

3

3

4

4

ei1

e

ei2

e

ei3

e

ei4

e

Strain

i+1

i+1 i+1 i+1 i+11 2 3 4

�i

�

Temperature

eeTemperature

Interpolated

yield curve

i+1 i+1P P2 3

Plastic

strain

1

1

2

2

3

3

4

4

t�

� i

�i+1

.6-2: Interpolation of multilinear yield curves with temperature

Figure 3


UGS Corp. 107

he • The expressions for plastic strain increments resulting from tflow theory are P t

ij ijde d sλ= for isotropic hardening and

( )tP td sij ij ijde λ α= − for kinematic hardening, in which dλ is the

from n tion tfy = 0. In the case of kinematic hardening, we

ield surface position in the form

where tC is the modulus as defined for temperature-independent plasticity but using now of course temperature-dependent moduli.

T perature is large. Hence, it mended to use this model only when the variation in ET as

on of temperature is small.

3.6.3 Evaluation of creep strains

creep law is obtained by setting TYPE = 222 in the e

plastic multiplier (positive scalar) which can be determined the yield co diexpress the change of the y

t Peα =ij ijd Cd

• Care should be exercised in the use of this model in kinematic hardening conditions. Namely, nonphysical effects can result when

e variation in E as a function of temthis recoma functi

• Two creep laws are currently available in Solution 601. The first called the Power creep law is obtained by setting TYPE = 300 in the CREEP material entry. The second creep law called the

xponential ECREEP material entry. They are currently supported only for thelastic-creep material model. • The effective creep strain is calculated as follows:

Power creep law (creep law 1) :

t C t b de a tσ= ⋅ ⋅ in which σ is the effective stress, t is the time, and a, b, d arematerial constants f

rom the CREEP material entry.



Exponential creep law (creep law 2) :

( ) ( )( ) ( )R t G t− σ ⋅1t Ce F e= σ ⋅ − + σ ⋅

with

( ) ( ) ( ) ( )tb t ( ) ( ); ;td fF a e R c G e e⋅ σσ = ⋅ σ = ⋅ σ ⋅ σσ = ⋅

in whicateria

ns rdening

rocedure for load and temperature variations, and the O.R.N.L.

ref. C.E. Pugh, J.M. Corum, K.C. Liu and W.L. Greenstreet,

"Currently Recommended Constitutive Equations for Inelastic Design of FFTF Components," port No. TM-3602, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 1972.

Hence, the incremental creep strains are calculated using

e

ultiplier

h a through f are material constants from the CREEP l entry. m

• The creep strai are evaluated using the strain haprules for cyclic loading conditions.

Re

C t tij ijt sγ= ∆

where the creep m tγ varies with stresses and creep strains, nd the sij are the deviatoric stresses (see Section 3.6.2 and ref.

KJB, p. 607). procedure used to evaluate the incremental creep strains is

summarized in the following:

en the total creep strains and the deviatoric stresses tsij at time t,

ta

The

t CijeGiv

1) Calculate the effective stress


UGS Corp. 109

123

2t

ij ijsσt t s⎡ ⎤= ⎢ ⎥⎣ ⎦

2) Calculate the pseudo-effective creep strain

( )( )122

3t C t C orig t C orig

ij ij ij ije e e e e⎡ ⎤= − −⎢ ⎥⎣ ⎦

3) Substitute tσ , t Ce and tθ into the generalized uniaxial creep law

( )t C t t

ce F tσ θ= , ,

t . 4) Solve for the pseudo-time

5) Calculate the effective creep strain rate from the generalized uniaxial creep law

( )t C

t tcFt t

σ θ∂∂ e t= , ,∂ ∂

6) Calculate tγ

t C

t tγ

⎜ ⎟∂3 ⎝ ⎠= tσ2

and the increment

e⎛ ⎞∂

t C t tije t γ∆ = ∆ s

Steps 1) through 4) correspond to a strain hardening procedure.


110

3.6.4 Com nal procedures

The stresses and strains at the integration points are evaluated

ion

o-Elasto-Plasticity and Creep," Int. J. Numer. Meth. Engng., Vol. 24, No. 8, pp. 1509-1532,

e p alc

he general constitutive equation

putatio

•using the effective-stress-function algorithm.

ref. M. Kojić and K.J. Bathe, "The Effective-Stress-FunctAlgorithm for Therm

1987.

Briefly, th rocedure used consists of the following c ulations.T

( )( ) ( ) ( )t t i t+∆ ( )i t t E t t i t t P t C i t t THσ +∆ +∆ +∆ +∆ +∆= − − −C e e e e (3.6-1)


Advanced Nonlinear Solution ⎯ Theory and Modeling Guide

d separately for the mean stress and for the deviatoric tresses. In this equation the index (i) denotes the iteration counter

sier writing this dex will be dropped in the discussion to follow. The mean stress

lated as

is solvesin the iteration for nodal point equilibrium. For eainis calcu

( )1 2

t t+∆t t t t t t TH

m mt t

E e eσ+∆ +∆ +∆+∆= −

− ν (3.6-2)

The deviatoric stresses

t t+∆ s depend on the inelastic strains and the can be expressed as y

( )1 1t t t t tt

E

ta t

ττ α γ

α γ λ+∆ +∆ ′′

t+∆⎡ ⎤= − − ∆ ⎣ ⎦+ ∆ + ∆

s

s e (3.6-3)

herew 1E t t+∆

t t E+∆t t a+∆ = , = deviatoric stress at the start of the ts

+ νtime step and α is the integration parameter used for stress evaluation ( )0 1α≤ < The creep and plastic multipliers τγ and

λ∆ are functions of the effective stress t tσ+∆ only, and they


UGS Corp. 111

account for creep and plasticity; also

t t t t t P t C+∆ +∆′′ ′= − −e e e e is known since the deviatoric strains t t+∆ ′e , plastic strains and reep strains are known from the current displacements and the

e step.

The following scalar function

t Pecstress/strain state at the start of the current tim

t Ce

( )t tf σ+∆ is obtained from Eq.

(3.6-3) ( ) 2 2 2 0t t t tf a b c dτ τσ σ γ γ+∆ +∆ 2 2= + − − = (3.6-4)

e solution for the effective stress The zero of Eq. (3.6-4) provides th

t tσ+∆

a a t τ

, where

t tE α γ λ+∆= + ∆ + ∆

( )3 1 t t tij ijb t e sα +∆ ′= − ∆

( )1 tc tα σ= − ∆

2 32

t t t tij ijd e e+∆ +∆′′ ′′=

with summa

tion on the indices i, j. Once the solution for has been determined from Eq. (3.6-4),

i ultaneously with the scalars τγ and λ∆s m from the creep and l sticity conditions, the deviatoric stress is calculated from q. (3.6-3), an end of the time tep are obtained as

t t+∆ sp aE d the plastic and creep strains at thes

(1

t t P t P t t

t t C t C t t t t τ

λ

α α γ

+∆ +∆

+∆ +∆

= + ∆

⎡ ⎤= + − ) + ∆⎣ ⎦

e e s

e e s s

The above equations correspond to isotropic hardening conditions and a general 3-D analysis. The s

olution details for



inematic hardening conditions and for special problems (for the

ref. M. Kojić and K.J. Bathe, "Thermo-Elastic-Plastic and

.

3.7 Hyper

erfoam material models. They are all defined using the ATHE command. In addition MATHP can be used to define a

al ion cted.

r hyperelastic materials.

• In Solution 701 only the Mooney-Rivlin and Ogden material models can be used, and only for 3-D solid elements. • The isotropic hyperelastic effects are mathematically described by specifying the dependence of the strain energy density (per unit original volume) on the Green-Lagrange strain tensor

kplane stress, shell, isoparametric beam elements) are given in the above cited references, and also in the following reference:

Creep Analysis of Shell Structures", Computers & Structures, Vol. 26, No 1/2, pp. 135-143, 1987

elastic material models

• The hyperelastic material models available in Advanced Nonlinear Solution are the Mooney-Rivlin, Ogden, Arruda-Boyce and HypMhyperelastic Mooney-Rivlin material.

• This material model can be employed with the 2-D solid and 3-D solid elements.

• It uses large displacement/large strain kinematics. A TotLagrangian (TL) formulation is employed. The same formulatis used if a large displacement/small strain kinematics is sele • Thermal strains can be included via a constant thermal expansion coefficient. Section 3.7.5 shows how they are computed fo

W ijε . • We now give a brief summary of the quantities and concepts used. For more information, refer to ref KJB, section 6.6.2. Here and below, we omit the usual left superscripts and subscripts for ease of writing. Unless otherwise stated, all quantities are evaluated at time and referred to reference time t 0 .

3.7: Hyperelastic material models

UGS Corp. 113

• Useful quantities are the Cauchy-Green deformation tensor given by

ijC ,

2ij ij ijC ε δ= + where ijδ is the Kronecker delta; the principal invariants of the Cauchy-Green deformation tensor,

1 kkI C= , ( )22 1

12 ij ijI I C C= − , 3 detI = C

the reduced invariants:

131 1 3I I I −= ,

232 2 3 I I I −= ,

123J I= ,

the stretches iλ iλ where the �s are the square roots of the principal stretches of the Cauchy-Green deformation tensor; and the reduced

stretches:

( )13λ λ λ λ λi i 1 2 3

−= Note that 1 2 3J λ λ λ= is the volume ratio (ratio of the deformed volume to the un • he strain e invon w

deformed volume).

energy density W is written in terms of thTariants or stretches. In many cases, the strain energy density is veniently ritten as the sum of the deviatoric strain energy c

density DW and the volumetric strain energy density .

he 2 Piola-Kirchhoff stress tensor is evaluated using

VW • With knowledge of how the strain energy density W depends on the Green-Lagrange strain tensor (through the invariants or stretches), t nd



12ij

ij ji

W WS ∂ ∂= +⎜ ⎟⎜ ⎟

⎛ ⎞

∂ε ∂ε⎝ ⎠

and the incremental material tensor is evaluated using

12

ij ijijrs

rs sr

S SC

∂ ∂⎛ ⎞= +⎜ ⎟∂ε ∂ε⎝ ⎠

ney-Rivlin material model • The Mooney-Rivlin material model is obtained by setting

3.7.1 Moo

odel=Mooney in the MATHE material entry. It can also be

) ( ) ( )( )( ) ( ) ( ) ( )( ( )

210 1 01 2 20 1 11 1 2

2 3 202

12 03 2

3 3 3 3

3

W C C I C I I

C I C I C I I

C I C I

= − + − + − − +

− + − + − − +

−-1)

Mobtained using the MATHP material entry. It is based on the following expression of the strain energy density:

) (3 C I− +( I

)( )2 30 1 21 1 2

2 31 2

3 3 3 3

3 3I− − +

(3.7 where Cij are material constants , and 1I and 2I are the first and second strain invariants at time t, referring to the original configuration (see ref. KJB, Section 6.6.2 for the definitions of thstrain invariants).

e

e

compressible material

Note that constants Aij used in the MATHP material entry aridentical to Cij constants used in MATHE and in the equation above.

• This strain energy density expression assumes a totally ( )3 1 .I =in It is modified to account for

material compressibility by: 1) substituting for the invariants 1 2,I I the reduced invariants

1 2,I I , 2) removing the condition 3 1I = , and



UGS Corp. 115

metric strain energy density

3) adding the volu

( )21 12VW Jκ= −

where κ is the bulk modulus given by K in the MATHE material

ry). This expression for the volumetric strain energy density yields the fol tio:

entry (or two times D1 in the MATHP material ent

lowing relationship between the pressure and the volume ra

( )1p Jκ= − − Th :

is results in the following strain energy density

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )

( )( ) ( ) ( )

202 2

12

3 3 3 3

3 3 3 1

W C

C I C I C I I

C I I C I Jκ

=

− + − + − − +

− − + − + −

No e ma rial is almost incompressible.

n is metric locking.

• madifferentiadisplacemseparately interpolated pressure.

ref.

uctures, Vol.

Selection rial descriptio κ. Strictly sp

210 1 01 2 20 1 11 1 2

2 3

3 3 3 3 3I C I C I C I I− + − + − + − − +

30 1 21 1 2

2 3 211 2 03 2 2

te that the bulk modulus can be set very large so that thteThe mixed displacement-pressure (u/p) element formulatio

always used for hyperelastic materials to avoid volu The terial stress-strain descriptions are obtained by

tion of 0tW to obtain stresses due to the element

ents and then taking into account the effect of the

T. Sussman and K.J. Bathe, "A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis," J. Computers and Str26, No. 1/2, pp. 357-409, 1987.

of material constants: The Mooney-Rivlin maten used here has 9 Cij constants and the bulk modulus eaking, this material law is termed a higher-order or



, .

rms of these constants as (assuming

generalized Mooney-Rivlin material law. Choosing only C10 ≠ 0 yields the neo-Hookean material law, and choosing only C10 ≠ 0C01 ≠ 0 yields the standard two-term Mooney-Rivlin material law

The small strain shear modulus and small strain Young�s modulus can be written in teκ = ∞ )

( )10 012G C C= + (3.7-5) ( )10 016E C C= + (3.7-6)

These moduli must be gr

The bulk modulus κ i he

than about 2000. This rule of thumb can be used to estimate the the absence of exp a. However, lower

odulus can be used to m pressible materials.

• Solution 601 a n small strain near-

eater than zero.

• s used to model the compressibility of tmaterial. As a rule of thumb, the material will behave in an almost incompressible manner if the bulk to shear modulus ratio is greater

bulk modulus invalues of the bulk m

erimental datodel com

ssumes a default for the bulk modulus based oincompressibility, i.e.,

( ) w

3 1 2E

νith κ ν= = 0.499 (3.7-7)

rain shear modulus G,

−

where E is the small strain Young's modulus or, in terms of the small st

( )( )

2 1500 for

3 1 2G

Gν

κ νν

+= = = 0.499

−

• Solution 701 a s based on small strain near-incompressibility reduce the stable to use one that

ssumes the same bulk modulu. However, this can significantly

time step. In such cases, is betterresults in ν=0.49.


UGS Corp. 117

e.

As the material deforms, the bulk to shear modulus ratio may hange, because the instantaneous shear modulus is dependent on

ount of deformation. A value of the bulk modulus that orresponds to near incompressibility for small strains may not be r ge

3.7.2 Ogd

odel=Ogden the MATHE material entry. It is based on the following

expression:

• When automatic time step calculation is used for a Mooney-Rivlin material in Solution 701, the critical time step is governed by the dilatational wave speed. This is most frequently an acceptable assumption since the material is almost incompressibl •cthe amcla ge enough to correspond to near incompressibility for larstrains.

en material model

• The Ogden material model is obtained by setting Min

9

α α αµ⎛ ⎞1 2 3

1D

n nα=

3n n nnW λ λ λ⎡ ⎤= + + −⎜ ⎟⎣ ⎦⎝ ⎠

where n

∑

µ and nα are Ogden material constants.

This strain energy density expression assumes a totally •

incompressible material ( )0 3 1 .t I = It is modified to account for

material compressibility by: 1) substituting for the stretches 1 2 3, ,λ λ λ the reduced stretches

1 2 3, ,λ λ λ , 2) removing the condition 11 2 3λ λ λ = , and

3) adding the volumetric strain energy density

( ) (2 21 11V 1 2 3W Jκ λ λ λ κ )12 2

= − = −


where κ is the bulk modulus. The relationship between the pressure and the volumetric ratio is the same as for the Mooney-Rivlin material description.



er the u/p formulation (default) or the displacement-based rmulation can be used. For comments about the u/p formulation,

Selection of material constants: The Ogden material description used here has 19 constants:

Eithfosee the corresponding comments in the Mooney-Rivlin material description.

, , 1,...,9n n nµ α = and the bulk modulus. Choosing only ,3 , 0, 1, 2n n nµ α ≠ = the standard three-term Ogden material description is recovered.

The small strain shear modulus and small strain Young�s modulus can be written as (assuming κ = ∞ )

9

1

12 n n

nG µ α

=

= ∑

9

1

32 n n

nE µ α

=

= ∑

hese moduli m

e step is governed by the

3.7.3 Arru

T ust be greater than zero. • When automatic time step calculation is used for an Ogden

aterial in Solution 701, the critical timmdilatational wave speed. This is most frequently an acceptable assumption since the material is almost incompressible. • For comments about the bulk modulus, see the corresponding comments about the bulk modulus in the Mooney-Rivlin material escription. d

da-Boyce material model

• The Arruda-Boyce model is obtained by setting MODEL = BOYCE in the MATHE material entry. It is based on the A

following expression:


UGS Corp. 119

( ) ( ) ( )

( ) ( )4 519 519+ − + −

ng the number of statistical links of the material chain.

model is described in the following

ref. E. M. Arruda and M. C. Boyce, �A three-dimensional f

2 31 1 12

1 1 11[ 3 9 272 20 1050D KTW N I I I

N N= − + − + −

1 13 481 243 ]7050 673750

I IN N

where NKT is a material constant and N is a material parameter representi • The Arruda-Boyce material reference:

constitutive model for the large stretch behavior orubber elastic materials�, J. Mech. Phys. Solids, Vol,. 41 (2), pp 389-412 (1993).

• This strain energy density expression assumes a totally incompressible material ( )3 1 .I = It is modified to account for material compressibility by: 1) substituting for the strain invariant 1I the reduced strain invariants 1I , 2) removing the condition 3 1I = , and 3) adding the volumetric energy term

( )2 1Jκ ln

2 2VW J⎡ ⎤−⎢ ⎥= −⎢ ⎥⎣ ⎦

where κ is the sm ip between the pressure

all-strain bulk m dulus. The relationshand the volume ratio is

o

1

2p J

Jκ ⎛ ⎞

= −⎜ ⎟⎝ ⎠




,

ulation is used, there should be at least one solution unknown. This is because the constraint equation used in the u/p formulation is nonlinear in the unknown pressures. Therefore equilibrium iterations are required for convergence, even when all of the displacements in the model are prescribed.

3.7.4 Hyperfoam material model

• The Hyperfoam material model is obtained by setting Model=Foam in the MATHE material entry. It is based on the following expression:

Either the u/p formulation (default) or the displacement-based formulation can be used. For comments about the u/p formulationsee the corresponding comments in the Mooney-Rivlin material description. When the u/p form

( )1 2 31n n n=

13 1n n n n n

Nn α α α α βµ λ λ λ

α β−W J

⎡ ⎤= + + − + −⎢ ⎥

⎣ ⎦

in which there are the material constants , , , 1,...,n n n n N

∑

µ α β = .

The maximum value of N is 9. • A material model similar to the hyper-foam material model is described in the following reference:

ref. B. Storåkers, �On material representation and constitutive branching in finite compressible elasticity�,J. Mech. Phys. Solids, Vol,. 34(2), pp 125-145 (1986).

In this reference, n

β is the same for all values of n. The strain energy density can•

v be split into deviatoric and

olumetric parts

/3

1 2 3 33n n n n

Nn α α α αµ λ λ λ

1n n=DW J

α⎡ ⎤+ + −= ⎣ ⎦∑


UGS Corp. 121

( ) ( )/ 33 3

1

13 1 1n n nnV

n n n

W J Jα α β

α β−

=

N µ ⎡ ⎤= − + −⎢ ⎥

⎣ ⎦∑

Notice that D VW W W= + . This decomposition of the strain energy density has the advantage that the stresses obtained from thedev

iatoric and volumetric parts separately are zero when there are

no deformations:

0

0

ij

ijij ji ε =

1 0D D DW WS⎛ ⎞∂ ∂

2ij ε == + =⎜ ⎟ , ⎜ ⎟∂ε ∂ε⎝ ⎠

0

0

1 02ij

ij

V V Vij

ij ji

W WSε =

ε =

⎛ ⎞∂ ∂= + =⎜ ⎟⎜ ⎟∂ε ∂ε⎝ ⎠

Notice that DW contains the volumetric part of the motion throughthe term /3

33 nJ α . Therefore

DW is not entirely deviatoric.

• The material is not assumed to be totally incompressible. Because both DW and VW contain the volumetric part of motion, the mixed u/p formulation cannot be used with the hypefoam material. A displacement-based formulation is used. Selection of material constants: The hyper-foam materiadescription used here has 27 constants: , , , 1,...,9n n n n

the r-

l µ α β = .

The small strain shear modulus and small strain bulk modulucan be written as

s

9

1

12 n n

nG µ α

=

= ∑

9

1

13n n

nnκ β µ

=

⎛ ⎞= +⎜ ⎟⎝ ⎠

∑ α



These moduli must be greater than zero, hence we note that nβ should be greater than �1/3. When all of the nβ are equal to each other β= , then the Poisson�s ratio is related to β using

1 2

νβν

=−

• The hyper-foam material model is generally used for highly compressible elastomers. If the ratio of the bulk modulus to shear modulus is high (greater than about 10), the material is almost incompressible and we recommend that one of the other hyperelastic materials be used.

3.7.5 Thermal strain effect

• When the material is temperature-dependent, you can include a

of thermal ated as

coefficient of thermal expansion. The coefficientexpansion is constant. The thermal strain is calcul ( )0thε α θ θ= −

0θwhere is the initial temperature, and it is assumed to be isotropic. This is similar to the formula as is used for the other thermo-elastic materials, see Section 3.1.5 assum

ermal expansion coefficient.

ent

ing a constant th • When the thermal strain is non-zero, the deformation gradiX is assumed to be decomposed into a thermal deformationgradient thX and a mechanical deformation gradient mX , using m th=X X X The thermal deformation gradient is (1 )th the= +X I therefore the mechanical deformation gradient is


UGS Corp. 123

the mechanical Cauchy-Green deformation tensor is

nge strain tensor is

1(1 )m the −= +X X

2(1 )m the −= +C C

and the mechanical Green-Lagra

( )2 2(1 ) 1 (1 )2m th the e− −= + − − +ε ε I

For small thermal strains, the last equation reduces to

1

m the≈ −ε I , so that the strains are nearly the sum of the mechanical and thermal strains, as in small strain analysis.

owever, we do not assume that the thermal strains are small.

ε

H • The strain energy densities are computed using the mechanical defstrCauchy-Green deformation tensor. ing the e the str ob

ormations. This is done by computing all invariants and etches using the mechanical deformations, e.g. the mechanical

The 2nd Piola-Kirchhoff stresses are obtained by differentiat strain energy density with respect to the total strains. Sincain energy density is a function of the mechanical strains, wetain

1 W WS⎛ ⎞∂ ∂

= +2 ) ( )

) )12 ) ) ( ) )

ij ji

m ab m ba

m ab ij m ba j

W W

ε

ε εε ε ε

∂( ∂ ε⎝ ⎠⎛ ⎞∂( ∂(∂ ∂

= +⎜ ⎟⎜ ⎟∂( ∂( ∂ ε ∂(⎝

( )( )2 112 ) ( )th

m ij m

W Weε ε

− ∂ ∂= + +⎜⎜ ∂( ∂⎝ ⎠

ij

i

ji

⎜ ⎟⎜ ⎟

⎠⎛ ⎞

⎟⎟

With this definition, the 2nd Piola-Kirchhoff stresses are conjugate o the Green-Lagrange strains. t



3.8 Gaske

Gaskets are relatively thin components placed between two

(see Fig. 3.8-1). While most gaskets are flat, any arbitrary gasket in Solution 601. The gasket material is

obtained using the MATG material entry to define the transverse

t material model (Solution 601 only)

•bodies/surfaces to create a sealing effect and prevent fluid leakage

geometry can be modeled

and through-thickness gasket properties together with an elastic isotropic MAT1 material entry to define the in-plane gasket properties.

Gasket thickness direction(material X axis)

Only one elementthrough the thickness

Gasket in-planedirections

Figure 3.8-1: Schem

The pplied t

e

r elasto-plastic material when ad-

atic of gasket

sealing effect is created when the compressive load, a• in he direction of the gasket thickness, exceeds the initial yield stress of the gasket. The sealing effect is maintained as long as thcompressive stress does not drop beyond a specified threshold value. The gasket ruptures if the compressive stress exceeds the gasket�s ultimate stress. Unlike rupture, if a gasket leaks it still maintains its load-deflection characteristics. • The gasket model can be used with 3-D solid elements. It can also be used with small displacement/small strain, large displacement/small strain kinematics. • The gasket behaves as a nonlineacompressed in the thickness or gasket direction. Its lo

3.8: Gasket material model

UGS Corp. 125

closure curves. Tensile stiffness can be assumed to be constant or zero. The closure strain is always measured as the change in gasket thickness divided by the original gasket thickness. The gasket�s uni-directional plasticity model speeds up computations, and allows more flexibility in defining the shape of the loading and unloading

The closure strain is always defined as the change in gasket ickness divided by the original gasket thickness. It is positive in

compression. The gasket pressure has units of stress, and it is also positive in compression.

• Fig. 3.8-2 pical pressure-closure relationship. It onsists of a main loading curve consisting of any number of

for different points on the loading curve. However, it is not necessary to define a loading/unloading curve for each point on the main loading curve.

deformation characteristics are typically represented by pressure-

curves. •th

shows a ty

celastic segments and any number of plastic loading segments. Loading/unloading curves can be provided

Loading curve

Leakagepressure

Rupturepoint

Closure strain

Gas

ket

pre

ssure

Multiple loading/unloading curves

Initial yieldpoint

Figure

3.8-2: Pressure-closure relationship for a gasket material



re.

losed: The gasket pressure is higher than the leakage pressure but has not yet caused plasticity.

Sealed: There has been plastic gasket deformation and the

Leaked: After plastic deformation, the gasket pressure has w gasket leakage pressure.

lements.

ns,

attempts to automatically efine the gasket�s material axes if they are not explicitly defined

The top and bottom surfaces of a gasket can be separate from

common surface with the tended mating surface. In this case, contact is not needed,

rom its target. Tied contact an he gasket and its mating surface.

• The number of points in all loading/unloading curves must be

main e.

• Each gasket can have one of the following five states:

Open: The gasket pressure is less than the leakage pressu C

current pressure is above the gasket leakage pressure.

dropped belo

Crushed: Gasket closure strain has exceeded the rupture value. Modeling issues • The gasket must be modeled as a single layer of 3-D eOnly linear elements are possible (6-node wedge and 8-node brickelements). • Since the gasket has different properties in different directioit is an orthotropic material. The material X-axis must be set to thegasket normal direction. Solution 601 dby the user. •those of the mating surfaces. In this case, they should be connected via contact. The gasket can also share ainhowever, the gasket cannot separate fc also be used between t

identical for efficiency. Also the last point in each loading/unloading curve must be one of the input point on theloading curv

3.8: Gasket material model

UGS Corp. 127

akage pressure is automatically set to 1% of the initial yield pressure.

The input: In addition to the pressure-closure relationships the

g -plane material properties are also required: Young�s modulus,

Note that all these output variables are scalar quantities.

• The le

following transverse material properties are needed for the gasket: transverse shear modulus, tensile Young�s modulus. The followininPoisson�s ratio, thermal expansion coefficient and density.

Output variables: The following gaskets output variables are available: Gasket pressure, Gasket closure strain, Gasket yield stress, Gasket plastic closure strain, Gasket status. •

Chapter 4: Contact conditions


4. Contact

• Contact conditions can be specified in Advanced Nonlinear olution to model contact involving solid elements, shell elements,

Very general contact conditions are assumed:

! The points of contact are assumed not known a priori.

s

d sliding can be modeled. ! Repeated contact and separation between multiple bodies is

permitted in any sequence.

contact can be modeled (Solution 601 only).

described in the following references:

the, K.J. and Chaudhary, A., "A Solution Method for Planar and Axisymmetric Contact Problems," Int. J.

ints Arising From Contact Conditions in Finite Element Analysis," J. Computers & Structures, Vol. 40, No. 2, pp. 203-209, July 1991.

.

Section 6.

conditions

Sand rigid surfaces.

•

! Friction can be modeled according to various friction law(only standard Coulomb friction for Solution 701). ! Both sticking an

! Self-contact and double-sided contact are permitted. ! Tied

! A small displacement contact feature is available.

Some of the contact algorithms used in Advanced Nonlinear Solution are

ref. Baref. KJB7

Num. Meth. in Eng., Vol. 21, pp. 65-88, 1985. ref. Eterovic, A. and Bathe, K.J., "On the Treatment of

Inequality Constra

ref. Pantuso, D., Bathe, K.J. and Bouzinov, P.A."A Finite Element Procedure for the Analysis of Thermo-mechanical Solids in Contact," J. Computers & Structures, Vol. 75, No. 6, pp. 551-573, May 2000

4. Contact conditions

UGS Corp. 129

of one or ore contact regions or surfaces. Contact pairs are then defined

• Table 4-1 lists the case control commable 4-2 lists the bulk data commands related to contact surface

commands related to

Contact Cas

• Contact in Advanced Nonlinear Solution is modeled using contact sets or groups. Each contact set is composed mbetween contact surfaces. Contact segments are the buildingblocks for contact surfaces.

ands related to contact, Tdefinition, and Table 4-3 lists the bulk data

ntact set definition. co e Control Command Description

BCSET Selects which contact set to use

BCRESULTS Selects which contact results to output

Table 4-1: Ad ntact

Contact Surfa

vanced Nonlinear Solution Case Control commands related to co

ce Command Description

BSURFS ement and nodes) Define contact surface on solid elements (by el

BSURF Define contact surface on shell elements (by element number)

BCPROP ements (by property ID) Define contact surface on shell el

BC Set parameters for contact surface RPARA

Table 4-2: Ad e def

Co Description

vanced Nonlinear Solution commands related to contact surfacinition

ntact Set Command

BCTSET Define contact sets

BCTADD Define union of contact sets

BCTPARA Set parameters for contact sets

Table 4-3: Addefinition

vanced Nonlinear Solution commands related to contact set



• Most of the features and tolerances needed for contact sets are provided in the BCTPAR ran Quick Refere(such as contaThese setting

4.1 Overv

ents, shell elements, or ttached to rigid nodes. It should be defined on areas that are itially in contact or that are anticipated to come into contact

e Fig. 4.1-1 for an illustration.

A entry. An explanation of this entry is provided in the NX Nastnce Guide. Some contact settings however apply to all contact sets ct convergence tolerances, suppression of contact oscillations).

s are provided in the NXSTRAT entry.

iew • A contact surface is made up of a group of 3-D contact egments (faces) either on solid elems

ainduring the solution. Se

3-D contact surface pair

Body 2

Body 1

Target surface

(top surface of

body 2)

Contactor surface

(surface of cylinder)

Figure 4.1-1: Typical contact surfaces and contact pair • A contact pair consists of the two contact surfaces that may ome into contact during the solution. One of the contact surfaces

in the pair is selected to be the contactor surface and the other contact surface to be the target surface. In the case of self-contact, the same surface is selected to be both contactor and target.

prevented

c

• Within a contact pair, the nodes of the contactor surface are

from penetrating the segments of the target surface, and

4.1: Overview

UGS Corp. 131

not vice ve In Solution 601 target surfaces may be rigid, while contactor

uld not also have all

if

t, or be

on

rsa.

•surfaces cannot be rigid. A contactor node shoof its degrees of freedom dependent. • In Solution 701 both contactor and target surfaces can be rigid the penalty algorithm is used. Otherwise, the same restriction mentioned above for Solution 601 applies. • Rigid surfaces have no underlying elements and therefore noflexibility apart from rigid body motions. All their nodal degrees of freedom must be either fixed, have enforced displacemenrigidly linked to a master node which is defined on the BCRPARAentry. • Fig. 4.1-2 shows the effect of contactor and target selectionthe different contact configurations.

Contactorsurface

No penetration

No penetration

Targetsurface

Targetsurface

Contactorsurface

Figur

ntact surface is expected to come into

d in e

e 4.1-2: Contactor and target selection • Self-contact is when a cocontact with itself during the solution. • Two types of contact surface representation are supporteAdvanced Nonlinear Simulation, an old and a new contact surfacrepresentation (set via the CSTYPE parameter in the NXSTRAT



. The ma differences between the two representations are:

In the new representation, contact is based on the actual faces of the contact se e contact onstraints. Quadratic contact segments in the new contact surface

lit up into multiple linear segments.

ate

features: - egment method algorithm - igid target algorithms - Double-sided contact • It is acceptable for the nodes on the contactor and target surfaces to be coincident (have identical coordinates). In this case, it is important to ensure that the two surfaces do not share the same nodes. • For single-sided contact, which is defined using NSIDE=1 parameter on the BCTPARA card (see Fig. 4.1-3), one side of the contact surface is assumed to be internal and the other side to be external. Any contactor node within the internal side of a target surface is assumed to be penetrating and will be moved back to the surface. This single-sided option is ideal for contact surfaces on the faces of solid elements since in that case it is clear that one side is internal to the solid while the other is external. In this case, the

entry). The new contact surface representation is the defaultin

- gments which results in more accurat

crepresentation are not sp - The new representation uses a more accurate contact traction calculation algorithm. - The new representation generates more accurate contact constraints for 3-D contact segments resulting from 10, 11 node tetelements and 20, 21 node brick elements. These elements generzero (10, 11 node tets) or negative (20, 21 node bricks) contact forces at their corner nodes when subjected to a uniform contact pressure. - Tractions are reported as nodal quantities in the new surface representation. The new contact surfaces cannot be used with the following

SR

4.1: Overview

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external side can usually be predicted from the geometry. This option is also useful for shells when it is known that contact will definitely occur from one direction. In this case, however, the program cannot intuitively predict the internal side of the contact surface.

Target surface

External side

Internal side(no contactor nodes allowed)

Figure 4.1-3: Single-sided contact surface

• In double-sided contact, which is defined using NSIDE=2 parameter on the BCTPARA card (see Fig. 4.1-4), there are no internal or external sides. The contactor surface nodes in this case are prevented from crossing from one side of the target contact surface to the other during solution. This option is more common for shell-based contact surfaces. If a contactor node is one side of

e target surfac e t + ∆t.

e at time t, it will remain on the same side at timth

Contactor node cannotpenetrate upper side

Contactor node cannotpenetrate lower side

Targetsurface

Figure 4.1-4: Double-sided contact



hed n

r-TPARA

ces.

et . The main difference is that the coefficients for the rigid

lements are automatically determined by the program and they are nly applied for the nodes that are initially in contact. The basic

idea is illustrated in Fig. 4.1-5.

• When the tied contact feature is selected for a contact set (TIEDparameter in the BCTPARA card), Solution 601 performs an initial contact check at the start of the analysis. All contactor nodes that are found to be in contact or overlapping are permanently attacto their respective target segments. Contactor nodes that are not icontact are also set to be tied if the contact gap is less than a usespecified contact tolerance (TIEDTOL parameter in the BCcard). This tolerance is useful when the contact gap is due to non-matching finite element discretizations of the contacting surfa The tied contact feature is conceptually similar to using Rigidelements or multipoint constraints to attach the node to the targsurfaceeo

Contactor surface

Target surface

Gap betweencontact surfaces(exaggerated)

Permanent rigidconnections

Figure 4.1-5: Tied contact option

Tied contact is not "real" contact because there can be tension between tied contact surfaces. Also no sliding can occur between tied contact surfaces.

The tied contact option can be used to connect two incompatible meshes. However, the mesh glueing feature described in Section 5.9 produces more accurate results.

If the contact surfaces initially overlap, they are not pushed back to eliminate the overlap. Similarly, if there is an initial gap it is not eliminated.

T the initial nodal positions only. The

he tied contact constraint equations are computed based onconstraints generated in tied

4.1: Overview

UGS Corp. 135

y may be

mal direction are determined. The local coordinates f the target point and the normal direction are then kept constant

remainder of the analysis. This is in contrast torge displacement contact, where the contact constraints are

little relative deformation around the contact region. For such problems, it is much more computationally effici t to perform only one detailed contacrepeating the search every

e general

contact are not updated during the analysis. Hence, theless accurate if the bodies experience large rotations. • If the small displacement contact feature is used (CTDISP parameter in the NXSTRAT entry or DISP entry in the BCTPARA entry), the contact constraints are generated once in the beginningof the analysis and are kept constant, as shown in Fig. 4.1-6. A target location is identified for each contactor node if possible, and its gap and norofor the the standard laupdated every iteration, and the contactor nodes can undergo anyamount of sliding.

This feature is useful when there is very

ent search at the beginning of the analysis, rather than

iteration. Also, in some cases, convergence can also be slow or unachievable with thalgorithm, for example as nodes oscillate between one target segment and another equally valid neighboring target segment.



Contactor surface

Small displacement contact acceptable

Small displacement contact notacceptable due to excessive displacement

Small displacement contact notacceptable due to excessive rotation

x1

N

x1

x1

x1

*

Original geometry: Determine targetpoint normal vectorx N1 and*

*

*

x1

x1

*x1

x1

Target surface

Figure 4.1-6. Small displacement contact feature

• Contact result output is controlled by the BCRESULTS Case Control command. The user can request output of nodal contact forces and/or nodal contact tractions. • The normal contact conditions can ideally be expressed as 0; 0; 0g gλ λ≥ ≥ = (4.1-1) where t force. Different algorithms may vary in the way they impose this condition.

ensional friction variable

g is a gap, and λ is the normal contac

• For friction, a nondim τ can be defined s a

TFτµλ

= (4

.1-2)

here FT is the tangential force and λ is the normal contact force.

• The standard Coulomb friction condition can be expressed as


w

4.1: Overview

UGS Corp. 137

( ) ( )

1and 1 implies 0 while 1 implies sign sign

uu

ττ

τ τ

≤< =

= =

""

(4.1-3)

ence, lems.

m fr .

ts are

o contact: the gap between the contactor node and target

Sliding: the gap between the contactor node and the target

to the

target segments (either due to frictionless contact to a frictional restrictive force less than the limit Coulomb force.

Sticking: as long as the tangential force on the contactor node that initiates sliding is less than the frictional capacity (equal to

cient), the

4.2 Contact algorithms for Solution 601

• Solution 601 offers three contact solution algorithms (set via the TYPE flag in the BCTPARA entry):

! Lagrange multiplier (segment) method, or ! Rigid target method

where u" is the sliding velocity.

• In static analysis, the sliding velocity is calculated by dividing the incremental sliding displacement by the time increment. Htime is not a dummy variable in static frictional contact prob • When (Coulo b) friction is used, the iction coefficient can beconstant or calculated from one of several predefined friction laws • The possible states of the contactor nodes and/or segmen

Nsegment is open.

segment is closed; a compression force is acting oncontactor node and the node kinematically slides along the

the normal force times the Coulomb friction coefficontactor node sticks to the target segment.

! Constraint-function method,



• Each contact set must belong to one of these three contact algorithms. However, different contact sets can use different algorithms.

4.2.1 Constraint-function method

ard), d the

onditions. In the constraint-function method, the inequality constraints of Eq. (4.1-1) are replaced by the following normal constraint function:

• All 3 contact algorithms can be used with or without friction.

• In this algorithm (selected using TYPE=0 on BCTPARA cconstraint functions are used to enforce the no-penetration anfrictional contact c

( )2g gλ λ,

2 2 Nw g λ ε= − +⎜ ⎟⎝ ⎠

where εN is a small user-defined parameter. The function is shown in Fig. 4.2-1. It involves no inequalities, and is smooth and differentiable. The parameter ε is set via the EPSN variable in

+ −⎛ ⎞

Nthe BCTPARA entry.

g

�

w(g, )�

Figure 4.2-1: Constraint function for normal contact

so used to regularize the rigid non-differentiable stick-slip transition of Eq. (4.1-3). This results in a smooth transition from stick to slip and vice versa, and

• The constraint function method is al

4.2: Contact algorithms for Solution 601

UGS Corp. 139

it also results in a differentiable friction law that is less likely to cause convergence difficulties. Two friction regularization algorithms are available in

default algorithm v the frictional constraints

and, in general, converges much faster than its predecessor. The older friction algorithm can still be accessed via the FRICALG parameter in the NXSTRAT entry.

l constraint function takes the form:

Advanced Nonlinear Solution. The newerin olves a more accurate linearization of

( )v , 0u τ =" The frictiona . In the old friction algorithm, the v function is defined implicitly via

vv arctan 0u⎛ ⎞2 −τ +

Tε− =⎜ ⎟

"

the

oulomb friction law as shown in Fig. 4.2-2.

π ⎝ ⎠

Here εT is a small parameter which provides some elastic slip toC

u.

�

-1

1

-10 -2 -1 21 10

2x1_ _

_

� T

T

n Figure 4.2-2: Frictional contact constraint function for old frictioalgorithm

A multilinear frictional constraint function is used in the new friction algorithm as shown in Fig. 4.2-3.



u.

�

-1

1

-10 -2 -1 21 10

1_

_

T

T

Figure 4.2-3: Frictional contact constraint function for new friction

algorithm

4.2.2 Lagrange multiplier (segment) method

selected using TYPE=1 on BCTPARA card),

tor

d sliding states. It also alculates this state for each contactor node based on the contact

This method should not be used with a force or energy/force

4.2.3 Rigid

act algorithm (selected using TYPE=2

n BCTPARA card). The algorithm is fully described in Section

4.2.4 Selec

Our experience is that in most frictionless contact problems the tion method is more effective than the segment

method. The constraint function method is the default.

• In this method (Lagrange multipliers are used to enforce the contact conditions ofEq. (4.1-1). The kinematic conditions are enforced at the contacnodes, and the frictional conditions are enforced over the contact segments. • This method involves distinct sticking ancforces on the target segment. •convergence criterion.

target method

• This is a simplified conto4.8. tion of contact algorithm

•constraint func

4.2: Contact algorithms for Solution 601

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Note that the target surface can be rigid in all three contact at the

4.3 Contac • lu n ms (set via the XTYPE flag in the BCTPARA entry): ! Kinematic constraint method, P a ! Rigid od

• ct algorithms. However, different contact sets can use different

s can be used with or without iction.

4.3.1 Kine

card). It is the default explicit contact algorithm for Solution 701.

• A predictor step is first done without applying contact

• For friction contact dominated contact problems, the performance of the constraint function and segment methods are comparable. The default is still the constraint function method. •algorithms. The presence of a rigid target does not mean thrigid target algorithm should be used.

t algorithms for Solution 701

So tio 701 offers three contact solution algorith

! en lty method, or target meth

Each contact set must belong to one of these three conta

algorithms. • All Solution 701 contact algorithmfr

matic constraint method

• This algorithm is selected by setting XTYPE=0 on BCTPARA

constraints or forces. Then displacements are evaluated andpenetration is detected and corrected. The exact correction of displacements requires the solution of a non-diagonal system ofequations. Instead, a good approximation is done. In this case, for each penetrating contactor node, a penetration force

*2

N N NC C C CM M

tδ

= =∆

F a N



C

to that of eacnode is projected to the target in the same way:

is calculated. This is the force required to remove the penetration at the contactor node. However, not all the penetration will be removed by moving the contactor. The target will get some motion depending on its mass relative to the contactor and how manycontactor nodes are touching it. So, the N

CF force above is projected to the target segment nodes:

i

N NT iN=F F

where Ni is the shape function relating the contactor displacement

h target node. Similarly, the mass of the contactor

iT i CM N M= nd this mass is added to that of ta he target node itself. Then the

T T T TM M

acceleration of the target node is determined as

N N+ =( )i i∑ ∑a F

then used to update the target displacements. The

ontactor acceleration is This correction isc

*N N NC C T iN= − ∑a a a

• For friction, a similar approach is used. A correction force is calculated

*T TC CM

t=

∆vF

where v is the tangential sliding velocity. However, this force Tcannot exceed the limit force based on the normal force and the coefficient of friction

( )*min ,T NC Cµ=F F T

CF

4.4: Contact set properties

UGS Corp. 143

,

in this case depends n whether the rigid target has natural or essential boundary

conditions.

4.3.2 Pena

The rest of the procedure is very similar to the case of normal contact. The form of the equations is different if there is dampingand is also different if the previous and current time steps are notthe same.

• A modification is also required for rigid targets, which are common in contact. The form of the equationso

lty method

• In this algorithm (selected using XTYPE=1 on BCTPARA card), contact conditions are imposed by penalizing the inter-penetration between contacting surfaces. When a penetration is detected, a normal force of

( )N N D NA A K Kδ δ= = +F P N " is applied to the contactor node, where KN is the normal stiffnessK

, D is a normal rate stiffness, δN is the penetration, Nδ" is the

penetration rate, N is the normal vector pointing towards the contactor, A is the contact area and P is the normal contact traction

n opposing force is distributed to the .

target nodes. A • Similarly, in the presence of friction, the relative sliding velocity between the two bodies is penalized as follows:

T TA K=F v where vT is the tangential sliding velocity. • The normal and tangential penalty stiffnesses KN and KT can be selected by the user, or determined automatically by the program

TPARA parameters: XKN, XKNCRIT, alty rate stiffness KD can be explicitly

selected by the user, or determined by the program as a ratio of act node (using the XDAMP and

XNDAMP parameters).

based on the following BCXKT, XKTCRIT. The pen

critical damping for the cont



time step.

ve

4.3.3 Rigid

sed in olution 601. It is selected using XTYPE=3 on the BCTPARA

4.3.4 Selec

01.

The penalty method is the simplest and fastest of the explicit

gets simultaneously.

• The main disadvantage of the penalty method is that contact onditions are not exactly satisfied and it usually shows oscillations

using penalty also sensitive to the choice of the enalty stiffness. If that stiffness is too large it leads to instability nd oscillations, and if it is too small it leads to excessive

penetrations. • The default penalty stiffness selected by Solution 701 is, in most cases, a suitable compromise.

4.4 Contact set properties

This section describes the main options available to contact sets.

• When penalty stiffnesses are automatically determined they are chosen based on the masses of the contactor nodes and the time step. They are selected such that they have a minimal effect on theexisting Note that unduly small penalty stiffnesses will lead to excessipenetrations, and while unduly large penalty stiffnesses will lead to excessive oscillations or unstable time integration. target method

• This algorithm is similar to the rigid target method uScard. The algorithm is fully described in Section 4.8.

tion of contact algorithm

• The kinematic constraint method is the default in Solution 7 •contact algorithms. It can also handle rigid contactor and target surfaces. It also allow a contactor node to be in contact with multiple tar

cin contact forces. These oscillations can usually be removed by

damping. It is pa


UGS Corp. 145

• Contact surface offsets Penefined by the contact segment nodes is penetrated. However, an

specified which causes the actual contact surface to be offset from the plane defined by the contact surface

two that

Note that the ffset distance should be small compared to the contact surface

ntact set offset.

etration of a contact surface occurs when the plane or line doffset distance can be

nodes. In the case of double-sided contact, the offset createsseparate surfaces above and below the reference surface. Note if the contact surface is on a shell then half the shell thickness canautomatically be used as the offset. Fig. 4.4-1 shows the possibilities for single and double-sided contact. olength. Offsets for a whole contact set are specified via the OFFSET parameter in the BCTPARA entry, while offsets for a specific contact surface are set via the OFFSET parameter in the BCRPARA entry. If one of the contact surfaces has a definedoffset, it will overwrite the co

Internal side Defined contactsurface

Defined contact surface

Actual contactsurface

Actual contactsurfaces

Offset

(a)

(b)

Offset

tt

t

Offset

Figure 4.4-1: Contact surface offsets for: (a) single-sided contact (using NSIDE=1, OFFSET=t on BCTPARA card), and

l direction to a contact segment will in general not be ontinuous between segments as illustrated in Fig. 4.4-2. This

(b) double-sided contact (using NSIDE=2, OFFSET=t on BCTPARA card)

• Continuous normals (Solution 601 only) The normac



ent. This leads to a uniformly varying ormal direction. The SGNORM parameter in the BCTPARA card

e ls

In modeling target surfaces with sharp corners, either use

the orners be computed correctly. See Section 4.7.2 for modeling tips

sometimes causes convergence difficulties due to the non-unique normals at nodes and segment edges. The continuous normals feature first calculates nodal normals as averages of all the normals from the attached segments, and then interpolates these nodal normals across the segmndetermines the setting for continuous normals. The default SGNORM=1 results in continuous normals(Fig. 4.4-2(b)), whilSGNORM=-1 results in discontinuous segment-based norma(Fig. 4.4-2(a)). discontinuous normal vectors, or use small segments near the corners, in order that the normal vectors for segments nearcrelated to this feature. Continuous normals give poor results with explicit time integration. Therefore, they are blocked from Solution 701.

Nodal normals

(a) Discontinuous normals (b) Continuous normals

(a) Figure 4.4-2: Contact surface normals • Contact surface depth By default, the contact region extends for an infinite distance belowthe contact surface (for single-sided contact). However, a contact surface depth can be defined (by setting PDEPTH=t in the BCTPARA card), below which the contact surface is no longer active. The default PDEPTH=0.0 results in an infinite contact depth extension. Fig. 4.4-3 shows some of the possibilities.


UGS Corp. 147

A B

C

D

EF

G

Contact surface is used in self-contact.

Continuum createdby segment A-B

A B

C

D

EF

G

a) Target depth option not used

Nodes F and G havepenetrated segment A-B.

Target depth

t

b) Target depth option used

Nodes F and G have notpenetrated segment A-B.

Figure 4.4-3: Contact surface depth • Initial penetration The treatment of initial penetrations in Solution 601 is governedthe INIPENE parameter in the BCTPARA entry. By default, if there is initial overlap (penetration) between a contact node and target segment in the first solution step, the program attempts to eliminate the overlap. Advanced Nonlinear Solu

by

a

tion can eliminate the overlap at the first step or over a user-specified time using the TZPENE parameter in BCTPARA. This feature is useful if the initial penetrations are too large to be eliminated in a single step. The program can also calculate initial penetrations at the start of

does not detect penetration for a contactor node if the amount of enetration is less than or equal to the recorded amount. Fig. 4.4-4

ection 4.7.2 for modeling tips related to this feature.

).

solution and ignore them in future steps. In this case the program

pshows some of the possibilities. See S

Initial penetrations can also be set to gap override (see below



Continuum

Target surface

Contactor node

Solution start, contactor nodeinitially penetrates target surface

First solution time,overlap is eliminated

Overlap is gradually eliminated First solution time,overlap is recorded

Overlapdistance

Eliminate penetration

INIPENE = 0, TZPENE = 0.0

INIPENE = 0, TZPENE > 0.0 INIPENE = 2

Ignore penetrationEliminate penetration

over time period

Figure 4.4-4: Initial penetration options

• Gap override In the gap override feature (set via the initial penetration flag INI

me as explained in the Section on Initial

enetration above). This feature is useful for problems involving curved meshes in close proximity, such as the shrink fit example shown in Fig. 4.4-5. The gaps and penetrations measured from the discretized finite

PENE=3 in BCTPARA), the gaps and penetrations calculated from the finite element mesh are replaced by a fixed user-specifiedvalue (GAPVAL parameter in the BCTPARA entry). A positive value represents an initial gap, zero means that the contact is touching the target, and a negative value represents an initial penetration (which can be removed either immediately or over auser-specified tip


UGS Corp. 149

element mesh are sometimes inaccurate for such problems (unless matching meshes are used). In some problems, such as that shown in the figure, a constant geometry based overlap should be applied to all nodes, which corresponds to a gap override value of �δ. Note that mesh refinement and quadratic elements reduce the error in the measured overlaps but frequently a very high mesh density would have to be used if gap override is not used. Note also that the error in mesh based gaps and penetrations for curved surfaces can be more significant when low precision numbers are used for the node coordinates (such as when short input file format is used). Gap override is also useful for such cases.

Proper initial overlap at

all nodes should be �

Two rings with a geometric overlap (shrink fit)�

Directly measured overlapsare incorrect at most nodes

Fi ure 4.4-5. Significance of gap override for curved non-matched geometries

g

is useful where the edge of the n

• Contact surface extension The target surface can be enlarged beyond its geometric bounds, sothat contactor nodes that slip outside the target can still be considered in contact (via the EXTFAC parameter on the

CTPARA card). This feature Bco tactor and target surfaces coincide, as shown in Fig. 4.4-6.



Targetsurface

Corner contactor nodemay slip outside target dueto numerical round-off

Figure 4.4-6: Contact requiring contact surface extension

• Contact surface compliance (Solution 601 only) Contact surface compliance is set via the CFACTOR1 parameter on the BCTPARA card and is only available with the constraint function algorithm in Solution 601. Contact surfaces are commonlyassumed to be rigid meaning that no interpenetration is allowed. In this case, the contact surface compliance is set to 0.0. However,

is feature can be used to

simulate soft or compliant surfaces. The es

thamount of allowed interpenetration between the contacting surfacin this case is penetration normal contact pressurepε= × (4.4 The constraint function in the presence of a compliance factor is modified as shown in Fig. 4.4-7.

-1)

1

P

g

�

.4-7: Constraint function for compliant contactFigure 4


UGS Corp. 151

only) The consistent contact stiffness feature is set via the parameter

ange in the normal direction. Therefore, higher convergence rates

he contact birth feature activates a contact set at a specific time,

of the analysis, and a death time less an or equal to the birth time means that the contact set does not

• Friction delay (Solution 601 only) When the friction delay feature is activated (FRICDLY parameter in the BCTPARA entry), frictional conditions are applied to a contactor node one time step after contact is established. This

Note that the relative sliding velocity cannot be uniquely ermined when a node was not in contact at time t, and is in

∆t (see Fig uivalent to assuming at contact was established close to time t+∆t, and hence the

liding velocity is zero and so is the frictional force.

• Consistent contact stiffness (Solution 601

CSTIFF on the BCTPARA card. Changes in the direction of the contact normal provide an additional contribution to the stiffnessmatrix that is proportional to the value of the contact force and the ch(closer to quadratic) can sometimes be obtained by selecting theconsistent contact stiffness option which accounts for these additional stiffness contributions. This results, however, in an increase in the size of the stiffness matrix which is detrimental for large problems. This option is more beneficial when discontinuouscontact normals are selected. • Contact birth/death Twhile the contact death feature disables a contact set at a specific time. They are set via the TBIRTH and TDEATH parameters onthe BCTPARA card. A 0.0 birth time means that the contact set starts active at the beginningthdie.

feature can be useful in many problems, since it delays the non-inearity associated with friction until contact is established. l

detcontact at time t+∆t. That velocity depends on the exact time at which contact started, which is somewhere between times t and

. 4.4-8). Delaying friction is eqt+ths



Time = t

Friction delay

non-uniqueunique

x1

x1 Rx1 R

x1 R= Relative sliding velocity.

..= 0

t+ t�

t

t+ t�t+2 t�

t

t+ t�

t+ t�

t+2 t�

t+ t�t+2 t�

t+2 t�

FNFN

FT

FT

FT

Figure 4.4-8. Friction delay feature

n

4.5 Frictio Advanced Nonlinear Solution has a general Coulomb type friction model, where the coefficient of friction µ can be a constant or calculated based on several pre-defined friction laws. Solution 701 however, only supports standard Coulomb friction.

4.5.1 Basic friction model

ecified foret

ent.

4.5.2 Pre-

ad

tact

By default, a constant coefficient friction is used. It is spach contact pair via the BCTSET entry. The Rigid Targ

ealgorithm can only use a constant Coulomb friction coefficiSolution 701 also can only use a constant Coulomb friction coefficient.

defined friction models (Solution 601 only)

One of the following predefined friction laws can be used insteof constant Coulomb friction. The friction law and its input parameters are set via the BCTPARA entry. The following variables are used in the friction laws: the magnitude of the relative sliding velocity ,u" the contact traction ,nT the consistent con

4.5: Friction

UGS Corp. 153

f sliding force ,nF the current nodal coordinates x, the direction ov, and the time t. The setting for the FRICMOD parameter equired for each friction law is given in parentheses. r

• Constant coefficient of friction (FRICMOD = 1)

1Aµ =

• Different static and dynamic friction coefficients (FRICMOD = 4)

1 ifA

3u Aµ

2 3A u A

if ≤⎧

= ⎨ >⎩ ""

• Friction coefficient varying with sliding velocity (FRICMOD = 5)

1 3 1 22

3 2

( ) if

if

uA A A u AA

A u Aµ

⎧ + − <⎪= ⎨⎪ >⎩

" "

"

• Anisotropic friction mo

del (FRICMOD = 6)

2 2 21 (1) 2 (2) 3 (3) 5

4 5

( ) ( ) ( ) if

if

A A A u A

A u Aµ

⎧ + + >⎪= ⎨≤⎪⎩

v v v "

"

where v(1), v(2) and v(3) are the x, y and z components of the sliding direction.

Friction

coefficient varying with consistent contact force RI D = 7)

A A F

•(F CMO

1 2 n , 0 1µ µ= + ≤ ≤

• Time varying friction model (FRICMOD = 8)



1 3 1 22

3 2

( ) if

if

tA A A t AA

A t Aµ

⎧ + − <⎪= ⎨⎪ >⎩

• Coordinate-dependent friction model (FRICMOD = 9)

in 2DA A A

µ µ+ +⎧

= ≤x x

iction model 1a (FRICMOD = 2)

1 3 (2) 4 (3)2

1 3 (1) 4 (2) 5 (3), 0

in 3DA

A A A A≤⎨ + + +⎩ x x x

• Fr

2

1nT Aµ 1 exp( )A Tn− −

=

• Friction model 1b (FRICMOD = 12)

2

1

1 exp( )n

n

A FF A

µ − −=

• Friction model 2a (FRICMOD =3)

2 2 1 3( ) exp( )nA A A A Tµ = + − −

Friction model 2b (FRICMOD = 13)

•

2 2 1 3( ) exp( )nA A A A Fµ = + − −

4.6 Conta

4.6.1 Dynam

or Solution 601 • Oscillations in velocities and accelerations can sometimes be present in implicit dynamic contact analysis especially for high speed impact problems. These oscillations can be reduced by

ct features

ic contact/impact F

4.7: Modeling considerations

UGS Corp. 155

- applying post-impact corrections, - setting the Newmark parameter α = 0.5, - adding compliance to the contact surfaces. • In post-impact corrections, the velocities and accelerations of the contactor and target can be forced to be compatible during contact (only in the normal contact direction). This feature is activated by setting IMPACT = 1 in the NXSTRAT entry. This is achieved by modifying the velocities and accelerations of the contact nodes once convergence is reached such that they satisfy conservation of linear and angular momentum. • Setting the Newmark α = 0.5 instead of the default α = 0.25 (trapezoidal rule � see Section 6.3) results in an accurate solution of rigid body impact problems, and frequently has a positive effect on reducing numerical oscillations in flexible body contact. This

entry, or by changing ALPHA to 0.5 also in the NXSTRAT entry.

ct surface can also significantly reduce the numerical oscillations that result from dynamic time

n the BCTPARA entry. In this case, the compliance factor must be

do not

s.

computations. • The post-impact correction feature should not be used together

s, ave

feature can be activated by setting IMPACT = 2 in the NXSTRAT

• Adding compliance to the conta

integration. This is done by setting a non-zero CFACTOR1 i

selected based on Eq. (4.4-1) such that the contact pressurescause excessive penetration. Allowing penetration of the order of 1% of the element size usually eliminates numerical oscillation • Only the post-impact correction option requires additional memory and

with any of the other two oscillation suppression features.

• If post-impact correction is activated, all target nodes, except those with all degrees of freedom fixed or enforced displacementmust have a positive non-zero mass. The contactor nodes can hzero mass.



present in explicit dynamic contact analysis especially for high th

hm. In that case, they can be reduced by

other sources of damping such as Rayleigh damping

can reduce contact oscillations by damping the high frequency hem.

tic

for

4.6.2 Cont

odes from penetrating the target segments. During each equilibrium iteration, the most current geometry of the

te the

• For single-sided contact, the calculation of overlap at a k consists of a contact search, followed by a

penetration calculation. The contact search starts by identifying all possible target surfaces where node k can come into contact. For each of these target surfaces:

- Find the closest target node n to node k. - Find all the target segments attached to node n.

- Determine if node k is in contact with any of these

For Solution 701 • Oscillations in velocities and accelerations can sometimes be

speed impact problems. These oscillations are more common withe penalty contact algorit

- reducing the normal penalty stiffness, - adding penalty contact damping. See Section 4.3.2 for details on the explicit penalty contact algorithm.

In addition,

modes that generate t

• Oscillations in results can also occur when using the kinemaconstraint algorithm. These oscillations can be due to a mismatch in the masses of the two contacting surfaces. See section 4.7.3more details. act detection

• As explained earlier in this chapter, the contact conditions prevent the contactor n

contactor and target surfaces is used to determine and eliminaoverlap at the contactor nodes.

contactor node


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segments. - If node k is in contact, update the information.

• For double-sided contact, the contact search algorithm uses time tracking and checks whether the contactor node penetrated a target segment between times t and t + ∆t. • Note that the contactor and target bodies must lie on opposite sides of the interface (see Fig. 4.6-1). Thus for admissible conditions of contact the dot product of the normals to the contactor surface and the target surface must be a negative number.

4.6.3 Suppression of conta

his is STRAT

entry. In this case, the program records the pairing target segment for each contactor node in the previous NSUPP iterations. Once this array is full, and the contactpairing targ s

its

ct oscillations (Solution 601)

• In some problems contactor nodes may oscillate during equilibrium iterations between several (usually two) neighboring target segments. Frequently, both solutions are acceptable. A special procedure can be used to prevent such oscillations. Tdone by selecting a non-zero NSUPP parameter in the NX

or node is still in contact, and the et segment is one of those recorded in previou

iterations, the suppression feature is activated. The contactor node from this iteration onwards is associated with only that target segment. It may remain in contact with the segment, or in contact with an infinite plane passing through the segment, or it can separate from contact completely. The node is released fromrestrictions once iteration ceases, either because convergence is reached, or due to non-convergence.



Contactor body

Contactor surface

Contact interface

Target surface

Target body

Normal vector(contactor)

Normal vector(target)

Figure 4.6-1: Admissible conditions of contact (dot product of

normals to contactor surface and target surface is a negative number)

mended

that NSUPP be set greater or equal to 5 and at least 5 less than the aximum number of iterations.

nodes.

4.6

be

• If this oscillation suppression feature is used, it is recom

m • Note that there is memory overhead associated with this feature, where an integer array of size NSUPP is defined for all contactor

.4 Restart with contact

• Changes in contact parameters are allowed between restarts, with some exceptions. Some restrictions exist, such as no restart from friction to frictionless and vice versa. • The contact algorithm itself for a certain contact group can also change in a restart. For this purpose, the contact algorithms can divided into two groups. The first group includes the constraint function (implicit), Lagrange multiplier segment (implicit),


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ese algorithms. owever, restarts are not allowed between the two groups.

4.6.5 Contact damping

• The contact damping feature allows the user to add normal and tangential grounded viscous dampers to all contactor and target nodes in the model. The damping is activated via the CTDAMP parameter in NXSTRAT, and the normal and tangential damping coefficients are CTDAMPN and CTDAMPT. Using the same value for both normal and tangential direction results in isotropic viscous damping. The damping force on each node is

kinematic constraint (explicit), and penalty (explicit). Restarts are possible between different algorithms in this group. The second group includes the implicit and explicit Rigid Target algorithms. Restarts are possible between thH

Damp N N T TC C= +F u" " u • This damping can be useful in static problems for stabilizing the model especially when there are insufficient boundary conditions to remove rigid body modes. It can also be useful in dynamic analysis to dampen out high frequency numerical oscillations. The damping can be set to act only at the initial time step, or to be constant throughout the analysis. • Using the initial damping option, the damping will be active at the beginning of the first time step, and will be reduced gradually (between iterations) until it fully dies out by the end of the first time step. Thus the final solution at the first time step will be free of any damping. Note that if contact is not established and nothing else stabilizes the model, the program will not converge and will

nt damping remains active throughout the analysis. In e

See Section 4.8.6 for modeling hints on using contact damping to

give an appropriate warning message. Consta•

this case, the program outputs the sum of all damping forces in thoutput file, and the user must check that these forces are significantly smaller than the sum of the reaction forces (also written to the output file).



4.7 Mode

4.7.1 Contactor and target selection

a contact pair can be interchanged without much effect on the is

• If it is more important for the nodes of one surface not to penetrate the other, then that surface should be the contactor. This

ld preferably be the target in this case. A related condition occurs around corners or edges as shown in Fig. 4.7-2. The upper

,

ce can also be e too

• If one surface is significantly stiffer than the other, it should preferably be the target, unless one of the two conditions above

handle improperly supported structures and how to choose the damping constants.

ling considerations

• For some contact problems, the contactor and target surfaces in

solution. However, for many cases, one of the two alternativesbetter.

factor is usually important when one surface has a much coarser mesh than the other as shown in Fig. 4.7-1. The coarse surface shou

surface should preferably be the contactor in this case. • If one of the surfaces has mostly dependent degrees of freedomit should be the target. This dependency can be due to boundary conditions, constraints or rigid elements. The surfarigid if its nodes are not attached to any elements. In that casit has to be the target (except in the explicit penalty algorithm where this is permitted).

also exist.


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Contactor nodesTarget segments

Body 1 (contactor)

Body 2 (target)

Figure 4.7-1: Effect of incorrect contactor-target selection due to

mesh density

Contactorsurface

Targetsurface

Figure 4.7-2: Target selection for surfaces of different sizes

4.7.2 General modeling hints

• Advanced Nonlinear Solution automatically defines the direction of the contact surfaces on the faces of solid elements using the BSURFS entry. For contact surfaces defined on shells

sing the BSURF entry) the user has to ensure that the correct (udirection is defined using the BCRPARA entry.



ments are moved from the model if they are found to be attached to a rigid

contact surface. A contact surface is deemed rigid if: ! it is the target of a contact pair in a contact set using the rigid target algorithm, or ! the TYPE flag in the BCRPARA entry is set to RIGID.

• If the contact surface is rigid the MGP parameter in the BCRPARA command can also be used to define a master node that will control the motion of the rigid surface. Internally, rigid links

the rigid

almost rigid

If the degrees of freedom of a node on a contactor surface are used in constraint equations or attached to a rigid element (see Section 5.7), the contactor node degrees of freedom should preferably be independent.

• Rigid target surfaces can be modeled using nodes with no degrees of freedom or nodes with enforced displacements for all active degrees of freedom. As a result, a fine discretization of a complex rigid surface geometry is possible with only a small increase in the solution cost. • The commands for contact surface definition all require the contact surface nodes to be connected with 3-D solid or shell elements. Therefore, to model a rigid target dummy, shell elements should be used to define the surface. These shell elere

are created between the master node and all the nodes onrget. ta

• In general it is recommended that the lengths of segments on thecontactor and target surfaces be approximately equal. This is particularly important if multiple contact surface pairs are considered in the analysis or if the contact surface geometries are omplex. c

• If required, a contactor surface can be modeled as

y choosing a reasonably high Young's modulus for the finite belements modeling the contactor surface. However, the stiffness of the surface elements should not be excessively high and make the model ill-conditioned. •


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• If a contactor node has all of its translational degrees of freedom dependent, the node is dropped from the contact surface. • If the contact surfaces are smooth (i.e., the coefficient of friction is small), the frictionless model is recommended as it is less costly to use. It is also recommended that prior to any contact analysis involving friction, the frictionless solution is first obtained, whenever possible. • It is not recommended that contact pairs with friction coexist

• A contactor node should preferably not belong to more than one

ntactor node oscillating between one target to ed. It is the user�s responsibility however, to

convergence since friction will only act once a converged contact solution is established. This feature is also very useful for many problems involving initial penetrations. In this case, the first time

• Restarting from frictionless contact to contact with friction and

• Ignoring initial penetrations is a useful option when these nt discretization,

with contact pairs without friction in the same contact group.

contact surface in a contact group, otherwise the contactor node may be over-constrained. • For problems in which the contactor and target surfaces are initially relatively close to each other and no significant sliding between these surfaces is expected throughout the analysis, the small displacement contact feature may be used. The analysis will be faster in this case, since the relatively time consuming contact search is only performed once, and convergence difficulties associated with a coanother are eliminatmake sure that the problem is suitable for small displacement contact. • The friction delay feature can sometimes lead to better

step during which these initial penetrations are removed will be frictionless.

vice versa is not possible. However, it can be done if the frictionless analysis is replaced by a frictional analysis with a very small friction coefficient.

penetrations are just a product of the finite eleme



meaning that they do not exist in the physical model. Fig. 4.7-3 illustrates one such case involving contact between concentric cylinders. In this situation, if initial penetrations are eliminated, the contact algorithm will try to push the penetrating contactor nodes to the target surface segments in the first step, creating initial prestressing. These initial penetrations and any prestressing that they might cause are unrealistic. Ignoring them is useful in this

ted at e

est

case. Note however, that if either cylinder is significantly rotathe initial penetrations calculated ach contactor node (in theinitial configuration) will no longer be valid. In this case, the balternative would be to use a much finer mesh.

Target surface,marked with

Contactor surface,marked with

Geometry beforediscretization

Overlap to ignore

Outer cylinder

Inner cylinder

4.7.3 Mod

analysis (see Section 6.2.2).

Figure 4.7-3: Analysis of contact between concentric cylinders, initial penetration is ignored

eling hints specific to Solution 601

• It is recommended that the ATS method be used in contact


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etimes be beneficial for contact problems.

tion rameter in BCTPARA entry). For most

roblems, this parameter should be one or two orders of magnitude y

ces or more

liding. Using an excessively small value enforces the Coulomb ce

of stick and slip, but is more likeldifficulties. • Friction in the Rigid Target algorithm is regularized by

an n

ds to a moother friction law, which generally converges faster but results

ore

e pplication. Thus, for problems involving such

atures, the load steps should be small. The time step ∆t should

ot

d with ontinuous contact normals, the normal vectors may be inaccurate

s

• Line search can som

• Frictional contact problems using the constraint function algorithm can be sensitive to the choice of frictional regularizaconstant (EPST papsmaller than the expected sliding velocity. Using an excessivellarge value leads to a smoother friction law, which generallyconverges faster but results in smaller frictional forslaw more accurately but is more likely to experience convergendifficulties. • Friction is not regularized or smoothed in the Lagrange multiplier segment algorithm. This results in accurate enforcement

y to experience convergence

replacing the perfectly rigid (no-slip) stick friction state byelastic stick state via the tangential stiffness parameter (TCMOD iBCTPARA entry). Using an excessively large value leasin smaller frictional forces or more sliding. Using an excessively small value enforces the Coulomb law more accurately but is mlikely to experience convergence difficulties. • Geometric and material nonlinearities can highly depend on thsequence of load afealso be small in dynamic analysis and when time dependent material constitutive relations (e.g., creep) are used. • If rigid elements are connected to contact surface nodes, the flexible option can be used. In this case, the rigid element does ncreate any dependent degrees of freedom. This feature is activated via the EQRBAR and EQRBE2 flags in the NXSTRAT entry. • If a contact surface with corners or edges is modelecas shown in Fig. 4.7-4(a). In this case, switch to discontinuou



normals, use different contact surfaces for each smooth part (Fig. 4.7-4(b)) or use a fine mesh close to the corners or edges (Fig. 4.7-4(c)).

Contact surfacewith sharp corners

Contact surface 1

Contactsurface 1

Contactsurface 3

Contact surface 2

Contact surface 1(with fine

corner mesh)

Arrows correspond to normalvectors pointing to exterior side

a) Single contact surface

b) Three separate contact surfaces c) Single contact surfaces withfine mesh at corners

Figure 4.7-4: Defining contact surfaces (with continuous normal vectors) in the presence of corners

4.7.4 Modeling hints specific to Solution 701

• The penalty algorithm is preferred when both surfaces are rigid or have many fixed or prescribed nodes. • Large oscillations in the contact forces may occur when using the penalty method even though the model is stable. These can be reduced by reducing adding a penalty damping term and/or


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reducing the penalty stiffness. • When using the penalty contact algorithm it is important to check that the contact stiffnesses are properly selected. Unduly small penalty stiffnesses will lead to excessive penetrations, while unduly large penalty stiffnesses will lead to excessive oscillations or unstable time integration. • Large mismatches between the masses of contacting surfaces should be avoided when using the kinematic constraint method. This mismatch is common when contact involves a rigid surface with a small mass and an applied force, as shown in Fig. 4.7-5. The best solution in such cases is to minimize the mismatch by increasing the mass of the rigid surface. The inaccuracy in this case results from the way the contact is enforced. The kinematic constraint method first predicts displacements without contact then applies a contact correction. The contact conditions are satisfied more accurately when the penetrations in the predicted configuration are small which is usually the case due to the small time step size of explicit analysis. However, some cases such as that mentioned above lead to large projected penetrations which results in incorrect contact conditions and tensile contact forces.

Lar contacting surfaces

e,

• c

ge mismatches between the masses ofan also lead to problems when using the penalty method. In this

case, the normal penalty stiffness required to avoid instability (without reducing the time step) can be unduly small leading to excessive penetrations. The best solution in such cases is to minimize the mismatch by increasing the mass of the rigid surfacor increase the penalty stiffness by setting it manually or by reducing the time step.



F

Tensile forces

One nodein contact

Small mass assigned to rigid surface

Correct solutionOriginal configuration

Final configurationProjected configuration

(before contact correction)

Wrong contactregion

Figure 4.7-5: Performance of kinematic contact algorithm when

contact surfaces have a large mass mismatch.

4.7.5 Convergence considerations (Solution 601 only)

• When Solution 601 fails to converge during the incremental analysis, the int 601 in the output listing can provide some useful information (see Fig. 4.7-6).

uthird line giving the maximum value. See Chapter 6 for definitions

ore det e

t

nd parameter CFNORM gives the norm of the contact force vector.

ermediate printout given by Solution

• Three non-contact related norms are given: first, the energy convergence criterion, the displacement and rotation convergence criterion (boxes b and c), and the force and moment convergence criterion (boxes d and e). Each box has 3 lines of output with the top one giving the norm of the quantity, the second one giving the equation number corresponding to the maxim m value, and the

and m ails on th se norms. • Box f of Fig. 4.7-6 shows the contact related norms. Parame er CFORCE indicates the norm of the change in the contact forces(between two iterations), a


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• The following additional convergence criterion is used when contact is present:

CFORCERCTOL=

max(CFNORM,RCONSM)

where RCONSM and RCTOL are set in the NXSTRAT entry. I N T E R M E D I A T E P R I N T O U T D U R I N G E Q U I ...

OUT-OF- NORM OF

BALANCE OUT-OF-BALANCE NORM OF INCREMENTAL ...

ENERGY FORCE MOMENT DISP. ROTN. CFORCE ...

(EQ MAX) (EQ MAX) (EQ MAX) (EQ MAX) CFNORM ...

VALUE VALUE VALUE VALUE

FOR ITE=0

.95E+04

ITE=1 -.49E-09 .17E-08 .00E+00 .66E+01 .00E+00 .67E+04 ...

( 85 ) ( 0 ) ( 6 ) ( 0 ) .67E+04 ...

-.81E-09 .00E+00 .12E+01 .00E+00

ITE=2 .76E+04 .15E+05 .00E+00 .17E+01 .00E+00 .22E+05 ...

( 67 ) ( 0 ) ( 81 ) ( 0 ) .21E+05 ...

.97E+04 .00E+00 .31E+00 .00E+00

ITE=3 -.24E-10 .42E-09 .00E+00 .56E+00 .00E+00 .25E+05 ...

( 65 ) ( 0 ) ( 9 ) ( 0 ) .19E+05 ...

-.17E-09 .00E+00 .14E+00 .00E+00

... I N G E Q U I L I B R I U M I T E R A T I O N S

... CONVERGENCE RATIOS CONVERGENCE RATIOS OUT-OF-BALANCE LOAD

... FOR OUT-OF-BALANCE FOR INCREMENTAL VECTOR CALCULATION

... ENERGY FORCE DISP. CFORCE BETA RATIO

... MOMENT ROTN. (ITERNS)

COMPARE WITH COMPARE WITH

... ETOL RTOL DTOL RCTOL

... (NOT USED)

... -.51E-13 .57E-13 .00E+00 .10E+01 .10E+01 .93E-13

... .00E+00 .00E+00 ( 1)

... .80E+00 .49E+00 .00E+00 .10E+01 .10E+01 -.43E+14

... .00E+00 .00E+00 ( 1)

... -.25E-14 .14E-13 .00E+00 .13E+01 .10E+01 -.77E-14

... .00E+00 .00E+00 ( 1)

box b box c box d box e box f

igure 4.7-6: Solution 601 Output listing of convergence criteria Fduring equilibrium iterations



out onvergence, and all norms are decreasing, the maximum number

ing b a s, a

• When CFNORM is stable but CFORCE changes rapidly during equilibrium iterations, the contact n be oscillating between 2 or

ping, or turn on the suppression of contact oscillations feature. When

.

r

4.7.6 Handling improperly supported bodies

ary lem (one in which there are

• When the maximum number of iterations is reached withcof iterations should be increased. • When the norms are rapidly chang efore convergence f ilit is commonly caused by applying the load too quickly or using large time step.

camore close solutions. In this case, try to change the time step

CFNORM varies rapidly, usually e other three norms also vary • The segment contact algorithm should not be used with force oenergy/force convergence criteria.

th

Many static problems depend on contact to provide the boundconditions necessary for a stable probno rigid body modes). Some examples are shown in Fig. 4.7-7. F

Contact pair 1

Contactpair 2

Contactpair 3

Blankholderpressure

Appliedload

Fixed die

Sy

mm

etry

Figure 4.7-7. Examples of improperly supported bodies n such cases, the stiffnessI

c matrix is singular if the contact

itions pressure). Weak springs can be added by the user to del stable. However, the selection of appropriate

onstraints are inactive. Even if the constraints are active the stiffness matrix is still not positive definite. The problem is more serious if natural boundary cond are applied (forces,moments, or

ake the momlocations and stiffnesses for such springs may not be feasible.


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r a dynamic or the low-speed ynamics feature. However, in many cases, this too is not a feasible

option. Therefore, several other modeling techniques are available in Solution 601 to handle such problems. These are stiffness stabilization, contact damping and limiting incremental isplacements. These techniques can be used separately or

ng .

ber. Typical values are between 10 and 10 .

Contact damp ous dampers to all contactor and target nodes. Setting the damping to be only at the initial time

When the first time step converges contact must be established and removed. This way, the converged solution

ill be free of any contact damping. Other problems however,

er to ensure that amping is not excessive.

The damping constants have units of force per unit velocity. Hence, their proper value is problem dependent. If initial contact damping is used to stabilize a problem involving two contact odies at least one of which is unsupported, and with a gap

analysis), and the tiffness term (since one or both bodies are initially unsupported),

Some models may be better suited fod

dcombined in the same model. Stiffness stabilization (see Section 8.5 for details).This feature provides a stabilizing effect by scaling all diagonal stiffness terms without affecti the right-hand-side load vectorThe outcome of each iteration will be affected, but the final converged solution will not be (within the bounds of the convergence tolerances). Since the stabilization constant in non-dimensional, it should always be a small num

-12 -9

Contact damping (see Section 4.6.5 for details). ing adds grounded visc

step is sufficient for some problems such as those in Fig. 4.7-7.

damping will have beenwrequire the damping to be constantly present. In this case, the program outputs the damping forces at every time step. These forces should be compared with the reactions in ordd

bbetween them, then a good estimate of the damping constants CN and CT is one in which the gap is nearly closed in the first iteration.Starting with the dynamic equations of motion (see, Equation 6.3-1) and canceling out the inertial term (static swe obtain

=C U R"



here C is the total damping matrix, which in this case is diagonal, r. We can assume the normal and

ngential damping constants to be equal, the total damping amping constant times the number of

ontact nodes on the unsupported contact surface N (the top

wand R is the applied load vectotacontribution to be the dccircular body in the example in Fig. 4.7-7), and the velocity to be approximately equal to the minimum initial gap between the two bodies, g, divided by the time step size ∆t. This leads to the following value of the damping constants

N TR tC CN g

∆= =

where R is the sum of the applied loads at the first time step. Note that this is only an estimate, but is frequently an acceptable one. Limiting maximum incremental displacement (see Section 6.2.1

ent per iteration is

on or viscous damping is used, the initial displacement away from the converged

e ent length size in this case would scale down the potentially huge displacement in the d sol ne matrix, so

many cases it may be necessary to use it together with stiffness

4.8 Rigid a

4.8.1 Introdu

Thapplications in which the target surfaces are considered to be rigid. Fig

for details). Limiting the maximum incremental displacemuseful when a load is applied to a body that is not initially in contact. The model at that stage is unstable and even when stiffness tabilizatis

can be excessive leading the programolution, and thus making the return to the proper solution difficult. s

S tting the limiting displacement to about the elem

first iteration so that the results remain close to the convergeution. Note that this feature does not stabilize the stiff ss

instabilization or viscous contact damping or both.

t rget contact algorithm

ction

e rigid target contact algorithm is intended for use in

. 4.8-1 shows a typical application in metal forming.

4.8: Rigid target contact algorithm

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Blank

Blank holder

Blank holder force

Prescribed punchdisplacement

a) Physical problem

Punch

Die

Drawbead

Contactsurface 1(contactor),offsets usedto model blankthickness

Contactsurface 2(target)

b) Modeling with contact surfaces

Contactsurface4 (target)

Contactsurface 3(target)

Figure 4.8-1: Sample metal forming analysis using the rigid target contact algorithm



A t body or can rotate as a rigid body.

Co

h

Adon nced Nonlinear Solution of NX 4 is

of NX 5 is

e

Throughout this section, the rigid target contact algorithm in Advanced Nonlinear Solution of NX 4 is referred to as the �NX4� rig

X on the NX4 ran 4 Advanced

Molg ed when using the current rigid

S tion 4.8.6.

e suggest that new models not be set up using the NX4 rigid

It i s usi t contact algorithms described earlier in this chapter. However, the rigid target contact alg ve, b ause the

gid target contact algorithm uses the assumption of rigid targets to

4.8.2 Basi co

4.8.2.1 Contactor surfaces

SimSolution, the contact surfaces are organized into contact sets. Each on e consists of 3- or 4-node contact segments. A

arget surface can either be stationary, can translate as a rigid

ntact can be frictionless or can include Coulomb friction.

T e rigid target contact algorithm is completely revised in vanced Nonlinear Solution of NX 5. However the rigid target tact algorithm in Advac

retained in Advanced Nonlinear Solution for backwardsompatibility. The revised rigid target contact algorithmc

th default.

id target contact algorithm. This section does not describe the 4 rigid target contact algorithm; for informationN

rigid target contact algorithm, see the NX Nastonlinear Theory and Modeling Guide. N

dels that were set up using the NX4 rigid target contact orithm may need to be revisa

target contact algorithm, see the conversion hints in ec

Wtarget contact algorithm.

s also possible to solve many problems involving rigid targetng the constraint function and segmen

orithm described here is frequently more effecti ecrisimplify the contact searching.

c ncepts

ilar to the other contact algorithms in Advanced Nonlinear

tact surfacccontact pair consists of a contactor surface and a target surface. In


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a contactor surface to be contact with more than one target surface simultaneously.

e contactor surface definition includes the

ossibility of offsets.

the contactor surface is escribed entirely by the contactor nodes. (Fig. 4.8-2(a)).

the rigid target algorithm, it is allowed for in

Contactor surface: Thp

When there are no offsets specified, d

Contactsegment 1

Contactsegment 2

Segment normals

a) Contactor segments without offsets. Segment normals are not used.

Lower surface

Radius of sphere= contactor offsetUpper surface

b) Contactor segments with spherical offsets. Contactor normals are not used.

4.8-2(b), or e, the

f t thi

Figure 4.8-2 Contactor segments

e offsets can either be described When there are offsets specified, thusing spheres centered around the contactor nodes (Fig. sing the contactor normals (Fig. 4.8-2(c)). In either casu

of set magnitude is either constant or taken from the currenckness of attached shell elements, as described below.



Lower contactor point

Contactornode normal

Contactor node normal = average of all segment normals

Offset vector = contactor node normal offset magnitude�

Offsetvector

Upper contactor point

c) Contactor segments with offsets, normals used to describe offsets.

ents (continued)

When the offsets are described using the contactor normals, offset

constructed from

re described using spheres, the center contactor

erations in static and implicit dynamic analysis will not converge,

Figure 4.8-2 Contactor segm

vectors are constructed using the averaged contactor normals and the offset magnitude. The upper and lower contactor points are

the contactor nodes and the offset vectors.

When the target surface is concave, it is possible for the contact situation to be similar to the one shown in Fig. 4.8-3. In this case,

hen the offsets awnode cannot be in contact with both target segments at once, hence the center contactor node will oscillate between them. The center contactor node cannot be in contact with target edge 1 since edge 1is farther away than either of the target segments. Equilibriumitbecause of the oscillation. However contact is correctly modeledwhen the offsets are described using normals, because the centercontactor node can be in contact with target edge 1.


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Target segment 1

Spherical offset

a) Spherical offsets, contact incorrectly modeled

Contactor and target surfaces viewed from the sidefor ease of visualization.

Contactor surface

This contactor node cannot be in contactwith both target segments at the same time.

In contact with target segment 1

In contact with target segment 2

Target segment 2

Figure 4.8-3 Concave target surface, contactor surface with offsets



Target segment 1

Normals offset

b) Normals offsets, contact correctly modeled

Target edge 1

In contact with target edge 1

In contact with target segment 1In contact with target segment 2

Target segment 2

Figure 4.8-3 Concave target surface, contactor surface with offsets (continued)

4.8.2.2 Target surfaces

Each target surface is either stationary, or can rigidly move (translate, rotate or a combination of translations and rotations).

4.8.2.3 Determination of contact between contactor and target

No contactor offsets: It is allowed for a contactor node to be in contact with a target segment, target edge or target node. The program searches for the target segment, edge or node for which the absolute value of the distance d between the contactor node and the target segment, edge or node is minimized, where the distance is measured in the direction opposite to the target normal (Fig. 4.8-4). A positive distance corresponds to a geometric gap; a negative distance corresponds to geometric overlap.


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a) Interaction between contactor node and target segment

Target segment

Target segment normal

Contactor node

d

n

Closest point ontarget segment

b) Interaction between contactor node and target edge

Target edge

Contactor node

d

nClosest point ontarget edge

c) Interaction between contactor node and target node

Target node

Contactor node

d

n

Figure 4.8-4 Interaction between contactor node and target sur

face

otice that, for interaction between a contactor node and target N



ween a conta target node, the normal the line segmsh

-5 shows two target segments with a common target edge.

es indicate which of the target entities anylosest

which the contactor node is closest to the target edge depends upon the angles between the ta .

edge, or bet ctor node anddirection is taken fromnode and the target, as

ent connecting the contactor own in Fig. 4.8-4.

Fig 4.8The shaded volumcontactor node is c

to. Notice that the shaded volume in

rget segments attached to the edge

Contactor node in this shaded volumeis closest to segment 1.

Contactor node in this shaded volumeis closest to segment 2.

Contactor node in this shaded volumeis closest to edge 1.

Segment 1

Segment 2

Edge 1

Figur nd edges

nce the target segment, edge or node is determined, then the gap is computed using

e 4.8-5 Interaction of contactor node with target segment a

Ocontact

GAPBIASg d= −

ot model contact where GAPBIAS can be chosen to, for example, n


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the corresponding gap is negative, and less than DEPTH, then

ther wo

even if there is geometric overlap. (The default for GAPBIAS is 0.)

If0gcontact occurs, in o rds DEPTH− ≤ ≤ is the contact

it the depth of the target exactly as in the o

Contactor offsets described using spheres:

is determined ex e p is computed us

condition. DEPTH can be chosen to limsurface, ther contact algorithms.

In this case, the actly as if there are no offsets. Then thing

distance dcontact ga

OFFSET GAPBIASg d= − −

where OFFSET is the offse c

assuming that GAPBIAS = 0 (Fig. 4.8-6). The same idea is used r interaction between a contactor node and target edge or node.

t magnitude. The process is illustrated ontactor node and target segment, for interaction between a

fo

Target segment

Contactor node

d - OFFSET

d

n

Closest point ontarget segment

OFFSET

Figure 4.8-6 Interaction between contactor node and target

segment, spherical offsets

per he

scillation checking: The search for the nearest target segment,

Contactor offsets described using normals: In this case, contact is detected using the up and lower contactor points instead of tcontactor nodes.

Oedge or node is performed every equilibrium iteration in Solution601. During the equilibrium iterations, it is possible for the



ontact he target

urface is concave. When the contactor node oscillates between

, then, when oscillation is detected between two eighboring segments, the contactor node is placed into contact ith the shared target edge. In many cases, this procedure allows

the iterations to converge, if in fact the contactor node �should� ave been in contact with the shared target edge.

e contact between the contactor the current equilibrium iteration.

So to the �wrong� target

verged solution.

t

contactor node to move in such a way as to be alternately in cwith two neighboring segments. This is especially true if tstwo neighboring segments, the solution cannot converge unless oscillation checking is turned on. When oscillation checking is turned onnw

h Oscillation checking only forces thnode and shared target edge for For the successive equilibrium iterations, the contactor node is

segment, edge or node.always in contact with the nearest target rce contactoscillation checking cannot fo

egment, edge or node in a cons

Contact normal force: The normal force corresponding to contacis computed as n nF k g= − where the normal force acts in the direction opposite to the target normal direction (Fig. 4.8-7). is

nkthe contact normal stiffness, entered as a parameter (see Section 4.8.3 for hints about choosing nk ). nk can be considered to be a penalty parameter.

Fn

g-DEPTH

Tensile contact curve

Slope -kn

Figure 4.8-7 Normal contact force vs. gap

Tensile contact: During equilibrium iterations in Solution 601, a node can tempo

rarily be in �tensile contact�. The basic ideas for


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nsile contact are illustrated in Figures 4.8-8 to 4.8-10.

t sed. For iteration ite-2, the contactor node and target segment

on f the contactor node and target segment, the contactor node and

eration ite, there is a large overlap because the contactor spring ng,

es large contact forces, which can cause

tro

te Fig. 4.8-8 shows the iteration history when tensile contact is nouoverlap. Hence contact is assumed between the contactor node and target segment. For iteration ite-1, because of the relative motiotarget segment do not overlap. For this iteration, no contact is assumed between the contactor node and target segment. For itunloads, since there are no forces acting on the contactor spriand the target does not provide any stiffness to the contactor node.This large overlap caus

uble in convergence in the successive iterations.

Contactor node

Spring is compressed

Spring is uncompressed

b) Iteration ite-1,no contact force

c) Iteration ite,contact force Fis large and compressive

ite

a) Iteration ite-2,contact force Fis compressive

ite-2

Target segment

Figure 4.8-8 Iterations when tensile contact is not used Fig. 4.8-9 shows the iteration histor

the verl p

y when tensile contact is used. Now, in iteration ite-1, tensile contact is assumed between the contactor node and target segment. In tensile contact; the target surface still provides stiffness to the contactor node. Henceo a in iteration ite is small.



Contactor node

b) Iteration ite-1,contact force Fis tensile

ite-1

c) Iteration ite,contact force Fis compressive

ite


ite-2

Target segment

Figure 4.8-9 Iterations when tensile contact is used Fig.4.8-10 shows the iteration history when tensile contact is used, and the gap is large. In iteration ite-1, tensile contact is assumed, and in iteration ite, no contact is assumed.

Contactor node

b) Iteration ite-1,contact force Fis tensile

ite-1

c) Iteration ite,no contact force


ite-2

Target segment

Figure 4.8-10 Iterations when tensile contact is used, converged

solution not in contact

t is seen th hen the onverged solution is in contact, and slows down the convergence

4.8.2.4

tive sliding velocity

at tensile contact speeds up the convergence wI

cwhen the converged solution is not in contact.

It is not permitted for a solution in which tensile contact is present to converge, unless the tensile forces are all less than the value of a user-input parameter (see Section 4.8.3). Hence the tensile contactfeature does not affect the converged solution.

Frictional contact

The friction force is calculated using the rela


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between the target and contactor. The relative sliding velocity fu" is calculated from the velocities of the contactor and target using

( ) ( )( )f c t c t= − − − ⋅u u u u u n n" " " " "

where cu" is the velocity of the contactor node, tu" is the velocity of the target and n is the target normal.

In static analysis, the contactor and target velocities are calculated using the nodal incremental displacements divided by the time steIn dynamic analysis, the contactor and target velocitie

p. s are taken

from the nodal velocities.

heT friction force magnitude is computed using

minmin

min, f fF uµ= ≥u" "

where nF is the normal contact force and

,n f ff

F uµ <u" "

ffF

u=

u""

µ is the Coulomb friction constant (Fig. 4.8-11). minfu" is the minimum sliding ve ocity, entered as a parameter (see Section 4.8.3 for hints about choosing minfu" ). The direction of the friction force is always

l

opposite to f

u" .



Sticking

Sliding

Sliding

Ff

�Fn

��Fn

u.u

.f�u

.f

Figure 4.8-11 Friction force vs. velocity

When minf fu<u" " , the friction is sticking, otherwise the friction is

sliding.

Oscillation checking with friction: During equilibrium iterations in Solution 601, it is possible for the contactor node to undergo �sliding reversals�. Namely, the contactor node slides in one direction for an equilibrium iteration, then reverses sliding direction for the next equilibrium iteration. When sliding reversals occur, the solution cannot converge unless oscillation checking is turned on. When oscillation checking is turned on, then, when sliding reversals are detected, the contactor node is placed into sticking contact, even if the sticking force is larger than the sliding force, and convergence is prevented for the current equilibrium iteration.

forces sticking friction for the current

sliding and st tion checking cannot converge to a solution in which the frictional state is wrong.

on

Oscillation checking onlyequilibrium iteration. For successive equilibrium iterations, the frictional state (sliding or sticking) is determined as usual from the

icking forces. So oscilla

4.8.3 Modeling considerations

Selection of rigid target contact: For Solution 601, set TYPE=2 the BCTPARA card. For Solution 701, set XTYPE=3 on the BCTPARA card.


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o r urface must be connected to a

master node�, either using constraint equations or using rigid r

Algorithm used: The current rigid target contact algorithm isselected by default. To select the NX4 rigid target contact algorithm, set RTALG=1 on the NXSTRAT card. M deling of target surfaces: If the target surface translates orotates, all of the nodes on the target s�links. For example, in Fig. 4.8-12, all of the nodes on the lowetarget surface are connected to a master node using rigid links.

Moving target surface

Rigid links

Master node, independent degrees of freedom

Slave nodes

Fixed nodes

Fixed target surface

Figure 4.8-12 Modeling of fixed and moving target surfaces

It is not allowed for the nodes on a target surface to have

the independent degrees of freedom. All degrees of freedom fornodes on a target surface must be fixed or constrained.



FDET. If OFFTYPE=0, there is no offset. If FFTYPE=1, a constant offset of value OFFSET is used. If

etermines the description of the offset. If OFFDET=0, then Advanced Nonlinear Solution determines the offset description (either spheres or normals, see Section 4.8.2.1). The criterion used by Advanced Nonlinear Solution is that an offset description of spheres is used for each target surface that is convex or flat, and an offset description of normals is used for each target surface that is concave. If OFFDET=1, then the offset description is spheres, and if OFFDET=2, then the offset description is normals. When normals are used for the offset description, small steps should be used in Solution 601. This is because the offset vectors are assumed to remain constant during the equilibrium iterations. In particular, at convergence, the offset vectors corresponding to the previous converged solution are used. Determination of contact, modeling issues: Contact is affected by variables GAPBIAS and DEPTH, as described in Section 4.8.2.3. GAPBIAS is set using BCTPARA parameter GAPBIAS (default =0) and DEPTH is set using BCTPARA parameter PDEPTH (default=0). It is possible for the closest target segment, edge or node to not be

e expected one. An example is shown in Fig. 4.8-13. In this

ecause the distance between a contactor node and a target egment is measured in the direction of the target segment normal, c t

Modeling of contactor surfaces: The amount and description of offset is determined by the BCTPARA parameters OFFTYPE, OFFSET and OFOOFFTYPE=2, an offset equal to half of the current shell element thickness is used. When there is an offset, then BCTPARA parameter OFFDET d

thexample, the rim of the wheel is modeled with target segments. Bsa on actor node interacts with the lower target surface, and the contact algorithm detects a large overlap between this contactor node and the lower target surface.


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a) Orthographic view of wheel b) Side view of wheel

c) DEPTH feature used

DEPTH

d) Wheel modeled with additional target surfaces

Contactor node is closestto upper target

Contactor node is closestto lower target. Contactornode is not closest to uppertarget, since there is noupper target segment withnormal that points in thedirection of the contactornode.

Contactor nodecannot be in contactwith lower target Contactor node is closest

to target edge

Upper target

Upper targetUpper target

Lower target

Lower target Lower target

Figure 4.8-13 Modeling of a wheel

An

closest target segment. This g

ution 601, the equilibrium iterations would most likely not n d

other example is shown in Fig. 4.8-14. In Fig. 4.8-14(a), there is a gap between the contactor node and the closest target segment,as expected. In Fig. 4.8-14(b), the punch has moved upward relative to the contactor node. Now there is a large overlap between the contactor node and these ment is the only segment with a normal that points in the direction of the contactor node.

In both Fig. 4.8-13 and Fig. 4.8-14, the large overlap is unintended. In Solco verge. In Solution 701, the large forces between contactor antarget would overdistort the elements attached to the contactor node.



ode and the incorrect target segment. Another way to avoid this issue is to create additional target segments as shown. Then the contactor

of the additional target segments.

e

ence. e that: increasing causes the m ximum overlap between the contactor and the

target to becom smaller. Also, increasing caconvergence difficulties.

. There is no aximum overlap is less than the geometric error.

So, if the mesh is coarse, a large m ximum overlap can be used.

One way to avoid the large overlaps is to use the DEPTH feature so that contact is not detected between the contactor n

node is closest to one

Choice of nk : nk is set using BCTPARA parameter NCMOD. Thdefault value of the normal contact stiffness nk is 1E11. However,

nk can be chosen for optimal converg Not

nk ae nk n lead to

We recommend that you use the smallest value of nk such that themaximum overlap is still acceptably small. For example, if the target surface is curved, there will be a geometric error associated with using a coarse contractor surface (Fig. 4.8-15)advantage if the m

a


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a) Intended interaction of contactornode and target surface; contactornode is not in contact with target

b) Unintended interaction of contactornode and target surface; contactor isin contact with target with large overlap

d) Additional target segments used, contactoris not in contact with target

c) DEPTH feature used, contactor nodeis not in contact with target

DEPTH

Contactornode

Contactornode

Closest pointon target surfacethat interactswith contactor node



Target surface

Figure 4.8-14 Modeling of a punch

Another consideration for the choice of is the following. Because the rigid target algorithm is node-based, and because the

s e

olving should be greater

than

nk

contact stiffness is the same for each node in contact, the stressecomputed in higher-order elements on the contactor surface will binaccurate, if nk is too small. For example, in a problem invpressing an element onto a contact surface, kn

100 EAnL

where is the Young�s

i is

E modulus, A is the contact

area, L s the element thickness (in the contact direction) andthe number of nodes on the contact area. The basic concept is illustrated in Fig. 4.8-16.

n



Target surface, fine mesh used

Contactor surface, coarse mesh used

Geometric error

Figure 4.8-15 Modeling of a curved target surface

a) Soft target surface, k small,element stresses are inaccurate

n

b) Hard target surface, k large,element stresses are accurate

n

Contact forces acting on contactor nodesare all nearly equal

Contact forces acting on contactor nodesare nearly equal to consistent forcescorresponding to pressure load

Uniform pressure load

Contact area A

Young’smodulus E

L

Uniform pressure load

id target contact

t

ccuracy in the element stresses.

Figure 4.8-16 Higher-order elements and rig

This issue also arises when lower-order elements are used, but when lower-order elements are used, the variation in the consistennodal point forces is much less, so nk can be smaller for the same a


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701: For Solution 701, the time step should Time step for Solution be smaller than

2n

mtk

∆ =

This formula is derived from the following considerations.

ocity normal to the target. If this ode just touches the target at time

Consider a single contactor node with mass m and no additional stiffness or damping, with a vel

t t− ∆ , and penetrates the target nat time t , the node should remain in contact at time t t+ ∆ . The choice of t∆ in the above equation satisfies this condition. Clearly, decreasing nk will increase the time step t∆ . A node that is out of contact at time t t− ∆ , in contact at time t

t t+ ∆and out of contact at time is said to have had a contact ig. 4.8-17).

reversal (F

t t�� u.

t t�� u.

Contactor node at t t��

Contactor node at t t��

Contactor node at t

Figure 4.8-17 Contact reversal due to too large time step in

frictional contact: The time step size will locities and hence the results. This is

because, in static analysis, the nodal velocities used in the friction

Solution 701

Time step selection inaffect the frictional ve

calculations are calculated as the incremental displacements divided by the time step.



those steps of the analysis in which friction is not important, a .

can e blank holder force, and during

pringback calculations.

hoice of for frictional contact: is set using BCTPARA parame

inimum sliding velocity is 1E-10. However can be

ntal

701, e

In those parts of the analysis in which friction is important, a �realistic� time step should be used. Inlarge time step can be used, which causes the velocities to be smallFor example, in metal forming analysis, a large time step size be used when establishing ths

minfu" minfu"ter SLIDVEL. The default value of the

C

minfu" minfu"m

chosen for optimal convergence. Decreasing minfu" can lead to convergence difficulties. We recommend that you choose minfu" either from experime

data, or use the largest value of minfu" acceptable in your application. Time step for Solution 701 for frictional contact: For Solution th time step should be smaller than

min2 f

n

mut

Fµ∆ =

"

to prevent reverse sliding. This formula is derived from the finitdifference equation corresponding to explicit time

e integration,

hw en applied to a single contactor node with mass m and no additional stiffness or damping, sticking to the target, but with a nonzero sticking velocity. If t∆ is larger than the value in the above equation, the velocity will increase, and eventually the node will slide. The sliding will then tend to �reverse�, that is, for a given time step, the sliding direction will be opposite to the sliding direction in the previous step (Fig. 4.8-18).


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tfu

. t tf

�� u.Contactor node at t t��Contactor node at t

Figure 4.8-18 Reverse sliding due to too large time step in

Solution 701

ote that when the time step is greater than min2 fmut∆ =

"N

nFµution is still stable.

, the

sol Au autreturned from is

tomatic time step selection in Solution 701: When using theomatic time step selection options in Solution 701, the time step

the rigid target contact algorithm

min imt∆ = i

nk

here the

on time step discussed above.

eath: Rigid target contact surfaces eath time, similar to other contact surfaces. The birth and death

ther user-input parameters:

h lowed (default value 0.001). All nodes in tensile

ontact must have a tensile force less than this value for the n to converge. Tensile contact is not

w minimum is taken over all contactor nodes. Notice that friction is not considered in the automatic time step selection; this is because the model remains stable even if the time step is larger

an the frictith Birth/d can have a birth and dtimes are set by BCTPARA parameters TBIRTH and TDEATH (default =0.0, corresponding to no birth and death). O BCTPARA TFORCE The maximum tensile force for a node in tensile contact for whicconvergence is alcsolutio used in Solution 701.



CTPARA OCHECK

after equilibrium iteration ITE. The default is . Oscillation checking is not used in Solution 701.

4.8.4 Rigid

Recommend: If the maximum overlap is too large, increase ;

um tensile contact gap during iterations for nodes in contact t convergence:

Meaning: A node that is in contact at the start of the time step m hen go back into contact before convergence. This report item

Recommend: Either reduce the time step or decrease to

ntact, this is the maximum friction velocity of a node (either sticking or sliding). Recommend: If the maximum velocity is less than , and

the corresponding node should be sliding, decrease .

B If OCHECK = 0, then oscillation checking (described in Section 4.8.2.3) is turned off. If OCHECK = ITE>0, then oscillation checking is activated5

target contact reports for Solution 601

The following messages are output at the end of each convergedsolution. Maximum overlap at convergence:

Meaning: self-explanatory

n

if the maximum overlap is too small, decrease nk . Maxim

k

a

ay temporarily move out of contact during the iterations, t

reports the maximum contact gap of all such nodes. When the tensile contact gap is large, then convergence may be difficult.

n

reduce the tensile contact gap. Maximum friction velocity at convergence:

Meaning: For nodes in frictional co

k

minfu"

minfu"


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in sticking contact,

is

hange of contact status during iterations:

ber of nodes that switch contact status (not in

Recommend: Either reduce the tim step or decrease

e iterations (meaning that the nodes were almost out of contact) and in contact in the converged solution. When there

educe the time step or decrease to reduce the likelihood that nodes go into tensile contact.

Change in frictional contact status during iterations:

Recomconta

ssages are output at the end of each solution that did not converge.

Ma

Number of nodes in contact, number of nodes number of nodes in sliding contact:

Meaning: Self-explanatory. Each node is counted once for each target surface that the node is in contact with. So a node that in contact with two target surfaces simultaneously is counted twice.

C

Meaning The numcontact to in contact, or vice versa), is reported. If there are many nodes that switch contact status, this may cause convergence difficulties.

nk e In contact at convergence, in tensile contact during iterations.

Meaning: The number of nodes which were in tensile contact during th

are many such nodes then convergence may be difficult. Recommend Either r nk

Meaning: The number of nodes that change frictional contact status (from sticking to sliding or vice versa) is reported.

mend: If there are many nodes that switch frictional ct status, reduce the time step or increase minfu" .

The following me

ximum change of contact force at end of iterations:



ecommend: Examine the model near that contactor node for

Ch

the iterations.

Slidi

nodes that are undergoing sliding reversals at the end of the iterations.

Recommend: Reduce the time step or increase

rget entity at end of iterations.

Meaning: The number of nodes that are oscillating between different target entities at the end of the iterations.

4.8.5 Rigid

ich results are printed or saved:

ntact, number of nodes in sticking contact, umbe

e Ma erlap since solution start; maximum overlap since last

Meaning: The contactor node for which the contact force had the largest change is output. Rhints about why the solution did not converge.

ange of contact status at end of iterations:

Meaning: The number of nodes that are changing contact statusat the end of

Recommend: Reduce the time step or decrease nk .

ng reversal at end of iterations:

Meaning: The number of

minfu"

Change of ta

Recommend: If you do not have oscillation checking turned on,turn it on. Otherwise refine the target surfaces, or reduce the time step.

target contact report for Solution 701

The following items are output for each time step in wh

Number of nodes in con r of nodes in sliding contact:

M aning: See the corresponding message in Section 4.8.4.

ximum ov


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Meaning: See the corresponding message in Section 4.8.4.

n

aximum friction velocity since solution start, maximum friction

message in Section 4.8.4. Recommend: See the corresponding recommendations in

Meaning: This is a count of the total number of contact ber of contact reversals for the node

ith the most contact reversals is given, along with the mass of the node. Recommend: To reduce the number of contact reversals, either reduce the time step or decrease

Sliding reversals since solution start, since last report:

Meaning: This is a count of the total numberreversals. Also the number of sliding reversals for the node with the most sliding reversals is given, alothe node.

her

4.8.6 Mod

,

report:

Recommend: See the corresponding recommendations iSection 4.8.4.

Mvelocity since last report:

Meaning: See the corresponding

Section 4.8.4.

Contact reversals since solution start, since last report:

reversals. Also the numw

nk

of sliding

ng with the mass of

Recommend: To reduce the number of sliding reversals, eit

ce the time step or increase u" . redu minf

eling hints and recommendations • For a time step in which contact is established over a large areamany equilibrium iterations may be required in Solution 601. This is because the solution cannot converge until the nodes in and out



o determine which nodes are in and out of contact. An xample is shown in Fig. 4.8-19. The ATS cutback method will

n

of contact are determined, and it may take many equilibrium iterations tenot be effective for this time step. Rather, you should set the maximum number of iterations very large, so that the program cafind the converged solution.

a) Solution step before contact with target surface 2

b) Solution step after contact with target surface 2

Contactor surfaces

Possible contact in this area, manyequilibrium iterations may be requiredto accurately determine contact here.

Prescribed displacement

Target surface 1

Target surface 1

Target surface 2

Target surface 2

Figure 4.8-19 Establishment of contact over a large area during a solution step

• When forming a part that is relatively thin, setting the

f larger time steps. • The contact search algorithm may take a relatively long time for

rch

PLASALG flag of the NXSTRAT card to 2 can allow the use o

the first iteration of the first time step. Similarly, the contact seaalgorithm may take a relatively long time for the first iteration of


UGS Corp. 201

in which a contact set is born.

keep the overlap reasonable.

s

any time step

• As the contactor surface is refined, keeping nk constant, the overlap and contact force will decrease at each contactor node. Hence nk may need to be adjusted as the mesh is refined. In general, as the mesh is refined, nk can be decreased in order to

• Convergence in Solution 601 may become difficult when contactor nodes that were not in contact with the target suddenly interact with the target. An example is illustrated in Fig. 4.8-20. Eventually the contactor nodes on the right will come into contact with the target, and convergence may be difficult. Alternate wayto model this situation are shown in Fig. 4.8-20.

a) Poor modeling

b) Good modeling

c) Good modeling

Contactor node in contact

Drawingdirection

Contactor node not in contact

Figure 4.8-20 Contactor nodes suddenly coming into contact

4.8.7 Conversion of models set up using the NXN4 rigid target alg

The following hints may be useful when running models that were

orithm

set up using the NXN4 rigid target algorithm:



ntact lgorithms will usually be quite different:

) In Solution 601, the NXN4 rigid target algorit only

ines the state of contact at iteration 0; the current rigid target

gid target algorithm only allows contact between a contactor node and one target surface; the current rigid target

t detects tensile forces, the current rigid target

lgorithm does not force ATS cutbacks.

In general, in Solution 601, the current rigid target algorithm rigid target algorithm to

thm

rithm do not exist in the current rigid target algorithm.

d

fect

ffect the critical time step.

• The results from the NXN4 and current rigid target coa

a hmdetermalgorithm determines the state of contact at every iteration. b) The NXN4 ri

algorithm allows contact between a contactor node and more than one target surface. c) In Solution 601, the NXN4 rigid target algorithm can force ATScutbacks when ia d) In Solution 701, the basic formulation used by the V83 rigid target algorithm is quite different than the basic formulation used by the current rigid target algorithm. •requires more iterations than the NXN4establish contact. This is because, in the NXN4 rigid target algorithm, the state of contact is only determined in iteration 0. • Once contact is established, the current rigid target algorican be used with much larger time steps than the NXN4 rigid target algorithm. The �excessive penetration� issues of the NXN4 rigid target algo • In the NXN4 rigid target algorithm, the ATS method is automatically turned on in Solution 601. However, in the current rigid target algorithm, the ATS method is not automatically turneon. You may want to explicitly turn on the ATS method. • In Solution 701, the NXN4 rigid target algorithm does not afthe critical time step. But the current rigid target algorithm can a • In the NXN4 rigid target algorithm, the amount of friction was based on the incremental displacements. In the current rigid target


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lgorithm, the amount of friction is based on the velocity

a(incremental displacements divided by time step). So in frictional analysis, the time step size will affect the results in the current rigidtarget algorithm.

Chapter 5: Loads, boundary conditions and constraint equations


5. Loads, bouequations

5.1 Introduc

Thavailable in Advanced Nonlinear Solution for the description of pplied loads, boundary conditions, and constraint equations.

Case

ndary conditions and constraint

tion

e objective of this chapter is to present the various options

a

• Table 5-1 lists the Case Control commands used for loading, boundary conditions, and initial conditions. Control Command Comments

DLOAD Select load set (time varying) LOAD Select load set (non-time varying)

SPC Set single-point constraint set (including enforced displacement)

MPC Select multipoint constraint set

IC Select initial conditions set (displacements and velocities)

TEM and applied thermal load sets PERATURE Select initial BOLTD Select bolt preload set

Table 5-1: Case Control commands in Advanced Nonlinear

Solution

r

• Note that the selected DLOAD set can be used for any time varying loads in both static and dynamic analysis. Similarly, the selected LOAD set can be used for defining constant loads in bothstatic and dynamic analyses.

• Table 5-2 lists the load, boundary condition, and initial ondition Bulk Data entries supported in Advanced Nonlineac

Solution.

5.1: Introduction

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Bulk Data t En ry Comments

FORCE, FO C centrated force on nodes R E1, FORCE2 Con

MOMENT OMOMENT2

, M MENT1, Concentrated moment on nodes

SPC1,2, SPC1, SPCADD Fixed or enforced degrees of freedom on nodes

SPCD Enforced displacement on nodes

PLOAD Uniform pressure on shell element or 3-D solid face

PLOAD1 Distributed load on beam element Concentrated force on beam nodes

PLOAD2 Uniform pressure on shell element

PLOAD or 4 Pressure or distributed load on shell3-D solid face

PLOADX1 Variable pressure on axisymmetric 2-D solid element

TEMP Applied temperature on nodes

TEMPD Applied default temperature

GRAV Mass proportional inertial load

RFORCE Centrifugal load

MPC, MPCADD Define multipoint constraints

TIC3 n Initial displacement and velocity onodes

BOLTF Preload force on bolts OR Table 5-2: Bulk data entries for defining loads, boundary

conditions and constraints Notes:1. SPC can also enforce displacement.

become a

3. Initial conditions are discussed in Section 8.1.

2. If enforced displacements are always 0.0 theyboundary condition.



• Table 5-3 lists the Bulk Data entries used for combining appliedloads and/or enforced displacements in Advanced Nonlinear Solution.

Bulk Data Entry Comments

LOAD Defines a linear combination of constant loads

DLOAD Defines a linear combination of time varying loads (by combining different TLOAD1 entries)

TLOAD1 Defines time varying loads and enforced motion

TABLED1, TABLED2 Defines the time functions used by the loads

Table 5-3: Bulk data entries for applying loads and enforced

displacements • The LOAD entry is used for combining loads that are constant throughout the analysis while DLOAD is used for combiningtime-varying loads. The DLOAD entry refere

nces a load defined

ro pe e table

ntry (TABLED1 or TABLED2) defining the time variation of the load. • Both LOAD and DLOAD can be used in static and danalyses in Advanced Nonlinear Solution.

me function is defined as a series of points in

which t is tim

th ugh a TLOAD1 entry. The TLOAD1 entry references the tyof load (applied load or enforced displacement), as well as the

ynamic

• A ti ( )( ), it f t

e and ( )if t is the value of time function i at that time. Between two successive times, the program uses linear interpolation nction. • Advanced Nsubcases are only used to change the applied load in a static

to determine the value of the time fu

onlinear Solution does not support subcases. If

5.1: Introduction

UGS Corp. 207

analysis, then theyNonlinear Solution as tim • A typical time-varying load such as the enforced displacement

n the y direction of

can be equivalently defined in Advanced e-varying loads in a single case.

shown below (o node 100) will be applied as follows:

8

Time functions:

Time 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

tR

0.0

40 40 40 0 -10 -20 0

Resulting load values for tR = 10f(t):

4

3

2

1

-1

-2

2 4 6 10

f(t)

t

20 30

DLOAD, 1, 1.0, 10.0, 5 TLOAD1, 5, 3,, DISP, 7

tput

Note that in Solution 701 with automatic time step selection, the bove input will not result in 8 steps. Instead, the critical time step

SPCD, 3, 100, 2, 1.0 TABLED2, 7, 0.0, ,0.0, 2.0, 2.0, 4.0, 4.0, 4.0, 5.0, 0.0, ,7.0, -2.0, 8.0, 0.0, ENDT TSTEP, 1, 8, 1.0, 4

Note that the TSTEP entry is used for both linear and nonlinear analyses. In this case, 8 steps of size 1.0 are selected with ouevery 4 steps.

a



soon as the sodetails.

or to individual loads, and similarly the DLOAD case control command can point to a DLOAD entry or directly to a TLOAD1 entry. The initial and applied temperature load sets must be selected by the TEMPERATURE case control command if needed.

ol

sential

ude all applied forces nd moments.

es ts and constraint equations.

• of applied loading available in Advanced Nonlinear Solution. • the de at which they act. • equ rom the var

are centrifugal loading

and mass proportional loading, Advanced Nonlinear Solution first oad vectors tual work is used)

d

various types of loading are described in the following ecti

for the model will be used and output of results will be done as lution time exceeds 4.0 and 8.0. See Section 8.1 for

• The LOAD case control command can point to a LOAD entry

The active initial conditions must be selected by the IC case contrcommand. • Boundary conditions can be grouped into two classes: esand natural boundary conditions (see ref. KJB, Section 3.3.2). Essential boundary conditions can be enforced displacements or otations. Natural boundary conditions inclr

a • Displacement boundary conditions include fixed nodal degreof freedom, enforced displacemen

Force and moment boundary conditions include numerous types

All displacement and force boundary conditions are referred to isplacement coordinate system at the no d

Tihe externally applied load vector used in the governinglibrium equations is established using contributions f

ious applied loads. For concentrated loads, the contributions of these nodal loads directly assembled into the externally applied load vector. For pressure loading, distributed loading,

calculates the corresponding consistent nodal l(consistent in the sense that the principle of viran then assembles these load vectors into the externally applied load vector. The evaluations of the consistent nodal load vectors or thef

s ons.

5.1: Introduction

UGS Corp. 209

• linear Solution are used in conjunction with material models which include temperature eff

• d displacement can be associated with a time function which defines the time variation of the load throughout the solution period (see example in Fig. 5.1-1).

s (such as me is

hich is used, via the associated time function of each applied load, to define the load intensity at a

tep increment directly establishes the load increments. So, in the example shown in Fig. 5.1-1, the same

me-dependent effects are els in a static analysis, time is used

a similar way to define the load intensity of an applied load at a step. However, in these cases, time is a "real" variable ecause the time step increment is employed i

otion in a dynament stresses in a creep analysis.

Temperatures in Advanced Non

ects. Each applied load or enforce

! In a static analysis in which time-dependent effectcreep or friction) are not included in the material models, tia "dummy" variable w

step. Thus, the time s

solution is obtained regardless of the size of the time step increment.

! In a dynamic analysis or if tiincluded in the material modin

b n the actual integration of the equations of m ic analysis, and in the integration of the elemHence, in these cases the choice of the time step increment is no onger arbitrary. l



tR

tR tR

� �

� �

200 200

100 100

1 2 2t t4

2 steps 2 steps

Run 1: t = 1.0� Run 2: t = 2.0�

Note: identical results are obtained in Run 1 and Run 2

for a linear static analysis.

150 150

Figure 5.1-1: Example of time varying loads • the load is dif C) method

e,

e load

onse scaled proportionally using

that

• The activation of the various applied loads can be delayed using the X1 field in the TABLED2 entry. The arrival/delay time option does not apply, however, to centrifugal and mass-proportional loading, see Section 5.4.

The specification of a nonzero arrival time corresponds to a time s to

using a timethethe

The effect of the time functions on the magnitude of ferent in the case of Load-Displacement Control (LD

of Solution 601 (arc length method, see Section 6.2.3). In this casthe applied loads are not associated with any time function and thetime variation of the loads cannot be specified by the user. Thcontributions from all the loads are assembled into a constant vector denoted as the reference load vector. During the respcalculation, this reference load vector isa load multiplier (in general different from one step to the next) is automatically computed by the program.

shifting of the associated time function forward in time. If the function is used by a force boundary condition, this correspond

function multiplier of zero for all times t smaller than arrival time; see illustrations given in Fig. 5.1-2. However, if time function corresponds to a enforced displacement/rotation

5.1: Introduction

UGS Corp. 211

ero prescribed value).

the associated degrees of freedom are assumed to be free prior to the arrival time (not having a z

Time functions:

4

3

2

1

-1

-2

-3

2 4 6 8 10Arrivaltime Load active

Solution period

Time at solutionstart=0.0

f(t)

t

Input time function Time function shiftedby arrival time

Figure 5.1-2: Example on the use of the arrival time optio

rated loads

n

5.2 Concent

specified nodes using the FORCE, FORCE1, or FORCE2 entries. oncentrated moments are also applied to specific nodes using the O

for .

t is

2 ce cannot be used in defining

mo

Concentrated loads are nodal point forces applied at the •

CM MENT, MOMENT1, or MOMENT2 entries. Concentrated

ces on beam nodes can also be applied using the PLOAD1 entry • The direction in which a concentrated load acts depends on thedisplacement coordinate system assigned to the node.

• Note that concentrated moments applied to shell nodes converthe shell nodes automatically to 6 degree of freedom nodes. Thisdone since the local V1 and V directions at shell nodes are nknown to the user, and henu

ments.

Chapter 5: Boundary conditions/applied loading/constraint equations


• h ENT1 entries are used in a large dis be follower loads, meaning that the dire e simnothe

follower load can be controlled using RBAR nts (see Section 5.7). An example is given in

W en the FORCE1 or MOMplacement analysis, they canction of the applied force or moment can be updated during thulation based on the current coordinates of the G1 and G2

des. This however is only possible if either G1 or G2 is set to be node of load application.

• The direction of aor RBE2 rigid elemeFig. 5.2-1.

b) Configuration at time t

Note: tk is the rotation at node k at time t.�

Thin cantilever

Rigid element

G1

Node of load

application

a) Configuration at time t=0

Rigid element

Follower force

at time t

tk�

G2

Follower force

at t = 0

G2

G1

Figure 5.2-1: Example of the use of a rigid element to establish the

follower load direction

5.4: Inertia loads � centrifugal and mass proportional loading

UGS Corp. 213

5.3 Pressu

PLOAD2 entry.

ymmetric 2-D solid elements using the PLOADX1 entry.

of 3-D elements EXA, CPENTA, CTETRA) using the PLOAD or PLOAD4

e/distributed load surface specified, a consistent nodal load vector is calculated to represent the pressure/distributed

re and distributed loading

• Distributed loads can be applied to beam elements using the PLOAD1 entry. This entry can also be used to apply concentrated forces on beam nodes. • Pressure loads can be applied to shell elements using the PLOAD or PLOAD4 entries, and to shell 3-node and 4-node elements only (CTRIA3 and CQUAD4) using the

• Pressure loads can be applied to axis

• Pressure loads can be applied to the faces (Hentries. • When applied through the PLOAD4 entry, the pressure can benormal to the face of the element, or along a specified direction. • For each pressur

loading.



a) Axisymmetric 2-D solid element

PLOADX1

PLOAD1

PLOAD, PLOAD4

PLOAD, PLOAD2, PLOAD4

b) 3-D solid element

�

�

�

�

�

��

c) Beam element d) Shell element

a) Figure 5.3-1: Examples of distributed and pressure loading

ent this

on the

lution 601, deformation dependent loading should only be sed in a large displacement analysis. Equilibrium iterations (see

Chapter 6) should in general be performed if deformation dependent loading is present. • The loading direction for distributed loads can be along the asic coordinate system or the element coordinate system. Loads

along element coordinate systems can be deformation dependent in large displacement analysis.

• For pressure loading on 2-D and 3-D solid elements, the

• In a large displacement analysis, the pressure/distributed loading can be specified as deformation dependent for all elemtypes via the LOADOPT parameter in the NXSTRAT entry. In case, the calculations of the consistent load vectors are based latest geometry and configuration of the loading surface. • In Sou

b


UGS Corp. 215

stent load vector consists of nodal forces acting on the translational degrees of freedom only. The calculation of this load

rs

s acting at the beam nodes as

shown in Fig. 5.3-2. The calculation of these consistent forces and moments also follows the equations in ref. KJB Section 4.2.1.

consi

vector is given in ref. KJB, Section 4.2.1. The same effect occuin shells since the nodal translations and rotations are interpolated independently. • The distributed loading on a beam element results in equivalentconcentrated forces and moment

� �

L

q1

q2

1 2

(a) Beam distributed loading

(b) Fixed-end forces/moments representation

� �

L

F =L20

(7q + 3q )2 2 1F =L20

(7q + 3q )1 1 2

M =L

60(3q + 2q )1

21 2 M =

L

60(3q + 2q )2

2

2 1

Figure 5.3-2: Representation of beam distributed loading

• Displacements and stresses in the model are calculated by representing the actual distributed loading using the consistent load vector defined above. Hence, the calculated solution corresponds only to these equivalent concentrated nodal forces and moments,



and may not correspond entirely to beam theory results taking account of the distributed loading more accurately, see Fig. 5.3-3. In order to capture the applied moment more accurately, more beam elements are required between sections A and B.

��

a) Actual beam structure bending moment diagram

b) Finite element bending moment diagram

using 3 beam elements

A

A

MA

MA

MB

MB

B

B

Bending

moment

Bending

moment

Figure 5.3-3: Beam element bending moments when subjected to distributed loading

5.4 Inertia loads ─ centrifugal and mass proportional loading

• Centri odel the effect of body forces which arise from accelerations to which the structure is subjected.

fugal and mass proportional loading can be used to m


UGS Corp. 217

• The mass matrix used in the calculation of centrifugal and mass proportional loading can be lumped or consistent depending on the ma putational eff d in the ation o ed mass matrix is, in gen less th effort f sistent ass matrix. • in ma calculation and in the dynamic response calculation are the sam . • loa(at all times) only of the contributionsalive. • rifugal and mass proportional loading cannot be applied with a delay/arrival time. The time function has to be shifted ma reate ct. • ces at fixed nodes are into account in the calculation of reaction forces.

Ce

• usi al dis e axis of rotation, and the specified angular velocity

• Centrifugal loading is generated using the RFORCE entry, and mass proportional loading is generated using the GRAV entry.

ss setting for the whole model. Note that the comort involve evalu f a lumperal, much an the or a con m

Centrifugal and mass proportional loading can both be present a static or dynamic analysis. In a dynamic analysis, the type ofss matrix employed in the load

e

When elements die (due to rupture), their contribution to the d vector is removed. Hence, the consistent load vector consists

from the elements currently

Cent

nually to c this effe

Centrifugal or mass proportional for taken

ntrifugal loading

The consistent load vector for centrifugal loading is computed ng the mass matrix of the entire finite element system, the raditance from th

(see Fig. 5.4-1), as follows:

( )( ) ( )t t t t tCF f t= × × ⋅ ⋅R M ω ω r (5.4-1)

where ( )OMEGAt =ω n is the angular velocity vector, is the radial vector from the axis of rotation to the node, OMEGA is the applied angular velocity in radians/second, FC is a scaling factor

tr



fro ction and n is the unit vector parallel to the axis of rotation.

m the load application commands, f(t) is the time fun

x

y

z

= angular velocity vector

G

Structure

Centrifugal loads

�t��

�t��

the rotation factor in the RFORCE entry. Note that the time function associated with the centrifugal

loa l forces and not the angular velocity. The magnitude of the effective angular velocity at time t is

Figure 5.4-1: Convention used for centrifugal loading

Note that the units of the OMEGA used here are different from

ding directly determines the time variation of the centrifuga

( )OMEGA CF f t⋅ .

• When centrifugal loading is used in nonlinear analysis in So additio linear t ma

ncludin ll nonlinear effects is (dropping the scaling factors)

lution 601, nal non erms are added to the stiffnesstrix and the load vector. The complete expression of the load vector R, i g a

( )( )( )( ) 0 ( 1) ( )t t i t t t t i i+∆ +∆ +∆ −= × × + + ∆R M ω ω r U U

ere 0r is the vector of initial nodal distances to the rotation axis ω is the rotation vector. From the expression (5.4-2), it can

see ontribution 1NLK to the

fness matrix is present, which is given by

(5.4-2)

whandbe t an additional nonlinear cstif

n tha


UGS Corp. 219

( ) ( )( )1

( )t t iNL

+∆ ∆ =

that a deformation-dependent load vector is present, given by

( )

1 ( )i t t i− +∆ × × ∆K U M ω ω U

and

( )( )( 1)it t t t t t i

NL+∆ +∆ +∆ −= × ×R M ω ω U

Nonlinear centrifugal loading can be used in static analysis andnamic analysis.

• dy • The correction to the stiffness matrix and the correction to the loa endent loading is requested (LOADOPT param ).

Mass-proportional loading

• The consistent load vector for mass proportional loading in direction i is computed using the mass m ix of the entire finite element system and the specified accelerations (only in the translational degrees of freedom), as follows:

i (5.4-3)

where is a direction vector with "1" in the portions of the translational degrees of freedom acting into the direction i and "0" in the other portions, and ai is the acceleration magnitude in the direction i.

oint forces

loading,

5.5 Enfor

ding are made when a deformation depeter in the NXSTRAT entry

atr

t t ta=R M d i i

id

Note that the consistent load vector includes nodal papplied to all nodes of the finite element model.

• Mass proportional loading is commonly used to model gravity ading and uniform ground acceleration. For gravity t

ia lois the acceleration vector due to gravity.

ced motion

• Enforced displacements at specified degrees of freedom can be applied in Advanced Nonlinear Solution using the single point



con ries (SPC, SPC1, or SPCA D) or tmo . The applied displacement can be constant or des me function. •

• degree of freedom is isys • A nodal point can be "fixed" by prescribing a zero displacement component for all degrees of freedom at this node. This is, however, different from imposing a permaconstraint on the GRID entry because the enforced degrees of fre retained in the system matrices (i.e., equation numbers are assigned) whereas the degrees of freedom at which permanent

eted from the system

Delay or arrival times can be used for applied displacements. In this case, the displacements are free before the arrival timthe arrival time is reached, the displacements arenforced values. However, the enforced value can be interpreted as n absolute or total displacement or as a relative displacement

rrival time. This is controlled by e DISPOPT flag in the NXSTRAT entry.

• A new enforced displacement can also be applied in a restart run. In this case as well, the displacem

lative to the configuration at the start time of the restart analysis.

5.6 Applied temperatures

the TEMP entry or to the whole odel using the TEMPD entry. Direct nodal values applied with EMP override the default TEMPD value.

straint ent D he enforced tion (SPCD) entrycribed by a tiEnforced velocities and accelerations are not supported.

Nodal point translations and rotations can be enforced. The n the direction of the displacement coordinate

tem assigned to the node.

nent single-point

edom are

GRID constraints are imposed are delmatrices.

• Note that enforced displacements are not recommended on contactor surfaces (see Chapter 4). •

e. Once e set to their

abased on the configuration at the ath

ent can be a total value or re

• An applied temperature in Advanced Nonlinear Solution can be applied directly to a node usingmT

5.6: Prescribed temperatures

UGS Corp. 221

case control command selects the initial eld.

by

OAD1 in the DLOAD command as explained in the beginning of this chapter.

he nodal temperature definition, then the time function value will be zero if the time is less th a

5.7 Bolt preloa • Ad elements. The bolt preloads or forces are applied at an extra solution s p to the res th • The b r h

• Bolt preloa le in Solution 601.

See Section 8.6 for more details on the bolt feature.

5.8 Constraint equations

d Nonlinear Solution supports single-point and multipoint constraints. The single-point constraints are defined

t

• d MPCADD pe elements (

• The TEMPERATURE or reference temperature fi

• The applied temperature can also be made time dependentusing the TLOAD1 entry, and referencing the TL

• If a non-zero arrival time option is used in t

an the rrival time.

d

vanced Nonlinear Solution supports the preloading of bolt

tep erformed at the very beginning of the analysis priort of e step-by-step solution.

olt p eloads are applied via the BOLTFOR entry whicshould be used together with a BOLTLD case control command.

d (and bolts in general) are only availab

•

• Advance

using the SPC, SPC1, SPCADD entries. Permanent single-poinconstraints can also be defined using the GRID entry. This case, however, totally removes the associated degree of freedom from the solution.

Multipoint constraints are defined directly using the MPC an entries. They can also be defined through R-tysee Section 2.7).



The following relationship holds for multipoint constraints:

0j jj

R u =∑

• Multipoint constraints are only approximately satisfied in an

For R-type elements in large deformation the multipoint

e deformation of the structure.

Constraints can be applied to static and dynamic

• It is important to note that adding constraint equations to the finite element model results in adding external forces (amoments) at the degrees of freedom specified by the constraint

s.

ts detailed in Section 5.5 are ternally enforced using single-point constraints.

For an R-type element to produce multipoint constraints with g coefficients that capture large deformations, the

ints must be between only 2 nodes. In addition, one of the ndent degrees of freedom

e other node should only possess depenhese large displacement multipoint constraints are internally

called rigid links.

5.9 Mesh

explicit analysis (Solution 701), since imposing the constraint exactly requires a non-diagonal mass matrix. •constraint can have variable coefficients that are updated based onth

• analyses.

nd possibly

equations. These forces are included in the reaction calculation • Note that enforced displacemenin •changinonstrac

nodes should possess all the indepe and th dent degrees of freedom. T

• Mesh glueing internally creates constraint equations that are treated using Lagrange Multipliers. All independent degrees of freedom associated with the glued mesh remain independent.

glueing

! The glueing feature is used to attach two surfaces together. These two surfaces usually involve different finite element meshes (see Fig. 5.9-1). The glueing procedure should lead to smooth

5.9: Mesh glueing

UGS Corp. 223

nd tractions between the glued urfaces. This feature is useful for several applications:

! When a fine mesh is desired in a certain region and coarser

fferent regions are meshed independently with nstructured free meshes.

tetrahedral mesh attached to a brick mesh).

transition of displacements as

meshes are desired in other regions. ! When diu ! When different regions are meshed with different element types (such as a

Mesh glueing along dashed lines

Figure 5.9-1. Examples requiring mesh glueing

! The proper glueing constraint between the two surfaces can be xpressed as e

1 2( ) 0u u dλ

Γ⋅ − Γ =∫ (5.9-1

)

ent of the first glued surface, u2 is the ent of the second surface and λ is the Lagrange Multiplier

aster and the ther as the slave. The Lagrange Multiplier field involves nodal

where u1 is the displacemdisplacemfield imposing the constraint.

One of the glued surfaces is designated as the m o



tion is also performed over the slave surface. Hence Eq. .9-1) becomes

0

degrees of freedom at the nodes of the slave surface, and the integra(5

( )S

S M S Su u dλΓ

⋅ − Γ =∫ (5.9-2)

ivial since the

ed

! stem.

! ontact and should be used in its

e glued lement faces can be triangles or quads, and they can have linear or

ust locate a target segment on the master surface. Therefore, if one surface is smaller than the other, as shown in Fig. 5.9-2, the smaller surface should be the slave.

The accurate integration of Eq. (5.9-2) is not trisplacements uM and uS are generally interpolated over different d

domains. This integration is automatically performed by AdvancNonlinear Solution. ! Mesh glueing is not available in explicit analysis.

Glued nodes cannot have a skew coordinate sy

Glueing is superior to tied cplace whenever applicable. ! Only 3-D solid elements can be used glueing. Thequadratic displacement interpolation. Shell elements are not supported. ! Each node on the slave surface m

Master surface

Master surface

Slave surface

Slave surface

Figure 5.9-2. Selection of master and slave glue surfaces

5.9: Mesh glueing

UGS Corp. 225

! The two glued surfaces should ideally be smooth surfaces (no sha better to ed

! If the two surfaces have different mesh densities, either one can

freedom

rp corners). If corners exist it is create multiple glumeshes.

be used as slave (unlike contact where the finer one should be the contactor). Using the finer meshed surface as a slave will produce more equations, since the Lagrange Multipliers degrees ofare on the nodes of the slave surface.

Chapter 6: Static and dynamic analysis

226

. Static and

ve

he available solvers. Most flags or n

6.1 Linea

em d are

• ulti-grid solver can be used to solve this system of equations. • The equation solvers assume that the system stiffness matrix is sym etric. • m stiffness matrix is po mmarized as follows: The Rayleigh quotient

6 implicit dynamic analysis This chapter presents the formulations and algorithms used to solstatic and dynamic problems using Solution 601. This includes onvergence checking and tc

co stants that need to be input in this chapter are in the NXSTRATbulk data entry.

r static analysis

• In linear analysis using Solution 601, the finite element systequilibrium equations to be solve

KU = R

A direct sparse solver or an iterative m

m

The equation solvers assume that the systesitive definite. This requirement can be su

ref. KJBections 8.2.1,.2.2 and 8.2.3

S8


( )T

T

Kρ =φ φ

φφ φ

must be greater than zero for any displacement vector . Since φ

( )ρ φ is equal to twice the strain energy stored in the system (for T =1φ φ

energy), this is equivalent to the requirement that the strain

dis

stored in the finite element system when subjected to anyplacement vector φ must be greater than zero.

Hence, the finite element system must be properly supported, so that the system cannot undergo any rigid body displacements or rotations.

!

6.1: Linear static analysis

UGS Corp. 227

It also follows that no part of the total finite element system ust represent a mechanism, see ref. KJB, Fig. 8.7, p. 704, for

! Nodal point degrees of freedom for which there is no stiffness must be restrained. A degree of freedom does not carry any stiffness if all of the elements connected to the nodal

oint do not carry stiffness into that degree of freedom. In this ry

which are not connected

tomatically deleted by the program.

6.2 Nonlin a

be olved are:

!

msuch a case.

pcase the degree of freedom must be restrained using boundaconditions.

Note that nodal degrees of freedomto any elements and are not used as dependent nodes in constraint equations are au

! More details on the solvers available in Solution 601 are provided in Section 6.5.

e r static analysis

• In nonlinear static analysis the equilibrium equations tos

t t t t+∆ +∆− =R F 0

e t+∆t.

ity may come from the material properties, the

ref. KJB

Section 8.4

where t+∆tR is the vector of externally applied nodal loads at time (load) step t+∆t, and t+∆tF is the force vector equivalent (in the virtual work sense) to the element stresses at tim • The nonlinearkinematic assumptions, deformation dependent loading, the presence of contact, or the element birth/death feature.

• The solution to the static equilibrium equations can be obtained in Solution 601 using ! Full Newton iterations, with or without line searches ! Load-displacement-control (LDC) method (arc length

ethod) m



ns and

tomatic time stepping algorithms do not require the stiffness matrix to be positive

ess

6.2.1 Solution of incremental nonlinear static equations

These methods are described in detail in the following sectioalso in Sections 6.1 and 8.4 of ref. KJB. • The same equation solvers are used for both linear and nonlinear analysis. However, the au

definite, thus allowing for the solution of bifurcation problems. • The stiffness stabilization feature can be used to treat some nonlinear static problems involving a non-positive definite stiffnmatrix. See Section 8.5 for details.

Full Newton iterations: In the full Newton iteration method, the following algorithms are employed:

! Without line search

( 1) ( ) ( 1)t t i i t t t t i+∆ − +∆ +∆ −∆ = −K U R F (6.2-1a)

( ) ( 1) ( )t t i t t i i+∆ +∆ −= + ∆U U U (6.2-1b)

!

With line search

( 1) ( ) ( 1)t t i i t t t t i+∆ − +∆ +∆ −∆ = −K U R F (6.2-1c)

where

( ) ( 1) ( ) ( )t t i t t i i iβ+∆ +∆ −= + ∆U U U (6.2-1d)

( 1)t t i+∆ −K is the tangent stiffness matrix based on the solution calculated at the end of iteration (i - 1) at time t+∆t.

t+∆ tor at time t + ∆t; t+∆tF(i-

1) is the consistent nodal force vector corresponding to the element stresses due to the displacement vector t+∆tU(i-1); ∆U(i) is the increm i

accelefull Newton iteration m atrix is

tR is the externally applied load vec

ental displacement vector in iteration (i) and β( ) is an ration factor obtained from line search. Note that, since the

ethod is employed, a new stiffness m

6.2: Nonlinear static analysis

UGS Corp. 229

an be

e whole incremental displacement vector is caled down to satisfy the upper bound.

his

t is properly established, or in the first unloading steps after a material has undergone plastic deformation.

6.2.2 Line Search

n

always formed at the beginning of each new load step and in each iteration. • An upper bound for the incremental displacements in ∆U cset by the user (via the MAXDISP parameter in the NXSTRAT entry). If the largest increment displacement component exceeds the limiting value, ths This feature is useful for problems where one or more iterations can produce unrealistically large incremental displacements. Tmay happen, for example, if a load is applied to contacting bodies before contac

The line search feature is activated by setting LSEARCH=1 iNXSTRAT. In this case, the incremental displacements obtainedfrom the solver are modified as follows

( ) ( 1) ( ) ( )t t i t t i i iβ+∆ +∆ −= + ∆U U U

where β is a scaling factor obtained from a line search in the direction ∆U(i) in order to reduce out-of-balance residuals, according to the following criterion

( ) ( )Ti t t t t i+∆ +∆∆U R( ) ( 1)i t t t t i+∆ +∆ −⎡ ⎤∆ −⎣ ⎦U R F

where STOL is a user-input line search convergence tolerance NXSTRAT), and

STOLT

⎡ ⎤−⎣ ⎦ ≤F

(6.2-2)

(in ent

e

t+∆tF(i) is calculated using the total displacemvector t+∆tU(i).

Th magnitude of β is also governed by the following bounds

LSLOWER LSUPPERβ≤ ≤ (6.2-3)

in where LSUPPER and LSLOWER are user-input parameters



The in suitable ter satisfying Equations (6.2-2) and (6.2-3) is found within a reasonable number of line search

pecified energy threshold ENLSTH.

Line search is off by default. It is useful for problems involving ement problems involving beams

nd shells. It is also helpful in many contact problems. In the case

the line search only scales down displacements.

n. This usually happens in the first few iterations of a time step, or when a major chaini

• ote that line search increases the computational time for each itet+∆t

me abisol

6.2.3 Low speed dynamics feature

feature eed

contact problemIn essence, this feature includes dy ics effects in an

otherwise static pro olves

NXSTRAT.

cremental displacements are not modified (i.e., β = 1) if noline search parame

iterations, or if the unbalanced energy falls below a certain user-s

•plasticity, as well as large displacaof contact problems, it is sometimes better to set LSUPPER to 1.0 so that

• The effect of line search is more prominent when the current displacements are far from the converged solutio

nge occurs in the model, due for example, to contact tiation/separation, or the onset of plasticity.

Nration. Most of the extra time goes towards the evaluation of F(i) in Equation (6.2-2). However, for the types of problems ntioned above the reduction in the number of iterations and thelity to use bigger time steps leads to an overall reduction in ution time.

• A low speed dynamics option is available for static analysis (can only be used with ATS or TLA/TLA-S s). Low spdynamics is a special technique developed to overcome convergence difficulties in collapse, post-collapse and certain

s. nam

blem. Solution 601 s

( ) ( ) ( 1) ( ) ( 1)t t i t t i t t i i t t t t iα +∆ +∆ +∆ − +∆ +∆ −+ + ∆ = −M U C U K U R F"" " (6.2.4)

where M is the mass matrix and α is a mass scaling factor that can vary from 0.0 to 1.0 (default 1.0), to partially account for the


UGS Corp. 231

dynamic inertia effect . The C matrix is evaluated using

β=C K where β is a user-specified parameter (default 10-4), and is the (initial) total stiffness matrix (corresponding to zero initial displacements). For more details on this dynamics equation refer to Section 6.4.

• When low speed th ATS, the time step size

p size r, it is recommended that the loads be held

onstant for a period of time of at least 104β so that the dynamic effects die out. • Note that mass effects may not be needed in the low speed dynamic analysis. In this case, set the α parameter in Eq. (6.2-4) to zero, or, alternatively, set the material densities to zero. That way, only structural damping effects will be present in the otherwise tatic analysis.

6.2.4 Auto

ls the time tep size in order to obtain a converged solution. If there in no onvergence with the user-specified time step, the program

bdivides the time step until it reaches convergence. some cases, the time step size may be increased to accelerate the

ctor that Solution 601 uses to subdivide the time step when there d, if

that the time step size is not smaller than a inimum value. This minimum value is set as the original step size

Note that the loads at any intermediate time instant created by

K

dynamics is used wiwill influence the results. It is recommended that the time stebe at least 105β . Oc

s matic-Time-Stepping (ATS) method

• The automatic-time-stepping (ATS) method controscautomatically suInsolution. • Parameter ATSDFAC in the NXSTRAT entry sets the division fais no convergence. Successive subdivisions can be performenecessary, providedmdivided by a scaling factor provided by the user (ATSSUBD in NXSTRAT). •the ATS method are recalculated based on the current value of the time functions.



es, any time step

ubdivisions without convergence. In this case, the solution output

once y

options, or request that Solution 601 does the selection TSNEXT in NXSTRAT).

1. Use the time increment that gave convergence

he time step ba

2. Return to original time step size In this case, the program continues the analysis using the original user-provided time increment. At the next time step, the

shows that in useful. The increased time step size cannot

be larger than a user-specified value (determined from ATSMXDT in NXSTRAT). Due to this increase the analysis may be completed in fewer time steps than requested. This

faul when contact is present.

3. Proceed through user-defined time points e program sets the tim

the user. Hence, the analysis always proceeds through all user-specified time steps.

an example to illustrate the basic options of ATS e we are in load step 15 of a particular problem

with initial time t = 15.0 and a time step of 1.0. The solution does not converge and the time step is set to 0.5 (assuming a time step division factor of 2.0). If that too does not converge, the time step is set to 0.25. If that converges, the results are not yet saved. Another sub-step is performed for load step 15. The size of this step depends on which of the three options above is selected:

• The solution output is only furnished at the user-specified timexcept when the solution is abandoned due to too msis also given for the last converged time instant. • There are three options for controlling the time step sizeconvergence is reached after ATS subdivisions. You can select anof these(A

In this case, the program continues the analysis using the last converged time step. Once the end of the user-specified time step is reached, the program may increase t sed on the iteration history. This option is the default in analyses without contact.

step size may be increased if the iteration historysuch an increase

option is the de t

In this case, th e step size such that the final time is that initially provided by

• Following issubdivisions. Assum


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will be performed within load step 15 they all converge).

! In option 2, the next sub-step will use a time step size of 1.0. If this converges load step 16 starts with t = 16.25. ! In option 3, the sub-step will use a time step size of 0.75. If this converges load step 16 starts with t = 16.0.

• Note that while options 1 and 2 may result in outputted solution times that are different from those initially specified by the user, there are certain time values that are not skipped. These are the time values at the end of �time step blocks�. In this case, the time step size is reduced such that the solution time at the end of the block is satisfied. The program determines these time step blocks based on the time step pattern input by the user. The final solution time is always assumed to be an end of a time step block. • Option 2 is useful for contact because of the highly nonlinear response (sudden changes in stiffness) that occurs when a nodes comes into contact, or is released from contact, or even moves from one contact segment to another. A small step size may sometimes be needed only in the vicinity of this contact incident. Once contact is established/released the problem is �less nonlinear� and the original time step size can be used.

6.2.5 Total Load Application (TLA) method and Stabilized TLA (TLA-S) method

• The Total Load Application method is useful for nonlinear static analysis problems where all applied loads do not require the user to explicitly specify the time step sequence. In this case, the user applies the full load value and Solution 601 automatically applies the load through a ramp time function, and increases or reduces the time step size depending on how well the solution converges. This method cannot be used in dynamic analysis. The TLA method automatically activates the following features that are suitable for this type of uniform loading static problems:

! In option 1, the next sub-step will use a time step size of 0.25. Two other sub-steps both of size 0.25 (assuming



s a size of 1/50 of the total time. equilibrium iterations is 30.

e TLA settings listed above can be modified by the user

eters in the NXSTRAT command).

of

tiffness proportional damping (see Section 6.2

dicators are provided in the output file after each converged the various stabilization effects

The TLA-S method can serve several purtabilization indicators are all within reasonable bounds, typically

S the

indicators are large, the TLA-S method proapproximate solution that can frequently be used to detect

The following features cannot be used with the TLA and TLA-S methods:

- All materials with creep effects - All materials with viscosity effects - Rigid target contact

6.2.6 Load ) me od

- The first time step ha- Maximum number of- Line search is used. - Limiting incremental displacements per iteration is set to

5% of the largest model dimension. - The maximum number of time step subdivisions is set to

64. - The time step cannot be increased more than 16 times its

initial size.

Most of th(via the TLA* param • The Stabilized TLA method (TLA-S) is identical to the regular TLA method with the addition of various stabilizing features to create a more stable system and aid convergence. The sources stabilization are low speed dynamics which adds inertia and s .3), contact damping (see Section 4.6.5), and stiffness stabilization (see Section 8.5). Insolution to show if the forces due to are excessive. • poses. If the sless than 1% of the internal force indicator, then the TLA-solution may be reasonably accurate. However, even when

vides a useful

modeling errors such as incorrect contact definition, applied load, or boundary conditions. •

-Displacement-Control (LDC th

• The load-displacement-control (LDC) method (arc length


UGS Corp. 235

mea mthe ulated. The main feature of the method is that the level of the externally applied loads is adjusted autom in me • ns used in the LDC method used in Solution 601 are described in ref. KJB Section 8.4.3 and the following reference:

J. Computers and Structures, Vol. 17, No. 5-6, pp. 871-

• he LDC method is activated via the AUTO flag in the NXSTRAT entry. An enforced displacement on a user-specified degree of freedom is used to evaluate the initial load vector, and analysis continues until a specified displacement is reached at a cer e, o at hed. The LDC meth ing a sta C method are commonly referred to as arc-length methods.

thod) can be used to solve for the nonlinear equilibrium path of odel until its collapse. If desired, the post-collapse response of model can also be calc

atically by the program. The LDC method can only be used in nonlinear static analysis

which there are no temperature or creep effects. The LDC thod can be used in contact problems.

The formulatio

ref. Bathe, K.J. and Dvorkin, E.N., "On the Automatic

Solution of Nonlinear Finite Element Equations,"

879, 1983. T

tain nod r a critical point on the equilibrium p h is reacod may also be used as a restart analysis follow

tic or dynamic analysis. Variants of the LD

• The equations employed in the equilibrium iterations are

( )

( )( ) ( 1) ( )

( ) ( ), 0

t t i t t i i

i if

λ λ

λ

( 1) ( ) ( 1) ( ) ( 1)t t i i t t i i t t i+∆ − +∆ − +∆ −

+∆ +∆ −

∆ = + ∆ −

= + ∆

∆ ∆ =

K U R F

U U U

U

(6.2-5)

where

( 1)t t i+∆ −K = tangent stiffness matrix at time t+∆t, end of iteration (i-1)

R = constant reference load vector t t ( 1)iλ+∆ − = load scaling factor (used on R) at the end of

iteration (i-1) at time t+∆t



( )iλ∆ = increment in the load scaling factor in iteration

(i)

The quantities ( 1)t t i+∆ −F and ( )i∆U are as defined for Eq. (6.2-1).Note that in Eq. (6.2-5), the equation f = 0 is used to constrain

the length of the load step. The constant sph

erical arc length constraint method is usually used and the constant increment of

ork method is used if the arc length method has difficulties to converge.

To start the LDC method, the load multiplier for the first step

a s are in the NXSTRAT

ntry. The direction of the displacement is given by its sign.

As an example, two entirely different solution paths will be ed for the same model shown in Fig. 6.2-1 if initial

displacem

ally traces the n

. tλ is analogous to

the scaling of the applied loads R using a user predefined time function when the LDC method is not used (see Chapter 5). In the case of the LDC method, the scaling function is determined

user-specified.

external w

The reference load vector R is evaluated from all the mechanical loads.

•∆tλ (used to obtain the corresponding load vector ∆tλ R) is calculated using a user-specified enforced displacement (LDCDISP) acting on a given degree of freedom (LDCDOF) onspecific node (LDCGRID). All parametere

As shown in Fig. 6.2-1, the input for the initial enforced displacement (in particular whether it is positive or negative) is critical in establishing successive equilibrium positions using the LDC method.

obtainents of different signs are enforced for the first solution

step.

• After the first step, the program automaticno linear response by scaling the external load vector R proportionally, subject to the constraint of Eq. (6.2-5), so that at any discrete time t in iteration (i), the external load vector is tλ(i)R

The scaling of the reference load vector using

internally by the program instead of being


UGS Corp. 237

b) Equilibrium paths

t�R

t�

0� specified positive

0� specified negative

0�R

a) Model considered

X

Y

Reference load = R, actual load at time t = Rt�Enforced displacement for first step = , displacement at time t =0 t� �

Figure 6.2-1: Example of the dependence of s

displacement enforced in the first step for the LDC method

ement must satisfy the following relation:

olution path on the

• The converged displac

2 2100 tα ∆≤U U

where t t t+∆= −U Uthe current step, α

U is the incremental displacement vector for is a displacement

DCIMAX parameter in NXSTRAT), and is the

convergence input tolerance t∆ U(L

displacement vector obtained in the first step. If the above inequality is not satisfied, an internal restart of the iteration for thecurrent step is performed by the program.



! A critical point on the equilibrium path has been passed and

! The maximum number of arc length subdivisions in NXSTRAT) has been attempted using different

to converge within the number of allowed iterations.

6.2.7 Convergence criteria for equilibrium iterations

tion

! energy only,

force/moment,

! force/moment only, and ! translation/rotation only.

nce criterion is always used in addition to the above criteria (see Chapter 4).

LDC method not used

ethod is not used, the convergence in equilibrium qualities are satisfied:

• The LDC solution terminates normally when any one of the following conditions is satisfied:

! The maximum specified displacement is reached (LDCMAXin NXSTRAT).

solution termination has been requested. ! The number of converged solution steps is reached.

(LDCSUBDstrategies but each time the solution has failed

• The following convergence criteria can be specified in Solu01: 6

! energy and ! energy and translation/rotation,

• If contact is defined in an analysis, the contact force converge

• If the LDC miterations is reached when the following ine


UGS Corp. 239

Energy convergence criterion For all degrees of freedom:

( ) ( 1)

(1)

ETOLT t t t+∆

⎣ ⎦ ≤⎡ ⎤∆ −⎣ ⎦U R F

(6.2-6

Ti t t t t i+∆ +∆ −⎡ ⎤∆ −U R F)

where ETOL is a user-specified energy convergence tolerance. Force and moment convergence criteria For the translational degrees of freedom:

( 1)

2 RTOLt t t t i+∆ +∆ −−

≤R F

RNORM

For the rotational degrees of freedom:

( 1)

2 RTOLRMNORM

t t t t i+∆ +∆ −−≤

R F

where RTOL, RNORM and RMNORM are user-specified tolerances. Translation/rotation convergence criteria For the translational degrees of freedom

:

( )

2 DTOLDNORM

≤

i∆U

al degrees of freedom:

For the rotation

( )i

2 DTOLDMNORM

∆≤

U

where DTOL, DNORM and DMNORM are user-specified

tolerances.



orm maximum residual value; for

example, the force criterion could be interpreted as

here RTOL × RNORM is equal to the user-specified maximum

ch ubdivision of time or load step when the ATS method of

ation is used.

If contact is present in the analysis the following additional

Note that in each of these convergence criteria the residual nis measured against a user-specified

(norm of out-of-balance loads) ≤ RTOL × RNORM

wallowed out-of-balance load.

Note also that these convergence criteria are used in easautomatic step increment

•criterion is always used in measuring convergence

( )( )

( 1) ( 2) ( ) ( 1)

2 2

( 2)

max ,RCTOL

max ,RCONSM

i i i ic c

ic

− − −

−

− −≤

R R λ λ

R

2

where ( 1)i

c−R is the contact force vector at the end of iteration

( )1i − , λ is the Lagrange multiplier vector at the end of iteratio

( )i , RCONSM is a reference contact force level used to prevepossible division by zero and RCTOL is a user-specified contact orce convergenc

( )i n

nt

e tolerance.

is

f Non-convergence: Convergence might not occur when the maximum number of iterations is reached or when the solution diverging. • If the specified convergence criteria are not satisfied within the allowed number of iterations, but the solution is not diverging, the following can be attempted: ! Check the model according to the guidelines in Section 6.2.8. ! Use a smaller time step.


UGS Corp. 241

ce tolerances. In most cases, looser tolerances help. However, in some problems, tighter tolerances

, perform better with line search.

• Divergence of solution terminates the iteration process before the maximum number of iterations is reached. It is sometimes detected when the energy convergence ratio in Eq. (6.2-6) becomes unacceptably large, or when the excessive displacements lead to distorted elements and negative Jacobians. In this case, the following can be attempted: ! Check the model according to the guidelines in Section 6.2.8.

! Use a smaller time step.

! Make sure there are sufficient constraints to remove rigid body modes from all components in the model. Presence of rigid body modes usually results in a large ratio of maximum to minimum pivot during factorization (with sparse solver).

LDC method used

• Convergence in the equilibrium iterations is reached when the following inequalities are satisfied:

Energy convergence criterion: For all degrees of freedom

! Increase the number of allowable iterations.

! Change the ATS parameters.

! Change convergen

help by not allowing approximate solutions that could potentially prevent convergence in future time steps.

! Change line search. Some problems, such as those involvingplasticity

! Change contact settings. The optimal contact settings and features depend on the model. See Chapter 4 for more details.



( ) (Ti t tλ+∆⎡∆ ⎣U 1) ( 1)

(1) (1)ETOL

i t t i

T λ

− +∆ − ⎤− ⎦ ≤⎡ ⎤∆ ∆⎣ ⎦

R F

U R

is a user-specified energy convergence tolerance.

ce criteria: For the translational degrees of freedom

where ETOL

Force and moment convergen

( 1) ( 1)

2 RTOLt t i t t iλ+∆ − +∆ −−

≤R F

RNORM

nal degrees of freedom

For the rotatio

( 1) ( 1)

2 RTOLRMNORM

t t i t t iλ+∆ − +∆ −−≤

R F

where RTOL, RNORM and RMNORM are user-specified parameters.

The translation/rotation convergence criteria, and the contact

t

ter the maximum restart ttempts, the program saves the required restart information and

run. A

he initial displacement can be enforced at a ifferent nodal point in the first step of the restart run. The

6.2.8 Selection of incremental solution method

• This section gives recommendations on which incremental

convergence criterion, are the same as when the LDC method is noused, see above.

Non-convergence: If convergence has not been reached from an established equilibrium configuration afaprogram execution is terminated. The solution can be continued by performing a restart run. Notethat in this case the LDC method must be used in the restartdifferent value for tdenforced initial displacement then corresponds to a displacementincrement from the last converged equilibrium position, that is, at the time of solution start for the restart analysis.


UGS Corp. 243

Every nonlinear analysis should be preceded by a linear anaTh ghlight many important factors such as the proper application of boundary conditions, deletion of all ite ele

• se of a sufficient number of load steps and equilibrium iterations with tight convergence tolerances at each load step is conside d basic aim is to obtain a response prediction close to this accurate one at a sm • incpla ar eff to hardening. Fig. 6.2-2 shows some examples. • ent controlled loading generally converges faster than force controlled loading. For example, in the model shown in Figwithe c) fordis . Displacement controlled loading (apply an increasing ∆) would work in this case. Note that this case is also sui • e step size, the ATS method will result in almost the same "iteration patwi e user-specified load levels. ence, in general, it is most convenient to use a reasonable nu • nges due to contact, or other discontinuities, consider using the low

solution method to use for a given analysis. •

lysis, if only to check that the model has been set up correctly. e linear analysis results will hi

degrees of freedom without stiffness, the quality of the finment mesh, etc.

If the u

re to yield an accurate solution of the model, then the

s all a solution cost as possible.

It is helpful to know if the model softens or hardens under reasing load. Structures can soften due to the spread of sticity, and they can soften or harden due to geometric nonlineects. Contact usually leads

Displacem

. 6.2-2(b), applying an increasing tip displacement to the beam ll converge faster than an applied load P, and both will follow same load displacement curve. For the model in Fig. 6.2-2(ce control would fail past the local maximum on the load-placement curve

table for the LDC method.

When the ATS method is used, together with a reasonable tim

h" as when not using the method. Namely, no step subdividing ll be performed if convergence is always directly reached at th

Hmber of load steps together with the ATS method.

If the problem involves localized buckling, or sudden cha



spe l dam t excessive.

pse of the structure occurs o capture the post-collapse

response. Note, however, that the solution at a specific load or

to

ed dynamic feature. Make sure that the selected structuraping is no

• The LDC method is useful if colladuring the (static) solution. It can als

displacement level cannot be obtained using the LDC method because the load increments are automatically calculated by the program.

• Note that usually it is quite adequate to employ the energyconvergence tolerance only. The need to use one of the other convergence criteria arises when the energy convergence is not tight (small) enough. In addition, there exist special loading conditions under which the denominator of the inequality (6.2-6) in Section 6.2.7 becomes small and hence the inequality is difficultsatisfy.


UGS Corp. 245

��

p

p

(a) Softening problem. Materially-nonlinear-only analysis,

elasto-plastic analysis of a cylinder

�

P

�

P

(b) Stiffening problem. Large displacement nonlinear elastic

analysis of a cantilever

�

PP

�

(c) Softening/stiffening problem. Large displacement analysis of a

thin arch

Figure 6.2-2: Different types of nonlinear analyses



6.2.9 Example

i

with line search tolerance STOL=0.5, and the energy and force

NORM=10, RMNORM=10.

We now present a worked example that illustrates some of the convergence issues described above. Fig. 6.2-3 shows the terationhistory for a certain load step. The line searches option was used

convergence criteria were selected with ETOL=10-8, RTOL=10-2, R


UGS Corp. 247

OUT-OF- NORM OF

BALANCE OUT-OF-BALANCE NORM OF INCREMENTAL ...

ENERGY FORCE MOMENT DISP. ROTN. CFORCE ...

(EQ MAX) (EQ MAX) (EQ MAX) (EQ MAX) CFNORM ...

VALUE VALUE VALUE VALUE

ITE= 0 0.35E+03 0.75E+04 0.26E-12 0.61E-01 0.11E-03 0.00E+00 ...

( 40) ( 29) ( 54) ( 45) 0.00E+00 ...

-0.45E+04 0.19E-12 -0.24E-01 -0.49E-04

ITE= 1 0.28E-03 0.58E+01 0.10E+01 0.13E-02 0.87E-04 0.00E+00 ...

( 23) ( 22) ( 1) ( 45) 0.00E+00 ...

-0.34E+01 0.50E+00 0.64E-03 -0.39E-04

ITE= 2 0.60E-04 0.79E-01 0.84E+00 0.10E-02 0.70E-04 0.00E+00 ...

( 15) ( 22) ( 1) ( 45) 0.00E+00 ...

0.34E-01 0.40E+00 0.52E-03 -0.31E-04

ITE= 3 0.27E-05 0.19E+00 0.18E+00 0.22E-03 0.15E-04 0.00E+00 ...

( 47) ( 22) ( 1) ( 45) 0.00E+00 ...

-0.12E+00 0.85E-01 0.11E-03 -0.66E-05

ITE= 4 0.26E-06 0.47E+00 0.33E-01 0.47E-04 0.32E-05 0.00E+00 ...

( 47) ( 22) ( 1) ( 45) 0.00E+00 ...

0.30E+00 0.16E-01 0.23E-04 -0.14E-05

ITE= 5 0.77E-07 0.67E-02 0.31E-01 0.37E-04 0.25E-05 0.00E+00 ...

( 47) ( 22) ( 1) ( 45) 0.00E+00 ...

-0.41E-02 0.15E-01 0.19E-04 -0.11E-05

... CONVERGENCE RATIOS CONVERGENCE RATIOS OUT-OF-BALANCE LOAD

... FOR OUT-OF-BALANCE FOR INCREMENTAL VECTOR CALCULATION

... ENERGY FORCE DISP. CFORCE BETA RATIO

MOMENT ROTN. (ITERNS)

... COMPARE WITH COMPARE WITH

... ETOL RTOL DTOL RCTOL

(NOT USED)(NOT USED)

... 0.10E+01 0.75E+03 0.00E+00 0.00E+00

... 0.26E-13 0.00E+00

... 0.78E-06 0.58E+00 0.00E+00 0.00E+00 0.10E+01 0.71E-03

... 0.10E+00 0.00E+00 ( 1)

... 0.17E-06 0.79E-02 0.00E+00 0.00E+00 0.10E+01 0.27E+00

... 0.84E-01 0.00E+00 ( 1)

... 0.76E-08 0.19E-01 0.00E+00 0.00E+00 0.40E+01 0.21E+00

... 0.18E-01 0.00E+00 ( 3)

... 0.72E-09 0.47E-01 0.00E+00 0.00E+00 0.40E+01 0.19E+00

... 0.33E-02 0.00E+00 ( 3)

... 0.22E-09 0.67E-03 0.00E+00 0.00E+00 0.10E+01 0.37E+00

... 0.31E-02 0.00E+00 ( 1)

Figure 6.2-3: Example of iteration history printout



Row ITE=0: This row shows the result of the initial iteration called iteration 0. For this iteration, the program performs the following steps:

ompute .

Compute and using

considering translational degrees of freedom of the

(0)t t t+∆ =U UC

(0)t t+∆ F (0)t t+∆ K (0)t t+∆ U .

Compute the out-of-balance force vector (0)t t t t+∆ +∆−R F . Only, the norm

out-of-balance force vector is (0)

20.75Et t t t+∆ +∆ 04+− =R F

and the largest magnitude in the out-of-balance force vector is −0.45E+04 at equation number 40. Only considering rotationadegrees of freedom, the norm of the out-of-balance force vector

l

is (0)

20.26E 12t t t t+∆ +∆ −− =R F and the largest magnitude in

of the

the out-of-balance force vector is 0.19E-12 at equation number 29.

Compute (1)∆U using (0) (1) (0)t t t t t t+∆ +∆ +∆= −K ∆U R F . Only considering translational degrees of freedom, the norm

(1)incremental displacement vector is 2

0.61E 01−∆U and

the largest magnitude in the incremental displacement vector is-0.24E-01 at equation number 54. Only considering rotational degrees

=

o m, the norm of the incremental displacement vector is

f freedo(1)

20.11E 03−=∆U and the largest magnitude in

vector is -0.49E-04 at equation number 45. the incremental displacement

Computes the �out-of-balance energy�

( )(1) (0) 0.35E 03T t t t t+∆ +∆ +− =∆U R F .

Compute the energy convergence criterion

(1) (0)

(1)0.10E 01

T t t t t

T t t t

+∆ +∆

+∆+

⎡ ⎤∆ −⎣ ⎦ =⎡ ⎤∆ −

U R F

U R F

⎣ ⎦


UGS Corp. 249

Compute the force and moment convergence criteria (0)

2 0.75E 03RNORM

t t t t+∆ +∆

+−

=R F

and

(0)

2 0.26E 13RMNORM

t t t t+∆ +∆

−−R F

-8

ration, the program performs the following teps:

e

= .

Since the energy convergence criterion is greater than ETOL=1.Eand the force convergence criterion is greater than RTOL=1.E-2, convergence is not satisfied.

Row ITE=1: This row shows the results of the first equilibrium iteration. In this ites

Compute (1) (0) (1)t t t t+∆ +∆= + ∆U U U , (1)t t+∆ F and the lin

search ratio (1) (1)

0.71E 03T t t t t

T

+∆ +∆

−⎡ ⎤∆ −⎣ ⎦ =

U R F. This ra

(1) (0)t t t t+∆ +∆⎡ ⎤∆ −⎣ ⎦U Rtio

is less than STOL=0.5, so the program sets the line search β

F

facto 1r (1) = . Compute (1) (0) (1) (1)t t t t β+∆ +∆= + ∆U U U .

Compute (1)t t+∆ F using (1)t t+∆ U . Since the modified Newton method is used, the program does not compute (1)t t+∆ K .

Compute the out-of-balance force vector (1)t t t t+∆ +∆−R F . Only considering translational degrees of freedom, the norm of the

(1)

20.58E 0out-of-balance force vector is 1t t t t+∆ +∆ +− =R F

s of freedom, the norm of the out of balance force vector

and the largest magnitude in the out-of-balance force vector is −0.34E+01 at equation number 23. Only considering rotational degreeis (1)

20.10E 01t t t t+∆ +∆ +− =R F and the largest magnitude in

the out-of-balance force vector is 0.50E-00 at equation number 2. 2



. Only Compute (2)∆U using (1) (2) (1t t t t t t+∆ +∆ +∆= −K ∆U R F )

considering translational degrees of freedom, the norm of the incremental displacement vector is (2)

20.13E 02−=∆U and

the largest magnitude in the incremental displacement vector is .64E-03 ion number 1. Only considering rotational

t 0 at equatdegrees of freedom, the norm of the incremental displacemenector is (2) 0.87E 04

2−=∆U and the largest magnitudv e in

n

Compute the �out-of-balance energy�

the incremental displacement vector is -0.39E-04 at equationumber 45.

( )(2) (1) 0.28E-03T t t t t+∆ +∆− =∆U R F .

Compute the energy convergence criterion

(2) (1)

(1)0.78E-06

T t t t t

T t t t

+∆ +∆

+∆

⎡ ⎤∆ −⎣ ⎦ =⎡ ⎤∆ −⎣ ⎦

U R F

U R F.

Compute the force and moment convergence criteria

(1)

2 0.58E 00RNORM

t t t t+∆ +∆

+−

=R F

and

(1)

2 0.10E 00RMNORM

t t t t+∆ +∆

−−

=R F

.

Since the energy

o

gain, convergence is not satisfied.

one

n. In row ITE=3, the line search proceeds as follows:

convergence criterion is greater than ETOL=1.E-8, the force convergence criterion is greater than RTOL=1.E-2 and the moment convergence criterion is greater than RTOL=1.E-2, convergence is not satisfied. R w ITE=2: This row shows the results from the second equilibrium iteration. This row is interpreted exactly as is row ITE=1. A

Row ITE=3: This row shows the results from the third equilibriumiteration. This row is interpreted exactly as is row ITE=2, with exceptio


UGS Corp. 251

Compute and the line (3) (2) (3)t t t t+∆ +∆= + ∆U U U , (3)t t+∆ F

search ratio (3) (3)

(3) (2)

T t t t t

T t t t t

+∆ +∆

+∆ +∆

⎡ ⎤∆ −⎣ ⎦⎡ ⎤∆ −⎣ ⎦

U R F

U R F. This ratio is greater

than STOL=0.5, so the program computes the line search factor (3) 0.40E 01β = + in three line search iterations. At the end of

is computed from and the line search ratio the third line search iteration, (3) (2) (3)4t t t t+∆ +∆= + ∆U U U ,

(3)t t+∆ F (3)t t+∆ U(3) (3)

(3) (2)0.21E 00

T t t t t

T t t t t

+∆ +∆

+∆ +∆+

⎡ ⎤∆ −⎣ ⎦ =⎡ ⎤⎣ ⎦

U R F

F, which is less than

STOL.

ows ITE=4, 5: These rows show the results from the fourth and

eration, the energy convergence criterion is

∆ −U R

Again, convergence is not satisfied.

Rfifth equilibrium iterations. At the end of the fifth equilibrium it

(6) (5)

(1)0.22E-09 ETOL

T t t t t

T t t t

+∆ +∆

+∆<

⎡ ⎤∆ −⎣ ⎦ =⎡ ⎤∆ −⎣ ⎦

U R F

U R F, and the force and

moment convergence criteria are

(5)

2 0.67E-03<RTOLRNORM

t t t t+∆ +∆−=

R F and

(5)

2 0.31E 02 RTOLRMNORM

t t t t+∆ +∆

− <−

=R F

.

to Convergence is satisfied. Five equilibrium iterations were used

obtain the converged solution.



6.3 Linea

ed by implicit integration using the Newmark method.

the following sections in

the descriptions of the equilibrium equations:

ment

= vectors of nodal point displacements at time t, t+∆t U" +∆t

- tU.

The governing equilibrium equations at time t+∆t are given by

(6.3-1)

hapter 9.

method or the composite time integration method. The ewmark method is explained in ref. KJB., Section 9.2.4, and the

composite method is explained in the following paper.

ref. K.J. Bathe, �Conserving Energy and Momentum in Nonlinear Dynamics: A Simple Implicit Time Integration Scheme� J. Computers and Structures, In press.

r dynamic analysis

• Linear dynamic analysis in Solution 601 is perform

• The notation given below is used in

M = constant mass matrix C = constant damping matrix K = constant stiffness matrix

,t t t+∆R R = external load vector applied at time t, t+∆t F = nodal point force vector equivalent to the ele

stresses at time t ,t t t+∆U U"" "" = vectors of nodal point accelerations at time t, t+∆t

t

,t t t+∆U U , t t+∆ U" = vectors of nodal point velocities at time t, tt

U = vector of nodal point displacement increments fromtime t to time t+∆t, i.e., U = t+∆tU

•

t t t t t t t t+∆ +∆ +∆ +∆+ + =M U C U K U R"" "

The procedures used in the time integration of the governing equations for dynamic analysis are summarized in ref. KJB, C

• The time integration of the governing equations can use the Newmark N

6.3: Linear dynamic analysis

UGS Corp. 253

posite method, the displacements, velocities, and e t + γ∆t, where γ ∈ (0,1) using

8 to s

that, for a given step size, the composite scheme is about expensive as the Newmark method due to the extra

e t + γ∆t.

• In the comaccelerations are solved for at a timthe standard Newmark method. The γ parameter is set to 0.585keep a constant effective stiffness matrix. The composite scheme ithen used to solve for the displacements, velocities and accelerations at time t + ∆t. Notewice ast

solution step at tim • The following assumptions are used in the Newmark method:

( )1t t t+∆ t t t tδ δ +∆⎡ ⎤= + − + ∆⎣ ⎦U U U U"" "" (6.3-2)

" "

212

t t t t t t tt tα α+∆ +∆⎡ ⎤⎛ ⎞= + ∆ + − + ∆⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦U U U U U" "" "" (6.3

where α and δ are the Newmark time integrati

-3)

on parameters.

(6.3-4)

a a= + +K K M C (6.3-5)

This transforms Eq. (6.3-1) to � �t t t t+∆ +∆=K U R where �

0 1

( ) (0 2 3 1 4 5 )t t t t t ta a a a a+ + + + + +M U U U C U U U" "" " ""

(6.3-6) and where a0, a1, ... , a5 are integration constants for the Newmark method (see Ref. KJB, Section 9.2.4).

e

The trapezoidal rule (also called the constant-average-cceleration method of Newmark) obtained by using δ = 0.5, α = 0.25 is recommended for linear dynamic analysis (when the

ref. KJBSections 9.2.4

and 9.4.4

� t t a+∆=R R

A similar procedure can be followed for the composite timintegration scheme.

•a



tics:

! It is an implicit integration method, meaning that equilibrium of the system is considered at time t+∆t to obtain the solution at time t+∆t. ! It is unconditionally stable. Hence, the time step size ∆t is selected based on accuracy considerations only, see ref. KJB, Section 9.4.4.

• The Newmark method is in general stable when the following

n be

Newmark method is used). • The trapezoidal rule has the following characteris

constraints are satisfied: 20.5, 0.25( 0.5)δ α δ≥ ≥ + .

• Newmark parameters different from the trapezoidal rule caused. Contact/impact problems sometimes perform better with

0.5, 0.5.δ α= = Other Newmark values can add some numeridamping for high frequencies which is useful for some models.

cal

on

consistent mass discretization can be effective.

y

• The Newmark method is effective for both structural vibratiand wave propagation problems. In these analyses, the use of higher-order elements, just as in static analysis, and the use of a

• The time step increment (∆t) recommended for dynamic analysis with the Newmark method is given b 0.20co tω ∆ ≤ where coω is the highest frequency of interest in the dynamic response. • Whether the mass and damping matrices are diagonal or banded

6.3.1 Mass m

(lumped or consistent discretization), the solution always requires that a coefficient matrix be assembled and factorized.

atrix

• The mass matrix of the structure may be based on a lumped or consistent mass calculation.

6.3: Linear dynamic analysis

UGS Corp. 255

ted

where

• The consistent mass matrix for each element ( )iM is calculausing

( ) ( ) ( ) ( )i i i T i dVρ= ∫M H H

( )iρ is the density, and is the displacement interpolation matrix specific to the element type. • The construction of the lumped mass matrix depends on the type of element used. Each of the elements in Chapter 2 detail how its lumped mass matrix is calculated. For elements with translational degrees of freedom only, the total mass of the element is divided equally among its nodes.

6.3.2 Dam

• Damping can be added directly to the model through Rayleigh

( )iH

ping

damping. Additional indirect damping results from the selected time integration parameters, plasticity and friction. • If Rayleigh damping is specified, the contributions of the following matrix ( )RayleighC are added to the total system damping

matrix C described in Section 6.3:

α β= +RayleighC M K

r tent, and K is the total system stiffness matrix.

where M is the total system mass matrix which can be lumped oconsis

• See Ref. KJB, Section 9.3.3, for information about selecting the Rayleigh damping constants α, β. In the modal basis, the damping ratio can be written as

2 2i

ii

βωαξω

= +

iξ is the damping ratio for mode iωwhere . It is clear that α tends

p lower modes and β tends to damp higher modes.


to dam


256

If α is not used, then a value of pTβ =

π will overdamp all

motions with periods smaller than otions with periods

smaller than can be suppressed by choosing

pT . Hence m

pT pTβ =

π. This may

be of interest when using damping to suppress numerical oscillations.

The above comments apply only when the stiffness matrix does not change significantly during the analysis, however.

6.4 Nonlinear dynamic analysis

• Nonlinear dynamic analysis in Solution 601 is performed by irect implicit integration using the Newmark method or the

c

damping

dcomposite time integration method, similar to linear dynamianalysis. • The use of Rayleigh ( )RayleighC is the same as

ed

described in Section 6.3.2. In this case, the total mass matrix and the initial total stiffness matrix are used to evaluate the Rayleigh damping matrix. • Since RayleighC is constant throughout the solution, it is formonly once in Solution 601, before the step-by-step solution of the

ref. KJBSection 9.5


equations. equilibrium

• The governing equations at time t t+ ∆ are

( ) ( ) ( )t t i t t i t t i t t t t i( 1)+∆ +∆ +∆ +∆+ + ∆ =M U C U K U"" " +∆ −−R F

tions ents obtained in

iteration (i) respectively. The vector of nodal point forces t+∆tF(i-1) is equivalent to the

lement stresses in the configuration corresponding to the isplacements t+∆tU(i-1).

where ( ) ( ) ( 1) ( ), ,t t i t t i t t i i+∆ +∆ +∆ − + ∆U U U U"" " are the approximato the accelerations, velocities, and displacem

ed

6.4: Nonlinear dynamic analysis

UGS Corp. 257

ems, it is sometimes better to set α = 0.5 as explained in Section 4.6.1.

me uding the ATS

method and line search, see Sections 6.2.1 and 6.2.2 for more details. However, the LDC method cannot be used in dynamics. • The energy and force/moment convergence criteria used in nonlinear dynamic analysis are:

r all degrees of freedom:

• The trapezoidal rule obtained by using δ = 0.5 and α = 0.25 isrecommended if the Newmark method is used. Note that for contact/impact probl

• The dynamic equilibrium equations are solved using the saiterative procedures used in static analysis, incl

Energy convergence criterion Fo

( ) ( 1) ( 1) ( 1)

(1) (0) (0)ET

Ti t t t t i t t i t t i

T t t t t t t t

+∆ +∆ − +∆ − +∆ −

+∆ +∆ +∆

⎡ ⎤∆ − − −⎣ ⎦ ≤⎡ ⎤∆ − − −⎣ ⎦

U R M U C U F

U R M U C U F

"" "

"" "OL

:

Force and moment convergence criteria For the translational degrees of freedom

( 1) ( 1) ( 1)

2 RTOLRNORM

t t t t i t t i t t i+∆ +∆ − +∆ − +∆ −− − −≤

R M U C U F"" "


( 1) ( 1) ( 1)

2 RTOLRMNORM

t t+∆ t t i t t i t t i+∆ − +∆ − +∆ −− − −≤

R M U C U F"" "

Translation/rotation convergence criteria For the translational degrees of freedom:

( )

2 DTOLDNORM

i∆≤

U




( )

2 DTOL≤ DMNORM

i∆U

where DTOL, DNORM and DMNORM are user-specified tolerances.

of the different ed in Section 6.2.4 for nonlinear static analysis.

me contact convergence criteria, guidelines for dealing of convergence, and s explained in Sections

4 and 6.2.5 are applicable to dynamic analysis.

mic analysis the solution is m e step size. Using a large step leads to inaccurate time integration

ess of the tightness of the convergence t ances.

6.5 Solvers

Two solvers are available in Solution 601. These are the direct sparse solver and the iterativ ls on parallel

n be found in S

6.5.1 Direct sparse solver

on metho hybrid ordering sc and the

s he amount of nd the tota formed in equations

• The sparse matrix solver ust and should erally be used for most pault solver.

The meaning tolerances is as stat

• The sa

ith lack w modeling tip nonlinear 6.2.

• In dyna ore sensitive to the tim

regardl

oler

e multi-grid solver. Detaiprocessing ca ection 8.6.

• The direct soluti d in Solution 601 is a sparse matrix solver. Aminimum

heme of the nested dissection degree algorithm is used to greatly reduce t

storage required alution of the

l number of operations perthe so

.

is very reliable and robgendef

roblems in Solution 601. It is the

6.5: Solvers

UGS Corp. 259

lver memor y y the rest of the program. It is also dynamically allocated er as needed. The total memory allocated by the

Nastran program for Solution 601 covers both the program�s lver�s me

The sparse solver can be used both in-core and out-of-core. It is

icient to run an out-of-core sparse solver using real ory than it is to run an in-core sparse solver using

memory. Therefore, for large problems, we recommend emory size (via the Nastran command) until it fits

em in-core, or it reaches approximately 85% of the real memory.

sitive defi .e. one with a

zero or negative diagonal ele during solution, the program may stop or continue, according to the following rules:

! If a diagonal element is ution 601 stops unless

on number corresponding to the zero diagonal element is only attached to inactive elements (elements that are dead due to rupture or the element death feature).

- The user requested that Solution 601 continue execution

using the NPOSIT flag in the NXSTRAT entry.

value of a diagonal element is sm ller than 10-12 but not equal to zero, or the value of a diagonal element is negative, Solution 601 stops unless one of the following options is used:

load-displa

tic time-step - Contact analysis - The user requested that Solution 601 continue execution

using the NPOSIT f

• The sparse so y is separate from that memorallocated bby the solv

memory and the so mory.

• more eff(physical) memvirtual increasing the mthe probl

• When a non-po nite stiffneme

ss matrix (int) is encountered

exactly equal to 0.0, Sol

- The equati

! If the a

- Automatic cement (LDC) - Automa ping (ATS)

lag in the NXSTRAT entry.



• When Solution 601 stops essages for ive diagonal elements.

Solution 601 continues execution and diagonal is smaller than 10-12, it assigns a very large number to the element, effectively pring to that

ree of freedom.

atrix can be non-positive definite due to error, for exampl

in static analysis. In lts obtained can be

multi-grid solver

• In the analysis of very lar of storage uired by a direct solution

vailable computer resources. F he use of the ethod of solution

rid solver ava braic solver, and can be used with all solution options of Solution 601.

i-grid solver is sensitive to the conditioning of the . It generally res fewer

for well-cond ll-conditioned ay require a large n ations or may not

ll.

sensitivity of the multi-grid solver makes it 3-D solid models compared to thin structural

m dels where the membrane stiffness is much higher than the a ynamic

pared to static), because of the stabilizing effect of atrix (inertia effect) on the coefficient matrix.

• Note that the multi-grid solver cannot recognize that the stiffness matrix is singular. olver will

thout converging.

• The multi-grid solver is som ient for

, it prints informational mthe zero or negat • When theelement

agonaldi attaching a very stiff sdeg • Note that the stiffness ma modeling

strained e if the model is not sufficiently

this case the resuremisleading.

6.5.2 Iterative

ge problems, the amountreq solver may be too large for the

or such problems, taiterative m is necessary.

• The multi-g ilable in Solution 601 is an alge

• The multcoefficient matrix

tions) performs better (requi

solver iterablems m

itioned problems. Ipro umber of iterconverge at a • The conditioning more suited for bulky

obending stiffness. It also manalysis (com

kes it more efficient in d

the mass m

For such problems, the siterate wi

etimes also less effic

6.5: Solvers

UGS Corp. 261

blems with

placement coordi icantly along the model

- a large number of ri

large number of ro

some contact proble

ases, the 3D-iter see Section

in practical differences between the use of the direct

ver and the multi-grid solve

e direct solver executes a predetermined number of rations after which the solution is obtain It is less

ensitive to the condition

rid solver performs a predetermined number of operations per iteration, butknown beforehand. The number of iterations depends on the condition number of the coefficient matrix. The higher the ondition number, the mo The number

erations required varie.

Regarding the convergen

that the system of equations of A , N is t

ion at solv idual vector is We can de

pro

- dis nate systems that vary signif

gid elements or constraint equations

- a d or beam elements

- ms. In such c ative solver might be used, 6.5.3.

• The masol r are as follows:

! Thope ed. s ing of the coefficient matrix. ! The multi-g

the number of iterations is not

c re iterations are needed. sof it from a few hundred to a few

thousand

• ce of the multi-grid method, assume to be solved is =Ax b , D is the

diagonal vector ap

he dimension of x, ( )kx is theproximate solut

( ) ( )k k= −r b Ax . er iteration k, and the resfine:



(minRDR RDA= )( ) (1) and, / ,kRDB RDB

( )kRDA r= ( )

2

( ) ( ) ( 1)

2

( ) ( )

2

( ) (1)

( ) ( )

,

,

,

/

/ .

k

k k k

k k

k

k k

D N

RDB x x

DC x

RDA

RDC

−= −

=

i-grid method converges when one of the following

eached:

R

RDX RDB=

The multcriteria is r

( )

( )

( ) ( 1)

EPSII and EPSB,≤

EPSA EPEPSA and EPSB

EPSII *0

,,

, ,

k

k

k k

RDA RDXRDR

R RDX−

≤

≤ ≤≤ ≤

− ≤

EPSIA, EPSIB, and EPS ances set . The defaults are EPSIA=10-6,

EPSIB=10-4, EPSII=10-8. However, for nonlinear analysis with equilibrium iterations, looser to

6.5.3 3D-iterative solver

• The 3D-iterative solver has been developed to efficiently solve large models containing mainly(e.g., 10-node CTETRA, 20-node CHEXA, etc.).

• The 3D-iterative solver is invoked if SOLVER=2 in the T entry.

• In addition to the higher o he models can contain other elements available in the program (e.g., shells,

, etc.), including contact con ions.

solver is static namic analysis.

SB and EPSB, ,RDX ≤

max x x .1

RDARD

wherevia the NXSTRAT entry

II are convergence toler

lerances can be used.

higher order 3-D solid elements

NXSTRA

rder 3-D solid elements, t

rods, beams, rebars

dit

• The 3D-iterativeanalysis and in nonlinear dy

effective in linear or nonlinear

6.5: Solvers

UGS Corp. 263

lver, like all iterative solvers, performs a

number of iterations until convergence is reached.

ly incompressible hyperelastic materials may slow down e solver. For these material

the bulk modulus κ should be restricted to a value ng to ν = 0.49, instead of the default 0.499 (see Eq.

• Convergence control in th ws:

Considering the linearized equation , let

• The 3D-iterative so

• Nearthe convergence of the 3D-iterativmodels, correspondi3.7-7)

e 3D-iterative solver is as follo

Ax b= ix be its approximate solution at inner-iteration and be its

corresponding residual. The convergence in the iterative solver is said obtained if any one of the following criteria is satisfied

i ( )i ir b Ax= −

10 0

i i ib or r or x xε ε −≤ ≤ − 0ε≤

Equation residual: 0.001 3

3ei

scalee

ir r

iσ

σ⎧ <⎪⎪≤⎨⎪ ≥⎪⎩

Solution residual: 1 0.001 33

vi iscale

v

ix x x

iσ

σ−

⎧ <⎪⎪− ≤⎨⎪ ≥⎪⎩

Solution norm: 11

ninx x= ∑

In the above, , the equation scale 16

0 10ε −≡ scaler b= , and the variable scale

( ){ }( ){ }

1 2 310 3

1 2 310 3

max ,

max , ,scale

x x x for linear problemsv

x x x x for nonlinear problems

ε

ε

⎧⎪ + +⎪⎪⎪=⎨⎪⎪ + + ∆⎪⎪⎩



where x∆ is the current solution increment in outer-iteration (Newton-Raphson iteration).

7.1: Formulation

UGS Corp. 265

7. Explicit dynamic analysis

This chapter presents the formulations and algorithms used to solve explicit dynamic problems using Solution 701 including time step calculation. Most flags or constants that need to be input in this chapter are in the NXSTRAT bulk data entry. The elements and material properties available for explicit analysis with Solution 701 are listed in Table 2-3.

7.1 Formulation

The central difference method (CDM) is used for time integration in explicit analysis (see ref. KJB, Section 9.2.1). In this case, it is assumed that

( )2

1 2t t t t t t

t−∆ +∆= − +

∆U U U U"" (7.1-1)

and the velocity is calculated using

( )12

t t t t t

t−∆ +∆= − +

∆U U U" (7.1-2)

The governing equilibrium equation at time t is given by

t t t t+ = −M U C U R F"" " (7.1-3)

Substituting the relations for t U"" and t U" in Eq. (7.1-1) and (7.1-2), respectively, into Eq. (7.1-3), we obtain

2 2 2

1 1 2 1 12 2

t t t t t t t

t t t t t+∆ −∆⎛ ⎞ ⎛ ⎞+ = − + − +⎜ ⎟ ⎜ ⎟∆ ∆ ∆ ∆ ∆⎝ ⎠ ⎝ ⎠

M C U R F M U M C U

(7.1-4)

from which we can solve for t t+∆ U .

• The central difference method has the following characteristics: ref. KJBSections 9.2.1,

9.4 and 9.5.1

Chapter 7: Explicit dynamic analysis


! It is an explicit integration method, meaning that equilibrium of the finite element system is considered at time t to obtain the solution at time t+∆t.

! When the mass and damping matrices are diagonal, no coefficient matrix needs to be factorized, see ref. KJB, p. 772. The use of the central difference method is only effective when this condition is satisfied. Therefore, only lumped mass can be used in Solution 701. Also damping can only be mass-proportional.

! No degree of freedom should have zero mass. This will lead to a singularity in the calculation of displacements according to Eq. 7.1-4, and will also result in a zero stable time step.

! The central difference method is conditionally stable. The time step size ∆t is governed by the following criterion

NminCR

Tt t∆ ≤ ∆ =π

where CRt∆ is the critical time step size, and TBNmin B is the smallest period in the finite element mesh.

• The central difference method is most effective when low-order elements are employed. Hence quadratic 3-D solid and shell elements are not allowed.

• The time step in Solution 701 can be specified by the user, or calculated automatically (via the XSTEP parameter in NXSTRAT). When the user specifies the time, Solution 701 does not perform any stability checking. It is the user�s responsibility, in this case, to ensure that an appropriate stable time step is used.

• When automatic time step calculation is selected, the TSTEP entry is only used to determine the number of nominal time steps and the frequency of output of results. The stable time step is used instead of the value in TSTEP (unless the value in TSTEP is smaller).

7.1: Formulation

UGS Corp. 267

For example, if the following TSTEP entry is used TSTEP, 1, 12, 1.0, 4 there will be 12 nominal steps each of size 1.0. If the stable time step is smaller than 1.0 it will be used instead and results will be saved as soon as the solution time exceeds 4.0, 8.0 and exactly at 12.0 since it is the last step of the analysis.

7.1.1 Mass matrix

• The construction of the lumped mass matrix depends on the type of element used. Details are provided in the appropriate section in Chapter 2. For elements with translational degrees of freedom only, the total mass of the element is divided equally among its nodes. For elements with rotational masses (beam and shell elements), the lumping procedure is element dependent. Note that the lumping of rotational degrees of freedom is slightly different in implicit and explicit analysis. The rotational masses in explicit analysis are sometimes scaled up so that they do not affect the element�s critical time step.

7.1.2 Damping

• Damping can be added directly to the model through Rayleigh damping. Additional indirect damping results from plasticity, friction and rate dependent penalty contact. • Only mass-proportional Rayleigh damping is available in explicit analysis. Hence, the damping matrix C in Eq. 7.1-3 is set to:

Rayleigh α=C M

where M is the total lumped mass matrix. See Ref. KJB, Section 9.3.3, for information about selecting the Rayleigh damping constant α.



7.2 Stability

• The stable time step for a single degree of freedom with central difference time integration is

2N

CRN

Ttπ ω

∆ = =

The stable time step for a finite element assembly is

min

max max

2 2NCR

N E

Tt tπ ω ω

∆ ≤ ∆ = = ≤

where maxNω is the highest natural frequency of the system, which is bound by the highest natural frequency of all individual elements in a model maxEω (see Ref. KJB, Example 9.13, p. 815). • When automatic time step is selected, the time step size is determined according to the following relationship

minmax

2E

E

t K t Kω

∆ = × ∆ = × (7.2-1)

where K is a factor (set via the XDTFAC parameter in NXSTRAT) that scales the time step. • For most element types the critical time step can be expressed in terms of a characteristic length and a material wave speed

ELtc

∆ = (7.2-2)

where the definition of the length L and the wave speed c depend on the element and material type. For all elastic-plastic materials the elastic wave is used. This condition is used in Solution 701 instead of actually evaluating the natural frequency in Eq. (7.2-1).


7.2: Stability

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• Note that the critical time step calculated for all elements is only an estimate . For some elements and material combinations it is exact, and for others it is slightly conservative. However, it may not be small enough for excessively distorted elements (3-D solid and shells), and it will therefore need scaling using K factor in Eq (7.2-1). • The time step also changes with deformation, due to the change in the geometry of the elements and the change in the wave speed through the element (resulting from a change in the material properties). Rod elements The critical time step for a 2-node rod element is

ELtc

∆ =

where L is the length of the element, and c is the wave speed through the element

Ecρ

=

Beam elements The critical time step for the (Hermitian) beam element is

2

12/ 1EL Itc AL

∆ = +

where L is the length of the element, A is the element area, I is largest moment of area, and c is the wave speed through the element

Ecρ

=

Shell elements The critical time step for shell elements is



ELtc

∆ =

where L is a characteristic length of the element based on its area and the length of its sides, and c is the planar wave speed through the shell, which for linear isotropic elastic materials is

2(1 )Ec

ρ ν=

−

The critical time step estimated here is only approximate, and may be too large for excessively distorted shell elements. 3-D solid elements The critical time step for the 3-D solid elements is

ELtc

∆ =

where L is a characteristic length of the element, based on its volume and the area of its sides, and c is the wave speed through the element. For linear isotropic elastic materials c is given as

(1 )

(1 )(1 2 )Ec ν

ρ ν ν−

=+ −

The critical time step estimated here is only approximate, and may be too large for excessively distorted 3-D solid elements.

Spring elements The critical time step for a spring element is

1 2

1 2

2 2( )E

N

M MtK M Mω

∆ = =+

where M B1B and MB2B are the masses of the two spring nodes and K is

7.2: Stability

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its stiffness. Massless springs are not taken into account in the calculation of the stable time step.

R-type elements These elements are perfectly rigid and therefore do not affect the stability of explicit analysis. Gap and Bushing elements These elements use the same criterion as spring element.

7.3 Time step management • The stable time step size has a major influence on the total simulation time. Since this time step is determined based on the highest eigenvalue of the smallest element, a single small or excessively distorted element could considerably increase the solution time, even if this element is not relevant to the full model. Note that the element having the smallest critical time step size is always provided in the output file. • Ideally, all elements should have similar critical time steps. If the material properties are uniform throughout the model this means that elements should approximately have the same lengths (see Eq. 7.2-2). • The evaluation of the critical time step for each element takes some computational time. Therefore, it does not need to be performed at every time step. The parameter XDTCAL in NXSTRAT determines how frequently the critical time step is reevaluated. • The time step size for explicit analysis can be unduly small for a realistic solution time. Three features are provided to deal with this problem.

• A global mass scaling variable can be applied to all elements in the model (the XMSCALE parameter in NXSTRAT). This scale factor is applied to the densities of all elements, except scalar elements where it is applied directly to their mass.



• Mass scaling could also be applied to elements whose automatically calculated initial time step is below a certain value (XDTMIN1 parameter in NXSTRAT). A mass scale factor is then applied to these elements to make their time steps reach XDTMIN1. The mass scaling ratio is then held constant for the duration of the analysis. • Elements with automatically calculated time step smaller than a specified value (XDTMIN2 parameter in NXSTRAT) can be completely removed from the model. This parameter is useful for extremely small or distorted elements that do not affect the rest of the model. • The three parameters explained above (XMSCALE, XDTMIN1 and XDTMIN2) should all be used with great care to ensure that the accuracy of the analysis is not significantly compromised.

8.1: Initial conditions

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8. Additional capabilities

8.1 Initial conditions

8.1.1 Initial displacements and velocities

• Initial displacements and velocities at nodes can be specified using the TIC entry together with the IC case control command. Any initial displacements or velocities specified in a restart run are ignored.

8.1.2 Initial temperatures

• Initial temperatures are specified via the TEMPERATURE(INITIAL) case control command. The actual temperature values are specified via the TEMPD and TEMP entries.

8.2 Restart

• Restart is a useful feature in Advanced Nonlinear Solution. It can be used when the user wishes to continue an analysis beyond its previous end point, or change the analysis type, loads or boundary conditions or tolerances. A restart analysis is set via the MODEX flag in the NXSTRAT entry. • All relevant solution data needed for a restart run are saved in a file in case they are needed in a restart analysis. The frequency of data writing to a restart file is set via the IRINT flag in the NXSTRAT entry. • Note that multiple restart data can be appended to the restart file. This enables the restart analysis to be based on a solution step different from the last converged solution. Saving multiple time step solutions to a restart file can be expensive, however, as it leads to a large restart file size.

Chapter 8: Additional capabilities


• The geometry, and most element data cannot be changed in a restart analysis. However, the following changes are allowed: ! Type of analysis can change. Static to dynamic and dynamic to static restarts are allowed. ! Solution type can be changed. Solution 601 (static or dynamic) to Solution 701 restarts are allowed and vice-versa. Some restrictions are imposed however when changing solution type:

- Features not available in either solution type cannot be used.

- Incompatible modes cannot be used. ! Solution control variables can change. The flags, constants and tolerances for the iteration method, convergence, time integrations, automatic time stepping and load-displacement-control can be changed. ! Externally applied loads and enforced displacements can be modified. ! The material constants can be changed. However, note that in a restart run the same material model (with the same number of stress-strain points and the same number of temperature points, if applicable) must be used for each element as in the preceding run. ! Boundary conditions can be changed. ! Constraint equations and rigid elements can be changed. ! Contact settings can be changed. This includes most contact set, contact pair and contact surface parameters. See section 4.6.4 for restrictions. ! Rayleigh damping coefficients can be changed. ! Time increment and number of solution steps can be modified.

8.2: Restart

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! Time functions describing the load variations can be changed.

8.3 Element birth and death feature

• The element birth and death option is available for modeling processes during which material is added to and/or removed from the structure (set via the EBDSET case control and bulk data commands). Such processes, for example, are encountered in the construction of a structure (structural members are added in succession), the repair of a structure (structural components are removed and new ones are added) or during the excavation of geological materials (a tunnel is excavated). If the element birth and death option is used, the corresponding element groups become automatically nonlinear. Fig. 8.3-1 illustrates two analyses that require the element birth and death options.

• The main features of element birth and death are as follows:

! If the element birth option is used, the element is added to the total system of finite elements at the time of birth and all times thereafter. ! If the element death option is used, the element is taken out of the total system of finite elements at times larger than the time of death. ! If both element birth and death options are used, the element is added to the total system of finite elements at the time of birth and remains active until the time of death. The time of death must be greater than the time of birth. The element is taken out of the total system of finite elements at all times larger than the time of death.

• Once an element is born, the element mass matrix, stiffness matrix and force vector are added to the mass matrix, stiffness matrix and force vector of the total element assemblage (until the death time, if any). Similarly, once an element dies, the element mass matrix, stiffness matrix and force vector are removed from the total assembled mass matrix, stiffness matrix and force vectors for



all solution times larger that the time of death of the element. • Note that an element is born stress free. Hence, even if the nodal points to which the new element is connected have already displaced at the time of birth, these displacements do not cause any stresses in the element, and the stress-free configuration is defined to occur at the nearest solution time less than or equal to the birth time.

Column to be

replaced

Installation of

temporary support

(a) Repair of a bridge

(b) Excavation of a tunnel

�

��

�

�

�

�

�

�

�

�

�

�

Tunnel to be

excavated

Ground level

Figure 8.3-1: Analyses that require the element birth and death

options

8.3: Element birth and death feature

UGS Corp. 277

• When the element birth/death option is used, the tangent stiffness matrix may at some solution times contain zero rows and corresponding columns. The equation solver disregards any zero diagonal element in the tangent stiffness matrix if no elements are attached to the associated degrees of freedom.

• Advanced Nonlinear Solution enables the user to set an element death decay time parameter (DTDELAY in NXSTRAT) which causes the gradual reduction of the element stiffness matrix to zero over a finite time rather than instantly. The reduction starts at the death time and progresses linearly with time until the decay time has passed. The element therefore totally vanishes at a time equal to the sum of the death time and the death decay time. This option is useful for mitigating the discontinuity that the structure may experience due to the death of some of its elements. • The element birth/death option applies to any mass effect i.e., gravity loading, centrifugal loading and inertia forces. The mass matrix, therefore, does not remain constant throughout the solution. • The time at which an element becomes active or inactive is specified by the parameters TBIRTH and TDEATH respectively (in the EBDSET entry). • If an element is required to be born at time t Bb B, i.e., the configuration of the element at time t Bb B corresponds to the stress-free configuration, you should input TBIRTH = t Bb B + ε where

1000tε ∆

= and ∆t is the time step between time t Bb B and the next

solution time. If an element is required to be inactive at and after

time t Bd B, you should input TDEATH = t Bd B - ε, where 1000

tε ∆= and

∆t is the time step between the previous solution time and time t Bd B . Birth option active: Fig. 8.3-2(a) shows the activity of an element for which the birth option is active. Note that if TBIRTH is input for the range shown (where TBIRTH > t Bb B and TBIRTH is ≤ t Bb B + ∆t), then the stress-free configuration of the element is at time t Bb B,



and the element is first active at time t Bb B + ∆t. Note that the stress in the element is based on the displacements measured from the configuration at time t Bb B, irrespective of the input value of TBIRTH, provided that TBIRTH is input in the range shown in Fig. 8.3-2(a). See also the example given below. Death option active: Fig. 8.3-2(b) shows the activity of an element for which the death option is active. Note that if TDEATH is input for the range shown (where TDEATH ≥ t Bd B - ∆t and TDEATH < t Bd B), then the element is first inactive at time t Bd B.

0Time

t

Stress freestate

t+ t�First solution time for whichthe element is active

(a) Birth option active

TBIRTH in this range causes the elementto be included in the stiffness matrix andthe force vector at time t+ t�

0Time

tt- t�First solution time for whichthe element is inactive

(b) Death option active

TDEATH in this range causes the element to beincluded in the stiffness matrix and the

force vector at time tnot

Figure 8.3-2: Use of element birth and death option

Birth then death option active: This is a direct combination of the birth and death options. Initially some elements in an element group are inactive. At a particular solution time determined by the time of birth TBIRTH, the elements become active and remain so

8.3: Element birth and death feature

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until a subsequent solution time determined by the time of death TDEATH, where TDEATH > TBIRTH. Example of the element birth option: Consider the materially linear truss element model shown in Fig. 8.3-3(a) in which the time of birth for element 2 is slightly larger than t. At time t, the length P

tPL corresponding to the load P

tPF is determined as shown in Fig. 8.3-

3(b). Note that P

tPL corresponds to the length at which element 2 is

stress-free.

Element 1

Element 1

(always active)

Element 2

tL

L

F(t)

u(t)

Linear 2-node

trusses

(b) Solution at time t

(TBIRTH = t + t/1000)�

tF

K =AE/ L

F =K ( L- L)1

1

0

t int t 0

1

(a) Model schematic

Figure 8.3-3: Example on the use of the element birth option

At the time of birth, element 2 with length P

tPL is added to the

system which was already in equilibrium, see Fig. 8.3-3(c). Note that the internal force in element 2 is exactly zero after its addition to the system.

At time t + ∆t, element 2 is now active. The force in element 2 is determined based on its deformation with respect to the stress-free state, see Fig. 8.3-3(d). Hence, the total increment in



displacement from time t to time t + ∆t determines the force in the truss. Identically, the same solution would be obtained using any value for TBIRTH which satisfies the relation TBIRTH > t and TBIRTH ≤ t + ∆t.

• The birth/death feature is available for contact pairs in contact analysis.

Element 2

tL

F = 0in t

Element 1

Element 2

t+ t� L

t+ t� F

(d) Solution at time t + t�

(c) Stress-free configuration for element 2

2

K =AE/ L

F =K ( L- L)1

1

0

t+ t int t+ t 0� �

1

K =AE/ L

F =K ( L- L)2

2

t

t+ t int t+ t t� �

2

Figure 8.3-3. (Continued)

8.4 Reactions calculation

• Output of the reaction forces and moments is governed by the SPCFORCES case control command.

• Note that loads applied to fixed degrees of freedom do not contribute to the displacement and stress solutions. However, these loads are accounted for in the reaction calculations. No reactions are calculated for degrees of freedom that are deleted from the model.

8.4: Reactions calculation

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• Reaction forces and moments at a node are computed using the consistent force vectors (calculated from the element internal stresses) of elements attached to the node. Hence, a check on the balance of the support reactions and the applied loads often provides a good measure on the accuracy of the solution (in terms of satisfying equilibrium in a nonlinear analysis). • Reaction calculations in dynamic analysis with consistent mass matrix take into account the mass coupling to the deleted degrees of freedom. The reactions exactly equilibrate the applied forces in all cases.

8.5 Element death due to rupture

• For the materials and elements that support rupture, element death is automatically activated when rupture is detected at any integration point of the element. The element is then considered "dead" for the remainder of the analysis, and, in essence, removed from the model (mass, stiffness, and load contributions). • When elements die, contactor segments connected to these elements are also removed from the model. • Dead elements may be gradually removed from the model in order to avoid sudden changes in stiffness and acceleration. This feature is activated by setting a non-zero DTDELAY time in the NXSTRAT entry.

8.6 Stiffness stabilization (Solution 601 only)

• Stiffness stabilization is a useful feature for stabilizing static problems (mainly involving contact) where there are insufficient constraints to remove rigid body motions. This is sometimes the case in contact problems right before contact starts and right after it ends. This feature is activated via the MSTAB flag in the NXSTRAT entry. • A stabilizing effect is added to the stiffness matrix by scaling all



diagonal stiffness terms (except those belonging to contact equations). At the same time, the right-hand-side load vector is not modified. The diagonal stiffness terms are modified as follows

(1 )ii STAB iiε= +K K

where εBSTABB is the stabilization factor set via the MSFAC parameter in the NXSTRAT entry. The default stabilization factor selected by Solution 601 is suitable for most stabilization problems.

• Since the internal force vector is not affected by this stabilization, the exact solution is the same as without stabilization. However, the iteration history, or the displacements at each iteration can be affected. A large stabilization factor will result in a slow convergence or no convergence, while a very small stabilization factor will not remove the singularities in the system.

• Insufficiently supported static problems can alternatively be treated by adding weak springs at various locations in the model. Stiffness stabilization is generally more beneficial for the following reasons:

! Determining the number, location and stiffness of the springs requires a lot of user intervention. ! There may be no suitable location for the springs. ! The stiffness of the spring has to be entered as an absolute value while the stiffness stabilization factor is normalized with respect to the stiffness matrix. ! The springs generate an internal force, which affects the exact solution while the stiffness stabilization does not.

• Note that even if no supports are present in the model, the stiffness matrix will be positive definite once the stabilization factor is applied. • Stiffness stabilization can only be used for nonlinear analyses.

8.6: Stiffness stabilization

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8.7 Bolt feature • Bolts in Solution 601 are modeled using beam elements with a user-specified initial bolt force or preload. The beam elements that make up a bolt are selected via the BOLT entry, and the bolt force is defined via the BOLTFOR entry. The bolt preload set must be selected via the BOLTLD case control command. • An iterative solution step is required to obtain the desired bolt force in all bolts. This solution step is performed at the very beginning of an analysis prior to the rest of the step-by-step analysis. External forces are not included in the bolt force iteration step. • Bolt force iterations can be performed in one step (default) or in a number of �bolt steps� (set via the BOLTSTP parameter in NXSTRAT). This feature should be used if the bolt conditions are too severe to converge in one step. • The bolt feature can only be used in static and implicit dynamic analysis. • In the time steps following the bolt force steps, the force in the bolts can vary depending on the loads applied to the rest of the model. • Both small and large displacement formulations can be used for the bolt�s beam elements. Any cross-section available for the beam element can be used, but only the isotropic elastic material model can be used. • In the bolt force calculations we iterate as follows: 0 ( 1) ( ) ( 1) 0 ( 1)

Ii i i i− − −∆ = −K U R F

and 0 ( ) 0 ( 1) ( )i i i−= + ∆U U U



where ( 1)Ii−R is the consistent nodal point force corresponding to the

forces in the bolt elements. • The bolt force convergence is checked for every bolt element m:

0

0 TOLLR R

Rm m

m

−<

where 0Rm is the user-input bolt force for element m, and Rm is the current force in bolt element m. • The ( 1)

Ii−R vector is calculated from

0

T( 1) ( 1) ( 1)Ii i i

V

dV− − −= ∫R B τ

8.8 Parallel processing

• Solution 601 supports parallel processing on all supported platforms, for the in-core and out-of-core sparse solvers. • Solution 701 also supports parallel processing on all supported platforms. • The HP, Linux and SGI platforms also support parallelized assembly of the global system matrices (excluding contact), and contact search for the rigid target contact algorithm.

8.9 Usage of memory and disk storage

Solution 601 • Depending on the size of the problem and the memory allocated to Solution 601, it can perform the solution either in-core (entirely within virtual memory) or out-of-core (reading from and writing to disk files). Whenever possible the solution is performed in-core.

8.9: Usage of memory and disk storage

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• Depending on the size of the problem and the available memory, there are 3 in-core/out-of-core settings for the program without the solver. These are: ! The stiffness (and mass) matrices and the element information are all stored in-core. ! The stiffness (and mass) matrices are stored in-core, while the element information is kept out-of-core. ! All matrices and element information are kept out-of-core.

• The sparse solver may be set to in-core or out-of-core. • When using the iterative multi-grid solver (Section 6.5.2), the solution is always performed in-core. The out-of-core solution procedure would take an unreasonably long time in most cases. Solution 701 • Solution 701 can only run in-core. Enough memory must be provided.

Additional reading


Additional reading

This section lists some additional readings (publications since 1993) related to Solution 601. References on the analysis of fluid flows and fluid-structure interactions are not given.

Books K.J. Bathe, �Finite Element Procedures�, Prentice Hall, 1996 D. Chapelle and K.J. Bathe, �The Finite Element Analysis of Shells � Fundamentals�, Springer, 2003

Papers N.S. Lee and K.J. Bathe, �Effects of Element Distortions on the Performance of Isoparametric Elements�, Int. J. for Numerical Methods in Engineering, U36 U, 3553-3576, 1993. M.L. Bucalem and K.J. Bathe, �Higher-Order MITC General Shell Elements�, Int. J. for Numerical Methods in Engineering, U36 U, 3729-3754, 1993. K.J. Bathe, J. Walczak and H. Zhang, �Some Recent Advances for Practical Finite Element Analysis�, J. Computers & Structures, U47 U, No. 4/5, 511-521, 1993. D. Chapelle and K.J. Bathe, �The Inf-Sup Test�, J. Computers & Structures, U47 U, No. 4/5, 537-545, 1993. K. Kato, N.S. Lee and K.J. Bathe, �Adaptive Finite Element Analysis of Large Strain Elastic Response�, J. Computers & Structures, U47 U, No. 4/5, 829-855, 1993. K.J. Bathe, �Remarks on the Development of Finite Element Methods and Software�, Int. J. of Computer Applications in Technology, U7 U, Nos. 3-6, 101-107, 1994.

Additional reading

UGS Corp. 287

N.S. Lee and K.J. Bathe, �Error Indicators and Adaptive Remeshing in Large Deformation Finite Element Analysis�, Finite Elements in Analysis and Design, U16 U, 99-139, 1994. P. Gaudenzi and K.J. Bathe, �An Iterative Finite Element Procedure for the Analysis of Piezoelectric Continua�, J. of Intelligent Material Systems and Structures, U6 U, No. 2, 266-273, 1995. G. Gabriel and K.J. Bathe, �Some Computational Issues in Large Strain Elasto-Plastic Analysis�, Computers & Structures, U56 U, No. 2/3, pp. 249-267, 1995. D. Pantuso and K.J. Bathe, �A Four-Node Quadrilateral Mixed-Interpolated Element for Solids and Fluids�, Mathematical Models and Methods in Applied Sciences, U5 U, No. 8, 1113-1128, 1995. K.J. Bathe, �Simulation of Structural and Fluid Flow Response in Engineering Practice�, Computer Modeling and Simulation in Engineering, U1U, 47-77, 1996. M.L. Bucalem and K.J. Bathe, �Finite Element Analysis of Shell Structures�, Archives of Computational Methods in Engineering, U4U, 3-61, 1997. A. Iosilevich, K.J. Bathe and F. Brezzi, �On Evaluating the Inf-Sup Condition for Plate Bending Elements�, Int. Journal for Numerical Methods in Engineering, U40U, 3639-3663, 1997. K.J. Bathe, O. Guillermin, J. Walczak, and H. Chen, �Advances in Nonlinear Finite Element Analysis of Automobiles�, Computers & Structures, U64U, no., 5/6, 881-891, 1997. K.J. Bathe and P. Bouzinov, �On the Constraint Function Method for Contact Problems�, Computers & Structures, U64U, no. 5/6, 1069-1085, 1997. D. Pantuso and K.J. Bathe, �On the Stability of Mixed Finite Elements in Large Strain Analysis of Incompressible Solids�, Finite Elements in Analysis and Design, U28U, 83-104, 1997.

Additional reading


D. Pantuso and K.J. Bathe, �Finite Element Analysis of Thermo-Elasto-Plastic Solids in Contact�, Proceedings, Fifth International Conference on Computational Plasticity, Barcelona, Spain, March 1997. K.J. Bathe, �On Reliability in Finite Element Analysis: Choosing the Mathematical Model, Finite Element Discretization and Time Integration�, Proceedings, Seventh Structural Engineering Arab Conference, Kuwait City, Kuwait, November 1997. D. Chapelle and K.J. Bathe, �Fundamental Considerations for the Finite Element Analysis of Shell Structures�, Computers & Structures, U66U, no. 1, 19-36, 1998. K.J. Bathe, J. Walczak, O. Guillermin, P.A. Bouzinov and H. Chen, �Advances in Crush Analysis�, Computers & Structures, U72 U, 31-47, 1999. K.J. Bathe, S. Rugonyi and S. De, �On the Current State of Finite Element Methods � Solids and Structures with Full Coupling to Fluid Flows�, Proceedings of Plenary Lectures of the Fourth International Congress on Industrial and Applied Mathematics (JM Ball and JCR Hunt, eds), Oxford University Press, 2000. K.J. Bathe, A. Iosilevich and D. Chapelle, �An Evaluation of the MITC Shell Elements�, Computers & Structures, U75 U, 1-30, 2000. K.J. Bathe, A. Iosilevich and D. Chapelle, �An Inf-Sup Test for Shell Finite Elements�, Computers & Structures, U75 U, 439-456, 2000. D. Chapelle and K.J. Bathe, �The Mathematical Shell Model Underlying General Shell Elements�, Int. J. for Numerical Methods in Engineering, U48 U, 289-313, 2000. D. Pantuso, K.J. Bathe and P.A. Bouzinov, �A Finite Element Procedure for the Analysis of Thermo-Mechanical Solids in Contact�, Computers & Structures, U75 U, 551-573, 2000.

Additional reading

UGS Corp. 289

W. Bao, X. Wang, and K.J. Bathe, �On the Inf-Sup Condition of Mixed Finite Element Formulations for Acoustic Fluids�, Mathematical Models & Methods in Applied Sciences, U11 U, no. 5, 883-901, 2001. K.J. Bathe, �The Inf-Sup Condition and its Evaluation for Mixed Finite Element Methods�, Computers & Structures, U79 U, 243-252, 971, 2001. N. El-Abbasi and K.J. Bathe, �Stability and Patch Test Performance of Contact Discretizations and a New Solution Algorithm �, Computers & Structures, U79 U, 1473-1486, 2001. J.-F. Hiller and K.J. Bathe, �Higher-Order-Accuracy Points in Isoparametric Finite Element Analysis and Application to Error Assessment�, Computers & Structures, U79 U, 1275-1285, 2001. X. Wang, K.J. Bathe and J. Walczak, �A Stress Integration Algorithm for J3-dependent Elasto-Plasticity Models�, in Computational Fluid and Solid Mechanics (K.J. Bathe, ed.), Elsevier Science, 2001. M. Kawka and K.J. Bathe, �Implicit Integration for the Solution of Metal Forming Processes�, in Computational Fluid and Solid Mechanics (K.J. Bathe, ed.), Elsevier Science, 2001. D. Chapelle and K.J. Bathe, �Optimal Consistency Errors for General Shell Elements,� C. R. Acad. Sci. Paris, t.332, Serie I, 771-776, 2001. K.J. Bathe and F. Brezzi, �Stability of Finite Element Mixed Interpolations for Contact Problems�, Proceedings della Accademia Nazionale dei Lincei, s. 9, U12 U, 159-166, 2001. P. S. Lee and K.J. Bathe, �On the Asymptotic Behavior of Shell Structures and the Evaluation in Finite Element Solutions�, Computers & Structures, U80 U, 235-255, 2002.

Additional reading


K.J. Bathe, J.F. Hiller and H. Zhang, �On the Finite Element Analysis of Shells and their Full Interaction with Navier-Stokes Fluid Flows� (B.H.V. Topping, ed.), Computational Structures Technology, 2002. K.J. Bathe, D. Chapelle and P.S. Lee, �A Shell Problem �Highly-Sensitive� to Thickness Changes�, Int. Journal for Numerical Methods in Engineering, U57 U, 1039-1052, 2003. J.F. Hiller and K.J. Bathe, �Measuring Convergence of Mixed Finite Element Discretizations: An Application to Shell Structures�, Computers & Structures, U81 U, 639-654, 2003. K.J. Bathe, P.S. Lee and J.F. Hiller, �Towards Improving the MITC9 Shell Element�, Computers & Structures, U81 U, 477-489, 2003. F.J. Montáns and K.J. Bathe, �On the Stress Integration in Large Strain Elasto-Plasticity�, in Computational Fluid and Solid Mechanics 2003 (K.J. Bathe, ed.), Elsevier Science, 2003. K.J. Bathe and F.J. Montáns, �On Modeling Mixed Hardening in Computational Plasticity, Computers & Structures, U82 U, 535-539, 2004. D. Chapelle, A. Ferent and K.J. Bathe, �3D-Shell Elements and Their Underlying Mathematical Model�, Mathematical Models & Methods in Applied Sciences, U14 U, 105-142, 2004. P.S. Lee and K.J. Bathe, �Development of MITC Isotropic Triangular Shell Finite Elements�, Computers & Structures, U82 U, 945-962, 2004. N. El-abbasi, J.W. Hong and K.J. Bathe, �The Reliable Solution of Contact Problems in Engineering Design�, International Journal of Mechanics and Materials in Design, U1 U, 3-16, 2004.

Index

UGS Corp. 291

Index

2

2-D solid elements, 12, 49 formulations, 54 mass matrices, 56 material models, 54 numerical integration, 56 plane strain, 49 recommendations for use, 56

2 P

ndP Piola-Kirchhoff stresses, 82

3

3-D solid elements, 12, 57 formulation, 64 mass matrices, 65 material models, 64 numerical integration, 65 recommendations for use, 66

3D-iterative solver, 262

5

5 degrees of freedom node, 41

6

6 degrees of freedom node, 42

A

Accumulated effective plastic strain, 99

Applied temperatures, 220 Arc length method, 234, 236, 244 Arruda-Boyce material model, 118 ATS method, 231, 243

low speed dynamics, 230

B

Beam elements, 20

cross-sections for, 24 elastic, 22 elasto-plastic, 23 mass matrices, 30 numerical integration, 27 off-centered, 22 shear deformations, 26 warping effects, 26

Bernoulli-Euler beam theory, 21 Bolt feature, 283 Bolt preloads, 221 Boundary conditions

displacement, 208 essential, 208 force, 208 moment, 208 natural, 208

Bubble functions, 62

C

Cauchy stresses, 78, 82 Centrifugal loads, 216, 217 Composite shell elements, 44 Composite time integration, 252 Concentrated loads, 211 Concentrated mass element, 74 Consistent contact surface stiffness,

151 Constraint equations, 221 Constraint-function method, 138, 141 Contact algorithm, 140, 144 Contact birth/death, 151 Contact compliance, 155 Contact damping, 159, 171 Contact detection, 156 Contact features, 154 Contact oscillations, suppressing, 157 Contact pairs, 130 Contact set properties, 144

Index


Contact surface compliance, 150 Contact surface depth, 146 Contact surface offsets, 145 Contact surfaces, 128, 130

constraint-function method, 138, 141

convergence, 168 initial penetration, 147 rigid target method, 140, 172 segment method, 140

Contact surfaces extension, 149 Convergence criteria, 238

contact, 240 energy, 239, 241, 244, 257 force and moment, 239, 242, 257 translation and rotation, 239, 244,

257 Creep laws, 107

exponential, 108 power, 107

Creep strains O.R.N.L. rules for cyclic loading

conditions, 108 strain hardening, 108

D

Damper elements, 12, 67 Deformation gradient tensor

elastic, 85 inelastic, 85 total, 83

Deformation-dependent distributed loads, 214

Deformation-dependent pressure loads, 214

Director vectors, 34 Displacement-based finite elements,

52, 60 Distributed loads

beam, 215 deformation dependent, 214

Dynamic analysis, 252, 256

E

Effective plastic strain, 99 Elastic-creep material models, 101 Elastic-isotropic material model, 89,

91 Elastic-orthotropic material model, 89,

91 3-D solid elements, 91

Elasto-plastic material model, 95 Element birth/death, 227, 275 Element death due to rupture, 281 Element locking, 48 Elements

2-D solid, 12, 49 3-D solid, 12, 57 beam, 12 concentrated mass, 74 dampers, 12, 67 gap, 73, 75 general, 67 line, 12, 18 masses, 12, 67 other, 73 rigid, 70 rod, 12 R-type, 12, 69 scalar, 12, 67 shell, 12, 30 solid, 12, 57 springs, 12, 67 surface, 12, 49

Enforced displacements, 219, 220 Enforced motion, 219 Engineering strains, 78, 80 Engineering stresses, 78, 82 Equilibrium iterations

full Newton method, 228 Explicit dynamic analysis, 265 Exponential creep law, 108

Index

UGS Corp. 293

F

Five degrees of freedom node, 41 Formulations for

2-D solid elements, 54 3-D solid elements, 64 rod elements, 19 shell elements, 38

Friction basic models, 152 pre-defined models, 152

Friction delay, 151 Full Newton iterations, 227, 228

line searches, 228

G

Gap element, 73, 75 Gap override, 148 Gasket material model, 124 General elements, 67 Green-Lagrange strains, 79, 80

H

Hermitian beam elements, 20 Hyperelastic material models, 112 Hyper-foam material model

3-D analysis, 121 axisymmetric analysis, 121 plane strain analysis, 121 selection of material constants, 121

I

Implicit time integration, 252 trapezoidal rule, 253, 257

Improperly supported bodies, 170 Incompatible modes finite elements,

62 Inelastic deformations, 85 Inertia loads, 216 Initial conditions, 273

Isotropic hardening, 95 Iterative multi-grid solver, 260

K

Kinematic hardening, 95 Kirchhoff stresses, 82

L

Large displacement formulation, 19, 24, 38, 89, 100

Large displacement/large strain formulation, 23, 55, 64, 79, 96, 112

Large displacement/small strain formulation, 23, 55, 64, 78, 96, 103

LDC method, 234, 244 Limiting maximum incremental

displacement, 172 Line elements, 12, 18 Line search, 228, 229 Linear dynamic analysis, 252 Linear formulation, 38, 55, 64, 89 Linear static analysis, 226, 269, 270 Loading

centrifugal, 216, 217 concentrated, 211 inertia, 216 mass-proportional, 216, 219

Logarithmic strains, 81 Low speed dynamics, 230

M

Mass elements, 12, 67 Mass matrices for

2-D elements, 56 shell elements, 48

Mass matrix, 254, 267 Mass scaling, 272 Mass-proportional loads, 216, 219 Material models, 76

Arruda-Boyce, 118

Index


elastic-creep, 101 elastic-isotropic, 89, 91 elastic-orthotropic, 89, 91 elasto-plastic, 95 gasket, 124 hyperelastic, 112 Mooney-Rivlin, 114 nonlinear elastic, 92, 93 plastic-bilinear, 95 plastic-multilinear, 95 temperature-dependent elastic, 100 thermal elasto-plastic, 101 thermal isotropic, 100 thermal orthotropic, 100 thermal strain effect, 122

Material models for 2-D solid elements, 54 3-D solid elements, 64 rod elements, 19, 54 shell elements, 38

Materially-nonlinear-only formulation, 38, 55, 64, 96, 100, 103

Matrices for 3-D solid elements, 65 beam elements, 30

Memory allocation, 259 in-core solution, 284, 285 out-of-core solution, 284, 285

Mesh glueing, 222 MITC, 32 Mixed Interpolation of Tensorial

Components, 32 Mixed-interpolated finite elements,

53, 61 Mixed-interpolation formulation, 97,

115 Modeling of gaps, 94 Mooney-Rivlin material model, 114

selection of material constants, 115 Multilayer shell elements, 44

N

Nominal strains, 81 Nonconvergence, 240, 242 Nonlinear dynamic analysis, 256 Nonlinear elastic material model, 92,

93 Nonlinear static analysis, 227

selection of incremental solution method, 242

Non-positive definite stiffness matrix, 259

Numerical integration for 2-D solid elements, 56 3-D solid elements, 65 beam elements, 27 rod elements, 19

O

O.R.N.L. rules for cyclic loading conditions, 108

Ogden material model selection of material constants, 118

P

Parallel processing, 284 rigid target contact algorithm, 284

Plastic strains, 105 Plastic-bilinear material model, 95 Plastic-multilinear material model, 95 Positive definite stiffness matrix, 226 Post-collapse response, 235 Power creep law, 107 Pre-defined friction models, 152 Pressure loads

deformation-dependent, 214

R

Rayleigh damping, 255, 256, 267 Reactions, 280

Index

UGS Corp. 295

Recommendations for use of shell elements, 48

Restart, 273 Restart with contact, 158 Rigid elements, 70 Rigid target, 284 Rigid target contact algorithm, 172 Rigid target method, 140, 172 Rod elements, 18

formulations, 19 material models, 19, 54 numerical integration, 19

Rotation tensor, 83, 84 R-type elements, 12, 69 Rupture conditions, 99

S

Scalar elements, 12, 67 Second Piola-Kirchhoff stresses, 82 Segment method, 140 Shell elements, 12, 30

4-node, 48 basic assumptions in, 32 composite, 44 director vectors, 34 formulations, 38 locking, 48 mass matrices, 48 material models, 38 MITC, 48 multilayer, 44 nodal point degrees of freedom, 39 recommendations for use, 48 shear deformations, 36 thick, 48 thin, 48

Six degrees of freedom node, 42 Small displacement contact feature,

128, 135 Small displacement formulation, 19,

23, 89, 100

Small displacement/small strain formulation, 22, 38, 55, 64, 78, 96, 103

Solid elements, 12, 57 Solvers, 258

3D-iterative solver, 262 iterative multi-grid solver, 260 sparse solver, 258

Sparse solver, 258 in-core, 259 memory allocation, 259 out-of-core, 259

Spring elements, 12, 67 Stabilized TLA method, 234 Static analysis, 226, 269, 270 Stiffness stabilization, 171, 228, 281 Strain hardening, 108 Strain measures, 80

engineering strains, 78, 80 Green-Lagrange strains, 80 logarithmic strains, 81 stretches, 81

Stress measures, 81 2 P

ndP Piola-Kirchhoff stress, 82

Cauchy stress, 82 Cauchy stresses, 78, 82 engineering stress, 82 engineering stresses, 78, 82 Kirchhoff stress, 82 Second Piola-Kirchhoff stresses, 82

Stretch tensor right, 83

Stretches, 79, 81 Structural vibration, 254 Suppressing contact oscillations, 157 Surface elements, 12, 49

T

Temperature-dependent elastic material models, 100

Thermal elasto-plastic material

Index


models, 101 Thermal isotropic material model, 100 Thermal orthotropic material model,

100 Thermal strains, 87, 104 Tied contact, 134 Time functions, 206, 209 Time step management, 271 TL formulation, 55, 64, 90, 96, 100,

103, 112 TLA method, 233 TLA-S method, 234 Total Load Application method, 233

Trapezoidal rule, 253, 257 True strains, 81

U

UL formulation, 90, 96, 100, 103 ULH formulation, 65, 96, 103 ULJ formulation, 96

V

von Mises yield condition, 29

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