Multi-Step Nonlinear User's Guide - Siemens PLM

SIEMENSSIEMENSSIEMENS

Multi-Step NonlinearUser’s Guide

Contents

Proprietary & Restricted Rights Notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Overview of nonlinear capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1Program architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1Nonlinear characteristics and general recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Supported inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Case control section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Bulk data section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3Nonlinear Parameters: NLCNTL entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4Iteration related output data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4Supported output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5Solver Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

Subcase Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

Subcase analysis type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Subcase sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Cyclic symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2Fourier harmonic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

Element support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

Element Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1Elements in nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2Generalized plane strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2Rigid elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5Error estimator for mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7Progressive failure analysis in solid composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8Chocking elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14Cohesive elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17Crack simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19Formulation of isoparametric elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21

Isoparametric coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21Shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24Example element matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-26Volume integration of element matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27Element loads and equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28Element coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30

Multi-Step Nonlinear User’s Guide 3

Contents

Stress data recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-31

Material support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

Material overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Support for plasticity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Overview of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3User defined materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7Creep analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-24Overview of the Creep Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-31Disable plasticity and creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Multipoint constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Enforced displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

Loads overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Mechanical loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Thermal loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4Defining solution time steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8Bolt preload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-11Initial stress-strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15

Contact conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

Contact Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1Contact Subcase Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1Contact Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2Contact Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6Contact kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11Contact Penalty Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22Contact Sliding and Geometry Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-23Contact and rigid body motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-24Contact Offsets and Initial Penetrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-25Contact Surface and Edge Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-25Contact Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-26Contact Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-29

Glue conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

Overview of Gluing Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1Glue Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2Defining and Selecting Glue Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3Glue Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5Glue preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7

Considerations for nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

4 Multi-Step Nonlinear User’s Guide

Contents

Contents

Discrete system for a nonlinear continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1Finite element formulation for equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-6Displacement sets and reduction of system equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8Nonlinear solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11

Geometric nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

Overview and user interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1Updated element coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

Concept of convective coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6Updated coordinates and net deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7Provisions for global operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-9

Follower forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-10Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11

Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1

Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1Adaptive Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2Newton’s method of iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2Stiffness update strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6

Update principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6Divergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7

Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-9Rudimentary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-9Convergence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10Error functions and weighted normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-11Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-12


Contents

Proprietary & Restricted Rights Notice

© 2016 Siemens Product Lifecycle Management Software Inc. All Rights Reserved.

This software and related documentation are proprietary to Siemens Product Lifecycle ManagementSoftware Inc. Siemens and the Siemens logo are registered trademarks of Siemens AG. NX is atrademark or registered trademark of Siemens Product Lifecycle Management Software Inc. or itssubsidiaries in the United States and in other countries.

NASTRAN is a registered trademark of the National Aeronautics and Space Administration. NXNastran is an enhanced proprietary version developed and maintained by Siemens Product LifecycleManagement Software Inc.

MSC is a registered trademark of MSC.Software Corporation. MSC.Nastran and MSC.Patran aretrademarks of MSC.Software Corporation.

All other trademarks are the property of their respective owners.

TAUCS Copyright and License

TAUCS Version 2.0, November 29, 2001. Copyright (c) 2001, 2002, 2003 by Sivan Toledo, Tel-AvivUniversity, [email protected]. All Rights Reserved.

TAUCS License:

Your use or distribution of TAUCS or any derivative code implies that you agree to this License.

THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY EXPRESSED ORIMPLIED. ANY USE IS AT YOUR OWN RISK.

Permission is hereby granted to use or copy this program, provided that the Copyright, this License,and the Availability of the original version is retained on all copies. User documentation of any codethat uses this code or any derivative code must cite the Copyright, this License, the Availability note,and "Used by permission." If this code or any derivative code is accessible from within MATLAB, thentyping "help taucs" must cite the Copyright, and "type taucs" must also cite this License and theAvailability note. Permission to modify the code and to distribute modified code is granted, providedthe Copyright, this License, and the Availability note are retained, and a notice that the code wasmodified is included. This software is provided to you free of charge.

Availability (TAUCS)

As of version 2.1, we distribute the code in 4 formats: zip and tarred-gzipped (tgz), with or withoutbinaries for external libraries. The bundled external libraries should allow you to build the testprograms on Linux, Windows, and MacOS X without installing additional software. We recommendthat you download the full distributions, and then perhaps replace the bundled libraries by higherperformance ones (e.g., with a BLAS library that is specifically optimized for your machine). If youwant to conserve bandwidth and you want to install the required libraries yourself, download thelean distributions. The zip and tgz files are identical, except that on Linux, Unix, and MacOS,unpacking the tgz file ensures that the configure script is marked as executable (unpack with tarzxvpf), otherwise you will have to change its permissions manually.


Chapter 1: Introduction

1.1 Overview of nonlinear capabilitiesThis book covers nonlinear structural analysis with the solution sequence, SOL 401 - NLSTEP. SOL401 is a multistep, structural solution which supports a combination of static (linear or nonlinear)subcases and modal (real eigenvalue) subcases.

SOL 401 is the structural solution used by the Simcenter Multiphysics environment within thePre/Post application. The Multiphysics environment supports all combinations of structural-to-thermaland thermal-to-structural coupling with the Simcenter Thermal solution. SOL 401 is also supported asa stand-alone NX Nastran solution.

Primary operations for nonlinear elements are updating element coordinates and applied loadsfor large displacements. The geometric nonlinearity becomes discernible when the structure issubjected to large displacement and rotation. Geometric nonlinear effects are prominent in twodifferent aspects: geometric stiffening due to initial displacements and stresses, and follower forcesdue to a change in loads as a function of displacements. The large deformation effect resulting inlarge strains has not been implemented.

Material nonlinearity is an inherent property of any engineering material. Material nonlinear effectsmay be classified into many categories. Included are plasticity, nonlinear elasticity, creep, andviscoelasticity. SOL 401 supports plasticity and creep.

The primary solution operations are time increments, iterations with convergence tests for acceptableequilibrium error, and stiffness matrix updates. The iterative process is based on variations ofNewton's method. The stiffness matrix updates are performed to improve the computationalefficiency, but may be overridden at your discretion.

1.2 Program architectureThe software has a modular structure to separate functional capabilities which are organized under anefficient executive system. The program is divided into a series of independent subprograms, calledfunctional modules. A functional module is capable of performing a pre-defined subset of operations.It is the Executive System that identifies every module to execute by MPL (Module Properties List).

The Executive System processes the input data by IFP (Input File Processor) and the generalinitialization, which are known as Preface,operations. It then establishes and controls the sequenceof module executions in the OSCAR (Operation Sequence Control Array) based on the user-specifiedDMAP (Direct Matrix Abstraction Program) or solution sequence. The Executive System allocatessystem files to the data blocks in the FIAT (File Allocation Table) and maintains a parameter table formodule interface. The Executive System is also responsible for the database management and allthe input and output operations by GINO (General Input/Output Routines).

The functional module consists of a number of subroutines. Modules communicate with each otheronly through secondary storage files, called data blocks (matrix or table). Each module performs acertain function with input data blocks and produces output data blocks. A module may communicate

Multi-Step Nonlinear User’s Guide 1-1


with the Executive System and with other modules through parameters, which may be input and/oroutput variables of the module. Modules utilize main memory dynamically. If the size of the mainmemory is insufficient to complete an operation, the module uses scratch files, which reside in thesecondary storage as an extension of the main memory. This is known as a spill operation.

DMAP is a kind of macro program using a data block oriented language. The solution sequence is acollection of module statements written in the DMAP language tailored to process a sequential seriesof operations, resulting in a specific type of structural analysis. A typical solution sequence consistsof three phases of functional operations: formation, assembly, and reduction of matrices; solutionof equations; and data recovery. Solution sequences that process superelements have built-insuperelement loops in the first and the last phases.

The nonlinear solution sequences have built-in loops in the second phase for subcase changes, loadincrements, and stiffness matrix updates. Nested in this DMAP loop, nonlinear solution processescomprise a number of internal iteration loops. Confining the discussion to SOL 401, the hierarchyof the nonlinear looping is shown in the table below. Central to the nonlinear processes is moduleNLTRD3. The module is self-contained to perform iterations for converged solutions.

Table 1-1. Hierarchy of Nonlinear LoopingName or Loop Type

1 Subcases (boundaries, temperatures, loads,outputs) DMAP Control

2 Time Steps (NLTRD3) Module Control

3 Stiffness Matrix Updates

The actual stiffness update is underDMAP control, but the request fora stiffness update in the middle ofa solution is under Module control.Decomposition is under modulecontrol.

4 Iterations (Vector Arithmetic) Module Control5 Elements (NLEMG) Subroutine Control6 Volume Integration (Gauss Points) Subroutine Control

1.3 Nonlinear characteristics and general recommendationsThe modeling guidelines for nonlinear analysis and linear analysis are summarized as follows:

• The analyst should have some insight into the behavior of the structure to be modeled; otherwise,a simple model should be the starting point.

• The size of the model should be determined based on the purpose of the analysis, the trade-offsbetween accuracy and efficiency, and the scheduled deadline.

• Prior contemplation of the geometric modeling will increase efficiency in the long run. Factorsto be considered include selection of coordinate systems, symmetric considerations forsimplification, and systematic numbering of nodal points and elements for easy classificationof locality.

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Introduction

• Discretization should be based on the anticipated stress gradient, i.e., a finer mesh in the area ofstress concentrations.

• Element types and the mesh size should be judiciously chosen. For example, avoid highlydistorted and/or stretched elements (with high aspect ratio).

• The model should be verified prior to the analysis by some visual means, such as plots andgraphic displays.

Nonlinear analysis requires better insight into structural behavior. First of all, the type of nonlinearitiesinvolved must be determined. The geometric nonlinearity is characterized by large rotations whichusually cause large displacements. Intuitively, geometric nonlinear effects should be significant if thedeformed shape of the structure appears distinctive from the original geometry without amplifying thedisplacements. There is no distinct limit for large displacements because geometric nonlinear effectsare related to the dimensions of the structure and the boundary conditions. The key to this issue is toknow where the loading point is in the load-deflection curve of the critical area.

Additional recommendations are important for nonlinear analysis:

• PARAM,LGDISP,1 must be defined to turn on geometry nonlinearity.

• Material nonlinear effects can also be included. See Support for plasticity analysis and Supportfor creep analysis.

• The nonlinear region usually requires a finer mesh. Use a finer mesh if severe element distortionsor stress concentrations are anticipated.

• The subcase structure should be utilized properly to divide the load or time history forconveniences in data recovery, and database storage control, not to mention changing constraintsand loading paths.

• Many options are available in solution methods to be specified on the NLCNTL and the TSTEP1bulk entries. The defaults should be used on all options before gaining experience.


Introduction

Chapter 2: User Interface

2.1 Supported inputsThe input data structure includes an optional header, executive control section, case control section,and the bulk data section. In general, features and principles for the user interface are consistent withother solution sequences. Any exceptions for SOL 401 are explained in this guide.

Mechanical design is dictated by the strength, dynamic, and stability characteristics of the structure.The software provides the analysis capabilities of these characteristics with solution sequences,each of which is designed for specific applications. The type of desired analysis is specified inthe executive control section by using a solution sequence identification. SOL 401 is designed forstatic and quasi-static.

The basic input data required for a finite element analysis may be classified as follows:

• Geometric data

• Element data

• Material data

• Boundary conditions and constraints

• Loads and enforced motions

• Solution methods

The first three classes of data may not be changed during the course of an analysis whereas the lastthree classes of data may be changed in midcourse via subcases under the case control section.

2.1.1 Case control section

The primary purpose of the case control is to define subcases. The subcase structure provides ameans of changing loads, boundary conditions, and solution methods by making selections fromthe bulk data. In SOL 401, loads and solution methods may change from subcase to subcase.Constraints can be changed from subcase to subcase. As a result, the subcase structure determinesa sequence of loading and constraint paths. The subcase structure also allows you to select andchange output requests. Any commands defined above the subcase specifications are applicable toall the subcases. Commands defined in a subcase supersede any made above the subcases. Thetable below lists the case control commands supported by SOL 401.



Table 2-1. Summary of CaseControlADAPTERRANALYSISBCRESULTSBCSETBEGIN BULKBGRESULTSBGSETBOLTLDBOLTRESULTSCKGAPCRSTRNCYCFORCESCYCSETCZRESULTSDISPLACEMENTDLOADDTEMPECHOEKEELSTRNELSUMESEFORCEGCRSTRN

GELSTRNGPFORCEGPKEGPLSTRNGROUNDCHECKGSTRAINGSTRESSGTHSTRNHARMONICSHOUTPUTINCLUDEINITSJINTEGLABELLINEMAXLINESMEFFMASSMETHODMONVARMPCMPCFORCESNLCNTLNSMOLOAD

OMODESOPRESSOSTNINIOTEMPPARAMPFRESULTSPLSTRNSEQDEPSETSETMCNAMESMETHODSPCSPCFORCESSTATVARSTRAINSTRESSSUBCASESUBTITLETEMPERATURETHSTRNTITLETSTEPWEIGHTCHECK

2.1.2 Bulk data section

Most of the input data is defined in the bulk data section. The table below lists the bulk entriessupported by SOL 401.

ACCELACCEL1BCRPARABCTPARMBCTSETBEDGEBGADDBGPARMBGSETBOLTBOLTFORBOLTFRCBOLTLDBOLTSEQBSURFSCCHOCK3CCHOCK4

CORD2SCORD3GCPENTACPENTCZCPLSTN3CPLSTN4CPLSTN6CPLSTN8CPLSTS3CPLSTS4CPLSTS6CPLSTS8CPYRAMCQUADX4CQUADX8CRAKTPCTETRA

FORCE2GRAVGRDSETGRIDGROUPINCLUDEINITADDINITSMAT1MAT9MAT11MATCIDMATCRPMATCZMATDMGMATFTMATS1

PLOTELPMASSPSOLCZPSOLIDRBARRBE2RBE3RFORCERFORCE1SLOADSPCSPC1SPCADDSPCDSPOINTTABLED1TABLED2



User Interface

CCHOCK6CCHOCK8CHEXACHEXCZCMASS1CMASS2CMASS3CMASS4CONM1CONM2CORD1CCORD1RCORD1SCORD2CCORD2R

CTRAX3CTRAX6CYCADDCYCAXISCYCSETDAREADLOADDTEMPDTEMPEXECHOOFFECHOONEIGRLENDDATAFORCEFORCE1

MATT1MATT9MATT11MPCMPCADDMUMATNLCNTLPARAMPCHOCKPCOMPSPGPLSNPLOADPLOAD4PLOADE1PLOADX1

TABLED3TABLED4TABLEM1TABLEM2TABLEM3TABLEM4TEMPTEMPDTEMPEXTLOAD1TSTEP1VCEV

2.1.3 Parameters

Parameters are used for requesting special features or specifying miscellaneous data. Parametersare initialized in the MPL, which can be overridden by a DMAP initialization. Modules may change theparameter values while the program is running.

There are two types of parameters: user parameters (V,Y,name in the DMAP) and DMAP (non-user)parameters. You can change the default value of user parameters by specifying PARAM data inthe bulk data section, or for some parameters, in the case control section. See the ParameterApplicability Tables in the NX Nastran Quick Reference Guide. The following table lists theparameters supported in SOL 401.

Table 2-2.COLPHEXACOUPMASSF56GRDPNTLGDISPMATNLMAXRATIONOFISROGEOM

OMAXROMPTOPGOUGCORDPOSTPOSTEXTPOSTOPTPRGPSTPROUT

RGBEAMARGBEAMERGLCRITRGSPRGKUNITSYSTINYWTMASS

2.2 Nonlinear EffectsThe parameter LGDISP turns the nonlinear large displacement capability on/off for the staticsubcases. If you define the parameter LGDISP for SOL 401, you must include it in the bulk dataportion of your input file. The single PARAM,LGDISP setting applies to all static subcases.

• PARAM,LGDISP,-1 (default) – Large displacement effects are turned off. Subcases which includeANALYSIS=STATIC are linear static subcases.


User Interface


• PARAM,LGDISP,1 – Large displacement effects are turned on. Subcases which includeANALYSIS=STATIC are nonlinear static subcases.

PARAM,LGDISP,1 turns on large displacement effects, but small strains are assumed.

Material nonlinear effects can also be included. See Support for plasticity analysis and Support forcreep analysis.

2.3 Nonlinear Parameters: NLCNTL entryThe NLCNTL bulk entry can be used to define strategies for the incremental and iterative solutionprocesses. It is difficult to choose the optimal combination of all the options for a specific problem.However, based on a considerable number of numerical experiments, the default option was intendedto provide the best workable method for a general class of problems. You should start with thedefault settings.

The NLCNTL bulk entry defines the parameters for SOL 401 control. The NLCNTL=n case controlcommand selects the NLCNTL bulk entry, and can be defined in a subcase or globally. You can definethe parameters on the NLCNTL bulk entry using the following format.

1 2 3 4 5 6 7 8 9 10NLCNTL ID Param1 Value1 Param2 Value2 Param3 Value3

Param4 Value4 Param5 Value5 -etc-

For example,

NLCNTL 1 EPSU 1E-3 EPSP 1E-3 EPSW 1E-7 ++ CONV PW KSTEP 5 MAXITER 25

See the NLCNTL bulk entry in the NX Nastran Quick Reference Guide for the list of parametersand descriptions.

2.4 Iteration related output dataAt the end of every iteration, the relevant data from the iteration process are printed under thefollowing heading:

TIME Solution time

ITERATION NO Iteration count for the current timestep

DISP Relative error in terms of displacements. See Error functions and weightednormalization.

LOAD Relative error in terms of loads. See Error functions and weightednormalization.

WORK Relative error in terms of work. See Error functions and weighted normalization.

TOTAL STIFFNESSUPDATES

Number of stiffness updates in the current time step.



User Interface

NO. OFBISECTIONS

Number of occurrences of bisection conditions during the iteration. SeeDivergence criteria.

NO. OF ITR DIV Number of occurrences of probable divergence during the iteration. SeeDivergence criteria.

STIFFNESSPARAMETERCURRENT

Value for the current stiffness parameter for the current iteration.

STIFFNESSPARAMETER %CHANGE

% Change in the value for the current stiffness parameter between and prioriteration.

2.5 Supported outputCase Control DescriptionADAPTERR Requests error estimates computed in a statics subcase.

BCRESULTS Requests contact forces, tractions, separation distance, and the totaland incremental slide distances.

BGRESULTS Requests glue forces and tractions.

BOLTRESULTS Requests the bolt force and the axial strain output in a bolt preloadsubcase.

CKGAP Requests gap result output for chocking elements.CRSTRN Requests grid point creep strains on elements.CZRESULTS Requests results output for cohesive elements.DISPLACEMENT Requests displacement output.EKE Requests element kinetic energy output.ELSTRN Requests elastic strain at grid points on elements.ESE Requests the output of the strain energy.FORCE Requests element force output.GCRSTRN Requests gauss point creep strains on elements.GELSTRN Requests elastic strain at gauss points.GPFORCE Requests grid point force balance output.GPKE Requests kinetic energy at grid points in a modal subcase.GPLSTRN Requests gauss point plastic strain output on elements.GSTRAIN Requests strain at gauss points.

GSTRESS Requests stress at gauss points.GTHSTRN Requests thermal strain at gauss points.

HOUTPUT Requests the harmonics for results output in the cyclic and Fouriernormal modes subcase types.

JINTEG Requests output of the j-integral for crack analysis.MEFFMASS Requests modal effective mass output in a modal subcase.MPCFORCES Requests multipoint constraint force output.OLOAD Requests the form and type of applied load vector output.OMODES Requests selects a set of modes for output.


User Interface


OPRESSRequests the solution pressures, which are from Simcenter Thermal inthe context of a coupled Simcenter multi-physics analysis, be includedin the SOL 401 output.

OSTNINI Requests initial strain output when an intial stress or strain is defined.OTEMP Requests solution temperatures output on grid points.

PFRESULTS Requests progressive failure results output for composite solidelements.

PLSTRN Requests grid point plastic strain output on elements.SPCFORCES Requests single-point force of constraint vector output.

STATVAR Requests state variable output computed by an external user definedmaterial routine.

STRAIN Requests element strain output.STRESS Requests element stress output.THSTRN Requests thermal strain at grid points on elements.

2.6 Solver SupportSOL 401 supports the sparse direct solver (default), the element iterative solver, or the PARDISOsolver (NLTRD3 nonlinear solution module). To select the SOL 401 solver type, supply a pair offields on the NLCNTL bulk entry of the form “SOLVER SPARSE”, “SOLVER ELEMITER”, “SOLVERPARDISO”, or “SOLVER MUMPS” . The default is SPARSE.

• The sparse direct solver is a robust and reliable option, well-suited to sparse models whereaccuracy is desired.

• The element iterative solver performs well with solid element-dominated models. It may be afaster choice if lower accuracy is acceptable. You can optionally define the SMETHOD casecontrol command and the ITER bulk entry to alter any of the default options available on theITER entry.

• For problems involving contact and 3D solid elements, the element iterative solver is generallyfaster as compared to the sparse direct solver.

• The PARDISO solver is a hybrid direct-iterative solver, potentially faster with larger numbers ofcores than the sparse solver but with slightly lower accuracy.



Chapter 3: Subcase Types

3.1 Subcase analysis typeThe ANALYSIS case control command defines the subcase analysis type. SOL 401 allows anycombination of the subcase types.

• Static subcase: You include ANALYSIS=STATIC in a subcase.

• Bolt Preload subcase: You include ANALYSIS=PRELOAD in a subcase.

• Modal subcase: You include ANALYSIS=MODAL in a subcase.

• Cyclic Normal Modes: You include ANALYSIS=CYCMODES in a subcase.

• Fourier Normal Modes: You include ANALYSIS=FOURIER in a subcase.

The ANALYSIS case control command does not have a default in SOL 401. You must define it inevery subcase, and it cannot be defined above the subcases (globally).

The modal subcase should include the METHOD case control command which selects the EIGRLbulk entry. The EIGRL entry defines the data needed to perform the real eigenvalue analysis with theLanczos method. The modal subcase automatically includes the stress stiffening from the previousstatic subcase, and can potentially include follower stiffness and spin softening depending on the typeof loading in the previous static subcase. The NLCNTL bulk entry has parameter inputs which allowyou to control the stiffness contributions for the modal subcase.

3.2 Subcase sequencingYou can use the SEQDEP case control command to define any subcase type as sequentiallydependent (SD), or non-sequentially dependent (NSD).

• SEQDEP=YES (default) – the subcase is a SD subcase.

SOL 401 uses time as the variable to increment temperatures and loads in a static subcase.An SD static subcase uses the final time from the previous static subcase for its start time.The start time is used to compute the solution time steps in a static subcase. See DefiningSolution Time Steps.

An SD subcase can receive the final state variables from the previous static subcase. Forexample, plastic strains, creep strains, and displacements.

• SEQDEP=NO – the subcase is a NSD subcase.

A NSD subcase is independent. The start time for a static NSD subcase is 0.0. See DefiningSolution Time Steps.

A NSD subcase does not use any data from a previous subcase, regardless of the parametersettings on the NLCNTL bulk entry.



3.3 Cyclic symmetricThe cyclic solution method takes advantage of cyclic symmetry to reduce the time needed to createand solve a full 360 degree model. To use this method, you create a 3D-solid element model thatrepresents a fundamental segment. The fundamental segment represents a structure that is made upof N repetitions, where each repetition can be obtained by rotating the fundamental segment an anglethat is an integer multiple of 2π/N.

An important feature of this cyclic solution method is the automatic coupling of the translational DOFon the symmetry faces. The CYCSET case control command, which selects the CYCSET bulkentry, or multiple CYCSET entries with the CYCADD bulk entry, defines the coupling. The couplingdefinition is required and must be defined globally. As a result, the MPC equations created by thesoftware are applied in every subcase.

To define the coupling, you select the cyclic source and target regions on the CYCSET bulk entry. Avery useful feature of the coupling definition is that the mesh on the source and target regions canbe dissimilar. In addition, features such as holes in one or both of the symmetry faces are alsopermitted. The software internally computes the correct coupling conditions between the grids onthe source and target faces.

The CYCAXIS bulk entry is also required to define the default cylindrical coordinate system forthe coupling. The origin of this cylindrical system must be at the center of the revolution, and theZ-axis must be consistent with the axial direction.

CYCMODES subcase

A cyclic modes subcase is available and designated with ANALYSIS=CYCMODES in the subcase.The cyclic modes formulation includes the harmonic index, k, which represents an additionaldimension of the vector space that is not present in an "ordinary" modal analysis. For cyclic modelswith an even number of sectors (N is even), the allowable set of harmonics is 0,1, ...., N/2. For cyclicmodels with an odd number of sectors (N is odd), the allowable set of harmonics is 0,1,…, (N-1)/2.

You request the harmonic index values in which you want modes to be computed with theHARMONICS case control command, and a cyclic modal solution occurs for each harmonic indexindependently. For example, if you request 10 modes on the EIGRL bulk entry, and you request amodal solution for the 0th, the 1st, and the 2nd harmonic, a discrete cyclic modal solution occursfor each of these harmonics.

When computing the cyclic modes, the software uses a duplicate sector method. For harmonics k=0and k=N/2, there are distinct eigenvalues, and only one eigenvector component associated with eacheigenvalue. For all other harmonics (0 < k < N/2), each eigenvalue is repeated, and the displacement

vector for each corresponding eigenvalue has two components; the cosine component and the

sine component .

Static, bolt preload, and modal (non-cyclic modes) subcases

The static, bolt preload, and modal (non-cyclic modes) subcases can also be included in the input,and are designated with ANALYSIS=STATICS, ANALYSIS=PRELOAD, or ANALYSIS=MODALdefined in the subcase. These subcases use the MPC equations automatically created by thesoftware, but the displacements in the static and modal subcases are not cyclic. That is, thedisplacements only represent the 0th harmonic, n=1 fundamental sector.



Subcase Types

Any of the subcase types (statics, preload, modal and cyclic modes) can be defined as sequentiallydependent. The parameters STRESSK, SPINK and FOLLOWK on the NLCNTL bulk entry can bedefined to request stress stiffening, spin softening, and follower stiffness, respectively.

Cyclic clocking and normalization for the CYCMODES subcase

As a result of the inherent symmetry with the cyclic modal solution, modes occur in pairs forharmonics 1 through N/2-1, where N is the total number of sectors.

Once NX Nastran computes the normal modes, it uses the initially computed global displacementvectors to do the following:

• The software clocks the eigenvector solution to the fundamental sector. This clocking ensuresthat, for the first mode in a mode pair, the maximum nodal displacement occurs on thefundamental sector.

• If you have selected either the AFNORM or DISP normalization options, the softwarerenormalizes using the maximum displacement relative to all sectors.

The clocking and normalization procedure is as follows.

The displacement result for a single mode and harmonic is represented by the equation:

The global displacement vectors and in a single mode are orthogonal to each other. In

addition, from one mode in a pair is related to from the same pair.

For a travelling wave with equal amplitude in any mode pair, every grid point traverses an ellipse inthree dimensional space. The maximum resultant displacement is the major axis of the ellipse. For agrid point i, the maximum resultant displacement is computed as follows.

is the cyclic cosine displacement vector (three components) at a specific grid point.

is the cyclic sine displacement vector (three components) at a specific grid point.

The software computes the following using the cyclic cosine and sine vectors:

The resultant displacement at each grid point i is computed as:

The software determines the grid point with the maximum resultant displacement. For this grid pointii, the phase angle is computed as:


Subcase Types


This phase angle will be used to clock the displacements to the fundamental sector.

The maximum displacement found at grid point ii is used to compute the normalization factor:

• For AF normalization, the factor is computed as:

where,

ω is the frequency for the mode, and

AFNORM is the parameter setting PARAM, AFNORM which defaults to 1.0.

For the modes considered as rigid body modes, the software sets ω = 1 when computing the AFnormalization factor. The software considers a mode to be a rigid body mode if its frequency isbelow the value of the parameter AFZERO (default=1.0 hz).

• For unit (MAX) normalization, the factor is computed as:

• For mass (MASS) normalization, the factor f=1.0 is used since the eigenvector was already massnormalized when the modes were computed initially.

The cyclic cosine and sine components are then clocked based on the computed values of .

The cyclic components for each mode are then reset to these values:

Cyclic modes subcase input summary

• The automatic coupling definition is required. The inputs for the coupling are described under the‘Automatic Coupling Details’ heading below.

• The ANALYSIS=CYCMODES case control command is defined in the specific subcases in whichyou are requesting the cyclic modes solution method.

• The HARMONICS case control command requests the specific harmonics in which modes arecomputed. "ALL" requests all possible harmonics. If you define the SID of a SET bulk entry,the SET entry lists the harmonic numbers to be computed, including "0" to request the zerothharmonic.



Subcase Types

The maximum harmonic for a model is related to the total number of segments which wouldtheoretically exist to represent the full model.

o For an even total number of segments:

Maximum harmonic = Total number of segments/2.

For example, if a 30 degree segment is modeled, the total number of segments to create afull model is 360/30 = 12. Since 12 is even, the maximum harmonic = 12/2=6.

o For an odd total number of segments:

Maximum harmonic = (Total number of segments-1)/2.

For example, if a 40 degree segment is modeled, the total number of segments is 360/40= 9. Since 9 is odd, the maximum harmonic = (9-1)/2 = 4.

o As a result of the inherent symmetry in the cyclic modal solution, mode pairs exist forharmonic numbers 1 through N/2 -1. The software automatically outputs the mode pairsfor these subcase types for the modes requested with the EIGRL entry. For example, ifyou request 10 modes on the EIGRL entry:

For harmonic index 0 and N/2, 10 modes are computed.

For harmonic numbers 1 through N/2 -1, 20 modes are computed (10 distinct modes).

This behaviour is consistent for modes requested with the OMODES case control command.See the remarks on the OMODES command for details.

• The HOUTPUT case control command optionally requests the harmonics to output modes. "ALL"requests output for every harmonic requested on the HARMONICS command. You can definean integer to select the SID of a SET bulk entry, which lists the harmonic numbers to be output.These IDs are a subset of the IDs requested on the HARMONICS command. The C, S, C*, andS* describers on the HOUTPUT command are not supported by SOL 401.

• The METHOD case control command selects the EIGRL bulk entry which then defines theeigenvalue solution options. For example, the lower and upper frequency ranges and the numberof modes. Since a single EIGRL entry is selected in a subcase, the same EIGRL options areused when the software computes the modes for each harmonic.

Automatic Coupling Details

• The symmetry faces are grouped into source and target regions. To do the automatic coupling,NX Nastran internally rotates the target region grids into the source region grids, it does a meshrefinement on both the source and target, and then creates MPC equations using the target asthe dependent DOF and the source as the independent DOF. The MPC equations are createdbetween any source and target region grids within the user defined search distance (SDISTi)using a weighted area method.

• The mesh on the source and target regions can be dissimilar. Features such as holes in one orboth of the symmetry faces are also permitted.

• It is recommended that the source and target faces have similar geometry. If the source andtarget geometry is different, the software will still couple the appropriate source and target grids,although, the solution accuracy will be comprimised.


Subcase Types


• You must define the automatic coupling globally. The resulting MPC equations are included inall subcases, including any static, preload, and modal (that is, a non-cyclic modes subcasewith ANALYSIS=MODAL).

Automatic coupling input summary

• The CYCAXIS bulk entry is required to define the default cylindrical coordinate system for thecoupling. The origin of this cylindrical system must be at the center of the revolution, and theZ-axis must be consistent with the axial direction.

• The Z-axis of every cylindrical coordinate system referenced by the CYCSET entry must havethe same origin and direction as the z-axis of the default coordinate system selected with theCYCAXIS bulk entry.

• The displacement coordinate system of grid points which are defined on the rotation axis musthave a Cartesian displacement coordinate system. For all other grid points, a cylindricaldisplcement coordinate system is recommended. See Rules for source and target DOF.

• The CYCSET case control selects the CYCSET or CYCSADD bulk entries. The CYCSETcase control must be defined above the subcase level. As a result, the MPCs generated bythe automatic coupling are used in every subcase (cyclic modes, static, and "normal" normalmodes subcases).

• The BSURFS and BCPROPS bulk entries define the regions. These are existing inputs usedto define glue and contact regions.

• The CYCSET bulk entry pairs the source and target face regions.

o The source region selected in a pair must have a smaller positive theta location than thetarget region.

o The software will use the number of segments (NSEG) field to compute the angle betweenthe source and target faces. For example, if a 30 degree segment is modeled, NSEGwould be 12 = (360/30).

o The SDIST field is used to pair source and target grids when creating the MPC equations.From each source grid, the search occurs in both the positive and negative theta DOFdirections. If the SDIST field is undefined, the software will automatically compute the searchdistance. The software computed value is reported in the f06 file.

• The CYCADD bulk entry can optionally be used to combine multiple CYCSET bulk entries. Thevalue defined in the NSEG field on all CYCSET entries referenced by a CYCADD entry must bethe same. A fatal error will occur if any are inconsistent.

• The CYCFORCES case control command optionally requests the MPC force output for the gridswhich are included in the automatic coupling. It can be defined above the subcases (globally)or in a subcase.

Rules for source and target DOF

• If you define SPC conditions on target region DOF with the SPC, SPC1, or SPCD entries, thesoftware reports a warning message that it is ignoring the SPC conditions on the target regionDOF, and the solution continues.



Subcase Types

• If you include a target region DOF on an RBE2, RBAR, or RBE3 element as a dependent DOF,the software reports a warning message that it is ignoring the rigid connections on the targetregion DOF, and the solution continues.

• If you include a source or target region DOF on an MPC bulk entry as a dependent DOF, thesolution ends with a fatal error.

• Grid points which are defined on the Z-axis of the default cylindrical coordinate system must havea Cartesian displacement coordinate system. For the grid points which are defined on the Z-axisand are included in a source or target region, in addition to any conditions that you defined, NXNastran automatically applies the following SPC conditions during the solution.

o For the harmonic index k=0, NX Nastran fixes DOF 1, 2.

o For the harmonic index k=1, NX Nastran fixes DOF 3.

o For all other harmonic index values, NX Nastran fixes all six DOF.

Post-processing the results

NX Nastran outputs results for the fundamental sector. Due to the symmetric nature of the problemand the orthogonal nature of the modes, the results for the entire structure (360 degree model) canbe inferred from the results of the fundamental sector.

• For the 0th harmonic:

=

Where,

n = sector for which results are to be inferred.

= Results corresponding to the fundamental sector at harmonic 0.

results for sector n at harmonic 0.

• For harmonic k (0 < k < N/2),

Where,

N = Total number of sectors.

n = Sector for which results are to be inferred.

k = Harmonic index

= Cosine cyclic component for the k harmonic of the mode being computed for thefundamental sector.

= Sine cyclic component for the k harmonic of the mode being computed for the fundamentalsector.

R = any output quantity of interest. For example, displacement or stress.


Subcase Types


• For harmonic N/2:

Where,

n = Sector for which results are to be inferred.

= results corresponding to the fundamental sector at harmonic N/2.

= results for sector n at harmonic N/2.

3.4 Fourier harmonic solutionA Fourier normal modes subcase is available in SOL 401 for models which include axisymmetricelements. The subcase is designated with the ANALYSIS=FOURIER and HARMONICS=N casecontrol commands in the subcase.

The conventional axisymmetric element includes radial and axial degrees-of-freedom with novariation in theta.

In the Fourier normal modes subcase, the axisymmetric element has radial, axial and thetadegrees-of-freedom. In addition, the degrees-of-freedom are represented with harmonic terms of aFourier series of the form:

where,

c=cos(kθ) and s=sin(kθ),

k is the harmonic number,

are symmetric displacements, and

are antisymmetric displacements.

Both symmetric and antisymmetric displacements are computed by NX Nastran for a particularharmonic k.

With the Fourier normal modes subcase, you request which harmonic numbers a modal solutionshould occur, and the harmonic terms for modal output. For each harmonic number in whichyou request modes and output, the software can compute the symmetric and antisymmetric



Subcase Types

displacements, stress, strain, SPC force and grid point forces. You can use the typical case controlcommands to request the output. You can then optionally use the NX post processor to display thephysical results on either a 3D segment, or on a full 360 degree model display.

The modal solution for each harmonic term is discrete, and independent of other harmonic terms. Forexample, if you request 10 modes on the EIGRL bulk entry, and you request a modal solution forthe 0th, the 1st, and the 2nd harmonic term, a discrete modal solution will occur for each of theseharmonics. You will have 10 modes for the 0th, 10 modes for the 1st, and 10 modes for the 2nd term,and there is no coupling of the mode results between the different harmonics.

Static and modal (non-Fourier normal modes) subcases can also be included in the input, and aredesignated with the case control commands ANALYSIS=STATICS or ANALYSIS=MODAL. Although,the conventional axisymmetric element formulation is used in the static and modal subcases.

The Fourier normal modes subcase can optionally be sequentially dependent on a static subcase.The parameters STRESSK, SPINK and FOLLOWK can optionally be defined on the NLCNTL bulkentry to request the additional stiffness terms computed in the previous static subcase.

In addition to axisymmetric elements, the plane stress and the chocking elements can also beincluded with the Fourier normal modes subcase. In the Fourier normal modes subcase, gausslocations on the chocking element use the axisymmetric Fourier formulation if the location isconsidered chocked. That is, it includes stiffness in the radial, axial and theta degrees-of-freedom,and all degrees-of-freedom are represented using harmonic terms of a Fourier series. To beconsidered chocked, the loads in a previous static subcase should result in the chocked condition,and the consecutive Fourier normal modes subcase should be defined as sequentially dependent. Bydefault, all gauss locations on the chocking elements are considered unchocked in a Fourier normalmodes subcase, and use the plane stress element formulation.

For grid points which are defined on the rotation axis, in addition to any conditions that you defined,NX Nastran automatically applies the following SPC and MPC conditions during the solution.

• For the harmonic index k=0, NX Nastran fixes DOF 1, 2.

• For the harmonic index k=1, NX Nastran fixes DOF 3, and it creates the MPC condition Ux = Uyfor the cosine terms, and the MPC condition Ux = -Uy for the sine terms.

Fourier normal modes subcase input summary

• The ANALYSIS=FOURIER case control command should be defined in the subcase in which youare requesting the Fourier normal modes subcase in SOL 401.

• The HARMONICS case control command requests the specific harmonics in which modes willbe computed. The SET entry then lists the harmonic numbers to be computed, including "0" torequest the zeroth harmonic. Since there is an infinite number of harmonics in the Fourier normalmodes analysis, the describer "ALL" is not supported in the ANALYSIS= FOURIER subcase.

• The HOUTPUT case control command optionally requests the harmonics to output modes. "ALL"requests output for every harmonic requested on the HARMONICS command. An integer can bedefined to select the SID of a SET bulk entry listing the harmonic numbers to be output. TheseID's typically represent a subset of the ID's requested on the HARMONICS command. The C, S,C*, and S* describers on the HOUTPUT command are not supported by SOL 401.


Subcase Types


• The METHOD case control command selects the EIGRL bulk entry, which then defines theeigenvalue solution options. For example, the lower and upper frequency ranges and the numberof modes.



Chapter 4: Element support

4.1 Element OverviewThe following summarizes all of the elements and materials supported in SOL 401.

• The 3D solids elements CTETRA, CHEXA, CPENTA and CPYRAM elements are supported forlinear, geometric nonlinear, and material nonlinear analysis.

• The axisymmetric elements CQUADX4, CQUADX8, CTRAX3, CTRAX6, the plane strainelements CPLSTN3, CPLSTN4, CPLSTN6, CPLSTN8, and the plane stress elements CPLSTS3,CPLSTS4, CPLSTS6, CPLSTS8 are supported for linear, geometric nonlinear, and materialnonlinear analysis.

The grid points on these elements must all lie in either the XZ plane, or all in the XY plane of thebasic coordinate system. The software automatically determines the orientation.

When axisymmetric elements are defined on the XZ plane, X is the radial direction, and Z is theaxial direction. The grid points defining these elements must have X ≥ 0.

When axisymmetric elements are defined on the XY plane, Y is the radial direction, and X is theaxial direction. The grid points defining these elements must have Y ≥ 0.

• A special, generalized plane strain formulation is available using the CPLSTN3, CPLSTN4,CPLSTN6, and CPLSTN8 element types. See Generalized plane strain analysis.

• The chocking element is available. The chocking elements is a special type of axisymmetricelement that are used to model regions in an axisymmetric analysis that can carry a compressivehoop stress, but cannot carry a tensile hoop stress. See Chocking elements.

• The cohesive element is available to model adhesively bonded interfaces. Cohesive elements canaccount for compliance in the connection and damage in the material. See Cohesive elements.

• The RBE2 and RBAR rigid elements are supported with optional large displacement effects andthermal expansion. The RBE3 rigid element is also supported, but it does not support the largedisplacement effects or thermal expansion. See Rigid element support.

• The mass elements CMASSi and CONMi are supported.

• The PSOLID or the PCOMPS bulk entries define the element properties. The PCOMPS isoptionally used to define a layered solid composite property.

You can model progressive ply failure in solid composites. See Progressive failure analysisin solid composites.

• The supported material types include the following.

The MAT1 and MATT1 (temperature dependent) bulk entries define isotropic materials.

The MAT3 and MATT3 (temperature dependent) bulk entries define isotropic materials.



The MAT9 and MATT9 (temperature dependent) bulk entries define anisotropic materials.

The MAT11 and MATT11 (temperature dependent) bulk entries define orthotropic materials.

Plastic and creep materials can optionally be assigned to the 3D solid elements, axisymmetricelements, the plane stress elements, and the plane strain elements. You can enable one or bothplasticity/creep in all subcases, or in specific subcases. See Support for plasticity analysis andSupport for creep analysis.

Externally computed, user defined material models are supported. You can define a materialmodel by developing and compiling an external routine. See User defined materials.

• You can request stress norm, stress error norm, strain energy norm, and strain energy errornorm output. The output is computed and stored on an individual element basis. The Pre/Postapplication uses the output for adaptive meshing. See Error estimator for mesh refinement.

• You can compute and output the j-integral in a crack simulation. The j-integral output canbe requested and used by third-party software like Zencrack to perform a fracture mechanicsanalysis. The CHEXA bulk entry allows for a collapsed element definition. See Crack simulation.

4.2 Elements in nonlinear analysisIn nonlinear finite element analysis, lower-order elements are often preferred over higher-order onesbecause of their robustness and reasonable accuracy at reduced costs. The software supports linearelements, rather than quadratic or cubic elements, to process nonlinearity. When using lower-orderelements, quadrilateral and hexahedral elements are generally preferred over triangular, pentahedralor tetrahedral elements. Triangular and tetrahedral elements can exhibit excessively stiff behavior,and caution is needed when using these elements.

Caution is also needed when different element types are combined in a model, and if theseelements are incompatible. In such cases, some provision (e.g., appropriate constraints) may benecessary at the interface boundary. Modeling the joints (such as bolted, riveted, or welded) isparticularly difficult. For lack of better information, the joints are usually modeled as rigid or free incertain degrees-of-freedom. If improved accuracy is required at such joints, the characteristics ofthe joint (stiffness and/or damping) may have to be identified from experiments or the local analysisof a detailed model at the joint. Modeling of the boundary conditions at the supports poses similardifficulties. Ideal boundaries are represented as free, clamped, pinned, roller or ball joints. Thereality tends to be in smeared condition.

Elements become actively nonlinear if the parameter LGDISP is tumed. As for geometric nonlinearity,the software does not currently support large strain capability. However, large displacement is treatedeffectively by computing element stresses and strains in the updated element coordinates.

4.3 Generalized plane strainA special plane strain formulation called generalized plane strain is available as an optional extensionto the standard plane strain formulation. Both formulations use the CPLSTN3, CPLSTN4, CPLSTN6,and CPLSTN8 plane strain element types. To invoke the generalized plane strain option, theplane strain elements need to reference a PGPLSN property bulk entry. The generalized planestrain option is only supported in SOL 401, and is only applicable to small strain, small deflection



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structural analyses. These structural analyses include linear static, creep, and plasticity analyses,and combination creep and plasticity analysis.

Analysis with the generalized plane strain formulation is highly specialized and typically used toevaluate the behavior of gas turbine compressor and turbine blades. For such an analysis, you meshthe cross section of the blade with CPLSTN3, CPLSTN4, CPLSTN6, or CPLSTN8 elements. All ofthe elements in the mesh should reference a single PGPLSN property bulk entry.

With the PGPLSN bulk entry, you can specify the following data:

• The material bulk entry that is referenced by the PGPLSN bulk entry. MAT1 and MAT3 materialbulk entries can be referenced.

• The control grid point. The control grid point is the location where out-of-plane loads or enforceddisplacements are applied to the set of elements that reference the PGPLSN bulk entry.

• The element thickness in the undeformed state.

• Optional user-defined additive normal stiffness and rotational stiffness values.

• You can specify time-varying added normal and rotational stiffness terms. To do so, in an addedstiffness field on the PGPLSN entry, enter the identification number of a TABLEDi bulk entry. Onthe TABLEDi entry,you will list tabular data to define how the added stiffness term varies with time.

For the generalized plane strain analysis, NX Nastran calculates the standard in-plane plane strainstiffness, but also calculates three net out-of-plane stiffness values relative to the displacementcoordinate system of the control grid point. Consequently, how you specify the displacementcoordinate system for the control grid point is very important. You should specify the displacementcoordinate system of the control grid point such that one axis is normal to the cross section and theother two axes are parallel to the principal axes of the cross section. By so doing, the three netout-of-plane stiffness values that NX Nastran calculates represent the normal stiffness of the crosssection, and the two bending stiffness for symmetrical bending of the cross section.

Because the CPLSTN3, CPLSTN4, CPLSTN6, and CPLSTN8 plane strain elements can only bedefined in the XY- or XZ-planes of the basic coordinate system, the direction normal to the crosssection is always in the Z- or Y-direction, respectively of the basic coordinate system. NX Nastranchecks that one of the axes of the displacement coordinate system of the control grid point coincideswith the correct normal direction and issues an error if one does not.

NX Nastran does not check the other two coordinate directions of the displacement coordinatesystem for the control grid point. It is your responsibility to assure that these directions are parallel tothe principal axes of the cross section.

If you specify additive stiffness, the normal stiffness is added to the normal stiffness that NX Nastrancalculates for the cross section . The additive rotational stiffness values are added to the bendingstiffness values as follows:

• If the model lies in the XY-plane of the basic coordinate system, the KR1 value on the PGPLSNbulk entry is added to the bending stiffness about the X-axis of the displacement coordinatesystem of the control grid point. the KR2 value on the PGPLSN bulk entry is added to the bendingstiffness about the Y-axis of the displacement coordinate system of the control grid point.

• If the model lies in the XZ-plane of the basic coordinate system, the KR1 value on the PGPLSNbulk entry is added to the bending stiffness about the X-axis of the displacement coordinate


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system of the control grid point. the KR2 value on the PGPLSN bulk entry is added to the bendingstiffness about the Z-axis of the displacement coordinate system of the control grid point.

You can apply loads to the control grid point and to the generalized plane strain element mesh. At thecontrol grid point, you can account for the centrifugal force that is attributable to the portion of theblade from the cross section you are modeling to the blade tip. To allow you to define a mechanicallyequivalent loading at the control grid point, you can specify not only a force that acts normal to thecross section, but also the bending moments that act on the cross section about axes parallel to theprincipal axes of the cross section.

To the generalized plane strain mesh, apply surface tractions, body forces, and in-plane enforceddisplacements that you want to include in the analysis. For example, you can apply aerodynamicforces to the grid points that lie on the periphery of the mesh.

From the net out-of-plane stiffness values and the loads that are applied to the control grid point, NXNastran calculates the thickness change over the cross section. Similar to planes remaining plane inpure bending of beams, NX Nastran enforces that the surface defined by the thickness change isplanar. From the thickness change over the cross section, NX Nastran calculates the out-of-planestrain of the elements at the grid locations. During the solution of the finite element model, NXNastran uses the out-of-plane strain and any surface tractions, body forces, and in-plane enforceddisplacements that you specified.

If an enforced displacement and enforced rotations are applied at the control grid point, the thicknesschange of the cross section is directly specified. From the thickness change, NX Nastran calculatesthe out-of-plane strain directly and the solution of the finite element model is as before.

Note the generalized plane strain element is not supported by glue or contact regions.

The following constitutive models are available with generalized plane strain elements:

• To model plasticity of an isotropic material, use the MAT1 and MATS1 bulk entries in combination.

• To model plasticity of an isotropic material with temperature-dependent properties, use somecombination of the MAT1, MATS1, MATT1, TABLEST, and TABLES1 bulk entries.

• To model plasticity of an orthotropic material, use the MAT3 and MATS1 bulk entries incombination. The elastic portion of the response is orthotropic, and the plastic portion of theresponse is isotropic.

• To model plasticity of an orthotropic material with temperature-dependent properties, use somecombination of the MAT3, MATS1, MATT3, TABLEST, and TABLES1 bulk entries. The elasticportion of the response is orthotropic, and the plastic portion of the response is isotropic.

• To model creep of an isotropic material, use the MAT1 and MATCRP bulk entries in combination.

• To model creep of an isotropic material with temperature-dependent properties, use the MAT1,MATT1, and MATCRP bulk entries in combination.

• To model creep of an orthotropic material, use the MAT3 and MATCRP bulk entries in combination.

• To model creep of an orthotropic material with temperature-dependent properties, use the MAT3,MATT3, MATCRP, and TABLEM1 bulk entries in combination. The elastic portion of the responseis orthotropic, and the creep portion of the response is isotropic.



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For additional information, see the PGPLSN bulk entry in the NX Nastran Quick Reference Guide.

4.4 Rigid elementsSOL 401 supports the RBE2, RBAR, and RBE3 elements. In the modal subcases in whichANALYSIS=MODAL, CYCMODES, or FOURIER, the software represents the rigid elements withMPC equations. In the static subcases in which ANALYSIS=STATICS or PRELOAD, the softwarealways represents RBE3 elements with MPC equations, although the RBE2 and RBAR elementshave the following additional options.

RBE2 and RBAR element options in static subcases

In subcases where ANALYSIS=STATICS or PRELOAD, the RBE2 and RBAR rigid elements supportlarge displacement effects and thermal expansion. The RBE3 rigid element is always representedwith MPC equations, and as a result, does not support the large displacement effects or thermalexpansion.

The RIGID case control command includes the AUTO and STIFF options to select the RBE2 andRBAR rigid element behavior. When RIGID=AUTO, which is the default for SOL 401, the behaviordepends on if large displacement effects are turned off with PARAM,LGDISP,-1 (default), or on withPARAM,LGDISP,1. The RIGID case control command must be defined globally, and it applies to allstatic subcases. The input combinations are as follows.

• When RIGID=AUTO and PARAM,LGDISP,-1, the software automatically applies theRIGID=LINEAR option. RBE2 and RBAR elements do not include large displacement effects orthermal expansion in static subcases.

• When RIGID=AUTO and PARAM,LGDISP,1, the software automatically applies the RIGID=STIFFoption. RBE2 and RBAR elements include large displacement effects and thermal expansion instatic subcases.

• When RIGID=STIFF and PARAM,LGDISP,-1, the large displacement effects are not included, ingeneral. RBE2 and RBAR elements include thermal expansion in static subcases.

• When RIGID=STIFF and PARAM,LGDISP,1, the RBE2 and RBAR element behavior is the sameas RIGID=AUTO and PARAM,LGDISP,1. RBE2 and RBAR elements include large displacementeffects and thermal expansion in static subcases.

• When RIGID=LINEAR, the RBE2 and RBAR elements do not include large displacement effectsor thermal expansion in static subcases. This behavior is independent of the PARAM,LGDISPsetting.

To compute large displacement effects and thermal expansion, the software internally replaces theRBE2 and RBAR elements with either a stiff beam element, or a stiff spring element. A coincident gridtolerance is used to determine if a beam or a spring is used. For the RBAR, if the distance betweenthe connecting grids is less than the tolerance, the stiff spring formulation is used. For the RBE2, ifthe distance between the grid defined in the GN field on the RBE2 entry, and any of the grids definedin the GM fields on the RBE2 entry, is less than the tolerance, the stiff spring formulation is used. Youcan optionally define the coincident grid tolerance explicitly with the parameter RGLCRIT. By default,it is automatically computed by the software:

Coincident Grid Tolerance = 1E-6 * LMODEL (units=length)


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where LMODEL is the largest dimension of the model determined by the software.

You can optionally define the beam stiffness and area explicitly using the parameters RGBEAME andRGBEAMA, respectively. By default, they are automatically computed by:

Beam Stiffness = 1e+2 * EMAX (units=force/length^2)

Beam Area = (LMODEL * 1e-2)^2 (units=length^2)

where EMAX is the largest Young’s modulus in the model. If no material is specified in the model,EMAX is set to 1.0E12.

You can optionally define the spring stiffness explicitly using the parameter RGSPRGK. By default, itis automatically computed by:

Spring Stiffness = EMAX * LMODEL (units = force/length)

Additional information:

• MPCFORCE and GPFORCE output are supported with all of the rigid elements. Since thesoftware internally replaces an RBAR or RBE2 with a stiff beam or spring element whenRIGID=STIFF, these elements are no longer represented as MPC equations. As a result,MPCFORCE output is not applicable to these elements. GPFORCE and FORCE output isapplicable.

• GPFORCE output will correctly account for large displacements, except for DOF which areincluded in MPC equations.

• In general, MPCFORCE output can be requested with large displacements (PARAM,LGDISP,1).Although, it is computed based on the initial, undeformed configuration. MPCFORCE output maynot be accurate in regions where large displacements occur.

• The TEMP(LOAD) and TEMP(INIT) value used on RBAR elements is an average calculatedfrom the grid point values. On RBE2 elements, an average TEMP(LOAD) and TEMP(INIT) iscalculated for each leg of the element using the values on the independent/dependent grid pairssuch that each leg can have a different thermal strain if the temperatures vary at the grids.

The rigid element thermal strains are calculated from

εthermal = α(AVGTEMP(LOAD) – AVGTEMP(INIT))

If TEMP(LOAD) or TEMP(INIT) are not defined, they are assumed to be zero.

• If you combine static subcases in which RBE2 and RBAR elements include large displacementeffects and modal subcases in your input file, the RBE2 and RBAR elements are treated as stiffbeams in a statics subcase, but are still treated as MPC equations in a modal subcase. If yourrigid elements experience large rotations in a statics subcases, a sequentially dependant modalsubcase will use the deformed state from the previous statics subcase, but the MPC equationsare still relative to the unchanging global coordinate system.

See the RIGID case control command.



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4.5 Error estimator for mesh refinementYou can request stress norm, stress error norm, strain energy norm, and strain energy error normoutput when using SOL 401. The output is computed and stored on an individual element basis.Pre/Post uses the output for adaptive meshing.

The output is supported for the following element types:

Solid elements CHEXA, CPENTA, CPYRAM, CTETRA (excludes CHEXA and CPENTAelements referencing PCOMPS bulk entries)

Axisymmetric elements CQUADX4, CQUADX8, CTRAX3, CTRAX6

Plane strain elements CPLSTN3, CPLSTN4, CPLSTN6, CPLSTN8 (includes elementsreferencing PGPLSN bulk entries)

Plane stress elements CPLSTS3, CPLSTS4, CPLSTS6, CPLSTS8

• The stress norm is calculated from:

• The stress error norm is calculated from:

• The strain energy norm is calculated from:

• The strain energy error norm is calculated from:

where

Ω is the element volume,

σunaveraged is the unaveraged stress vector,

σaveraged is an averaged stress vector computed at a grid point using the stress vectors from elementsconnected to the grid point,

D matrix is the constitutive relation.

When computing σaveraged, stress values are not averaged across different element families, materialproperties, material coordinate systems, orientation angles in 2D solid elements, and thicknesses inplane stress elements.

You use the STRESS, STNERGY and STEP describers on the ADAPTERR case control commandto request the output.


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• The ADAPTERR case control command must be defined above the subcases (globally).

• The STRESS describer requests the stress norm and the stress error norm.

• The STNERGY describer requests the strain energy norm and strain energy error norm.

• You can specify both the STRESS and STNERGY describers to request stress norm, stress errornormal, strain energy norm and strain energy error norm output.

• The software always outputs the maximum value on each element, for each output typerequested, by comparing the values from all solution steps. In addition, if you specify the STEPdescriber, the software will output what you have requested at the output increment steps definedwith the TSTEP1 entries.

For additional information, see the ADAPTERR case control command.

4.6 Progressive failure analysis in solid compositesYou can model progressive ply failure in composite laminates that are meshed with solid elements.NX Nastran supports a unidirectional ply failure model that is based on a model developed byLadeveze and Le Dantec (Damage modeling of the elementary ply for laminated composites,Composites Science and Technology 43, 1992) in which damage is linked to the transverse normalstress and in-plane shear stress. The model that is supported in NX Nastran can also account for:

• Damage in the fiber direction.

• Damage linked to the stresses in the out-of-plane direction.

• Damage linked to time delay effects.

• Coupling with plasticity.

• Stress/strain-dependent elastic modulus.

The procedure for estimating ply damage is iterative. Using the material properties in the undamagedstate, NX Nastran makes an initial calculation of the stress state. NX Nastran uses this stress state inthe unidirectional ply model to calculate an initial estimate of ply damage. Based on these damagevalues, NX Nastran calculates the material properties in the damaged state. With these values for thematerial properties, NX Nastran solves the model to obtain the updated stress state. NX Nastranuses the updated stress state in the unidirectional ply model to calculate a second estimate of the plydamage, and continues iterating until the ply damage values converge.

The converged damage values are termed the static damage. If you optionally include time delayeffects, the final damage values for the time step are the static damage values adjusted for thetime delay.

To use the progressive ply failure capability, model the laminate with CHEXA and CPENTA solidelements that reference PCOMPS bulk entries. In the MIDi fields of the PCOMPS bulk entries, enterthe MID of MAT11 bulk entries to define the linear elastic properties of the plies in the undamagedstate. To define the material properties and parameters that are related to progressive ply failuremodel, include MATDMG bulk entries that have the same MID as the MAT11 bulk entries. To obtainply failure results output, include a PFRESULTS case control command.



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Because progressive ply failure is applicable to composite laminates that are meshed with solidelements, you can optionally use CHEXCZ and CPENTCZ cohesive elements to model the interfacebetween the solid elements.

Unidirectional ply model

The unidirectional ply model uses the following equation for the strain energy density at a point in aply. This equation accounts for damage to the ply and is used to formulate expressions for elementstiffness and thermodynamic force.

Equation 4-1.

where:

• The 1-direction is the fiber direction.

• The 2-direction is the in-plane transverse direction.

• The 3-direction is the out-of-plane transverse direction.

• d11, d22, and d12 are the damage variables.

• EI0, Gij0, and νij0 are the elastic modulus, shear modulus, and Poisson’s ratio, respectively forthe undamaged material.

• λ is a parameter whose value is either zero or one that controls whether or not damage is linkedto out-of-plane stresses.

Note

‹x›+ means use the value for x when x > 0, and use x = 0 when x ≤ 0. Similarly, ‹x›- meansuse the value for x when x < 0, and use x = 0 when x ≥ 0.

The unidirectional ply model uses thermodynamic forces to predict ply damage. Thermodynamicforces are derived from the strain energy density as indicated in the following equation.


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Equation 4-2.

The criteria that the unidirectional ply model uses to predict ply damage are indicated in the followingequation.

Equation 4-3.

where:

• dijs are the static damage values.

• Y11lim+ and Y11lim- are energy threshold values in tension and compression, respectively.

• Y120 is the energy threshold for shear damage d12.

• Y12C is the critical value of energy for shear damage d12.

• b3 is the coupling coefficient between damage variables.

In Equation 4-3, Ymax is defined from Equation 4-4.

Equation 4-4.

where t is the time at the end of the current time step.

In Equation 4-4, Y(t) is referred to as the equivalent thermodynamic force, and it is given by Equation4-5.

Equation 4-5.



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where b2 is also a coupling coefficient.

In Equation 4-3, Y12F and Y22F are referred to as the thermodynamic forces in fragile behavior.They are defined by Equation 4-6.

Equation 4-6.

where η is the transition thickness of the ply, h is the thickness of the ply, and Y12S and Y22S arethe transverse fissuration thresholds.

Note

sup (supremum) evaluates to the least upper bound of the arguments.

Time delay effects

The unidirectional ply model can optionally include time delay effects. Time delay effects smooth theoccurrence of damage. Equation 4-7 shows how the rate of damage accumulation is calculated.

Equation 4-7.

where τc is a time constant, ac is a parameter for delay, and dmax is the maximum allowable valueof damage.

dijS are the static damage values calculated from Equation 4-3.

Coupling with plasticity

The unidirectional ply model includes coupling with plasticity. The plasticity calculations use theeffective stress definitions in Equation 4-8.


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Equation 4-8.

The software uses the effective stresses in Equation 4-9 to predict when yielding occurs.

Equation 4-9.

where:

• a is the coupling coefficient.

• p is the cummulative plastic strain.

• R0 is the initial plasticity threshold.

• R(p) is the yield function.

The form of the yield function that the software uses is given by Equation 4-10.

Equation 4-10.

where K and γ are empirically-derived material constants.

The software calculates the rate of plastic strain accumulation from Equation 4-11.



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Equation 4-11.

where:

Equation 4-12.

Nonlinear traction and compression

The unidirectional ply model can optionally include a nonlinear elastic modulus in the fiber directionas indicated in Equation 4-13.

Equation 4-13.

where ζ+ and ζ- are nonlinearity coefficients in tension and compression, respectively.

Specifying material properties and parameters for the unidirectional ply model

Table 4-2 shows where you specify the various material properties and parameters used in theunidirectional ply model.

Material property or parameter Bulk entry (field name)Ei0, Gij0, νij0 MAT11λ MATDMG (PE field)Y11lim+ MATDMG (Y11LIMT field)Y11lim- MATDMG (Y11LIMC field)Y120 MATDMG (Y012 field)Y12C MATDMG (YC12 field)b2 MATDMG (B2 field)


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Material property or parameter Bulk entry (field name)b3 MATDMG (B3 field)η MATDMG (HBAR field)h PCOMPS (TRi field)Y12S MATDMG (YS12 field)Y22S MATDMG (YS22 field)τc MATDMG (TAU field)ac MATDMG (ADEL field)dmax MATDMG (DMAX field)a MATDMG (A field)R0 MATDMG (LITK field)K MATDMG (BIGK field)γ MATDMG (EXPN field)ζ+ MATDMG (KSIT field)ζ- MATDMG (KSIC field)

4.7 Chocking elementsChocking elements are a special type of axisymmetric element that are used to model regions inan axisymmetric analysis that can carry a compressive hoop stress, but cannot carry a tensile hoopstress. Chocking elements behave like axisymmetric elements when a compressive hoop stress ispresent, and behave like plane stress elements otherwise.

Chocking elements complement the existing modeling capabilities of axisymmetric elements andplane stress elements as follows:

• Axisymmetric elements are used to model regions that carry a hoop stress.

• Plane stress elements may be used to model regions in an axisymmetric analysis that do notcarry a hoop stress.

Like plane stress elements in axisymmetric analysis, you use chocking elements to model regionswhere the axisymmetric geometry is violated by regularly-spaced features like holes or keyways.However, you use chocking elements where the potential for a compressive hoop stress exists.

Tip

As a rule, only use chocking elements in combination with axisymmetric elements. A meshcomprised solely of chocking elements may lead to singularities, convergence issues, anderroneous results.

An example is the shrouding around the periphery of turbine blades in an aircraft engine.



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A generic turbine assembly is shown in the figure. You can see that the shroud is constructed fromdiscrete segments that are attached to each turbine blade. A small gap exists between each segment.If the combination of mechanical and thermal loads is such that these gaps close, the shroud cansustain a compressive hoop stress. To account for this behavior in an axisymmetric model, you canmesh the shroud cross section with chocking elements.

The behavior of chocking elements depend on whether large displacements are enabled.

Linear analysis

When large displacements are not enabled, the gap status at the beginning of the analysis is usedthroughout the analysis. At any Gauss point location where there is no initial gap, the contributionto the elemental stiffness matrix from that Gauss point is based on the axisymmetric formulation ofthe chocking element.

Note

The axisymmetric formulation of chocking elements sustains tensile, as well ascompressive hoop stress. Because the gap distance does not update in a linear analysis,overly stiff results can occur for loadings that tend to increase the gap distance.

At any Gauss point location where there is an initial gap, the contribution to the elemental stiffnessmatrix from that Gauss point is based on the axisymmetric formulation of the chocking element withthe Eθ, νθr, and νθz elastic constants reduced by a factor of 1 x 106. Doing so causes the elementstiffness to be essentially identical to a plane stress formulation with σθ as the out-of-plane normalstress.

When large displacements are not enabled, the stiffness does not reformulate unless plasticity isenabled. When this occurs, the stiffness reformulates to account for plasticity effects only.

Geometric nonlinear analysis

When large displacements are enabled, the solution is iterative. For the initial iteration, at any Gausspoint location where there is no initial gap, the contribution to the elemental stiffness matrix from thatGauss point is based on the axisymmetric formulation of the chocking element. At any Gauss pointlocation where there is an initial gap, the contribution to the elemental stiffness matrix from thatGauss point is based on the axisymmetric formulation of the chocking element with the Eθ, νθr, andνθz elastic constants reduced by a factor of 1 x 106. Doing so causes the element stiffness to be


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essentially identical to a plane stress formulation with σθ as the out-of-plane normal stress. From thisstiffness formulation and the prevailing mechanical and thermal loads, the software calculates theout-of-plane engineering strain, εθ, at each Gauss point for the initial iteration.

An expression for the gap size at the end of the iteration for each Gauss point is obtained as follows:

1. In the deformed configuration, the circumferential distance occupied by the chocking elementis given by

2πr – Ng

where N is the number of gaps, g is the gap size at the end of the iteration and r is the radiusat the end of the iteration.

2. Because εθ is an engineering strain, and engineering strains are based on undeformed lengths,the circumferential distance occupied by the chocking element in the deformed configurationis also given by

(1 + εθ) (2πr0 – Ng0)

where r0 is the initial radius and g0 is the initial gap size.

3. Equating the above terms for the circumferential distance occupied by the chocking element inthe deformed configuration and solving for the gap size at the end of the iteration yields

g = (1 / N) [2πr – (1 + εθ) (2πr0 – Ng0)]

Using this expression for the gap size at the end of the iteration, the software calculates whether ateach Gauss point the element is chocked (g ≤ 0) or unchocked (g > 0).

The second and all successive iterations use the gap status at the end of the previous iteration toformulate the stiffness for the current iteration. If the element is chocked at a Gauss point, thecontribution to the elemental stiffness matrix from that Gauss point for that iteration is based on theaxisymmetric formulation of the chocking element. If the element is unchocked at a Gauss point, thecontribution to the elemental stiffness matrix from that Gauss point for that iteration is based on theaxisymmetric formulation of the chocking element with the Eθ, νθr, and νθz elastic constants reducedby a factor of 1 x 106. Doing so causes the element stiffness to be essentially identical to a planestress formulation with σθ as the out-of-plane normal stress.

For applications of chocking elements like the turbine shroud, the mesh of chocking elements isconnected to axisymmetric elements. Because the stiffness in the radial direction for such a modelis relatively large, even when a positive gap exists, the incremental radial displacements that thesoftware calculates during the iterative solution are relatively small and the solution is able toconverge.

However, if the model is extremely compliant in the radial direction, the radial coordinates that thesoftware calculates to reformulate the stiffness may be negative, which is physically impossible.When the software detects a negative radial coordinate, it does not reformulate the stiffness matrixand it uses the initial circumferential distance for the deformed configuration at the next iteration.

To enable large displacements, specify PARAM,LGDISP,+1 in the input file.



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Surface tractions on chocking elements

Use the PLOADX1 bulk entry to apply surface tractions to the edges of chocking elements. When thesoftware calculates the equivalent grid point forces from the surface traction data on a PLOADX1bulk entry, it accounts for the presence of gaps.

For example, suppose you use a PLOADX1 bulk entry to apply a pressure along the edge of achocking element. If the edge is directed in the axial direction, the total force applied to the chockingelement over 2π radians is:

p (2πr – Ng) L

where p is the pressure, r is the radial coordinate of the edge, N is the number of gaps, g is the gapsize, and L is the length of the edge.

Chocking element types

Four elements support the chocking capability. They are:

• CCHOCK3 – A triangular chocking element

• CCHOCK4 – A quadrilateral chocking element

• CCHOCK6 – A triangular chocking element with midside nodes

• CCHOCK8 – A quadrilateral chocking element with midside nodes

All four chocking elements must reference the PCHOCK property bulk entry. On the PCHOCK bulkentry, you specify the material property for the chocking element and the number of gaps. You canspecify the initial gap thickness on either the PCHOCK bulk entry or on the chocking element bulkentries that reference the PCHOCK bulk entry. If the initial gap thickness is specified on both, thespecification on the chocking element bulk entry takes precedence. If the gap is of uniform thickness,use the GAPT field on the PCHOCK bulk entry to specify the gap thickness. If the gap thicknessvaries through the cross section, use the GAPi fields on the chocking element bulk entries, or use acombination of GAPT specifications and GAPi overrides, to specify the gap thickness.

To request gap results output, use the CKGAP case control command.

4.8 Cohesive elementsYou can use cohesive elements to model adhesively bonded interfaces. The advantages of cohesiveelements over traditional glue connections in NX Nastran are that with cohesive elements, youcan account for:

• Compliance in the connection.

• Damage in the material.

You can define cohesive elements with the CHEXCZ and CPENTCZ elements. You can definethese elements as occupying a solid volume or a planar area. To define the element geometrysuch that it occupies a planar area, define coincident grid points on each edge that connects thetop and bottom faces.


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For example, to define the CPENTCZ element as a planar area, G1, G10, and G4 must have thesame coordinates. G1 is also included in the connectivity of an element that is part of the mesh onone side of the interface and G4 is also included in the connectivity of an element that is part of themesh on the other side of the interface. Mid-side grid points on edges that connect the top andbottom faces like G10 are exclusive to the cohesive element mesh.

CHEXCZ and CPENTCZ elements must reference a PSOLCZ property. With the PSOLCZ bulk entry,you reference the corresponding MAT1, MAT11, or MATCZ bulk entry, the material coordinate system,and, under certain circumstances, the thickness of the element.

There are three stiffness values associated with cohesive elements: K01 and K02 are the transverseshear stiffness; K03S is the out-of-plane normal stiffness. You specify these stiffness values asfollows:

• If the PSOLCZ bulk entry references a MAT1 bulk entry, NX Nastran calculates the stiffness ofthe cohesive elements to be K01 = G / THICK, K02 = G / THICK, and K03S = E / THICK, whereTHICK is the value you specify in the THICK field of the PSOLCZ bulk entry.

• If the PSOLCZ bulk entry references a MAT11 bulk entry, NX Nastran calculates the stiffness ofthe cohesive elements to be K01 = G13 / THICK, K02 = G23 / THICK, and K03S = E3 / THICK,where THICK is the value you specify in the THICK field of the PSOLCZ bulk entry.

• If the PSOLCZ bulk entry references a MATCZ bulk entry, you specify the stiffness directly inthe K01, K02, and K03S fields of the MATCZ bulk entry. For this case, the THICK field of thePSOLCZ bulk entry is ignored.

In all three cases, NX Nastran does not use the physical thickness of cohesive elements as definedby the geometric coordinates of the grids to determine the stiffness of the cohesive element.

You use the MATCZ bulk entry when you want to obtain material damage estimates. To activatedamage estimation, include PARAM,MATNL,1 in your input file. On the MATCZ bulk entry, specify thedamage estimation model. You can choose from the following options:

• In the polynomial model (the default), the Mode I, Mode II, and Mode III damage variables aretaken to be equal, and the damage is modeled as a function of thermodynamic force. Anevolution equation is used to estimate the damage.

• In the bi-triangular model, for Mode I, the relationship between stress and displacement ismodeled as bilinear.

• In the exponential model, for Mode I, the relationship between stress and displacement ismodeled as exponential.

NX Nastran also does not use the physical thickness of cohesive elements as defined by thegeometric coordinates of the grids in damage calculations.

Results for cohesive elements are calculated at the corner grid points. The results include:

• Damage values

• Surface tractions

• Relative displacements



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To request result output for cohesive elements, use the CZRESULTS case control command.With the CZRESULTS case control command, you can obtain the damage for all three modes,components of the surface tractions, and components of the relative displacements. Relativedisplacement is defined as the displacement of the top surface of the cohesive element relativeto the bottom surface of the cohesive element.

Four values are reported for surface tractions and relative displacements. One value is the magnitudeof the surface traction or relative displacement in the direction normal to the element. The otherthree values are the components relative to the basic coordinate system of the surface traction orrelative displacement in the plane of the element. Surface tractions have units of force per unitarea. Relative deformation has units of length.

4.9 Crack simulationYou can compute and output the j-integral for a given crack geometry. This capability is onlysupported for SOL 401. The j-integral output can be used by third-party software like Zencrack toperform a fracture mechanics analysis.

• You can use the JINTEG case control command to control the computation and output of thej-integral. With the JINTEG case control command, you can direct the j-integral output to either.op2 or .f06 files.

For additional information, see the JINTEG case control command.

• The creation of the CRAKTP bulk entry. You can use the CRAKTP bulk entry to specifyinformation related to the crack tip.

For additional information, see the CRAKTP bulk entry in the NX Nastran Quick Reference Guide.

• The creation of the VCEV bulk entry. You can use the VCEV bulk entry to define virtual cracktip extension vectors.

For additional information, see the VCEV bulk entry in the NX Nastran Quick Reference Guide.

• The modification of the CHEXA bulk entry to allow for collapsed CHEXA element definition. Notethat the collapsed CHEXA element is not supported in a glue or contact region.

• The creation of the COLPHEXA parameter. You can allow collapsed CHEXA elements to bypassinput file checks with the COLPHEXA parameter. To do so, specify PARAM,COLPHEXA,YESin the bulk section of the input file.

Collapsed CHEXA elements

Any face of a CHEXA element can be collapsed to an edge. The edge of the collapsed facerepresents the crack front.

Figure 4-1 shows the connectivity for a standard CHEXA element.


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Figure 4-1. Standard CHEXA Element

Figure 4-2 shows the CHEXA element of Figure 4-1 with the G2–G14–G6–G18–G7–G15–G3–G10face collapsed so that the G2–G14–G6 edge and the G3–G15–G7 edge become the crackfront. Alternately, the G2–G14–G6–G18–G7–G15–G3–G10 face could be collapsed so that theG2–G10–G3 edge and the G6–G18–G7 edge would become the crack front.

Figure 4-2. Collapsed CHEXA Element

Two options are available for specifying a CHEXA element with a collapsed face:

• In Format 1, 15 unique grid IDs are specified in the 20 grid ID fields of the CHEXA bulk entry.Format 1 is typically used for elastic material models. With Format 1, mid-side grids can move tothe quarter-span locations closest to the crack front.

For the collapsed CHEXA element shown in Figure 4-2, the Format 1 specification is as follows:

1 2 3 4 5 6 7 8 9 10CHEXA EID PID G1 G2 G2 G4 G5 G6

G6 G8 G9 G2 G11 G12 G13 G14



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1 2 3 4 5 6 7 8 9 10

G14 G16 G17 G6 G19 G20

where the same grid ID is entered in the G2, G3, and G10 fields, another grid ID is entered inboth the G14 and G15 fields, and another grid ID is entered in the G6, G7, and G18 fields.

• In Format 2, 20 unique grid IDs are specified in the 20 grid ID fields of the CHEXA bulk entry.However, eight of the grid IDs do not have unique coordinates. Format 2 is typically used forelasto-plastic material models. With Format 2, mid-side grids remain at the mid-span locations.

For the collapsed CHEXA element shown in Figure 4-2, the Format 2 specification is as follows:

1 2 3 4 5 6 7 8 9 10CHEXA EID PID G1 G2 G3 G4 G5 G6

G7 G8 G9 G10 G11 G12 G13 G14

G15 G16 G17 G18 G19 G20

where the grids entered in the G2, G3, and G10 fields would share the same coordinates, thegrids entered in the G14 and G15 fields would share the same coordinates, and the grids enteredin the G6, G7, and G18 fields would share the same coordinates. Unlike Format 1 where grids inthe CHEXA element connectivity are merged, Format 2 does not merge coincident grids in theCHEXA element connectivity. Thus, these grids can move independently of one another.

4.10 Formulation of isoparametric elementsIn the finite element method, parametric mapping is frequently used to map an irregular region into aregular one. The coordinate system used in the parametric mapping is a natural coordinate systemsuitable for the geometry. For instance, a natural coordinate system (ξ,η) is used for a quadrilateralsurface in which each corner node has an extremum value of +1 or -1 in ξ and η. Then, the geometryof the internal points of the finite element can be described in terms of the nodal coordinates by theparametric mapping. The mathematical expression for this parametric mapping or interpolation iscalled a shape function. The displacement field inside the element should also be interpolated interms of nodal displacements. The mapping is isoparametric if the same shape function is used tointerpolate the displacement field as well as the geometry. The merit of isoparametric mapping is thatthe displacement field is invariant to the orientation of the Cartesian coordinate system x and y.

Most of the elements are isoparametric elements, in which the shape functions are expressed interms of isoparametric coordinates. In what follows, derivations are shown for the isoparametriccoordinates, shape functions, element matrix describing the strain-displacement relations, volumeintegration for stiffness and mass matrices by Gauss quadrature, and element loads are derived fora tetrahedron element to illustrate element related operations.

4.10.1 Isoparametric coordinates

Cartesian coordinates are not convenient to describe the geometry or the displacement field of atetrahedron element. Let us introduce a set of volume coordinates (L1 L2 L3 L4), such that


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as shown in the figure Volume Coordinates for Tetrahedron. The relation between volume andCartesian coordinates can be established for a linear tetrahedron (without midside nodes) as follows:

x = L1x1 + L2x2 + L3x3 +L4x4y = L1y1 + L2y2 + L3y3 +L4y4z = L1z1 + L2z2 + L3z3 +L4z41 = L1 + L2 + L3 +L4

Equation 4-14.

It is obvious from Equation 4-14 that the shape functions are simply the volume coordinates, i.e.,

N1 = L1, N2 = L2, ..., etc.

Equation 4-15.

because x = Σ Nixi.

Shape functions for the quadratic tetrahedron can be derived using Lagrangian interpolation. Theseare

N1 = (2L1 – 1)L1, etc. for corner nodes

and N5 = 4L1, L2, etc. for midside nodes.

Equation 4-16.

Notice that these shape functions, Equations 4-15 and 4-16, satisfy element convergence criteria:integrability (Cn-1 continuity for n-th derivative) and completeness (no straining by a rigid body mode,constant strain condition, and continuous displacement field).

With the isoparametric element, the same shape functions are used to describe the displacementfield as well as the geometry, i.e.,

Equation 4-17.

The strains are calculated as

Equation 4-18.

where strain vector

Equation 4-19.

element matrix



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Equation 4-20.

nodal displacement vector

Equation 4-21.

with

Equation 4-22.

Equation 4-23.

Because the shape functions are defined in terms of local coordinates, a coordinate transformation isrequired to obtain global derivatives. Volume coordinates involve four coordinates (one of which isdependent), and the Jacobian matrix will become rectangular. To avoid this difficulty, let us introducelocal coordinates (ξ,η,ζ) as follows:

Equation 4-24.


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Derivatives with respect to the local coordinates can be expressed in terms of global derivativesusing the chain rule, i.e.,

Equation 4-25.

where the Jacobian matrix is expressed in terms of shape functions as

Equation 4-26.

This Jacobian matrix must be inverted to obtain global derivatives, from which the element matrix isformed. Notice that the determinant of the Jacobian matrix is called Jacobian which represents avolume change, i.e.,

Equation 4-27.

4.10.2 Shape functions

To make tetrahedron element compatible with other solid elements (HEXA and PENTA), deletionof any or.all of.the midside nodes is permitted. The shape functions are modified with Kronekerdeltas (δ5 - δ10) where

δi = 0 if the midside node i is deleted

= 1 if the midside node i is not deleted

The goal is to construct functions which are unity at the associated node and zero at all other nodes,regardless of any combination of deleted midside nodes, by the following scheme. At the corner nodes



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At the midside nodes

Shape functions are identified as follows:

Derivatives of the shape functions with respect to the local coordinates are obtained as follows:


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These shape functions and derivatives may be reduced to those for 4-noded and 10-nodedtetrahedron elements. It can be verified that, for any combination of deleted midside nodes, thereexist a condition

which satisfies the constant strain requirement.

4.10.3 Example element matrix

To illustrate the computational procedure, an element matrix for a linear tetrahedron (4-noded)element is explicitly derived here. The shape functions and the derivatives are tabulated below:

Node Ni δNi/δξ δNi/δη δNi/δς1 ξ 1 0 02 η 0 1 03 ς 0 0 14 1 – ξ – η – ς –1 -1 –1

Then the Jacobian matrix may be found as

where (xij = xi - xj) is used for convenience. Upon inverting the Jacobian matrix, we have

where



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The global derivatives of the shape functions are

Hence the element matrix will be

Notice that the rank of [B] matrix is 6.

4.10.4 Volume integration of element matrices

By virtue of variational principles, the element stiffness matrix is derived as follows:

Equation 4-28.

where D is a (6x6) material tangential matrix.

While analytical integration is possible, there are some advantages in using numerical integration.The Gaussian quadrature is used for the tetrahedron as usual. The minimum number of intergrationpoints required for non-singular stiffness matrix may be determined based on


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Equation 4-29.

These numbers are found to be 1 and 4 for linear and quadratic tetrahedron, respectively. Using theintegration formulas shown in the figure Gaussian Quadrature for Tetrahedron, the element stiffnessmatrices for linear and quadratic tetrahedron may be computed as

for 4–noded TETRA,

for 5–10 noded TETRA,

Equation 4-30.

where the scaling factor 1/6 is introduced to compensate IJI =6x (Volume). One-point scheme shouldbe used for 4-noded tetrahedron (all the midside nodes deleted) and four-point scheme otherwise.

It is difficult to determine lumped masses when some of the midside nodes are deleted. However, theconsistent mass matrix may be obtained using the same shape functions, i.e.,

Equation 4-31.

where [Ň] = [N1I N2I ..... N10I], with I being (3x3) identity matrix. Again the Gaussian quadrature4-point formula is used to find

Equation 4-32.

Notice that 4-point scheme is to be used even for 4-noded tetrahedron. For computationalconvenience, the consistent mass may be converted to the lumped mass. One way to achieve this isto take the diagonal terms and scale them so that the total mass is preserved.

4.10.5 Element loads and equilibrium

Nodal forces are computed from the element stresses using element matrix, i.e.,

Equation 4-33.



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The Gaussian integration is performed as

for 4-noded tetrahedron and

for (5-10)-noded tetrahedron.

Equation 4-34.

Thermal load is like an initial strain. Therefore, nodal forces due to thermal load are obtained by

Equation 4-35.

where {εo}T = αΔT < 1 1 1 0 0 0 >. With anisotropic thermal properties, {εo}T becomes ΔT < α1,α2 ... α6 > in general.

The pressure load applied to any surface of the tetrahedron may be distributed to the grid points usingthe shape functions applicable to the 6-noded triangular element with removable midside nodes, i.e.,

Equation 4-36.

where {n} is a unit direction vector associated with a pressure (p) and

with I being (3 x 3) identity matrix. Shape functions (Ni’) are shown in the figure Shape Functions forPressure Load. Area integration should be performed using Gaussian quadrature formulas shown inthe figure Gaussian Quadrature for Pressure Load, i.e.,

Equation 4-37.

During the nonlinear iteration process, the equilibrium is not reached until convergence is achieved.The equilibrium is sought in the global level when the residual load R approaches zero. The residualload vector is defined as

Equation 4-38.


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where {P} is the applied load vector including the thermal load and Σ implies assemblage in the globalcoordinate system (coordinate transformation required).

4.10.6 Element coordinates

The element coordinate system for the tetrahedron element is defined with the initial elementgeometry such that

• The origin is at the first grid point in the connectivity, G1.

• The x-axis is determined by connecting the origin G1 to node G2, i.e.,

Equation 4-39.

• The y-axis is determined by orthonormalization (Gram-Schmidt process) of the edge direction(V13) with respect to x-axis,

Equation 4-40.

• The z-axis is orthogonal to x and y according to the right-hand rule, i.e.,

Then, the transformation from element coordinates to the basic coordinates is simply

Equation 4-41.

where

Equation 4-42.

and < xe, ye, ze > Tbasic is the position vector of the element coordinate system with respect to thebasic coordinate system. Notice that [Tbe] is an orthogonal matrix, i.e.,



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4.10.7 Stress data recovery

In case of linear analysis, the stresses at the integration point are recovered as

Equation 4-43.

with

where {σ}, [B], {u} and {α} are defined in the element coordinate system, and the shape function{Ni} interpolates Gauss point temperatures from the nodal temperatures. In case of nonlinearanalysis, stresses are computed again upon convergence, starting from the last converged state (lastconverged solution of σ and u), i.e.,

Given {σold} and

Equation 4-44.

with

the nonlinear material routine computes updated stresses {σnew}, which are stored in ESTNL.

To output grid point stresses, the stresses, the integration points must be extrapolated to the nodalpoints. A linear extrapolation can be applied using stresses at the element c.g. and the corner Gausspoint associated with a grid point. For a tetrahedron, referring to Figure 4-4,

Equation 4-45.

where

This yields


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Equation 4-46.

In matrix form for all the grid points in the element

Equation 4-47.

where

This process must be operated on every component of stress vector. No extrapolation is requiredfor 1-point integration.



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Figure 4-3. Volume Coordinates for Tetrahedron


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Figure 4-4. Gaussian Quadrature for Tetrahedron



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Figure 4-5. Shape Functions for Pressure Load


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Figure 4-6. Gaussian Quadrature for Pressure Load



Chapter 5: Material support

5.1 Material overviewSOL 401 supports the following material types.

• The MAT1 and MATT1 (temperature dependent) bulk entries define isotropic materials.

• The MAT3 and MATT3 (temperature dependent) bulk entries define isotropic materials.

• The MAT9 and MATT9 (temperature dependent) bulk entries define anisotropic materials.

• The MAT11 and MATT11 (temperature dependent) bulk entries define orthotropic materials.

• Plastic and creep materials can optionally be assigned to the 3D solid elements, axisymmetricelements, the plane stress elements, and the plane strain elements. You can enable one or bothplasticity/creep in all subcases, or in specific subcases.

5.2 Support for plasticity analysisYou can perform a plasticity analysis in SOL 401. The constitutive model is a standard elastic-plasticmodel that allows for bilinear and multilinear stress-strain curve representations. For multilinearrepresentations, tabular data is used to define the stress-strain curve. You can specify that thestrain values in the tabular data are either total strain or plastic strain. You can also define materialproperties as temperature-dependent. At present, the von Mises yield criterion is the only yieldcriterion that is supported.

For bilinear stress-strain representations, you can select either isotropic, kinematic, or combinedhardening. For multilinear stress-strain representations, isotropic hardening is the only hardening ruleavailable.

You can selectively enable and disable plasticity effects in subcases. If plasticity-enabled subcasesare sequentially dependent, the plasticity state variables at the end of one subcase are used asthe plasticity state variables at the beginning of the next subcase. If a plasticity-disabled subcaseis placed between plasticity-enabled subcases, and the subcases are all sequentially dependent,the plasticity state variables at the end of the preceding plasticity-enabled subcase are used as theplasticity state variables at the beginning of the later plasticity-enabled subcase.

However, with the exception of special situations, we recommend you avoid placing a sequentiallydependent plasticity-disabled subcase after a plasticity-enabled subcase. Doing so eliminates thepossibility that the analysis does not account for changes to the plasticity state variables that mightresult from the loading in the plasticity-disabled subcase.

Creep analysis is also supported in SOL 401. You can enable one or both plasticity/creep in allsubcases, or you can enable one or both in specific subcases.

For more information on creep analysis in SOL 401, see Support for creep analysis.



User interface

• The MATNL parameter. With the MATNL parameter, you can globally switch the plasticityanalysis capability on or off.

• The PLASTIC parameter is available on the NLCNTL bulk entry to optionally turn off the plasticitycapability in a subcase.

• The MATS1 bulk entry allows you to define stress versus plastic strain tabular data.

To activate the plasticity analysis capability in SOL 401, do the following:

1. Reference both the MAT1 and MATS1 bulk entries in the regions where plasticity occurs.

2. Specify PARAM,MATNL,1.

3. Include a NLCNTL case control command that points to a NLCNTL bulk entry.

4. On the NLCNTL bulk entry, specify any applicable parameters.

If your input file contains subcases, and you want to include the effects of plasticity in specificsubcases, but not others, you have two options.

Option 1: Use a global NLCNTL case control command


2. Include a NLCNTL case control command above the subcases that points to a NLCNTL bulk entry.

3. On the NLCNTL bulk entry pointed to by the global NLCNTL case control command, specifyany applicable parameters.

4. In the subcases that you want to disable the plasticity analysis capability, include a NLCNTL casecontrol command that points to a NLCNTL bulk entry.

5. On the NLCNTL bulk entry pointed to by the NLCNTL case control commands in the subcases,specify “PLASTIC” in a PARAMi field and “NO” in the corresponding VALUEi field.

Option 2: Include NLCNTL case control commands in every subcase


2. Include NLCNTL case control commands in each subcase. Multiple NLCNTL case controlcommands can point to a single NLCNTL bulk entry.

3. In subcases that you want to enable the plasticity analysis capability, have the NLCNTL casecontrol command point to a NLCNTL bulk entry with any applicable parameters specified.

4. In subcases that you want to disable the plasticity analysis capability, have the NLCNTL casecontrol command point to a NLCNTL bulk entry that has “PLASTIC” specified in a PARAMi fieldand “NO” specified in the corresponding VALUEi field.

In a SOL 401 plasticity analysis, the property bulk entry referenced by all non-rigid elements mustreference a MAT1 bulk entry and a MATS1 bulk entry that have the same material identification



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number. If the properties on the MAT1 bulk entry are temperature-dependent, include a MATT1 bulkentry with the same material identification number.

On the MATS1 bulk entry, specify TYPE = “PLASTIC” or “PLSTRN” to select the strain type in tabulardata used to describe a multilinear stress-strain curve. Specify TYPE = “PLASTIC” if you want touse total strains. Specify TYPE = “PLSTRN” if you want to use plastic strains. Total and plasticstrains are related as follows:

where

To describe a bilinear stress-strain curve, specify either TYPE = “PLASTIC” or “PLSTRN” and enterthe work hardening slope, H, directly.

For additional information, see the MATS1 bulk entry in the NX Nastran Quick Reference Guide.

Time step control

Unlike creep analysis in SOL 401, there is no adaptive time stepping for plasticity analysis. The timesteps are defined directly by the solution times. To define solution times for the plasticity analysis,include a TSTEP case control command in your input file that points to a TSTEP1 bulk entry. On theTSTEP1 bulk entry, specify the solution times and the solution times you want results output.

5.3 Overview of PlasticityFor plasticity, SOL 401 includes a von Mises yield function with an associated flow rule. Availableoptions include isotropic hardening, kinematic hardening, and combined hardening. TheZiegler-Prager hardening rules are available for kinematic and combined hardening.

The plasticity model must also be calibrated with uniaxial stress-strain data. The strain informationmust be in the unitless form of length/length. Classical plasticity models include the following threefundamental ingredients.

1. A yield function or yield criterion defines the limit of elastic behavior for a general state of stress.

The yield function may be thought of as a surface in a six dimensional stress space. It dividesthe stress space into two regions. Points inside the yield surface are characterized by elasticstress-strain behavior while stress states on the yield surface are at the limit of elastic behavior.The yield function may be written as a function of stress and a hardening parameter k.

The von Mises yield function for an isotropic hardening material may be stated as:

J2 is the second invariant of the deviatoric stress tensor. The deviatoric stresses, sij, are given by:


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where σm is the mean or average normal stress, and δij is the Kronecker delta.

Isotropic hardening assumes that the uniaxial yield stress, σy, is a function of plastic straining.

For kinematic or combined hardening, the von Mises yield function is

J 2 is the second invariant of the shift stress that is defined as the deviatoric stress minus theback stress. The back stress may also be thought of as the position of the center of the yieldsurface in the six-dimensional stress space. For initially isotropic materials, the back stresstensor components are initially zero.

2. A flow rule determines the relative magnitudes of the components of the plastic strain incrementtensor.

The software uses an associated flow rule in which the plastic potential function, g, is the sameas the yield function, f, and the components of the plastic strain increment are given by:

In vector form, for the von Mises yield surface with kinematic hardening, the plastic flow vector isgiven by:

For isotropic hardening, this equation reduces to the well-known Prandtl-Reuss equations.

3. A hardening rule defines the changes in the yield function as a result of plastic straining.

Isotropic Hardening

When you select isotropic hardening, the software uses a piece-wise linear stress-strain curve.The isotropic hardening assumption isn't very realistic for most materials subjected to cyclicloading. However, it is relatively simple and efficient.

Isotropic hardening assumes that the yield surface expands uniformly as a result of plasticstraining. This assumption is achieved by making the yield stress a function of the integratedeffective plastic strain increments, which for a von Mises material is:



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The slope of the stress plastic strain curve, Ep, is called the plastic modulus. It can be obtained fromthe uniaxial stress-strain curve and is defined by:

For a von Mises material, the effective stress is given by:

Kinematic Hardening

When you select kinematic hardening, the software assumes a bilinear stress-strain curve. If thematerial database contains a multilinear representation, only the yield point and the tangent modulusof the first segment beyond the yield point are used to characterize the stress-strain behavior.

Kinematic hardening assumes that the yield surface translates in the stress space but doesn't changesize or shape. The yield stress, σy, doesn't change, but the back stress, αij, is a function of plasticstraining. The Ziegler-Prager kinematic hardening is one of the most widely used models. This modelassumes that the back stress increment is in the direction of the stress minus the back stress.

The factor, dμ, depends on the plastic strain history.

Ziegler-Prager Combined Kinematic Isotropic Hardening

When you select combined kinematic isotropic hardening, the software assumes a bilinearstress-strain curve. If the material database contains a multilinear representation, only the yieldpoint and the tangent modulus of the first segment beyond the yield point are used to characterizethe stress-strain behavior.

Combined hardening assumes that the yield surface both expands and translates in the stress space.The plastic strain increment is composed of two components shown in the following equations:

where,

is associated with the isotropic expansion of the yield surface,

and is associated with the translation of the yield surface.


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The reduced effective plastic strain associated with isotropic hardening is related to the effectiveplastic strain by the following:

The back stress increment for Prager combined hardening is:

The back stress increment for Ziegler-Prager hardening is:

where M=0.5 in SOL 401.

The software computes plastic strain increments using a backward Euler technique withoutsub-incrementation. For the isotropic hardening model:

where De is the elastic modulus matrix, εp is the plastic strain, is the effective plastic strain, andTYF is the von Mises yield function.

The software will compute a consistent tangent modulus for use in generating the tangent stiffnessmatrix. When this procedure is used with a full Newton-Raphson iteration, quadratic convergence canbe obtained. A detailed description of this procedure may be found in Crisfield, 1991.

References:

• Crisfield, M. A., Non-linear Finite Element Analysis of Solids and Structures (Chichester: JohnWiley & Sons, 1991).

• Chakrabarty, J., Theory Of Plasticity (New York: McGraw-Hill Book Company, 1987), 55-119.

• Chen, W. F. and Han, D. J., Plasticity for Structural Engineers, (New York: Springer-Verlag,1988), 239-281.

• Lemaitre, J. and Chaboche, J. L., Mechanics of Solid Materials, (Cambridge: CambridgeUniversity Press, 1990), 161-240.



Material support

5.4 User defined materialsSolution 401 supports externally computed, user defined material models. You can define a materialmodel by developing and compiling an external routine. The external routine can optionally includemultiple material models.

Source code examples are included with the NX Nastran installation for you to begin your ownexternal material development. A Ready-to-run routine and test cases are also included todemonstrate the input file requirements and the general workflow.

You can develop an external material routine using FORTRAN or C, and compile for Windows or Linuxoperating systems. You must use the NX Nastran ILP-64 executable with an external material routine.

NX Nastran Inputs

The MUMAT bulk entry is available to define the material data in the NX Nastran input file. NXNastran passes this data to the external routine. The MUMAT entry in your input file is the triggerNX Nastran uses to call the external routine.

The elements referencing the MUMAT entry material ID will use an associated material law defined inthe user defined material routine NXUMAT. See NXUMAT interface for the NXUMAT API description.The elements referencing the MUMAT entry must also reference MAT1, MAT9 or MAT11 entries. NXNastran uses the MATi properties to compute the initial elastic stiffness. Temperature dependentmaterials are also supported and are used by NX Nastran when computing the initial elastic stiffness.The initial elastic stiffness computed by NX Nastran and the data defined on the MATi, MATTI, andTABLEMi entries are all passed from NX Nastran to the external routine.

All of the data defined on the MUMAT bulk entry is also passed to the NXUMAT routine. You caninclude a variety data types on the MUMAT entry. For example, real, integer, tables and table oftables can all be included.

The following data is supported on the MUMAT entry:

• MODNAME1 and MODNAME2 fields - Optional character fields.

• MATNAME field - Can be used to request a specific material model.

• NUMSTAT field - Defines the total number of state variables, if they exist.

The following tabular data is supported on the MUMAT entry:

Note: When you reference TABLES1, TABLEM1, TABLEST entries on the MUMAT entry, NX Nastrandoes no interpolation or extrapolation of the data before passing it to the external routine.

• TABLES1 - This is a collection of real data pairs. You reference the ID of TABLES1 entriesdefined in your NX Nastran input file.

• TABLEM1 - This is a collection of real data pairs. You reference the ID of TABLEM1 entriesdefined in your NX Nastran input file.

• TABLEST - This is a collection real data values versus table ID’s. The table ID's reference othertables with a collection of real data pairs. You reference the ID of TABLEST entries defined inyour NX Nastran input file.

NX Nastran also optionally stores, retrieves, and outputs state variable data computed by theexternal routine. For example, stress, creep strain and plastic strain at each solution increment


Material support


can be stored as state variables. You define the number of state variables in the NUMSTAT fieldon the MUMAT entry, and NX Nastran will initialize the appropriate storage. The flow of statevariable data is as follows:

• Your external routine provides NX Nastran with updated state variables at the end of a time step.

• NX Nastran stores the variables in the database.

• NX Nastran provides the data back to the external routine at the beginning of the consecutivetime step.

You can request NX Nastran to output the state variables using the STATVAR case control command.Regardless of what the data is originally, for example, vector or tensor components, NX Nastranoutputs all of the state variable data as scalar values. The GRID/GAUSS output option is alsoavailable on the STATVAR command.

You can pass parameters defined in the NX Nastran input file to the external routine. This includesparameters defined with the PARAM case control, the PARAM bulk entry, and the PLASTIC, CREEP,and the MATNL parameter settings defined on the NLCNTL bulk entry. You can use these settings inyour external routine, for example, to turn on/off specific material computations in a subcase.

NXUMAT interface

The subroutine NXUMAT directs NX Nastran to a specific material model. A shared library (DLL/SOfile) is built from this routine and used by NX Nastran to model the material behavior. The buildingprocess of the shared library is described in Compiling instructions.

The arguments of NXUMAT are detailed below. NX Nastran expects that the real, integer andcharacter values passed from the argument list are or precision REAL*8, INTEGER*8 andCHARACTER (LEN=8).

SUBROUTINE NXUMAT(IOPER, MODNAME1, MODNAME2, MATID, HOOK, TANSTIFF,MATIR, MATIN, NMATI, MUDATAR, MUDATAI, NMUDATA ,DFGRDT0, DFGRDT1, EPSTOTT1, EPSMT1, EPSTHT1, EPSDELM,DELTAT, TIMET1, TEMPT0, TEMPT1, NB, INTVALS,REALVALS, XYZT1, ROT, NXPARAM, STATEVAR, NSVAR,SIGMA, EPSPL, EPSCR, DTCRPRAT, VOID1, VOID2, VOID3, IRET)

FUNCTION : USER MATERIAL DEFINITIONTHIS ROUTINE IS DIVIDED IN THREE STEPS (IOPER)

IF (IOPER.EQ.0) THENNXUMAT DLL VERSION NUMBER

ELSEIF (IOPER.EQ.1) THENINITIALIZATION OF STATEVAR

ELSECOMPUTATION OF MATERIAL LAW

ENDIFEND

Each of the arguments of the NXUMAT routine is described below.

IOPER (Input)

IOPER defines the operational step for which NXUMAT has been called for.



Material support

IOPER = 0: this is used to facilitate versioning for the NXUMAT library. This number is read fromthe argument STATEVAR(1). This float is printed to the F06 file as a user information message23209 along with the value In STATEVAR(1).

Example*** USER INFORMATION MESSAGE 23209 (IFPDRV)

VERSION 1.00 OF NXUMAT DLL LOADED.

IOPER=1: this is used to initialize state variables. This step can be used to assign initial values tothe STAEVAR array, which will be stored by NX Nastran and returned in the computational step.

IOPER>1: this is considered the computational step.

MODNAME1 (Input)

MODNAME1 contains the eight-character name from the MUMAT entry.

MODNAME2 (Input)

MODNAME2 contains the eight-character name from the MUMAT entry.

MATID (Input)

MATID is the material ID given in the MUMAT entry.

HOOK(NB,NB) (Input)

HOOK is an NBxNB-size matrix containing the hook’s matrix. This is pre-computed byNX Nastran based on the MAT1/MAT9/MAT11 entries associated with the MUMAT entry.

TANSTIFF(NB,NB) (Input/Output)

TANSTIFF contains the tangent stiffness matrix computed at the previous time step when enteringNXUMAT and should be updated with the tangent stiffness matrix for the current time stepupon convergence.

MATIR(NMATI) (Input)

MATIR contains the real data from the associated MAT1/MAT9/MAT11 entries. Its format isexplained in MATIN/MATIR array format.

MATIN(NMATI) (Input)

MATIN contains the integer data from the associated MAT1/MAT9/MAT11 entries. Its format isexplained in MATIN/MATIR array format.

NMATI (Input)

NMATI is the size of the MATIR/MATIN array.

MUDATAR (NMUDATA) (Input)

MUDATAR contains the real data from the MUMAT entries. Its format is explained inMUDATAI/MUDATAR array format.

MUDATAI (NMUDATA) (Input)


Material support


MUDATAI contains the real data from the MUMAT entry. Its format is explained inMUDATAI/MUDATAR array format.

NMUDATA (Input)

NMUDATA is the size of the MUDATAR/MUDATAI array.

DFGRDT0(3,3) (Input)

Deformation gradient at the previous time step. This is not defined in NX Nastran 11.

DFGRDT1(3,3) (Input)

Deformation gradient at the previous time step. This is not defined in NX Nastran 11.

EPSTOTT1(NB) (Input)

Total strain tensor (including mechanical and thermal strain).

EPSMT1(NB) (Input)

Mechanical strain.

EPSTHT1(NB) (Input)

Thermal strain.

EPSDELM(NB) (Input)

Mechanical strain increment.

DELTAT (Input)

Time step.

TIMET1 (Input)

Current time.

TEMPT0 (Input)

Temperature at the previous time step.

TEMPT1 (Input)

Temperature at the current time step.

NB (Input)

Number of tensor components.

INTVALS(*) (Input)

The INTVALS array contains various integer data, including the following:

INTVALS(1) contains the element ID.

INTVALS(2) contains the gauss ID.



Material support

INTVALS(3) contains the current time step number.

INTVALS(4) contains the iteration number for the current time step.

INTVALS(5) contains the subcase ID.

REALVALS(*) (Input)

The REALVALS array contains various real data, including the following:

REALVALS(1) contains the element thickness for shell elements.

XYZT1(3) (Input)

XYZT1 contains the updated coordinates of the current gauss point.

ROT(3,3) (Input)

ROT contains the rotational matrix between the structural and material coordinate systems.

NXPARAM(*) (Input)

NXPARAM contains the list of PARAM entry values that are in the input testcase. You can passparameters defined in the NX Nastran input file to the external routine. This includes parametersdefined with the PARAM case control, the PARAM bulk entry, and the PLASTIC, CREEP, andMATNL parameter settings defined on the NLCNTL bulk entry. You can use these settings in yourexternal routine, for example, to turn on/off specific material computations in a subcase. Refer tothe included utility routine PARAMQRY to understand how to use this functionality.

STATEVAR(NSVAR) (Input/Output)

STATEVAR contains the state variables that you specified. It contains the previous time stepstate variables as input and must be updated with current state variables upon convergence.

NSVAR (Input)

NSVAR is the number of state variables.

SIGMA(NB) (Input/Output)

SIGMA contains the stress tensor from the previous time step and must be updated with thecurrent time step value upon convergence.

EPSPL(NB) (Input/Output)

EPSPL contains the plastic strain tensor from the previous time step and must be updated withcurrent time step value upon convergence. This strain tensor is stored inside NX Nastran andis output when requested.

EPSCR(NB) (Input/Output)

EPSCR contains the creep strain tensor from the previous time step and must be updated withcurrent time step value upon convergence. This strain tensor is stored inside NX Nastran andis output when requested.

DTCRPRAT (Output)


Material support


DTCRPRAT is the creep time step ratio which can be used as the time stepping control for creep.

VOID1, VOID2, VOID3

These are empty slots for future use.

IRET (Output)

IRET is the return code. Any returned value other than 0 (>0) will stop the solution and issue afatal message with the return code.

The user data specified on the MUMAT entry and corresponding MAT1, MAT9, or MAT11 entry ispassed to the NXUMAT routine via the arrays MUDATAI/MUDATAR and MATIN/MATIR respectively.This section describes the layout of the input data arrays supplied to the NXUMAT subroutine.

Each entry (that is, the MUMAT entry and the MAT* entry) is stored in two arrays each. One array hasthe integer data and the other has the real data. Some cross-referencing between the two arraysmust be done to extract the desired data. The format of these arrays is described in the followingsections. These arrays are declared as INTEGER*8 for the integers and REAL*8 for the real portion.

MATIN/MATIR array format

MAT1,MAT9, and MAT11 data along with corresponding MATT1,MATT9, and MATT11 entries are sentto the NXUMAT routine through two arrays: MATIN and MATIR. Both MATIN and MATIR representthe MAT* entry data in the format laid out in the following table. The integer data of the MAT* andMATT* entries exist in MATIN and the real data exist in the MATIR array.

The first word in the array is the total number of words in the array. The second word is the offset tothe mapping array. The mapping array has information on the location of each table of data. Thethird word identifies the type of the array: 1 corresponds to the MATIN array and 2 corresponds tothe MUDATAI array. The fourth word is the material ID, and the fifth word contains the type of theMAT* entry (1, 9, or 11). Based on the type of MAT* entry for which this array contains data, thenext ‘N’ words from the sixth word contain the bulk entry data (N=11, 20, and 15 for the MAT1,MAT9, and MAT11 entry respectively). After that, in the next word (if any), corresponding MATT*entries are present, including its ID. If no MATT* entries are present, this word is zero. If the MATT*entry is present, after the ID the MATT* entries are laid out next. If MATT* entries are present, thatmeans TABLEM1 data is present. This is laid out next in the array. Each table has a fixed formatas shown in the following table and is ended by a -1. Multiple tables can exist. The mapping arrayis laid out in the end.

Word Description1 Total length of data used in this array (LENGTH).2 Offset to the mapping array (array index).3 The ID code for a MATIN array is always 1.

The ID code is used by the interpolation routines to identify the data structure.4 Material ID used to setup the array.5 MAT table type:

1: The record contains MAT1 data.





Material support

Word Description6 Number of entries in the record (N):

If the MAT table type is 1, N = 11.

If the MAT table type is 9, N = 20.

If the MAT table type is 11, N = 15.7 through N+6 MAT* constants (real).N+8 MATT* ID (0 if no MATT* entry).N+9 through 2N+8 MATT* entries (integers).N+6 throughLENGTH

TABLEM1 data tables (CODEX/CODEY: 0=Linear, 1=Log):

1. Table identification number (if 0, no table).

2. NUMPAIR: Number of X-Y Pairs.

3. CODEX: Type of interpolation for the x-axis (integer).

4. CODEY: Type of interpolation for the y-axis (integer).

5. EXTRAP: Extrapolation option.

6. X tabular value (real).

7. Y tabular value (real).

- Words 6 through 7 repeated NUMPAIR.

(-1) end of TABLEM1 tables.Mapping

1. Table ID.

2. Array index pointing to the table data.

For example, consider a MAT9 entry as show below along with a MATT9 entry and correspondingTABLEM1 entries.

MAT9 1 1.+7 2.5+6 1.+6 0. 0. 0. 1.+7++ 1.+6 0. 0. 0. 3.+7 0. 0. 0.++ 3.75+6 0. 0. 1.75+6 0. 1.75+6 0.1 1.-5++ 1.-5 1.-5 1.-5 1.-5 1.-5 0.

MATT9 1 1 1++ 3 +

+ 1 2++ 2 2 2 2 2

TABLEM1 2 LINEAR LINEAR +


Material support


+ 50. 1.-5 200. 1.-5 ENDT


+ 50. 1e+7 200. 7e+6 ENDT


50. 3e+7 200. 1e+7 ENDT

Using the information above, these entries are formatted into the array shown below.

Description MATIN IntegerData MATIR Real Data

1 Length MATIN(1) 104 MATIR(1) 0.000000000000000D+0002 Offset MATIN(2) 99 MATIR(2) 0.000000000000000D+0003 MAT* Identifier MATIN(3) 1 MATIR(3) 0.000000000000000D+0004 MID MATIN(4) 1 MATIR(4) 0.000000000000000D+0005 MAT* type MATIN(5) 9 MATIR(5) 0.000000000000000D+000

6 No. of MAT9 dataEntries MATIN(6) 30 MATIR(6) 0.000000000000000D+000

7 MAT9 data (refer toMATIR) MATIN(7) 0 MATIR(7) 10000000



10 MAT9 data (refer toMATIR) MATIN(10) 0 MATIR(10) 0.000000000000000D+000















Material support







28 MAT9 data (refer toMATIR) MATIN(28) 0 MATIR(28) 0.1

29 MAT9 data (refer toMATIR) MATIN(29) 0 MATIR(29) 1.000000000000000D-005








37 MATT9 ID MATIN(37) 1 MATIR(37) 0.000000000000000D+00038 MATT9 entries MATIN(38) 1 MATIR(38) 0.000000000000000D+00039 MATT9 entries MATIN(39) 0 MATIR(39) 0.000000000000000D+00040 MATT9 entries MATIN(40) 0 MATIR(40) 0.000000000000000D+00041 MATT9 entries MATIN(41) 0 MATIR(41) 0.000000000000000D+00042 MATT9 entries MATIN(42) 0 MATIR(42) 0.000000000000000D+00043 MATT9 entries MATIN(43) 0 MATIR(43) 0.000000000000000D+00044 MATT9 entries MATIN(44) 1 MATIR(44) 0.000000000000000D+00045 MATT9 entries MATIN(45) 0 MATIR(45) 0.000000000000000D+00046 MATT9 entries MATIN(46) 0 MATIR(46) 0.000000000000000D+00047 MATT9 entries MATIN(47) 0 MATIR(47) 0.000000000000000D+00048 MATT9 entries MATIN(48) 0 MATIR(48) 0.000000000000000D+00049 MATT9 entries MATIN(49) 3 MATIR(49) 0.000000000000000D+00050 MATT9 entries MATIN(50) 0 MATIR(50) 0.000000000000000D+00051 MATT9 entries MATIN(51) 0 MATIR(51) 0.000000000000000D+00052 MATT9 entries MATIN(52) 0 MATIR(52) 0.000000000000000D+00053 MATT9 entries MATIN(53) 1 MATIR(53) 0.000000000000000D+000


Material support



54 MATT9 entries MATIN(54) 0 MATIR(54) 0.000000000000000D+00055 MATT9 entries MATIN(55) 0 MATIR(55) 0.000000000000000D+00056 MATT9 entries MATIN(56) 0 MATIR(56) 0.000000000000000D+00057 MATT9 entries MATIN(57) 0 MATIR(57) 0.000000000000000D+00058 MATT9 entries MATIN(58) 0 MATIR(58) 0.000000000000000D+00059 MATT9 entries MATIN(59) 0 MATIR(59) 0.000000000000000D+00060 MATT9 entries MATIN(60) 2 MATIR(60) 0.000000000000000D+00061 MATT9 entries MATIN(61) 2 MATIR(61) 0.000000000000000D+00062 MATT9 entries MATIN(62) 2 MATIR(62) 0.000000000000000D+00063 MATT9 entries MATIN(63) 2 MATIR(63) 0.000000000000000D+00064 MATT9 entries MATIN(64) 2 MATIR(64) 0.000000000000000D+00065 MATT9 entries MATIN(65) 2 MATIR(65) 0.000000000000000D+00066 MATT9 entries MATIN(66) 0 MATIR(66) 0.000000000000000D+00067 MATT9 entries MATIN(67) 0 MATIR(67) 0.000000000000000D+00068 MATT9 entries MATIN(68) 0 MATIR(68) 0.000000000000000D+00069 TABLEM1 ID MATIN(69) 1 MATIR(69) 0.000000000000000D+00070 No. of XY data MATIN(70) 2 MATIR(70) 0.000000000000000D+00071 CODEX MATIN(71) 0 MATIR(71) 0.000000000000000D+00072 CODEY MATIN(72) 0 MATIR(72) 0.000000000000000D+000

73 EXTRAPOLATIONoption MATIN(73) 0 MATIR(73) 0.000000000000000D+000

74 X(1) MATIN(74) 0 MATIR(74) 5075 Y(1) MATIN(75) 0 MATIR(75) 1000000076 X(2) MATIN(76) 0 MATIR(76) 20077 Y(2) MATIN(77) 0 MATIR(77) 700000078 TABLEM1 End MATIN(78) -1 MATIR(78) 0.000000000000000D+00079 TABLEM1 ID MATIN(79) 3 MATIR(79) 0.000000000000000D+00080 No. of XY data MATIN(80) 2 MATIR(80) 0.000000000000000D+00081 CODEX MATIN(81) 0 MATIR(81) 0.000000000000000D+00082 CODEY MATIN(82) 0 MATIR(82) 0.000000000000000D+000


84 X(1) MATIN(84) 0 MATIR(84) 5085 Y(1) MATIN(85) 0 MATIR(85) 3000000086 X(2) MATIN(86) 0 MATIR(86) 20087 Y(2) MATIN(87) 0 MATIR(87) 1000000088 TABLEM1 End MATIN(88) -1 MATIR(88) 0.000000000000000D+00089 TABLEM1 ID MATIN(89) 2 MATIR(89) 0.000000000000000D+00090 No. of XY data MATIN(90) 2 MATIR(90) 0.000000000000000D+00091 CODEX MATIN(91) 0 MATIR(91) 0.000000000000000D+00092 CODEY MATIN(92) 0 MATIR(92) 0.000000000000000D+000



Material support



94 X(1) MATIN(94) 0 MATIR(94) 5095 Y(1) MATIN(95) 0 MATIR(95) 1.000000000000000D-00596 X(2) MATIN(96) 0 MATIR(96) 20097 Y(2) MATIN(97) 0 MATIR(97) 1.000000000000000D-00598 TABLEM1 End MATIN(98) -1 MATIR(98) 0.000000000000000D+00099 Mapping array starts.

Table ID, offset givenbelow

MATIN(99) 1 MATIR(99) 0.000000000000000D+000

100 Index to Table ID above MATIN(100) 69 MATIR(100) 0.000000000000000D+000101 Table ID MATIN(101) 3 MATIR(101) 0.000000000000000D+000102 Index to Table ID above MATIN(102) 79 MATIR(102) 0.000000000000000D+000103 Table ID MATIN(103) 2 MATIR(103) 0.000000000000000D+000104 Index to Table ID above MATIN(104) 89 MATIR(104) 0.000000000000000D+000

MUDATAI/MUDATAR array format

The data defined on the MUMAT bulk entry is passed to the NXUMAT routine via the MUDATAI andMUDATAR arrays. Both MUDATAI and MUDATAR represent the MUMAT entry data in the format laidout in the following table. The integer data of MUMAT exist in MUDATAI and the real data exist inthe MUDATAR array.

The first word in the array is the total number words in the array. The second word is the offset tothe mapping array. The mapping array has information of the location of each table of data. Thethird word identifies the type of array: 1 corresponds to the MATIN array and 2 corresponds to theMUDATAI array. The next six words are the data present on the first line of the MUMAT entry. Afterthis, the next word contains a code that identifies the type of data that is to follow (1 for real, 2 forinteger, 3 for TABLES1, 4 for TABLEST, 5 for TABLEM1, and 6 for a mapping array). After the code,the next word is the size of the array for this ‘code data’ followed by the data. Note that multiple typesof data can exist, so expect multiple codes followed by their corresponding data.

After this, the table data is laid out for each TABLES1, TABLEST, and TABLEM1 on the MUMATentry. The format is shown in the following table.

Word Description1 Total length of data used in this array (LENGTH).2 Offset to the mapping array (array index).3 The ID code for a MUDATAI array is 2 for MUMAT.

The ID code is used by the interpolation routines to identify the data structure.4-10 First line of the MUMAT Bulk Data Entry.


Material support


Word Description11 Code descriptor :

0: End of Data, or no data if the first word is 0.

1: REAL Data.

2: INTEGER Data.

3: TABLES1 Data.

4: TABLEST Data.

5: TABLEM1 Data.

6: Mapping.Code = 1 REAL data

Size of array

2-n) Real dataCode = 2 INTEGER data

Size of array

2-n) Integer dataCode = 3 TABLES1 data

Size of array

2-n) Table IDsCode = 4 TABLEST data

Size of array

2-n) Table IDsCode = 5 TABLEM1 data

Size of array

2-n) Table IDsCode = 6 Mapping

1. Size of the mapping array.

2. Table ID.

3. Table Type (code 3, 4, 5)

4. Array Index pointing to the table data



Material support

Word DescriptionTABLES1 data 1. Table identification number (if 0, no table) (integer).

2. NUMPAIR: Number of X-Y pairs.



Words 3 and 4 repeated NUMPAIR.TABLEST data 1. Table identification number (if 0, no table) (integer).


3. EXTRAP: Extrapolation option (0=no extrapolation, 1=extrapolation).


5. TID: Table ID (integer).

Words 4 and 5 repeated NUMPAIR.TABLEM1 data (CODEX/CODEY: 0=Linear, 1=Log)

1. Table identification number (if 0, no table) (integer).


3. CODEX: Type of interpolation for the X-axis (integer).

4. CODEY: Type of interpolation for the Y-axis (integer).

5. EXTRAP: Extrapolation option (0=no extrapolation, 1=extrapolation).



Words 6 and 7 repeated NUMPAIR.

For example, consider the MUMAT entry below along with a TABLES1 entry.

MUMAT 1 NLPLAST

$+ INTEGER 1 1 2 0

$ YF HR METHOD NLTYPE

$ 2 3 4 5 6 7 8 9 10

+ REAL 0.0 9.+2

$ H/EP LIMIT1 E NU RHO A TREF


Material support


+ TABLES1 80

TABLES1 80 +

+ 0.0 0.0 7.5-4 9.+2 17.5-4 1.5+327.5-42.0+3 +

+ ENDT

Using the information above, these entries are formatted into the array shown below.


1 Length of thearray MUDATAI(1) 38 MUDATAR(1) 0.000000000000000D+000

2 Offset MUDATAI(2) 24 MUDATAR(2) 0.000000000000000D+000

3 MUMATidentifier MUDATAI(3) 2 MUDATAR(3) 0.000000000000000D+000

4 MUMAT data MUDATAI(4) 1 MUDATAR(4) 0.000000000000000D+0005 MUMAT data MUDATAI(5) 2.31E+18 MUDATAR(5) 0.000000000000000D+0006 MUMAT data MUDATAI(6) 2.31E+18 MUDATAR(6) 0.000000000000000D+0007 MUMAT data MUDATAI(7) 0 MUDATAR(7) 0.000000000000000D+0008 MUMAT data MUDATAI(8) 0 MUDATAR(8) 0.000000000000000D+0009 MUMAT data MUDATAI(9) 7 MUDATAR(9) 0.000000000000000D+00010 MUMAT data MUDATAI(10) 2.31E+18 MUDATAR(10) 0.000000000000000D+00011 Code

descriptor

1 = Real data

MUDATAI(11) 1 MUDATAR(11) 0.000000000000000D+000

12 Size of realdata MUDATAI(12) 2 MUDATAR(12) 0.000000000000000D+000

13Value(1) inMUDATARarray


14Value(2) inMUDATARarray

MUDATAI(14) 0 MUDATAR(14) 900

15 Codedescriptor

2 = Integer data


16 Size of Integerdata MUDATAI(16) 4 MUDATAR(16) 0.000000000000000D+000

17 Value(1) MUDATAI(17) 1 MUDATAR(17) 0.000000000000000D+00018 Value(2) MUDATAI(18) 1 MUDATAR(18) 0.000000000000000D+00019 Value(3) MUDATAI(19) 2 MUDATAR(19) 0.000000000000000D+00020 Value(4) MUDATAI(20) 0 MUDATAR(20) 0.000000000000000D+00021 Code

descriptor

3 = Tables1data




Material support


22 Number ofTables1

Table IDs


23 Tables1 ID MUDATAI(23) 80 MUDATAR(23) 0.000000000000000D+00024 Code

descriptor

6 = Mappingdata


25 Size ofMapping data MUDATAI(25) 3 MUDATAR(25) 0.000000000000000D+000

26 Table ID MUDATAI(26) 80 MUDATAR(26) 0.000000000000000D+00027 Table Type

3 = Tables1


28 Index toTables1 data MUDATAI(28) 29 MUDATAR(28) 0.000000000000000D+000

29 Tables1 ID MUDATAI(29) 80 MUDATAR(29) 0.000000000000000D+00030 No. of XY pairs MUDATAI(30) 4 MUDATAR(30) 0.000000000000000D+00031 X(1) MUDATAI(31) 0 MUDATAR(31) 0.000000000000000D+00032 Y(1) MUDATAI(32) 0 MUDATAR(32) 0.000000000000000D+00033 X(2) MUDATAI(33) 0 MUDATAR(33) 7.500000000000000D-00434 Y(2) MUDATAI(34) 0 MUDATAR(34) 90035 X(3) MUDATAI(35) 0 MUDATAR(35) 1.750000000000000D-00336 Y(3) MUDATAI(36) 0 MUDATAR(36) 150037 X(4) MUDATAI(37) 0 MUDATAR(37) 2.750000000000000D-00338 Y(4) MUDATAI(38) 0 MUDATAR(38) 2000

Source code examples

Source code examples are included with the NX Nastran installation at the following locations.

• Source code written in C can be found at:installation_location\nxn11\nxumat\democ\

• Source code written in FORTRAN can be found at:installation_location\nxn11\nxumat\demof\

Compiling instructions

The source code examples are located with the installation at the following locations.

• The FORTRAN source code and compile procedures are located at:installation_location\nxn11\nxumat\demof

• The C source code and compile procedures are located at:installation_location\nxn11\nxumat\democ


Material support


The example source code has been tested with the Intel Compiler 13. The resulting .dll for Windowsand the resulting .so for Linux, both generated from this compiler version, have been tested andverified to work with NX Nastran 11. You can use any compiler as long as you adhere to the standardNXUMAT description in the provided source code, and the compiled .dll or .so file uses one of thefollowing handles as entry points:NXUMAT, NXUMAT_, nxumat or nxumat_.

Template make files are provided in the democ and demof folders for both Windows and Linuxoperating systems.

If you run “nmake libumat.dll” in the demof or democ folders on a Windows machine, a Windowsshared library (libnxumat.dll) will be produced from the FORTRAN or C code, respectively.

If you run “make libumat.so” in the demof or democ folders on a Linux machine, a Linux shared library(libnxumat.so) will be produced from the FORTRAN or C code, respectively.

This assumes that you have an Intel Fortran or C compiler, and the nmake or make utility on Windowsor Linux, respectively. You will need to point to proper compilers and linkers in the makefiles tosuccessfully compile.

Ready-to-run routine and test cases

Compiled example routines and NX Nastran input files are included with the NX Nastran installationto demonstrate the input file requirements and the general workflow.

• On Windows:Compiled Fortran code: installation_location\nxn11\em64tntl\libnxumat_demofCompiled C code: installation_location\nxn11\em64tntl\libnxumat_democ.dll

• On Linux:Compiled Fortran code: installation_location\nxn11\x86_64linuxl\libnxumat_demof.soCompiled C code: installation_location\nxn11\x86_64linuxl\libnxumat_democ.so

Before using the compiled examples, you will need to follow the instructions in Material library pathto point NX Nastran to a compiled example library. You can point NX Nastran to either the Fortranor the C compiled example.

Nine material models are included in the compiled example. You can select a specific model in yourNX Nastran input file with the MODNAME1 field on the MUMAT bulk entry.

The following table summarizes the nine material models, the MODNAME1 input definition youuse to select a specific material model, and a ready-to-run NX Nastran input file nxumatex*.dat foreach material model.

You can find the ready-to-run input files at:installation_location\nxn11\nxumat\demodat\

MODNAME1 Input File DescriptionEISO nxumatex1.dat Isotropic, temperature independentEORTHO nxumatex2.dat Orthotropic, temperature independentEANISO nxumatex3.dat Anisotropic, temperature independentETISO nxumatex4.dat Isotropic, temperature dependentETORTHO nxumatex5.dat Orthotropic, temperature dependentETANISO nxumatex6.dat Anisotropic, temperature dependent



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MODNAME1 Input File DescriptionNLPLAST nxumatex7.dat Plasticity, temperature dependentNLCREEP nxumatex8.dat Creep, temperature dependent

PLASCR nxumatex9.dat Plasticity and creep combined, temperaturedependent

Material library path

There are three ways, in the following order of precedence, in which you can point NX Nastran tothe location of your material library.

• You can define the keyword umatlib on either the command line or in your RCF file. For example,umatlib=D:/scratch/mymaterial.dll

• You can define the environment variable NXN_LOCAL_LIB_NAME. For example,NXN_LOCAL_LIB_NAME D:/scratch/mymaterial.dll

• If you are run NX Nastran on a Windows machine, you can replace the following file with yourcompiled routine:installation_location\nxn11\em64tntl\libnxumat.dll

If you are run NX Nastran on a Linux machine, you can replace the following file with yourcompiled routine:installation_location\nxn11\x86_64linuxl\libnxumat.so

Debugging

You can debug a .dll file using Visual Studio on Windows as long as you build the .dll file with theoptions required for debugging. The .dll file you use for debugging can be built with the make_dll.bator the nmake utility.You can use the following procedure to debug.

1. NX Nastran must be configured so that it reads its input options from a file instead of thecommand line. You will create an *.asg file which contains the configuration options. The *.asgfile will include options such as the input file name, memory settings, and optionally the umatlibkeyword defining the path to the .dll.You can use the following procedure to create an *.asg file.

a. Set the environment variable NXN_NOEXE=1.set NXN_NOEXE=1

b. Define your Material library path. Run an NX Nastran job using the input file which includesyour MUMAT bulk entry. When the variable NXN_NOEXE=1 is defined, NX Nastran willgenerate the .asg file and stop. No solution will be performed. The .asg file generated willhave the name of the input file along with some process id information appended to it.

c. Reset the environment variable NXN_NOEXE.set NXN_NOEXE

2. You can start debugging with the command:devenv installation_location\nxn11\em64tntL\analysis.exe


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This will start a Visual Studio debug session for the NX Nastran executable, analysis.exe. Enter aproject name including the path when Visual Studio makes this requests. For example:

D:\umat\demof\analysis.sln

3. Select Save.

4. Visual Studio will start with a solution named analysis. Right-click on the solution and chooseproperties.

5. When the property form appears, enter the location of your *.asg file in the Arguments fieldin the Parameters section.

6. Select the File Open command, open the top level driver for the UMAT .dll. In this environment, itwill be the “nxumat.F” file. Scroll down to the first executable line of code, and set a break.

7. Select the “Start Debugging” icon (green triangle) or press the F5 key. NX Nastran should launchand the execution should stop at the break point which was set in the previous step.

8. The next time you want to debug, you can use the following command which uses the savedsolution and bypasses several of the initial steps.

devenv analysis.sln

9. If the run is terminated prematurely from within the debugger, you must delete the temporaryfiles in the work directory, for example, D:\workdir. Failure to do so will cause subsequentdebugging runs to fail.

5.5 Creep analysisYou can perform creep analysis in SOL 401 using the Bailey-Norton model. All elements supported inSOL 401, except for the rigid elements, support the creep material defined using the MATCRP bulkentry. The Bailey-Norton model represents isotropic creep with optional temperature-dependency.

You can selectively enable and disable creep effects in subcases. If creep-enabled subcases aresequentially dependent, the total accumulated creep strain at the end of one subcase is used as theinitial creep strain for the next subcase. If a creep-disabled subcase is placed between creep-enabledsubcases, and the subcases are all sequentially dependent, the total accumulated creep strainat the end of the preceding creep-enabled subcase is used as the initial creep strain for the latercreep-enabled subcase because no incremental creep strain arises in creep-disabled subcases.

Creep analysis implementation

The Bailey-Norton model relates creep strain to stress and time as follows:

Equation 5-1.

where εec is the effective creep strain, σe is effective stress, t is time, and A, B, and D are user-definedcoefficients. Because the model uses effective stress and effective creep strain, the values for thecoefficients are directly relatable to results from uniaxial testing.



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In the Bailey-Norton model, temperature is not accounted for explicitly. To account fortemperature-dependence, you can define the coefficients as tabular functions of temperature.

For some very simple cases, you can use Equation 5-1 directly to calculate the effective creep strainas a function of time. However, for the general case where temperature and stress vary, and computersimulation is required, Equation 5-1 is applied incrementally over a finite number of time steps.

During the creep analysis, the incremental creep strain for each time step is calculated by numericallyintegrating the instantaneous creep strain rate. The formula for creep strain rate is obtained fromthe following flow rule:

Equation 5-2.

where έijc are the components of the creep strain rate tensor, έec is the effective creep strain rate, andSij are the components of the deviatoric stress tensor.

The effective creep strain rate is obtained by differentiating Equation 5-1 with respect to time.

To evaluate each increment of creep strain, the software performs a numerical integration basedon the generalized trapezoidal rule as follows:

Equation 5-3.

where Δtn = tn – tn-1 is the duration of the subinterval, and β is a user-defined numerical integrationparameter. Generally, the default value for β of 0.5 is appropriate.

User interface

• With the MATNL parameter, you can globally switch the creep analysis capability on or off.

For more information, see the MATNL parameter.

• Parameters are available for use with the NLCNTL bulk entry. These parameters allow you toturn off the creep capability in subcases, control adaptive time stepping or define a constant timestep, and define the integration factor in Equation 5-3.

For more information, see the NLCNTL bulk entry in the NX Nastran Quick Reference Guide.

• With the MATCRP bulk entry, you define parameters related to the creep constitutive model.

For more information, see the MATCRP bulk entry in the NX Nastran Quick Reference Guide.

To activate the creep analysis capability in SOL 401, do the following:

1. Reference both the MAT1 and MATCRP bulk entries in the regions where creep occurs.


3. Include a NLCNTL case control command that points to a NLCNTL bulk entry.


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4. On the NLCNTL bulk entry, specify any applicable parameters.

If your input file contains subcases, and you want to include the effects of creep in specific subcases,but not others, you have two options.

Option 1: Use a global NLCNTL case control command


2. Include a NLCNTL case control command above the subcases that points to a NLCNTL bulk entry.

3. On the NLCNTL bulk entry pointed to by the global NLCNTL case control command, specifyany applicable parameters.

4. In the subcases that you want to disable the creep analysis capability, include a NLCNTL casecontrol command that points to a NLCNTL bulk entry.

5. On the NLCNTL bulk entry pointed to by the NLCNTL case control commands in the subcases,specify “CREEP” in a PARAMi field and “NO” in the corresponding VALUEi field.

Option 2: Include NLCNTL case control commands in every subcase


2. Include NLCNTL case control commands in each subcase. Multiple NLCNTL case controlcommands can point to a single NLCNTL bulk entry.

3. In subcases that you want to enable the creep analysis capability, have the NLCNTL case controlcommand point to a NLCNTL bulk entry with any applicable parameters specified.

4. In subcases that you want to disable the creep analysis capability, have the NLCNTL case controlcommand point to a NLCNTL bulk entry that has “CREEP” specified in a PARAMi field and “NO”specified in the corresponding VALUEi field.

To directly define solution times for the creep analysis, include a TSTEP case control command inyour input file that points to a TSTEP1 bulk entry. On the TSTEP1 bulk entry, you can specify thesolution times and specify which solution times you want results output.

The solution times you specify on the TSTEP1 bulk entry may result in time steps that are either toocoarse to produce accurate results, or too fine to produce results efficiently. To assist you in avoidingsuch problems, the software uses adaptive time stepping by default. You can tweak the adaptive timestepping algorithm or override adaptive time stepping altogether with the parameters on the NLCNTLbulk entry. For more information on adaptive time stepping, see Time step control.

In a SOL 401 creep analysis, the property bulk entry referenced by all non-rigid elements mustreference a MAT1 bulk entry and a MATCRP bulk entry that have the same material identificationnumber. If the properties on the MAT1 bulk entry are temperature-dependent, include a MATT1 bulkentry with the same material identification number.

You use the MATCRP bulk entry to specify:

• The stress threshold below which creep does not occur.

• The hardening rule to apply.



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• The coefficients in the Bailey-Norton creep model.

You can specify the coefficients in the Bailey-Norton creep model as either constant or as a functionof temperature. To specify a coefficient as temperature-dependent, enter the identification numberof a TABLEM1 bulk entry in the corresponding A, B, or D field of the MATCRP bulk entry. On theTABLEM1 bulk entry, enter tabular data that describes how the coefficient varies with temperature. Atpresent, a MATCRP bulk entry cannot reference a TABLEM2, TABLEM3, or TABLEM4 bulk entry.

Time step control

During a SOL 401 creep analysis, the solution times depend on:

• How you specify the TSTEP1 bulk entry.

• How you specify the time stepping parameters.

The time steps that result from the TSTEP1 bulk entry specification may be too coarse to produceaccurate results, or too fine to produce results efficiently. By default, the software uses an adaptivetime stepping algorithm to avoid such problems.

You can tweak the adaptive time stepping algorithm or override adaptive time stepping altogether anduse a constant time step with new parameters for the NLCNTL bulk entry. The new parameters are:

CRCERAT Ratio of maximum creep increment to elastic strain that is used to adaptivelyspecify the next time step.

CRCINC Maximum creep increment that is used to adaptively specify the next time step.

CRICOFF Creep strain increment below which the next time step is the product of thecurrent time step and the maximum time step multiplying factor.

CRINFAC Numerical integration parameter.

See Equation 5-3.

DTINIT Initial time step for adaptive time stepping, or the constant time step if adaptivetime stepping is overridden.

CRMFMN Minimum time step multiplying factor.

CRMFMX Maximum time step multiplying factor.

DTSBCDT Controls whether the first time step in a sequential subcase uses the initial timestep or the time step calculated at the end of the previous subcase.

CRTEABS Maximum absolute truncation error that is used to adaptively specify the nexttime step.

CRTECO Effective creep strain below which CRTEABS is used, and above whichCRTEREL is used.

CRTEREL Maximum relative truncation error that is used to adaptively specify the nexttime step.

TSCCR Specifies the time stepping method.

DTMIN Minimum time step.

DTMAX Maximum time step.


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The TSCCR parameter controls the overall time stepping behavior. Use the TSCCR parameter to:

• Select the adaptive time stepping algorithm that the software uses to calculate the next time step.

• Override adaptive time stepping altogether and have the software use the value of the DTINITparameter as a constant time step.

The adaptive time stepping algorithm options include the following:

• The next time step is based on the maximum creep strain increment criterion. You specify themaximum creep strain increment with the CRCINC parameter.

• The next time step is based on the ratio of maximum creep increment to elastic strain criterion.You specify the ratio of maximum creep increment to elastic strain with the CRCERAT parameter.

• The next time step is based on the maximum truncation error criterion. For this option, youhave three sub-options.

o Use the maximum absolute truncation error. You specify the maximum absolute truncationerror with the CRTEABS parameter.

o Use the maximum relative truncation error. You specify the maximum relative truncationerror with the CRTEREL parameter.

o Use the maximum absolute truncation error if the creep strain is less than the value specifiedby the CRTECO parameter, and use the maximum relative truncation error if the creep strainis greater than the value specified by the CRTECO parameter.

• The next time step is the shortest time step calculated by any combination of the maximum creepstrain increment, ratio of maximum creep increment to elastic strain, and maximum truncationerror criteria.

When the creep simulation begins, the value of the DTINIT parameter is always used as the firsttime step. If adaptive time stepping is overridden, the value of the DTINIT parameter is used as aconstant time step throughout the simulation.

If adaptive time stepping is not overridden, after each time step the software compares the calculatedcreep strain increment to the value of the CRICOFF parameter. If the creep strain increment isgreater than the value of the CRICOFF parameter, the software uses the adaptive time steppingalgorithm to calculate the next time step. If the creep strain increment is less than the value ofthe CRICOFF parameter, the software uses the product of the current time step and the value ofthe CRMFMX parameter as the next time step.

If the software uses the adaptive time stepping algorithm to calculate the next time step, the next timestep is compared to the product of the current time step and the value for the CRMFMN parameter. Ifthe next time step is smaller than the product of the current time step and the value for the CRMFMNparameter, the software halves the current time step, recalculates the current creep strain increment,and reenters the algorithm at the point the creep strain increment is compared to the value of theCRICOFF parameter. If the next time step is larger than the product of the current time step and thevalue of the CRMFMN parameter, the software keeps the next time step.

The next time step is then compared against the values of the DTMAX and DTMIN parameters. First,the software checks to see if the value of the DTMAX parameter is 0.0. If so, the software accepts thevalue for the next time step and uses it to compute the next creep strain increment. If the value of



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the DTMAX parameter is not set to 0.0, the next time step is compared to the value of the DTMAXparameter. If the next time step is larger than the value of the DTMAX parameter, the software usesthe value of the DTMAX parameter as the next time step and uses it to compute the next creepstrain increment. If the next time step is smaller than the value of the DTMAX parameter, the nexttime step is compared to the value of the DTMIN parameter. If the next time step is smaller thanthe value for the DTMIN parameter, the software halves the current time step, recalculates thecurrent creep strain increment, and reenters the algorithm at the point the creep strain increment iscompared to the CRICOFF parameter. If the next time step is larger than the value of the DTMINparameter, the software accepts the value for the next time step and uses it to compute the nextcreep strain increment.

The adaptive time stepping algorithm is summarized by the following flowchart. In the flowchart, thenotation for the value of a parameter is Vparameter name.


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Figure 5-1. Flowchart of adaptive time stepping algorithm

Regardless of whether you are using adaptive time stepping or a constant time step, the solutiontimes you specify with the TENDi and NINCi fields on TSTEP1 bulk entries are always honored. At alltimes during the creep simulation, if the next time step would result in skipping over a TSTEP1-definedsolution time, the software truncates the next time step so that a solve occurs at that solution time.

As a best practice, consider using the TSTEP1 bulk entry to specify only the times you want theresults output, and let the adaptive time stepping algorithm determine all the other solution times.

For additional information on the creep-related parameters, see the NLCNTL bulk entry in the NXNastran Quick Reference Guide.



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5.6 Overview of the Creep ModelClassical creep models consist of the following three fundamental parts.

• A uniaxial creep equation expresses the effective creep strain or creep strain rate as a function ofeffective stress, temperature, and time. In a state of uniaxial stress, the effective creep strain andeffective stress reduce to the uniaxial creep strain and the uniaxial stress. The effective creepstrain in terms of the creep strain tensor components is given by:

The effective stress in terms of the stress tensor components ij and deviatoric stress tensorcomponents sij is given by:

• A flow rule determines the creep strain rate tensor components for a general state of stress.

• A hardening rule determines creep strain rates from the uniaxial rate equation under changingstress and temperature.

Overview of the Hardening Rules for Creep Models

The hardening rule provides the mathematical means to determine the effective creep rate underchanging temperature and/or stress. The creep hardening rule is used to determine the currentmaterial state relative to further creep straining. It plays a role similar to the yield surface in plasticitytheory.

You can select one of the following hardening rules for the creep model using the HARD field on theMATCRP bulk entry.

• HARD = STRAIN (default) selects strain hardening.

• HARD = TIME selects time hardening.

Strain Hardening

The strain hardening rule assumes that the material state is determined by the accumulated effectivecreep strain. When stress or temperature changes, the shift from one creep curve to another is basedon the accumulated effective creep strain. This shift is illustrated in the following figure.


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The strain hardening rule is generally considered to be superior to the time hardening rule and is,therefore, the default for the software's creep equations. Under conditions of constant stress andtemperature, the time hardening and strain hardening assumptions produce identical results.

Time Hardening

The time hardening rule uses time as a material state variable. It assumes that the material state isdetermined by the length of time the material has been creeping.

When stress or temperature changes, the shift from one creep curve to another is based on theaccumulated creep time. This shift is illustrated in the following figure.



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The time hardening rule is usually not very realistic when very large changes in temperature orstress occur over the interval of interest.

Maximum Truncation Error Criterion

With the maximum truncation error criterion, NX Nastran also calculates the next time step by scalingthe current time step. The value of the CRTECO parameter on the NLCNTL bulk entry determineswhether the maximum absolute truncation error or the maximum relative truncation error is used toadaptively specify the next time step.

• If the effective creep strain is less than the value of the CRTECO parameter, the maximumabsolute truncation error is used. The maximum absolute truncation error is specified with theCRTEABS parameter on the NLCNTL bulk entry.

• If the effective creep strain is greater than the value of the CRTECO parameter, the maximumrelative truncation error is used. The maximum relative truncation error is specified with theCRTEREL parameter on the NLCNTL bulk entry.

The next time step is calculated by scaling the current time. When the maximum absolute truncationerror is applied, the software computes the next time step as:

When the maximum relative truncation error is applied, the software computes the next time step as:


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where e is the calculated truncation error, and is the current effective creep strain increment.There are two approaches to calculate the truncation error e corresponding to the two differentscenarios:

1. At the first time step or if β=0 or β=1, the two point rule truncation error is used.

2. From the second time step and 0 < β< 1, the three point rule truncation error is used.

Maximum creep strain increment criterion

With the maximum creep strain increment criterion, NX Nastran calculates the next time step byscaling the current time step as follows:

Equation 5-4.

where CRCINC is the value of the CRCINC parameter and Δεec is the current effective creep strainincrement.

Ratio of maximum creep increment to elastic strain criterion

With the ratio of maximum creep increment to elastic strain criterion, NX Nastran calculates the nexttime step by scaling the current time as follows:

Equation 5-5.

where CRCERAT is the value of the CRCERAT parameter, Δεec is the current effective creep strainincrement, and εeE is the current total effective elastic strain.

5.7 Disable plasticity and creepThe MATNL parameter allows you to switch all creep and/or plasticity effects on/off for all relatedelements.

When the MATNL parameter is set to 1, PARAM,MATNL,1 is defined to turn creep and/or plasticityeffects on, the MATOVR bulk entry allows you to optionally disable the creep and plasticity effects offfor the elements selected GROUP entry.



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• Use TYPE = “ELEM” to reference a GROUP bulk entry that includes a list of elements. TheMATOVR specification applies to the listed elements.

• Use TYPE = “PROP” to reference a GROUP bulk entry that includes a list of properties. TheMATOVR specification applies to all elements that reference the properties listed in the GROUPbulk entry.

For more information, see the MATOVR bulk entry in the NX Nastran Quick Reference Guide.


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Chapter 6: Boundary conditions

6.1 ConstraintsIn the Case Control section, an SPC entry is used to select a single point constraint set (s-set) whichwill be applied to the structural model. The specified set identification must be identical to the SIDfield of an SPC, SPC1 or SPCADD bulk entry. Notice that SPCADD entries take precedence overSPC or SPC1 entries. If both have the same SID, only the SPCADD entry will be used.

A significant application of SPC is the imposition of boundary conditions. The PS field in the GRIDentry is also able to specify single-point constraints associated with a grid point. Although, theseconstraints are so-called permanent constraints which can not be changed during the analysis. Anadvantage of using SPC to specify boundary conditions is that these boundary conditions can bechanged from subcase to subcase by selecting a different SPC set inside each subcase.

SPC input summary

• The SPC=n case control command selects either the SPC, SPC1, or SPCADD bulk entry.

The SPC condition can change between subcases.

The SPC entry can optionally be used to define a time-unassigned enforced displacement. Itcannot be defined as time-assigned. That is, it cannot be selected with the EXCITEID on theTLOAD1 entry. For a time-assigned enforced displacement, you can use the SPCD entry. SeeMechanical Loads for information on the SPCD entry.

The SPCFORCES case control command is supported to request the SPC force output.

6.2 Multipoint constraintWhile a single point constraint (SPC) is used to constrain the motion of a degree-of-freedom,a multipoint constraint (MPC) is used to tie the motion of one degree-of-freedom to otherdegrees-of-freedom. The MPC command in the Case Control section is used to select a multipointconstraint set in the Bulk Data. The specified MPC set identification must appear at least in oneMPC or MPCADD bulk entry. Each MPC bulk entry may be used to define a constraint equationinvolving a group of degrees-of-freedom in which the first degree-of-freedom is assumed to be thedependent degree-of-freedom and included in the m-set. All the degree-of-freedom in m-set will becondensed out prior to the matrix operations. Their response will be directly recovered from those ofthe independent degrees-of-freedom according to the specified constraint equation. MPC conditionscan change from subcase to subcase.

MPC input summary

• The MPC=n case control command selects either the MPC or MPCADD bulk entry.

MPC conditions can change between subcases.

MPCs do not update for large displacements (PARAM,LGDISP,1).



6.3 Enforced displacementsEnforced displacements may be specified in the Bulk Data section using SPC or SPCD entries. AnSPC can be used to define time-unassigned enforced displacements. An SPCD can be used todefine time-assigned or time-unassigned enforced displacements. See Boundary conditions andMechanical Loads for input details.

Each SPC entry may define enforced displacements for up to two grid or scalar points. Several SPCentries which reference the same SID may be used if enforced displacements for more than two gridor scalar points are desired. The only disadvantage of this method is that the entire s-set must beredefined if the enforced displacement conditions vary among subcases.

If a time-assigned or time-unassigned enforced displacement condition is defined with the SPCDentry, a constraint must also be defined with the SPC entry on the same DOF referenced by theSPCD entry.

If multiple enforced displacement conditions are applied to the same DOF, the software uses thefollowing precedence.

• A time-assigned enforced displacement defined with the SPCD entry, which is referenced bythe EXCITEID on the TLOAD1 entry, will overwrite time-unassigned enforced displacementsdefined with the SPCD or SPC entries.

• A time-unassigned enforced displacement defined with the SPCD entry, which is referenced bythe LOAD=n case control command, will overwrite a time-unassigned enforced displacementdefined with the SPC entry.



Chapter 7: Loads

7.1 Loads overviewThe solution strategy in nonlinear is to apply the loads in an incremental fashion until the desired loadlevel is reached. The algorithms remember the loads from one subcase to the next.

The methods employed to define loads in SOL 401 are similar to those used in a time history solutions.A single degree-of-freedom or a set of GRID points may be loaded with force pattern that varies withtime. Functions may be tabular such as an earthquake or a booster liftoff, or they may be simpleanalytic functions such as a sine wave. Simple static load sets may be used to create the dynamicloads. They may be scaled and combined with other loads to simulate complex loading problems.

Time history loads define the loadings as functions of time and the location. They can be a loadapplied at a particular degree-of-freedom, pressure over the surface area, or the body force simulatingan acceleration. The time history is provided by TLOADi bulk entry.

7.2 Mechanical loadsMechanical loads can be defined in SOL 401 as time-assigned or time-unassigned. SOL 401 is astatic solution, and time is only used as the mechanism to increment loads. Time-assigned andtime-unassigned loads can be combined in the same static subcase.

• Load selection in Case Control:

o Time-unassigned loads are selected with the LOAD case control command,

LOAD=n

where n points to a DAREA, FORCE, FORCE1, FORCE2, GRAV, PLOAD, PLOAD4,RFORCE, RFORCE1, SLOAD, SPCD, or LOADSET entry.

The LVAR parameter on the NLCNTL bulk entry controls if time-unassigned mechanical loadsare ramped (default), or not ramped for each subcase. The ramping helps convergence byreducing the load increments. You can optionally turn the ramping off by setting LVAR=STEPon the NLCNTL bulk entry.

o Time-assigned loads are selected with the DLOAD case control command,

DLOAD=n

where n points to a load set defined by a TLOAD1 bulk entry, or a DLOAD bulk entry if youwant to combine multiple TLOAD1 entries into a single load set.

• Time-assigned load definition in Bulk Data:

o TLOAD1 - Defines a time-assigned load.

o TABLEDi (i=1,2,3,4) - Table that defines the load variation with time.


Chapter 7: Loads

o DLOAD - Combines several TLOAD1 entries.

• Defining the TLOAD1 entry:

o The EXCITEID field on the TLOAD1 entry selects the static load set IDs.

o The supported static load inputs are the DAREA, FORCE, FORCE1, FORCE2, GRAV,PLOAD, PLOAD4, RFORCE, RFORCE1, SLOAD, and SPCD bulk entries.

o The TYPE field on the TLOAD1 entry should be “0” for all load inputs selected by theEXCITEID field, except for the SPCD entry. The SPCD entry requires “1” in the TYPE field.

o A real value is supported in the DELAY field on a TLOAD1 entry to optionally shift the timesteps used to compute the associated loads.

o A temperature load cannot be selected on the EXCITEID field. See Thermal Loads.

o The TID field selects a TABLEDi, which defines a load scaling versus time function.

o The figure below demonstrates how the DLOAD, TLOAD1, FORCE (for example), andTABLEDi bulk entries relate to one another.

o Load Input Example 1:

When there is more than one time-assigned load set, the DLOAD bulk entry is required:

$2345678$2345678$2345678$2345678$2345678$2345678$2345678$2345678$2345678$$ DLOAD COMBINES MULTIPLE TLOAD1 (102 AND 105)DLOAD 17 1. 1. 102 1. 105$$ TIME-ASSIGNED FORCE, EXCITEID=125, TYPE=0 (DEFAULT), TIME FUNCTION TID=13TLOAD1 102 125 13$FORCE 125 80 0 1. 3. 0. 0.$$ TIME FUNCTION 13 USED FOR FORCE LOADTABLED2 13 0. ++ 0. 0. 1. 100. 2. 0. ENDT$$ TIME-ASSIGNED FORCE EXCITEID=3, TYPE=0 (DEFAULT), TIME FUNCTION TID=12TLOAD1 105 3 1 12


Chapter 7: Loads

Loads

$FORCE 3 73 0 2. 8. 0. 0.$$ TIME FUNCTION 12 USED FOR FORCE LOADTABLED2 12 0. ++ 0. 0. 2. 1. ENDT

o Load Input Example 2:

When there is only one time-assigned load set, the DLOAD entry is not required:

$2345678$2345678$2345678$2345678$2345678$2345678$2345678$2345678$2345678$$ TIME-ASSIGNED FORCE, EXCITEID=125, TYPE=0 (DEFAULT), TIME FUNCTION TID=13TLOAD1 102 125 13$FORCE 125 80 0 1. 3. 0. 0.PLOAD 125 100.0 21 30 18 10PLOAD 125 100.0 10 18 22 25$$ TIME FUNCTION 13 USED FOR LOADTABLED2 13 0. ++ 0. 0. 1. 100. 2. 0. ENDT$

• Additional Information about mechanical loads:

o Enforced displacements may be specified using SPC or SPCD entries. See Enforceddisplacements.

o Loads in any subcase are total loads as opposed to incremental loads from the previoussubcase. In other words, the ending load from a previous subcase does not become theinitial loading for the consecutive subcase.

o If no load is applied in a subcase, the total load is zero.

o LOAD=n or DLOAD=n defined at the global level is used in all statics subcases unlessa different LOAD=n or DLOAD=n is defined in a subcase.

o If a time-assigned and time-unassigned enforced displacement condition is defined withthe SPCD entry, a constraint must also be defined with the SPC entry on the same DOFreferenced by the SPCD entry.

o The TSTEP1 bulk entry defines the time step intervals in which a solution will be generatedand output in a static subcase. If your time steps defined by the TSTEP1 entry exceed thetime values defined in your TABLEDi entry, by default, the software will extrapolate the datadefined in the TABLEDi entry. The software will issue a warning if extrapolation occurs. Ifyou do not want the software to extrapolate the data, you can enter “1” in the EXTRAPfield on the TABLEDi entry.

o In SOL 401, when RFORCE or RFORCE1 entries are referenced by the EXCITEID field ona TLOAD1 entry, the data on the associated TABLEDi, along with the scale factors S and


Loads

Chapter 7: Loads

Si on a DLOAD entry (if defined), scale the angular velocity (ω) and acceleration (α), whichare used to compute an inertia force in the equation F = [m] [ω x (ω x r)) + α x r]. Since ω issquared in the force computation, the resulting scaling is not linearly related to the computedforce (F). All other solutions scale the computed force (F).

7.3 Thermal loadsA thermal load requires a load temperature (Tload), an initial temperature (Tinit), and a referencetemperature (Tref).

Thermal strain is calculated by

ε = αload(Tload – Tref) – αinit(Tinit – Tref)

• Tload is the temperature load which induces a thermal strain.

• Tinit is the strain free temperature used in the analysis.

• Tref is the initial temperature used when computing the temperature dependent coefficient ofthermal expansion, and is defined on the MATi entry. See Computing the coefficient of thermalexpansion.

• If either Tload or Tinit are defined, they both must be defined.

• If the coefficient of thermal expansion is defined as temperature dependent with the MATTientries, αload is evaluated at Tload, and αinit is evaluated at Tinit .

If the coefficient of thermal expansion is not defined as temperature dependent, αload and αinit areassigned the single value defined on the MATi entry.

• Tinit is defined using the TEMP(INIT) case control command, and must be the same for allsubcases. Typically, the TEMP(INIT) command is defined globally, and selects one of thefollowing.

o The TEMP(INIT) can select the TEMP and TEMPD entries in the bulk data.

For example,

...TEMP(INIT) = 100...BEGIN BULK...$ Initial temperatures defined in the bulk dataTEMP,100,5,232.0,6,354.4,...etc...$ TEMPD defines a temperature for grid points not included on a TEMP entryTEMPD,100,450.0...

o The TEMP(INIT) command can select the TEMPEX and TEMPD bulk entries. The TEMPEXentry references an external BUN file using the unit number defined with an ASSIGNstatement. The unit number must be unique to other BUN files, and to other reserved unitnumbers. The BUN file used to define Tinit must only include a single set of temperature data.


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Loads

If the BUN file only defines temperatures for a portion of the model (subset), the TEMPDentry must be included in the bulk data to define a temperature for the grid points notincluded in the BUN file.

For example,

...ASSIGN BUN=‘temperature0.bun’ UNIT=21...TEMP(INIT) = 100...BEGIN BULK...$ Initial temperatures defined in the BUN fileTEMPEX,100,21$ Temperature for grid points not in the BUN fileTEMPD,100,630.2...

• There are a variety of options to define Tload. These options can be defined globally and in asubcase. Any subcase definition will override any global definition. For example, if you define atime-unassigned Tload globally using the TEMP(LOAD) command, and you define a time-assignedTload in a subcase using the DTEMP command, the time-assigned Tload is used for that subcase.

o You can define a time-unassigned Tload with all temperatures defined in the bulk data. TheTEMP(LOAD) case control command selects the TEMP and TEMPD entries in the bulk data.

For example,

...SUBCASE 5

TEMP(LOAD) = 150BEGIN BULK...$ time-unassigned grid point load temperatures for subcase 5TEMP,150,74,232.0,23,354.4,...$ TEMPD defines a temperature for grid points not included on a TEMP entryTEMPD,150,450.0...

o You can define a time-unassigned Tload with temperatures defined in an external BUN file.The TEMP(LOAD) case control command selects the TEMPEX bulk entry and optionally theTEMPD entry. The TEMPEX entry references the external file using the unit number definedwith an ASSIGN statement. The unit number must be unique to other BUN files, and to otherreserved unit numbers. The BUN file selected with the TEMPEX bulk entry must only includea single set of temperature data.

If the BUN file only defines temperatures for a portion of the model (subset), the TEMPDentry must be included in the bulk data to define a temperature for the grid points notincluded in the BUN file.

TEMPEX example:

...ASSIGN BUN=‘temperature1.bun’ UNIT=22


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$...SUBCASE 10

TEMP(LOAD) = 200BEGIN BULK...$ Time-unassigned load temperatures for subcase 10TEMPEX, 200, 22$ Temperature for grid points not in the BUN fileTEMPD,200,630.2...

o You can define a time-assigned Tload with temperatures defined in the bulk data or in a BUNfile. The DTEMP case control command selects the DTEMP bulk entry, which defines a list oftime points versus set IDs. The set IDs are either the IDs of TEMP and TEMPD entries in thebulk data, or the IDs of TEMPEX and TEMPD entries in the bulk data. You cannot combineTEMP and TEMPEX entries with the same set ID.

Example with TEMP and TEMPD entries in the bulk data:

Note: This example assumes the TEMP entries for temperature sets 500 and 501 define temperatures for all grid points in the model, but set 502 defines temperatures for a subset. As a result, a TEMPD is only required for set 502....SUBCASE 15

DTEMP(LOAD) = 250...BEGIN BULK...$ DTEMP is a list of time points versus set IDsDTEMP,250,,,,,,,,++,.2,500,.4,501,.6,502...$ Load temperatures at t=.2TEMP,500,5,232.0,6,354.4,7,284.2...$ Load temperatures at t=.4TEMP,501,5,234.1,6,356.3,7,287.8...$ Load temperatures at t=.6TEMP,502,5,237.3,6,358.4,7,292.4$ Temperature for grid points not defined with TEMP entry 502.TEMPD,502,630.2...

Example with TEMPEX and TEMPD entries in the bulk data:

Note: This example asssumes the BUN files for temperature sets 501 and 502 define temperatures for all grid points in the model, but the BUN file for temperature set 502 only defines for a subset. As a result, a TEMPD is only required for set 500....ASSIGN BUN=‘temperature1.bun’ UNIT=22ASSIGN BUN=‘temperature2.bun’ UNIT=23ASSIGN BUN=‘temperature3.bun’ UNIT=24...SUBCASE 15

DTEMP(LOAD) = 250...BEGIN BULK...$ DTEMP is a list of time points versus set IDs


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Loads

DTEMP,250,,,,,,,,++,.2,500,.4,501,.6,502...$ Load temperatures at t=.2TEMPEX,500,22$ Temperature for grid points not defined in BUN fileTEMPD,500,345.4...$ Load temperatures at t=.4TEMPEX,501,23...$ Load temperatures at t=.6TEMPEX,502,24...$ If the BUN file for t=.2 and t=.4 includes data for all grid points, the TEMPD is not needed

o You can define a time-assigned Tload with temperatures defined in a single, external BUNfile. The DTEMP case control command selects the DTEMPEX bulk entry, which referencesthe external file using the unit number defined in the ASSIGN statement. The unit numbermust be unique to other BUN files, and to other reserved unit numbers. The single BUN fileselected with the DTEMPEX bulk entry must include temperature data for all grid points, andfor multiple time points. The BUN file can include temperatures for grids which are not in themodel, but unlike the TEMPEX example above, the BUN file selected with the DTEMPEXcannot define temperatures for only a portion of the model (subset). The TEMPD entrycannot be combined with the DTEMPEX entry.

DTEMPEX example:

...ASSIGN BUN=‘temperature.bun’ UNIT=23$...SUBCASE 20

DTEMP = 300BEGIN BULK...$ Time-assigned load temperatures for subcase 20DTEMPEX, 300, 23...

Additional information:

• The specification of TEMP(MATERIAL) or TEMP(BOTH) are unsupported and will cause afatal error if defined.

• The TVAR parameter on the NLCNTL bulk entry controls if time-unassigned temperature loadsselected with the TEMP(LOAD) case control command are ramped, or not ramped for eachsubcase.

o When TVAR=RAMP, the software ramps the load temperatures from the final Tload definedfor the previous static subcase to the Tload defined for the current subcase. The softwaredetermines the load temperature increments using the total number of time increments


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defined for that subcase. If Tload is not defined in the previous subcase, the software rampsfrom Tinit to the current Tload.

o When TVAR=STEP, the load temperatures are not ramped.

The default is “RAMP” except when Tend1 = 0.0 is defined on the TSTEP1 entry in the firststatic subcase. “STEP” occurs in this case.

• For the time-assigned temperature data, the software will interpolate the grid point temperatureswhen times are defined between the time points in the data. Although, if a solution time isoutside the data range, the software will use the data at the closest time point, and a warningwill be written to the f06 file.

• You can turn off the thermal strain computation by defining the parameter setting THRMST=NO(default=YES) on the NLCNTL bulk entry. This is useful for temperature dependent materialevaluation without thermal loading.

• When temperature dependent material properties are defined with the MATTi entries for a staticsubcase, the properties are evaluated at Tload selected with either the TEMP(LOAD) or DTEMPcase control. Both Tload and Tinit must be defined when temperature dependent propertiesare defined.

• A modal subcase which is not sequentially dependent (SEQDEP=NO) can include temperaturedependent material properties defined with the MATTi entries. The properties are evaluated atTload selected with the TEMP(LOAD) case control. The DTEMP case control command is notsupported in a modal subcase. Both Tload and Tinit must be defined when temperature dependentproperties are defined.

• The OTEMP case control command can be included to request solution temperature output.

Computing the coefficient of thermal expansion

You use temperature versus strain (length) test data to compute the temperature dependentcoefficient of thermal expansion (α). This data begins with the test specimen of initial length L at areference temperature (Tref). The axial strain (Li) is then measured at consecutive temperatures Ti.To calculate αi:

7.4 Defining solution time stepsMechanical and thermal loads can optionally be defined as a function of time in a static subcase.These time-assigned loads only use time as the mechanism to increment the loads.

The TSTEP1 bulk entry defines the time step intervals in which a solution will be generated andoutput in a static subcase. You include the TSTEP case control command in the static subcase toselect a specific TSTEP1 definition in the bulk data.


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Loads

The TSTEP1 entry includes the end times (Tendi), the number of increments (Ninci), and theincrement for computing output (Nouti). The start time for a particular subcase depends if it issequentially dependent (SD) or not sequentially dependent (NSD).

1 2 3 4 5 6 7 8 9 10TSTEP1 SID Tend1 Ninc1 Nout1

Tend2 Ninc2 Nout2

Tend3 Ninc3 Nout3

-etc-

TSTEP1 Input Example:

TSTEP1 1 10.0 5 2

50.0 4 3

100 2 ALL

In this example, assuming a start time=0.0 for the subcase, the resulting time steps are as follows.The time steps in which output occurs are highlighted. Output always occurs at the end time.

The 1st row has an end time of 10.0, 5 increments, and output at every 2nd time step.

Time Step 1 Time Step 2 Time Step 3 Time Step 4 Time Step 52.0 4.0 6.0 8.0 10.0

The 2nd row has an end time of 50.0, 4 increments, and output frequency at every 3rd time step.

Time Step 1 Time Step 2 Time Step 3 Time Step 420.0 30.0 40.0 50.0

The 3rd row has an end time of 100.0, 2 increments, and output at all time steps.

Time Step 1 Time Step 275.0 100.0

In the same example, assuming a start time=5.0 for the subcase, the resulting time steps for thefirst row are as follows.

Time Step 1 Time Step 2 Time Step 3 Time Step 4 Time Step 56.0 7.0 8.0 9.0 10.0

The 2nd and 3rd row are the same:

Time Step 1 Time Step 2 Time Step 3 Time Step 420.0 30.0 40.0 50.0

Time Step 1 Time Step 275.0 100.0


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Additional Information about TSTEP1:

• No output occurs when Nouti=0.

• Output always occurs at Tendi when Nouti≠0.

• Tendi must be increasing (Tendi < Tendi+1).

• When Tend1=0.0,

o No other times are allowed. This is the only time for the associated subcase.

o Ninci can be defined.

o Results will output at time = 0.0 if Nouti≠0.

• Nouti controls the frequency of results output. The table below summarizes the input options.

Nout Output frequencyYES Output occurs at all increments defined on TSTEP1.END Output occurs at the end time.

ALL

Output occurs at all increments on TSTEP1 and any softwaresubincrements.

Note: When Nouti=ALL in the context of the Simcenter Multiphysicssolution, the result output time steps will be a combination of the structuraloutput steps as well as the coupled time steps.

Integer = 0 No output occurs.

Integer > 0 Output is computed at every Nout increment specific with TSTEP1.

CPLD Output occurs only at coupling times. This option can only be defined bythe Simcenter Multiphysics environment.

• The start time (Tstart) for a static subcase is determined as follows:

o If a static subcase definition in the case control includes SEQDEP=NO, that subcase is notsequentially dependent (NSD). The start time for an NSD subcase is 0.0.

o For a sequentially dependent (SD) static subcase (default), the final Tendi from a previous SDor NSD static subcase is the start time (Tstart) for the current SD subcase. If an SD subcasehas no previous SD or NSD static subcases, the start time is 0.0 for that SD subcase, andTend1=0.0 is permitted. Otherwise, Tend1 > Tstart for all other SD subcases.

• If a creep material is included, the software uses adaptive time stepping by default. The adaptivetime stepping can result in additional solution time steps which are not defined by the TSTEP1entry. See Creep analysis.

Similarily, when running SOL 401 in the context of the Simcenter Multiphysics environment,additional solution time steps beyond what is defined on the TSTEP1 entries are possible.

For both of these cases, the Nout field on the TSTEP1 bulk entry still determines the frequency ofresults output.


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Loads

7.5 Bolt preloadThe bolt preload capability allows you to model bolts with either the 3D solid elements CHEXA,CPENTA, and CTETRA, or the 2D plane stress elements CPLSTS3, CPLSTS4, CPLSTS6, andCPLSTS8. A bolt preload subcase is required, and it can include geometric and material nonlinearconditions.

The input requirements for bolt preloads in SOL 401 are as follows.

• A bolt preload subcase requires the ANALYSIS=PRELOAD and BOLTLD=n case controlcommands. The BOLTLD command references a BOLTSEQ, BOLTFRC, or a BOLTLD bulk entry.Multiple bolt preload subcases can be defined, they can be defined as sequentially dependent ornonsequentially dependent. As a result, you can define bolt preload subcases to apply or removebolt forces in any sequence. For example, you can apply a tightening sequence of many boltsbefore and even after service loads are applied.

Note that a subcase with ANALYSIS=STATICS and the BOLTLD=n will cause a fatal error.

• You use the BOLTSEQ bulk entry to optionally define a sequence of preload steps in a singlesubcase. Each sequence step results in an axial bolt strain. The software then applies theresulting axial strain as an initial condition in consecutive preload steps in a sequence, and inconsecutive sequentially dependent static subcases. A preload step in a sequence includes thefollowing:

o The step number in the sequence.

o The ID of a BOLTLD and BOLTFRC bulk entry to select the bolts and the preloads.

o You use the BOLT bulk entry to select the elements that represent each bolt. The BOLTFRCentry select the BOLT bulk entry.

o The optional number of increments (Ninc) on the BOLTSEQ entry can be defined toincrement the bolt preloads. This is useful to reduce the bolt preloading steps and helpto solve convergence problems.

Note that the number of increments (Ninc) on the TSTEP1 entry in a bolt preload subcaseonly increments temperature loads and contact offsets. It does not increment the boltpreloads. If you set both the Ninc on the BOLTSEQ entry and Ninc on the TSTEP1 entry,all of the increments for temperature loads and contact offsets will occur in the first boltpreload increment.

• You use the BOLTLD bulk entry to optionally combine and scale bolt preload sets defined withthe BOLTFRC bulk entry..

Bolts in SOL 401 are defined with the ETYPE = 3 format on the BOLT bulk entry. This format requiresthat you list all of the elements that are used to model the bolt shaft on the BOLT entry. The grid pointyou enter in the GP field on the BOLT entry indicates to the software where to calculate the crosssectional area of the bolt. For this calculation, NX Nastran uses the direction that you define with theCSID and IDIR fields on the BOLT entry as the bolt axis. To avoid any cross sectional effects at thebolt ends, it is best to select a grid point in the GP field closer to the middle of bolt length.

NX Nastran uses the cross sectional area and the bolt preload force to estimate the initial bolt stressand strain. To account for compliance in the structure that is being bolted together, the softwareiterates on the bolt strain until convergence is satified.


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Bolt preload convergence

You can adjust the bolt preload convergence tolerance with the EPSBOLT parameter (default=1.0E-3),which is defined on the NLCNTL bulk entry. For each bolt preload iteration, the software computesthe difference between the current bolt preload and the user-defined preload. If the difference is lessthan value of the EPSBOLT parameter, the bolt preload calculation is considered converged. Ifthe difference is greater than EPSBOLT, the preload strain is recomputed for the next bolt preloaditeration. The iterations continue until either convergence is satisfied, or the number of iterationsreaches the value of the ITRBOLT parameter (default=20). The ITRBOLT parameter is also definedon the NLCNTL bulk entry.

The software issues a fatal error message if the bolt preload iterative solution fails to converge. Theconvergence information related to bolt preload is listed in the f06 file.

Note: In solution 401, the software iterates until the resulting bolt force matches, within a tolerance,the bolt preload you requested. The final bolt strain required to achieve the desired bolt force includesthe effects of any other defined loads (thermal loads, contact forces, and conditions from a previoussubcase). For example, if you compare the results from a solution with only bolt preloads defined withthe results from the same model with the addition of thermal loads and contact, the bolt force will bethe same, but the final bolt strain needed to achieve the bolt force will be different.

Zero bolt preload force

The option to define a zero bolt preload force is available. The zero preload is useful, for example, topredict the plastic axial bolt strain after applying a bolt preload, applying service loads, then finallyunloading the bolt to a zero axial force condition.

When you define a zero bolt preload force, the software iterates on the bolt strain until the zero boltforce condition is satisfied. The software checks for convergence by checking if P/AE < ZERBOLT,where

P is the resulting axial bolt force from an applied axial strain value,

A is the area of the bolt cross section,

E is the modulus of the material selected for the elements on the bolt, and

ZERBOLT is a parameter on the NLCNTL bulk entry and defaults to 1.0E-7.

Note that the software considers a bolt preload force to be zero when F/AE < ZERBOLT, where F isthe preload force you requested on the BOLTFRC entry. A preload of 0.0 always satisfies this, butit is also possible for the preload to be nonzero, yet A and E are large enough for the software todetermine that the zero preload convergence checking should be used.

If the software determines that your requested preload is nonzero, it uses the EPSBOLT parameter todetermine convergence, and not ZERBOLT. See the remarks on the NLCNTL bulk entry.

Loads in a preload subcase

The bolt preload subcase includes the bolt preload forces and can optionally include a temperatureload. No service loads can be defined directly in a bolt preload subcase. A service load is any loadselected with the LOAD=n or DLOAD=n case control commands. Although service loads cannot bedefined directly in a bolt preload subcase, a sequentially dependent bolt preload subcase will maintainthe service loads used in a previous static subcase.

For example, if a dload=n is defined in the previous static subcase, then this definition is used in theprevious subcase and in the bolt preload subcase. If the previous subcase does not include a service


Chapter 7: Loads

Loads

load, but a dload=n is defined globally, then this definition is used in the previous subcase andin the bolt preload subcase.

The service loads can change the strain and the resulting axial bolt force by either compressing orseparating the bolted joint. The abillity to include service loads in a bolt preload subcase is useful, forexample, if you define your bolt preloads in the first subcase, then you define your service loads in asingle or in consecutive subcases, then you reapply the bolt preloads.

Note that the maintaining of a service load from a previous static subcase is unique to the boltpreload subcase. For example, if two consecutive sequentially dependent static subcases aredefined and service loads are defined in the first but not in the second, the software will remove theload in the second subcase.

Constant time bolt preload subcase

A sequentially dependent bolt preload subcase must include a TSTEP1 bulk entry defined with aTend1 which is the same as the start time for that subcase.

Although the bolt preload subcase does not use time to increment bolt preloads, this enhancementprovides the ability to continue a time sequence through an intermediate bolt preload subcase. Aconsecutive sequentially dependent static subcase can then continue the time sequence to incrementservice loads.

A nonsequentially dependent bolt preload subcase still requires a TSTEP1 bulk entry defined withTend=0.0.

Bolted joints with a modeled gap

The MISFBLT parameter is available on the NLCNTL bulk entry to limit the bolt strain from onepreload increment to the next. It is useful, for example, if you model a bolted joint with a gap condition.

The bolt preload algorithm increments the bolt strain as the joint compresses. When a gap ispresent, before the joint is compressed, the axial bolt force will be relatively low. As a result, thepreload algorithm will increase the strain increments. Once the gap is closed and the joint begins tocompress, the bolt and contact forces will change quickly, possibly causing the contact conditionsto destabilized. By limiting the bolt preload strain increments, you can reduce the chance ofdestablization, and utlimately help achieve convergence.

At the start of a bolt preload solution, the software uses your requested preload force and the boltgeometry to estimate a bolt strain. The estimated strain (εes) is computed with the assumption thateverything is rigid except for the bolt. The software then computes the maximum allowable strainincrement using the MISFBLT parameter.

Δεmax = εes * MISFBLT

At each preload increment, the software compares Δεmax with the next computed preload incrementΔεi+1. The software uses the smaller of the two at each preload increment.

Bolt preload diagnostic messages in .f06

Bolt preload diagnostic messages are written to the .f06 file. The MSGLVLB parameter on theNLCNTL bulk entry controls the message level. The MSGLVLB options are as follows:

MSGLVLB=0: Bolt summary messages are written to the .f06 file upon convergence.

MSGLVLB=1 (default): Bolt summary messages are written to the .f06 file at every bolt preloaditeration.


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Chapter 7: Loads

MSGLVLB=2: Bolt summary messages and load/tolerance messages are written to the .f06 file atevery bolt preload iteration.

MSGLVLB=1 and 2 also require that the MSGLVL parameter, which is also defined on the NLCNTLbulk entry, be set to 1 (default). When MSGLVL= 0 is defined, the software also forces MSGLVLBto 0, even if you had defined MSGLVLB=1 or 2. The MSGLVLB 1 and 2 settings will also output atable of bolt data showing the forces, moments and strains per bolt. This is only written after all boltpreloads have converged.

The following examples demonstrate the MSGLVLB output options.

MSGLVLB = 0 – The following is output upon convergence.

----- BOLT COUNT ------ ----------- BOLT PRELOADING STATUS-----------ITERATION TOTAL PRELOADING WITHIN TOL ERROR TOL MAX ERROR4 1 1 1 1.00E-03 8.80E-05

MSGLVLB = 1 (Default) – The following is output at every bolt iteration.


MSGLVLB = 2 – The following is output at every bolt iteration.


TARGET ACTUAL ERROR WITHINBOLT ID PRELOAD PRELOAD RATIO TOL101 2.00E+08 7.58E+07 6.21E-01 -

MSGLVLB, 1 and 2 – The following is output upon convergence

-----BOLTS SUMMARY UPON CONVERGENCE-----BOLT ID AREA PRELOAD AXIAL SHEAR1 SHEAR2 MOM1 MOM2 STRN101 1.00E+02 2.00E+08 2.00E+08 -3.63E-07 1.10E-06 -2.79E+08 -2.79E+08 2.64E-02

Bolt preload results

The BOLTRESULTS case control command is available in SOL 401 to request the bolt force and theaxial strain output. The output is relative to the coordinate system used to define the bolt axis. Theoutput includes the axial, bending moment, shear forces, and axial strain. It can be written to the .f06,.op2, and .pch files. The data is written to the OBOLT1 data block in the .op2 file.

The bolt force output is a summation of the forces across the bolt cross section. The bolt forcecomputation for each force component is similar to cutting the bolt at a point along the axis, thensumming the forces on the faces of one side of the cut. Although the total force components arenot computed on a per element basis, the constant force values computed for the bolt are writtento every element defining the bolt. The result of this is that each bolt will display as a constantforce inside a post processor.


Chapter 7: Loads

Loads

7.6 Initial stress-strainThe option to define an initial stress or strain condition is available on all elements in SOL 401except for plane strain elements, generalized plane strain elements, solid composite elements, andrigid elements.

You define the initial stress or strain with the INITS case control command, which selects the INITSbulk entry. The INITS case control command must be defined above the subcases (globally). Itis reapplied in every static subcase.

The initial stress or strain available in NX Nastran is consistent with other static loads since it resultsin an initial unbalanced load condition in a static subcase. That is, it deforms a body where it isunconstrained, and produces a stress state where it is constrained.

The first row on the INITS bulk entry includes the following fields:

• The TYPE field defines the data type: TYPE=STRESS or TYPE=STRAIN

• The CSYS field selects the coordinate system relating the stress or strain components. Thedefault is the basic coordinate system. CSYS = -1 can also be defined to select the materialsystem.

• The LOC field defines the location:

LOC= GRID: Specifies that data is defined at grid points.

LOC= NOE: Specifies that data is defined on an element at grid locations. This can includecorner and/or midside grid locations.

You define the stress or strain data on the consecutive rows on the INITS entry. The softwareassumes the data is either engineering stress or engineering strain. The format of these rowsdepends on the data location defined in the LOC field, and the element type.

Format for the 3D solid elements CTETRA, CHEXA, CPENTA and CPYRAM:

Stress at grid points (TYPE=STRESS, LOC=GRID):GRID ID Sxx Syy Szz Sxy Syz Szx

....

Strain data at grid points (TYPE=STRAIN, LOC=GRID):GRID ID Exx Eyy Ezz Exy Eyz Ezx

...

Stress data at the element corners (TYPE=STRESS, LOC=NOE):ElemID GRIDID Sxx Syy Szz Sxy Syz Szx

...

Strain data at the element corners (TYPE= STRAIN, LOC=NOE):ElemID GRIDID Exx Eyy Ezz Exy Eyz Ezx

...

For the plane stress elements CPLSTS3, CPLSTS4, CPLSTS6, CPLSTS8, both in-plane andout-of-plane initial strain values are used by the software. Although, only in-plane initial stress valuesare used. For example, the following formats should be used when the plane stress elements aredefined on the XY plane, and the basic coordinate system (default) is used. For elements defined onthe XZ plane, Sxx, Szz, Szx or Exx, Eyy, Ezz, Ezx would be defined.


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Stress data at grid points (TYPE=STRESS, LOC=GRID):GRID ID Sxx Syy Sxy

...

Strain data at grid points (TYPE=STRAIN, LOC=GRID):GRID ID Exx Eyy Ezz Exy

...

Stress data at the element corners (TYPE=STRESS, LOC=NOE):ElemID GRIDID Sxx Syy Sxy

...

Strain data at the element corners (TYPE= STRAIN, LOC=NOE):ElemID GRIDID Exx Eyy Ezz Exy

...

For the axisymmetric elements CQUADX4, CQUADX8, CTRAX3, CTRAX6, in-plane (radial andaxial) and out-of-plane (theta) initial stress or strain values are used by the software. For example,the following formats should be used when the axisymmetric elements are defined on the XY plane,and the basic coordinate system (default) is used. For elements defined on the XZ plane, Sxx,Szz, Szx or Exx, Ezz, Ezx should be defined.

Stress data at grid points (TYPE=STRESS, LOC=GRID):GRID ID Sxx Syy Szz Sxy

...

Strain data at grid points (TYPE= STRAIN, LOC=GRID):GRID ID Exx Eyy Ezz Exy

...

Stress data at the element corners (TYPE=STRESS, LOC=NOE):ElemID GRIDID Sxx Syy Szz Sxy

...

Strain data at the element corners (TYPE= STRAIN, LOC=NOE):ElemID GRIDID Exx Eyy Ezz Exy

...

For the plane stress and axisymmetric elements, if you select a coordinate system other than thebasic system in the CSYS field on the INITS entry, the software first transforms the data into the basicsystem, then uses the components consistent with the formats described above.

The option to output the initial strains using the OSTNINI case control command is available. Theoutput can be requested at either the grid or corner gauss locations on elements. The OSTNINIcommand must be defined globally, and the output occurs once at the beginning of the solution. Thestrains are output in the basic coordinate system.

Additional Information:

• Initial stress and strain can be defined on a subset of the model. The software assumes a valueof 0.0 at the locations where data is undefined. An exceptiion is when data is undefined at amid-side grid point, and data is defined at both or either related corners. In this case, the softwarewill interpolate a value for that mid-side grid point.

• The option to apply an initial stress or strain condition before applying other loads in an initialsubcase is available to help convergence. The first subcase should have Tend=0.0 on the


Chapter 7: Loads

Loads

TSTEP1 entry, and no load set selected. The number of increments can optionally be definedwith NINC on the TSTEP1 entry to increment the initial stress or strain. When NINC=1 (default),the initial stress or strain is applied in a single step. When NINC>1, the initial stress or strain isramped. A service load cannot be defined when ramping initial stress or strain with NINC>1.

• The software converts an initial stress to an initial strain using the elastic modulus defined onthe MATi entries. If you define MATTi bulk entries to define the elastic modulus as temperaturedependent, the software will use the initial temperatures selected by the TEMPERATURE(INIT)case control command to evaluate the temperature dependent elastic modulus. Data on theMATS1 bulk entry, if defined, is not used to convert stress to strain.

• Multiple INITS bulk entries can be defined, each with a unique ID, then combined with theINITADD bulk entry. The INITADD entry is selected with the ID on the INITS case control. TheINITS entries selected by the INITADD entry must be all TYPE=STRESS or all TYPE=STRAIN.As a result, you cannot mix initial stress and initial strain definitions in the same input file.

• If data is defined on the same grid or element corner location, a fatal error will occur.


Loads

Chapter 8: Contact conditions

8.1 Contact OverviewSOL 401 supports surface-surface and edge-edge contact. The algorithm supports largedeformations and finite sliding between contact pairs. For large deformation problems (LGDISP>-1),the contact stiffness and contact forces are rotated with the geometry consistent with other elementformulation in SOL 401. In addition, the SOL 401 contact algorithm updates the geometry and pairingwhen there is finite sliding between the contacting surfaces.

Contact is modeled using a contact set, contact regions, contact segments and contact pairs.

The table below lists the case control commands and bulk entries used to define contact.

Contact definition summary

Case control commands:BCSET Selects a contact set.BCRESULTS Selects contact results to output.

Bulk entries related to edge or surface definition:BSURFS Defines contact region on 3-D solid elements.BCPROPS Defines contact region on 3-D solid elements by property ID.

BEDGE Defines contact region on 2-D axisymmetric, chocking, plane strainand plane stress elements.

BCRPARA Defines parameters for contact region.

Bulk entries related to contact set definition:BCTSET Defines the contact sets.BCTADD Defines a union of contact sets.BCTPARM Defines parameters for contact sets.

8.2 Contact Subcase ControlThe BCSET case control command selects the contact conditions defined in the bulk data. TheBCSET command can be defined in any static or bolt preload subcase type. For example, youcan remove or add contact regions and pairs, and change contact settings including parameters,from one static or preload subcase to the next.

The modal, cyclic, and Fourier subcase types which are sequentially dependent (default) use thefinal stiffness from a previous static or bolt preload subcase. The final stiffness from the static orpreload subcase includes the contact stiffness.



Note that if a subcase is sequentially dependent and it includes a new contact set, any contacttractions from a previous subcase will be used as an initial condition for the current subcase. Theprevious tractions define the initial condition for the newly formed contact elements.

8.3 Contact DefinitionEdge Contact Regions

An edge contact region is a collection of axisymmetric, chocking, plane strain, and plane stresselement free edges in a section of the model where you expect contact to occur. You create edgeregions with the BEDGE bulk entry.

• To define a contact edge using the BEDGE entry, you enter the element ID along with the cornerGRID IDs. You can define the edges on the BEDGE entry in any order.

Edge-to-edge contact can be defined on the edges of the following elements:

• Axisymmetric elements CTRAX3, CQUADX4, CTRAX6, CQUADX8.

• Chocking elements CCHOCK3, CCHOCK4, CCHOCK6, CCHOCK8.

• Plane stress elements CPLSTS3, CPLSTS4, CPLSTS6, CPLSTS8.

• Plane strain elements CPLSTN3, CPLSTN4, CPLSTN6, CPLSTN8.

The axisymmetric, chocking, plane stress, and plane strain elements can be defined in either the XZplane or in the XY plane. Edge-to-edge contact is supported in either orientation. The generalizedplane strain element is not supported by glue or contact regions.

Surface Contact Regions

A surface contact region is a collection of element free faces in a section of the model where youexpect contact to occur. These regions are created using the solid element free faces (BSURFSand BCPROPS).

• The BSURFS entry is defined by its own unique ID and is a list of solid element IDs each followedby 3 grid points defining which face of the 3-D element to include in the contact region.

• The BCPROPS entry is defined by its own unique ID and is a list of solid element property IDs.The free faces of the solid elements selected with a property ID are automatically determinedby the software.

Surface-to-surface contact can be defined on the faces of the following elements.

• 3-D Solid elements CHEXA, CPENTA, CPYRAM, and CTETRA.



Contact conditions

A contact surface can be defined with any face of a solid element. Although parabolic faces withomitted midside grid points are permitted, their use could affect accuracy. All region IDs defined withthe BEDGE, BSURFS, or BCPROPS entries must be unique.

Contact Region Parameters

The contact region parameter OFFSET can be defined using the BCRPARA bulk entry. The CRIDfield on BCRPARA must match the ID used on one of the BSURFS, BCPROPS, and BEDGE bulkentries to be considered by the solution. The OFFSET parameter is supported with surface-to-surfaceand edge-to-edge contact definitions.

Use the OFFSET field to account for a rigid layer between contact face or edge regions. For example,a model which has two metal surfaces coming into contact, and one of these has a ceramic coating.If the ceramic material stiffness is not significant enough to be included in the analysis, it may nothave been specifically modeled, but the thickness it adds to the face of the metal may be importantwhen considering the contact problem.

You can also use the OFFSET field to analyze an interference fit problem if unconnected elementsare modeled coincident. The offset value in this example can represent the theoretical interference.

• The option to increment contact offsets is available. For subcases which have a constant time*,the software automatically increments the contact offset using the number of increments. Thenumber of increments is defined with either the Ninc field on the TSTEP1 entry, or with theNinc field on the BOLTSEQ entry.

Note that the number of increments also increments loads and temperatures. The incrementingof the contact offsets, loads, or temperatures helps the solution converge by reducing thechanges which occur in an increment.

*A constant time subcase has a TSTEP1 bulk entry defined with either Tend=0.0, or a Tend whichis the same as the start time for that subcase.


Contact conditions


• When a sequentially dependent subcase is defined with SEQDEP=YES, the final contact offsetfrom a previous subcase, if it exists, is included at the start of the current subcase. The goal is tohelp convergence when contact offsets change from one subcase to the next. In this case, theoffset for the current subcase is calculated as:

OFFSET = OC * LF + OP * (1 - LF)

Where:OC = Contact offset for the current subcaseOP = Contact offset from a previous subcaseLF = Load factor incremented in Ninc steps. Initial value is 1/Ninc and the final value is 1.0.

• General recommendations for offset definitions:

- A separate subcase is recommended to resolve the contact offset.

- When bolt preloads and contact offsets are defined together, be aware that both can result inaxial bolt strain. If you do not want the contact offset to result in an additional bolt strain, thecontact offset should be modeled with a slight gap.

• You can optionally change the contact offset definition from one subcase to the next by selectingunique contact sets in each subcase which reference different BCRPARA bulk entries.

Contact Pairs

A contact pair combines two contact regions, source and target, in which contact will be analyzedduring the solution. Each contact pair can have its own unique friction value (if desired) and searchdistance.

The BCTSET bulk entry is used to define both edge-to-edge and surface-to-surface contact pairs.The CID field will need to match the value of ‘n' on the BCSET case control entry for the solutionto recognize this contact definition.

The SIDi and TIDi fields on the BCTSET bulk entry are used to define source and target regionsrespectively for a pair. As many pairs as desired can be included on a single BCTSET entry. Eachpair can have a unique friction value (optional), a minimum search distance, and a maximum searchdistance. You can define the optional Coefficient of Friction field (FRICi) for each contact pair. Whencontact is detected, the solver uses this value to calculate any tangential contact forces.

The regions you select with SID and TID depend on the type of contact:

• For edge-to-edge contact pairs, SID and TID are contact regions defined with the BEDGE entry.

• For surface-to-surface contact pairs, SID and TID are contact regions defined with the BCPROPSand BSURFS entries.

The minimum and maximum search distance fields (MINDi and MAXDi) define a range in which thesolver can initially determine if the distance between element edges or faces in a particular pair arewithin the threshold for creating contact elements. These values are used to determine where contactelements are created. The minimum distance can be negative if there is an interference conditionmodeled as overlapping regions.

The contact condition can update when large sliding occurs when large displacement effects areturned off (PARAM,LGDISP,-1) or on (PARAM,LGDISP,1).

Combining Contact Sets – BCTADD



Contact conditions

You can optionally define multiple BCTSET/BCTPARM bulk entry sets, each set with uniquecontact set IDs (CSID), and then combine them with a single BCTADD bulk entry. The multipleBCTSET/BCTPARM bulk entry sets are created to adjust certain contact parameters locally. Contactparameters can also be adjusted globally with a BCTPARM bulk entry having the same CSID asthe BCSET case control command.

The following example demonstrates the inputs.

CASE CONTROL$CSID on the BCSET case control matches CSID on BCTADDBCSET = 108...BULK DATA$Local Contact Set definitionsBCTSET 1 1 2 0.0 1.0BCTSET 2 3 4 0.15 0.0 0.15 6 0.15 0.0 0.1...$Local Contact ParametersBCTPARM 1 PENN 10 PENT 1BCTPARM 2 PENN 1.0 PENT 0.1...$Local Contact Sets are combined with BCTADDBCTADD 108 1 2...$Global Contact ParametersBCTPARM 108 MAXS 30 NCHG 0.02

See the section “Contact Control Parameters - BCTPARM” for more information on contactparameters.

Contact with Composite Solid Faces

Defining contact regions and pairs on composite solid faces which are perpendicular to the stackdirection (edge faces) may produce poor stress continuity. If the contact definition is between edgefaces belonging to different PCOMPS definitions, and if the number of plies on each PCOMPSdefinition is small and the same, and the ply thicknesses are similar, the stress continuity should befairly smooth. This also applies to the results requested with the BCRESULTS case control command.

Additional Recommendations

When defining contact regions and pairs on geometry which are not tangent continuous, creatingsingle contact regions which cross corner transitions can result in non-uniform stress results aroundthe corners. It is recommended to break these areas into multiple regions and pairs as shown below.


Contact conditions


When defining contact regions and pairs, it is recommended to not include the same element face inmultiple regions. In “A” below, an element is repeated in regions 2 and 3. In “B”, the same elementonly exists in region 2. “B” is recommended. Repeating element faces multiple times in the same ordifferent regions can significantly increase memory requirements and degrade performance.

8.4 Contact Control ParametersThe contact control parameters on the BCTPARM bulk entry (optional) can help you adjust the contactalgorithm when you are having problems getting a solution to converge and complete, or when thecontact results are not as expected. For many solutions, the default settings are appropriate, andthe BCTPARM entry is not required.

• For surface-to-surface contact definitions, all of the parameters listed below are supported.

• For edge-to-edge contact definitions, all of the parameters listed below are supported exceptfor REFINE and INTORD.

You can optionally define multiple BCTSET/BCTPARM bulk entry sets, each set with uniquecontact set IDs (CSID), and then combine them with a single BCTADD bulk entry. The multipleBCTSET/BCTPARM bulk entry sets are created to adjust certain contact parameters locally. Contactparameters can also be adjusted globally with a BCTPARM bulk entry having the same CSID asthe BCSET case control command.

Global and local contact parameters have the following definition and rules:

• Global Contact Parameters

The BCTPARM bulk entry, which uses the same CSID entered on the BCSET case controlcommand, defines global parameters.



Contact conditions

Any of the parameters on the BCTPARM bulk entry can be defined globally. A parameter’s defaultvalue is used if it is not defined globally or locally.

• Local Contact Parameters

The BCTPARM bulk entries associated to individual BCTSET bulk entries, which are thencombined with a BCTADD bulk entry, define local parameters.

Following is a description of the BCTPARM input parameters. Local contact parameters have “*”next to their names.

Table 8-1. Primary parameters supported by SOL 401:

Name Description

CNTCONV Contact convergence criteria.

1 – The contact convergence criteria is based on PTOL. (Default)

2 – The convergence criteria is based on CTOL.

PTOL* Contact penetration tolerance used when CNTCONV=1. If the contact penetrationsexceed the penetration tolerance, an extra augmentation loop is performed. Ifthe penetrations are below this tolerance, the augmentation loop is consideredconverged. In addition, if the global solution convergence criteria is satisfied, thenthe time step is considered converged. (REAL≠0.0; Default = 0.01 *characteristiclength)

A positive value is scaled by the characteristic length computed for the contact pair.

If a negative value is defined, the absolute value is used, but the value is notscaled by the characteristic length.

CTOL Contact augmentation traction convergence used when CNTCONV=2. Theaugmentation loop convergence criteria can be based on traction convergence.The contact force ratio FRAT is determined as:

where k is the augmentation loop id. If FRAT < CTOL, the contact augmentationloop is considered converged. (Default = 0.05)

RCTOL Iterative contact force convergence. (Default = 0.05; Real>0.0)

MAXS Maximum number of augmentation (outer) loops. If the augmentation loop hasnot converged in MAXS number of iterations, the solution will proceed to the nextstep if the global convergence criteria has been met. Setting MAXS=1 selects apure penalty formulation. (Default = 20; INTEGER≥1)


Contact conditions



Name Description

INIPENE* Use when the goal is for a pair of contact regions to be touching withoutinterference, but due to the faceted nature of finite elements around curvedgeometry, some of the element edges or faces may have a slight gap orpenetration.

0 or 1 - Contact is evaluated exactly as the geometry is modeled. No correctionswill occur for gaps or penetrations (Default).

2 - Penetrations will be reset to a new initial condition in which there is nointerference.

3 - Gaps and penetrations are both reset to a new initial condition in which thereis no interference.

INIPENE is applied when contact elements are initially created, and if they arerecreated as a result of large displacement effects when PARAM,LGDISP,1 isdefined.

OPNSTF* Open contact stiffness scale factor. The open contact stiffness is computed byOPNSTF * closed stiffness. (OPNSTF default = 1.0E-6)

OPNTOL* Open gap tolerance scale factor. The open contact stiffness (OPNSTF * closedstiffness) is applied to the contact elements that have a gap value less than orequal to OPNTOL * characteristic length, but greater or equal than GAPTOL *characteristic length. The contact element stiffness is 0.0 if the gap is greater thanOPNTOL * characteristic length. (OPNTOL default = 1.0)

GAPTOL* Closed gap tolerance scale factor. The closed contact stiffness is applied to thecontact elements that have a gap less than GAPTOL * characteristic length.(Default = 1.0E-10)

NOSEP* No separation contact option.

NOSEP=0 (default): When contact stiffness is recomputed in a consecutivenonlinear iteration, contact elements which are inactive as a result of normaltractions=0.0 and no penetration, and which have a gap greater than GAPTOL *characteristic length will remain inactive in the consecutive iteration.

NOSEP=1: The open contact stiffness (OPNSTF * closed stiffness) is applied tothe inactive contact elements that have a gap value less than or equal to OPNTOL* characteristic length, but greater or equal than GAPTOL * characteristic length..The contact elements with a gap greater than OPNTOL * characteristic lengthremain inactive. While sliding is permitted with this option, the magnitude of thesliding can be controlled by the tangential penalty factor. To define frictionlesssliding, set the coefficient of friction=0.0 or tangential penalty factor (PENT)=0.0.(Default=0)



Contact conditions


Name Description

GUPDATE Geometry update flag. (Default=0 when PARAM,LGDISP,-1; Default=2 forPARAM,LGDISP,1; INTEGER)

0 – Contact geometry updates will not be done during the analysis.

1 – Contact geometry updates occur when SLIP > (GUPTOL * Average elementlength).

2 - Contact geometry updates occur at the start of each step and when SLIP >(GUPTOL * Average element length).

3 - Contact geometry updates occur once a step.

4 - Contact geometry updates occur every iteration.

GUPTOL* Geometry update tolerance. If the relative sliding distance between the source andtarget regions exceeds this tolerance, a geometry update will be initiated with largedisplacement. (Default = 0.25 * characteristic length; REAL>0.0)

FRICMOD Tangential stiffness selection.

0 - The tangential stiffness if fixed throughout the subcase.

1 - The tangential stiffness is adaptively modlfied at every iteration as a function ofcontact pressure. (Default) The tangential stiffness is then computed as:

(FRICi * contact pressure) / SCRIT

where FRICi is the coefficient of friction defined on the BCTSET entry, and SCRITis the critical slip parameter.

SCRIT* Defines the critical slip when FRICMOD=1, where the critical slip = SCRIT *characteristic length for each pair. (Default=0.005; REAL)

A negative SCRIT value is treated as an absolute value and is not scaled by thecharacteristic length.

DISCAL Displacement scaling option.

0 – No scaling will be done.

1 – Scaling will be done if required during every iteration. A check will be performedafter every displacement increment to see if the incremental displacements wouldcause penetration between the source and target regions. If the penetrationsexceed DISTOL, the entire incremental displacements will be scaled back to limitthe penetrations in the model. (Default)

DISTOL Tolerance for displacement scaling feature. (Default = 1.0* characteristic length)


Contact conditions



Name Description

KSTAB Stiffness stabilization for contact.

0 – Stiffness stabilization is off. (Default)

1 – The stiffness matrix is stabilized when it is singular due to inactive contactconstraints. The stabilization adds a factor (1.0) to the diagonal terms of thestiffness matrix. KSTAB=1 is only supported with the sparse solver, and willdisable any open contact stiffness specified through the OPNSTF parameter.

AUTOSCAL* When PENN is not defined explicitly, scales the automatically calculated normalpenalty factor PENN either up or down. AUTOSCAL can be used to scalethe stiffness of specific contact pairs if convergence issues occur (Real>0.0;Default=1.0).

TANSCL* When PENT is not defined explicitly, scales the automatically calculated tangentialpenalty factor PENT either up or down. TANSCL can be used to scale the stiffnessof specific contact pairs if convergence issues occur (Real>0.0; Default=1.0).

FRICDLY * Option to delay contact friction to help alleviate convergence problems.

0 – Friction is not delayed. (Default)

1 – Friction is delayed. It is not included in the solution until the second contactiteration.

Table 8-2. Secondary parameters supported by SOL 401:

The following parameters are available for special cases.

Name Description

PENN* Penalty factor for normal direction. PENN and PENT are automatically calculatedby default. When PENT is defined but PENN is undefined, PENN = 10 * PENT.

PENT* Penalty factor for transverse direction. PENN and PENT are automaticallycalculated by default. When PENN is defined but PENT is undefined, PENT =PENN / 1000.

PENTYP* Changes how contact element stiffness is calculated (Default=1).

1- PENN and PENT are entered as units of 1/Length.

2 - PENN and PENT are entered as units of Force/(Length x Area).



Contact conditions

Table 8-2. Secondary parameters supported by SOL 401:

The following parameters are available for special cases.

Name Description

REFINE Requests that the software refine the mesh on the source region during thesolution to be more consistent with the target side mesh.

0 - Refinement does not occur.

2 - Refinement occurs (default).

INTORD Determines the number of contact evaluation points for a single element edge orface on the source region. The number of contact evaluation points is dependenton the value of INTORD, and on the type of element face.

1 – The reduced number of contact evaluation points is used.

2 – Use an increased number of contact evaluation points (default).

3 – Use a high number of contact evaluation points.

8.5 Contact kinematicsConsider the contact element between the points S and T in the figure below. S and T are theparametric locations of a contact element on the source and target faces respectively.

S1, S2, S3, S4,...Sj are the grid points defining one solid element face which is included in thesource region.

T1, T2, T3, T4,...Tj are the grid points defining one solid element face which is include in the targetregion.

Figure 8-1. Contact Source and Target Example

The global system of equations including contact at a given iteration i is:


Contact conditions


Equation 8-1.

Where:

is the assembled contact stiffness,

is the assembled nodal contact forces,

NCE is the total number of contact elements created,

Kc is the contact stiffness from a single contact element,

and Fc are the contact forces from a single contact element,

The total traction for a contact element c at an iteration i can be split into normal and tangentialcomponents as:

Equation 8-2.

where is the normal vector, and the tangential vector is .

Together , t1, and t2 form an ortho-normal basis.

Normal contact conditions

The normal contact conditions can be expressed as:

Equation 8-3.

Equation 8-4.

Equation 8-5.

Equation 8-3 imposes the condition that the penetration of the hitting surface into the target surface

can’t be greater than zero. Thus, surfaces can’t interpenetrate. The contact pressure, , is definedas the negative of the normal component of the surface traction.

Equation 8-4 states that the contact pressure can’t be less than zero or tensile. Or normal tractionsbetween surfaces can’t be tensile.

Equation 8-5 imposes the condition that:



Contact conditions

and

SOL401 uses a penalty method to enforce the contact constraints with the ability to augment thetractions (Augmented Lagrangian formulation) to keep the penetrations to within a specified tolerance(PTOL).

The pure penalty method can be activated by setting MAXS=1 on BCTPARM bulk data entry. Purepenalty method is computationally in-expensive and in most cases produces acceptable solution.However, since there is no check on penetration control this method can sometime cause largepenetrations in the model that could go undetected.

Augmented Lagrangian formulation is the default. With this method, first a converged solution isobtained with a penalty method. If the convergence criteria have not been achieved, the tractionsare augmented and another series of iterations is performed until convergence is achieved. Thetractions could be augmented several times until either the desired level of penetration tolerance(PTOL) is achieved when the convergence criteria is based on penetrations or alternately thetractions between two augmentations are converged to a value below CTOL for convergence criteriabased on augmented tractions. MAXS (default=20) parameter specifies the maximum number ofaugmentations that are performed for every time step. If the augmentation convergence is notachieved within MAXS number of iterations the solution will proceed to the next step if the usualglobal convergence criteria have been satisfied.

The normal traction is:

Equation 8-6.

is the normal traction from previous augmentation.

Note that =0 for the pure penalty method.

εn is the normal penalty stiffness (unit=force/length),

is the contact element normal gap evaluated at iteration i,

K is the augmentation loop ID.

The relative displacement at a contact element location for iteration i is computed as:

Equation 8-7.The contact element normal gap at iteration i is calculated as

Equation 8-8.


Contact conditions


Equation 8-9.

Nj = Standard Source Face shape functions evaluated at the contact point

N'k = Standard Target Face shape functions evaluated at the contact point

Ns = Number of nodes on source face

Nt = Number of nodes on target face

Xs=X0s + UsXt=X0t + UTX0s = Undeformed coordinates of the nodes on the source face

X0t = Undeformed coordinates of the nodes on the target face

Us = Nodal displacements on the source face

Ut = Nodal displacements on the target face.

Note that a positive value of indicates an overlap and a negative value indicates a separationbetween the source and target faces at the contact element location.

Consequently a contact element is considered CLOSED if > 0 and OPEN if < 0.

The tangential traction is

Equation 8-10.

where,

λnt is the tangential traction from previous step.

εt is the tangential penalty factor (unit=force/length),

is the relative displacement increment in the tangential direction since last time step where j= 1, 2.

The relative displacement increment in the tangential direction is calculated as:

Equation 8-11.

where,



Contact conditions

Equation 8-12.

(ΔUs) = Nodal displacement increment since last time step on the source side DOF.

(ΔUt)= Nodal displacement increment since last time step on the target side DOF.

A contact element can have the following states:

Inactive when < 0.

Active when > 0.

Coulomb Friction

If friction is present (μ > 0 where μ is the coulomb friction coefficient), an active contact element canhave either sticking or sliding state.

Similar to the normal contact conditions, the frictional contact conditions can be expressed as:

Equation 8-13.

Equation 8-14.

Equation 8-15.

Equation 8-16.

Equation 8-13 imposes the constraint that the magnitude of the in-plane friction traction, , cannotexceed the coefficient of friction μ, times the normal contact traction. When the magnitude of frictionforce reaches its maximum allowable value, the function Φ will be equal to zero.

Equation 8-14 relates the relative tangential displacement increment between the source and

the target surface, , to the magnitude of the relative slip increment, , which must be anon-negative quantity.


Contact conditions


Equation 8-16 implies that if ≥ 0 (there is slipping between the surfaces), then Φ= 0, and if

= 0 (the surfaces are sticking), then Φ ≤ 0.

Therefore, for Coulomb friction:

• The maximum possible tangential traction equals the coefficient of friction times the normaltraction force.

• Contacting surfaces will "stick" if the tangential traction is less than the coefficient of friction timesthe normal traction force.

• Contacting surfaces will "slide" in the direction of the tangential traction if the tangential tractionequals the coefficient of friction times the normal traction force.

Figure 8-2. Classical Coulomb Friction Model

Assume

A contact element state is Sticking if

Equation 8-17.

and Sliding if

Equation 8-18.



Contact conditions

Bilinear Coulomb Friction

Figure 8-3. Bi-Linear Coulomb Friction Model

By default, SOL401 uses a bi-linear coulomb model specified by FRICMOD=1 on BCTPARM. Thismodel converges better than the classical model (FRICMOD=0) by providing a more distinct stick slipzone. Here, the tangential penalty factor εT is adaptively computed based on the current contactpressure, coefficient of friction and a critical slip value Scrit provided by the user. The default for Scritis set at 0.5% of the characteristic element length in the pair. A larger value of Scrit helps witheasier convergence but leads to more sliding, while a smaller value represents the exact coulombbehavior more closely but also leads to slower convergence rates. The tangential stiffness is updatedevery iteration using:

Equation 8-19.

The incremental tangential relative displacement can be expressed as a sum of elastic and plasticparts.

Equation 8-20.

or

Equation 8-21.

where is the elastic slip at the end of previous step.

The predictor slip is


Contact conditions


Equation 8-22.

The trial tangential traction is evaluated from the predicted slip as:

Equation 8-23.

This predicted slip is elastic as long as the magnitude of the tangential traction is less than the

critical stress (i.e.) .

If the magnitude of the tangential traction exceeds the critical stress then slip must be taken intoaccount.

The magnitude of the plastic slip in this instance is given by where isthe magnitude of the predicted slip.

The incremental slip is computed as:

Equation 8-24.

When the slip output is requested by defining SYS642=0, the total slip at a contact node at an

output time point represents the total accumulated value of the plastic slip . The accumulatedslip will reset to zero when a contact node goes inactive. Slip is reported as a vector in the basiccoordinate system.

where relates the two local tangential basis vectors of the closestcontact element to the node in terms of basic coordinate system.

The total slide distance on the other hand represents the relative displacement between the sourceand target faces from the un-deformed configuration at any given time and includes the elasticdeformations as well. The slide distance does not reset to zero when a contact element goes inactive.The slide distance is computed by subtracting the normal component of the relative displacementfrom the total relative displacement between the source and target regions.

Thus, , where gic and gicn are total relative and the normal component of the relativedisplacements respectively computed using equations 8-7 and 8-8.

Contact Force

The contact nodal force contribution due to all active contact elements at iteration i is computedas follows:



Contact conditions

Equation 8-25.

Equation 8-26.

Contact Stiffness

From the normal component of traction we have:

Equation 8-27.

For a sticking contact element with friction,

Equation 8-28.

For a slipping contact element,

Equation 8-29.


Contact conditions


The contact stiffness contribution after ignoring the variation of friction coefficient with pressureand sliding velocity is:

Equation 8-30.

Equation 8-31.

Equation 8-32.

The global contact stiffness is obtained by summing the stiffness contribution from all active contactelements:

Equations 8-31 and 8-32 include non-symmetric stiffness terms. Using an un-symmetric solverwill significantly increase the memory requirements. The software automatically activates theun-symmetric solver if the coefficient of friction is greater than FSYMTOL. The parameter FSYMTOL(specified on NLCNTL) has a default value of 0.2.

Contact Iterations

The contact problem is solved by a dual loop algorithm with the augmentation loop serving as theouter loop and the regular NR iterations constituting the inner loop. In addition to the global criteriato determine the inner iterations convergence, contact algorithm also checks for contact forceconvergence. The inner loop is said to be converged if the change in contact force between twoconsecutive iterations is < RCTOL. After the convergence of the inner loop (i.e.) NR iterations,



Contact conditions

the tractions are augmented if needed and another round of inner iterations is done until either ofthe following 2 criteria is met:

1. CNTCONV=1. In this case the outer loop convergence criteria are based on penetrationtolerance. The augmentations are performed until the max penetration in each contact pair isbelow the tolerance PTOL.

2. CNTCONV=2. In this case the outer loop convergence criteria are based on contact tractionconvergence ratio CTOL. The contact tractions are augmented until the augmented force ratioFRAT change in the tractions between two consecutive augmentation loops is converged to avalue smaller than CTOL.

Contact Algorithm Flowchart

Steps 1 through 4 occur for each time step (n+1).

1. Loop over contact augmentations (K=1 to MAXS).

a. Initialize λkn .

b. Compute stiffness K and Kc.

2. Do Newton-Raphson iterations (i=1 to MAXITER).

a. Compute internal forces and where is the force contribution from contact.

b. Compute Residual

c. Solve (see equations 8-30 through 8-32 ).

d. Check for the solution convergence based on global criteria and RCTOL. The global solutionconvergence criteria are defined by the CONV parameter on the NLCNTL bulk entry.

A. If the solution convergence criteria are satisfied, go to step 3.

B. If the solution convergence criteria is not satisfied:

i. If (i < MAXITER) then do more iterations, go to 2a or if stiffness update is required,go to step 1b.

ii. If (i = MAXITER) then convergence is not achieved and end problem. GO TO step 4.

3. Check outer loop convergence for contact based on either PTOL or CTOL.

a. If the contact outer loop is converged, end the problem. Go to step 4.

b. If the outer loop is not converged, increment the outer loop counter, augment the tractions,and go to step 1b.


Contact conditions


4. End Problem.

8.6 Contact Penalty FactorsThe normal penalty stiffness εn and the tangential penalty stiffness εT are computed for each contactelement as follows:

εn = Ac * E * PENN

εT = Ac * E * PENT

PENN and PENT are the normal and tangential penalty factors respectively. Ac is the area associatedwith the contact element and E is the modulus of the softer region in the contact pair.

The normal penalty is computed by the contact algorithm or can be input by the user. The automaticpenalty factor calculation estimates geometry characteristics using element edge lengths in thevicinity of the contact regions. The solver computes a characteristic length Lc for each contact pairand estimates the normal penalty factor PENN as PENN = 1 / Lc. The default value of PENT isan order of magnitude smaller than the PENN.

When the option to adaptively compute the tangential stiffness is selected, the tangential penaltyfactor is adjusted based on current contact pressure, coefficient of friction and critical slip parameter.

PENT = μ * Tn / Scrit

The penalty factors influence the rate of convergence, and to a lesser extent, the accuracy of thecontact solution. The automatic penalty factor calculation works well for most instances, but manualadjustments may be necessary, particularly if a contact problem fails to converge. When the problemfails to converge or takes a lot of iterations to converge, reducing the default value of PENN by anorder or two of magnitude will generally help with convergence.

A large value of PENN will make the system of equations ill-stabled leading to convergence problems.They could also produce spotty contact pressure results. In addition, a small value of PENN maycause excessive penetrations which will require more traction augmentations. So the right choice ofpenalty factors is the key to getting good quality contact stress results in a reasonable number ofiterations.

The software provides an alternate way (PENTYP=2) to input PENN and PENT as spring rate perarea. That is, Force/(Length x Area). The contact element stiffness in this case is calculated as:

εn = Ac * PENN

εT = Ac * PENT

The spring rate input is a more explicit way of entering contact stiffness since it is not dependenton the modulus.

The auto-calculated values for PENN and PENT can be scaled using the AUTOSCAL and TANSCLparameters respectively. The range for AUTOSCAL is usually 0.01 - 1.0 with the default being 1.0.The default works well for bulk deformation dominant problems. For bending dominant or finite slidingproblems, a smaller range 0.01 - 0.1 is recommended.



Contact conditions

8.7 Contact Sliding and Geometry UpdateThe contact element locations on the element faces or edges are updated based on the value ofthe GUPDATE parameter and a threshold value defined with the GUPTOL parameter. GUPDATEcontrols the frequency of the update and the GUPTOL determines whether a geometry update isneeded depending on the amount of relative sliding between the contact regions. The default forGUPTOL is set at 25% of the characteristic element length in the pair. The default for GUDPATEis 2 for large displacement analysis (LGDISP=1). By default, for a small displacement analysis(LGDISP=-1), the contact pairing is updated at the start of each step.

If large enough sliding occurs within the defined tolerance, contact elements are recreated in thecurrent deformed configuration, and the relevant contact data like the contact stresses and slip forexample, are all mapped from the old to the new contact elements.

The following table summarizes the geometry update options for both small and large displacementanalysis.

GUPDATE Behavior0 No update to contact element locations.

1 Contact elements are updated if the relative sliding exceeds threshold set byparameter GUPTOL.

2Contact elements are updated if the relative sliding exceeds threshold setby parameter GUPTOL, and are updated at the start of each step. Defaultwhen LGDISP=1.

3 Update occurs only at the start of each step. Default for LGDISP=-1.4 Update occurs every iteration.

The following table illustrates the geometry update process.

Initial configuration

Sliding occurs during a step.The slide in this exampleis greater than the value ofGUPTOL. This will result inan updated configuration asshown in the next step below.

Contact configuration isupdated after the magnitudeof the slide exceeds the valuedefined on the parameterGUPTOL.


Contact conditions


8.8 Contact and rigid body motionMany static problems depend on contact to provide the boundary conditions necessary for a stableproblem. With these problem types, the stiffness matrix can be singular when the contact constraintsare inactive. You can use the following options to avoid singularities.

• Open and closed contact stiffness

The contact algorithm automatically adds a small stiffness at the start of the solution which isoften enough to prevent rigid body motion. In some instances, the default initial stiffness may beinsufficient to prevent large displacements. In such cases, you can redefine an initial stiffnesswith the OPNSTF parameter on the BCTPARM bulk entry.

The OPNSTF parameter works in conjunction with the OPNTOL parameter which specifies thegap up to which OPNSTF will be enforced. The default for OPNSTF is 1.0E-6 * the closedstiffness, and the OPNTOL default is the characteristic length of the contact pair. If the gap closesin the subsequent iteration, the closed stiffness will be applied to the contact element. Note thatthe open stiffness is only applied for the very first iteration unless the no-separation feature(NOSEP=1) is activated, in which case, the open stiffness will be applied every time a contactelement gap opens and the opening is smaller than OPNTOL.

The no-separation feature can be helpful during an unloading scenario when the contact surfacesbegin to separate as the loading is ramped down. This helps prevent the surfaces from flyingapart.

The following graph demonstrates the open and closed contact stiffness change. The horizontalaxis represents the contact gap, and the vertical axis represents the contact element stiffness.

go = open stiffness gap defined by the OPNTOL parameter.

gc = closed stiffness gap defined by the GAPTOL parameter.

Kopen = open stiffness is used when the gap is smaller than OPNTOL, but greater than GAPTOL.

Kclosed = closed stiffness is used when the gap is smaller that GAPTOL.

• Stiffness stabilization



Contact conditions

This feature provides a stabilizing effect by scaling all diagonal stiffness terms without affectingthe right-hand-side load vector. You can request this feature by defining KSTAB=1 on theBCTPARM bulk entry.

• Displacement scaling

Limiting the maximum incremental displacement per iteration is useful when a load is applied to abody that is not initially in contact. A model can be unstable even when stiffness stabilization oropen stiffness is used resulting in excessive initial displacement. Setting the limiting displacementto about the element size in this case would scale down the potentially huge displacement inthe first iteration so that the results remain close to the converged solution. The displacementscaling feature can be thought of as a form of line search. This option is on by default and can beturned off by setting DISCAL=0 on the BCTPARM bulk entry. You can also adjust the limitingfactor with the DISTOL parameter on the BCTPARM bulk entry.

8.9 Contact Offsets and Initial PenetrationsContact Surface Offsets

An offset distance can be specified on a contact region in the BCRPARA entry which causes theactual contact surface to be offset from the plane defined by the contact surface nodes. Initialinterferences can be modeled using surface offsets. It is recommended that any user defined offsetbe resolved in the first step prior to the application of service loads. Additionally, these offsets can beresolved incrementally based on the value of NINC parameter on the TSTEP1 bulk entry.

Initial Penetrations

The treatment of initial penetrations is governed by the INIPENE parameter on the BCTPARM entry.By default, if there is an initial overlap (penetration) between the contact source and target segmentsin the first solution step, the program attempts to eliminate the overlap.

• INIPENE = 0 (default): The contact element gap is equal to the geometric gap. No correctionsare applied to initial penetrations or gaps.

• INIPENE = 2: The contact element gap is equal to the geometric gap minus the initial penetrations.

• INIPENE = 3: Both the initial penetrations and gaps are ignored in the contact element gapcalculation.

8.10 Contact Surface and Edge RefinementThe refinement of the source and target occurs by default. The contact search algorithm identifies foreach source segment a set of closest target segments that could potentially come into contact.


Contact conditions


Each potential target segment is projected on to the source face or edge and a polygon clippingalgorithm computes an overlapping area. The overlapped area is divided into triangles creating a“refined” source face/edge [3]. Integration points are created on each triangle based on gaussquadrature rules and the value of the INTORD entry on the BCTPARM bulk entry.

The contact conditions are enforced at these integration points. The refinement process captures thecontact domain accurately and is an important piece in passing the constant stress patch test.

The refinement feature can optionally be turned off by setting REFINE=0 on the BCTPARM entry.

8.11 Contact ConvergenceThe software prints all of the relevant information pertaining to each contact pair at the start of thesolution. Contact max penetration (PRATIO) and change in forces (RCTOL) are printed as part of theiteration summary. PRATIO is the ratio of max penetration in the current iteration for all pairs and the



Contact conditions

corresponding penetration tolerance (PTOL) in that pair. In additional to global convergence criteriabeing met, the PRATIO and RCTOL criteria should also be met for contact problems.

Convergence considerations

• A high value of RCTOL (>1.0) or a large percentage change in contact status in each iterationusually indicates any combination of the following.

o The penalty factor is high.

o The load is being applied too quickly.

o The time step is large.

To correct this situation, you can adjust AUTOSCAL to 0.1 or lower, or you can reduce theload and time steps.

• If the PRATIO values are consistency higher than 1.0 but RCTOL and other convergencenorms are decreasing, increasing the penalty factor by an order can help achieve penetrationconvergence.

• Most convergence issues can be attributed to either loss of contact or the penalty factors beingtoo high or low. If there is rigid body motion present, setting open stiffness or stiffness stabilizationwould help at the start of the analysis.

• If the displacements are constantly getting scaled, either the loading is excessive or the contacthas been lost. If there are initial gaps/penetrations in the geometry, setting INIPENE to ignorethese would help.

• If there are excessive numbers of contact pairing updates in a step, then the load or time step istoo large.


Contact conditions


• If contact is lost due to difficult geometry or sharp changes in the geometry, reducing the step sizeor geometry update tolerance (GUPTOL) can help.

• The contact algorithm monitors these variables for abrupt changes and triggers a bisection if thenumber of adverse changes within a step exceeds the value of CNTMDIV.

• Use FRICDLY to delay friction effects until after the first step can also alleviate convergenceproblems. With this option turned on, the frictional conditions are applied one time stepafter contact is established. This feature can be useful in many problems, since it delays thenonlinearity associated with friction until contact is established.

• If the convergence is slow or cannot be achieved for models with friction, using the un-symmetricsolver can improve the convergence rates.

8.12 Contact OutputContact results can be requested with the BCRESULTS case control command. Forces, tractions,separation distance, total and incremental slide or slip distance, and contact status can be output.

The SEPDIS describer requests the final separation distance for grids on both the source and targetregions. It is computed based on the current deformed configuation. For grids on the source region,the separation distance is a scalar quantity representing the source side normal distance to the target.For grids on the target region, the separation distance is a scalar quantity representing the target sidenormal distance to the source. During the solution, the separation distance is known at the elementintegration points, but is written to the grids when output. The result at each source and target gridis the value of separation distance at the closest contact element. If there are two or more contactelements equidistant from the grid, then the minimum value of separation distance is used at the gridsrather than the average, since the average gives unexpected results for coarse meshes.

In addition for SOL 401, the SEPDIS describer requests the total and incremental slide or slipdistance for grids on both the source and target regions. By default, it requests slide output. If you setthe system cell 642 to 0, the SEPDIS describer requests the total and incremental slip.

SOL 401 computes slide as:

The slide distance is a relative displacement in the tangential direction between the source andtarget faces. It is computed without regard to the status of the contact condition. For example,a source and target may not be in contact but their locations are changing relative to one another.These changes are included in the slide distance output. The tangential slide distances are outputin the basic coordinate system. The incremental slide distance is the sliding which occurred sincethe last output step. The total slide distance is computed in the current deformed configuation andrelative to the initial, undeformed configuration. For example, if a model is loaded then unloadedin several subcases causing a source and target to slide a distance and then return to their initialrelative positions, the total slide distance in this example is zero.

The slip distance is computed using the following equation. See Contact kinematics.



Contact conditions

Slip distance will be reset to zero when a node goes out of contact. The total slip reported is the totalaccumulated plastic slip and does not include any elastic deformations unlike the slide distanceoutput. The tangential slip distances are output in the basic coordinate system. The incremental slipdistance is the slipping which occurred since the last output step.

When the contact status is requested with the STATUS describer on the BCRESULTS command,an integer value indicating the contact status is output on each grid point included in a contactsource or target region. The status values are:

0: No contact exists.

1: A sticking contact condition exist.

2: A sliding contact condition exist.

8.13 References1. T.A. Laursen and J.C. Simo. "Algorithmic Symmetrization of Coulomb Frictional Problems Using

Augmented Lagrangians". Computers Methods in Applied Mechanics and Engineering. Vol. 108,No. 1& 2. 133-146. 1993.

2. P. Wriggers. “Computational Contact Mechanics”. Second Edition.

3. M. Puso, T. Laursen. “A Mortar Segment-to-Segment Frictional Contact Method for LargeDeformations”, Computer Methods in Applied Mechanics and Engineering, 2003.


Contact conditions

Chapter 9: Glue conditions

9.1 Overview of Gluing ElementsSOL 401 supports the option to glue elements together. Glue is a simple and effective methodto join meshes which are dissimilar. It correctly transfers displacement and loads resulting in anaccurate strain and stress condition at the interface. The grid points on glued edges and surfaces donot need to be coincident.

By default, the glue formulation in SOL 401 creates a connection which prevents relative motion in alldirections. You can also optionally turn on the sliding glue formulation for both the surface-to-surfaceand the edge-to-edge glue conditions. Sliding glue includes a normal stiffness but no tangentialstiffness, and is requested by defining the parameter setting SLIDE=1 on the BGPARM entry.

The following table summarizes the supported glue conditions in SOL 401.

Table 9-1. Glue SummaryType Description

Edge-to-Edge

You can define edge-to-edge glue between the edges of theaxisymmetric elements CTRAX3, CQUADX4, CTRAX6, CQUADX8,plane stress elements CPLSTS3, CPLSTS4, CPLSTS6, CPLSTS8,plane strain elements CPLSTN3, CPLSTN4, CPLSTN6, CPLSTN8,chocking elements CCHOCK3, CCHOCK4, CCHOCK6, CCHOCK8.

Surface-to-Surface You can define surface-to-surface glue between the faces of thesolid elements CTETRA, CHEXA, CPENTA and CPYRAM.

Defining Glue Conditions

• Select the glue condition with the BGSET case control command. The BGSET case controlcommand must be above the subcases. As a result, the glue conditions apply to all subcases.See Defining and Selecting Glue Pairs.

• Define source and target regions. See Glue Regions.

• Pair the source and target regions. See Defining and Selecting Glue Pairs.

• Optionally adjust the glue algorithm using glue control parameters. See Glue Control Parameters.

• Optionally request the glue force and traction output with the BGRESULTS case control command.

Edge-to-Edge Glue Summary

• The axisymmetric, plane stress, plane strain, and chocking elements can be defined in either theXZ plane or in the XY plane. Edge-to-edge glue is supported in either orientation.

You create element edge regions with the BEDGE bulk entry. You then pair the regions using thesource and target fields on the BGSET bulk entry.



From the source region, the software searches in the outward normal direction. In addition, thesoftware searches a small distance in the inward normal direction in order to glue edges that mayinterfere due to meshing irregularities. A glue condition is created when a projected normal hitsthe target side, and the distance between the edges is equal to or less than the search distancewhich you specify for the glue pair on the BGSET entry

Surface-to-Surface Glue Summary

Surface-to-surface glue can be defined on the faces of the solid elements CHEXA, CPENTA,CPYRAM, and CTETRA.

You create solid element face regions with the BSURFS or BCPROPS bulk entries. You then pairthe regions using the source and target fields on the BGSET bulk entry. From elements in thesource region, the software searches in the outward normal direction. In addition, the softwaresearches a small distance in the inward normal direction in order to glue faces that may interferedue to meshing irregularities.

The software creates a glue condition if:

• Any of the source element normals intersect with an element in the target region.

• The distance between the two faces is equal to or less than the search distance which youspecify for the glue pair on the BGSET entry.

Additional information regarding glue in SOL 401.

• When PARM,LGDISP,1 is defined, the glue stiffness orientation will update as a result of largedisplacement effects. The glue stiffness is only computed once, at the beginning of a solution.

• The source side element characteristics are used to define the glue stiffness. Therefore itis possible for differences depending on which element faces or edges are selected as thesource region.

• The generalized plane strain element is not supported by glue or contact regions.

9.2 Glue RegionsThe following describes how edge and surface regions are defined. An element should not appearmore than once in the regions that are part of a glue/contact pair. All region IDs defined with theBEDGE, BSURFS, or BCPROPS entries must be unique.

Edge Glue Regions

The edge glue region in SOL 401 is a selection of axisymmetric, plane strain, plane stress, andchocking element edges in a section of the model where you expect glue to occur. An element edgeis selected on the BEDGE bulk entry by entering the element ID along with the corner GRID IDs.

Surface Glue Regions

A surface glue region is a selection of solid element free faces in a section of the model where youexpect glue to occur. The element faces are selected with the BSURFS and BCPROPS bulk entries.

• The BSURFS entry is a list of solid element IDs each followed by 3 grid points defining which faceof the 3-D element to include in the glue region.



Glue conditions

• The BCPROPS entry is a list of solid element property IDs. The free faces of the solid elementsselected with a property ID are automatically determined by the software.

9.3 Defining and Selecting Glue PairsA glue pair is a way to combine two regions, source and target, in which gluing will be analyzedduring the solution. You use the BGSET bulk entry to define each glue pair. The fields on the BGSETbulk entry are described as follows:

• GSID (glue set id) will need to match the value of ‘n’ on the BGSET case control entry for thesolution to recognize the glue definition.

• SIDi and TIDi are used to select source and target regions for a pair. For surface-to-surfacegluing, they select the regions created by the BSURFS and BCPROPS entries. For edge-to-edgegluing, they select the regions created by the BEDGE entries. As many pairs as desired canbe included on a single BGSET bulk entry.

• SDIST (search distance) defines the distance in which the solver can initially determine if thedistance between element edges or faces in a particular pair are within the threshold for creatingglue elements. The default value of SDIST of 10 is large enough to handle most geometrysituations, but can be adjusted as needed. This value is used once, at the beginning of thesolution, to determine where glue elements are created.

Combining Glue Sets – BGADD

You can optionally define multiple BGSET/BGPARM bulk entry sets, each set with unique glue set IDs(GSID), and then combine them with a single BGADD bulk entry. The multiple BGSET/BGPARM bulkentry sets are created to adjust certain glue parameters locally. Glue parameters can also be adjustedglobally with a BGPARM bulk entry having the same GSID as the BGSET case control command.

The following example demonstrates the inputs.

CASE CONTROL$GSID on the BGSET case control matches GSID on BGADDBGSET = 108...BULK DATA$Local Glue Set definitionsBGSET 1 1 2BGSET 2 3 45 6...$Local Glue ParametersBGPARM 1 PENN 80 PENT 80BGPARM 2 PENN 110 PENT 110...$Local Glue Sets are combined with BGADDBGADD 108 1 2...$Global Glue ParametersBGPARM 108 REFINE 0GLUETYPE 1


Glue conditions


See the section Glue Control Parameters – BGPARM Bulk Entry in this chapter for more informationon glue parameters.

Gluing Composite Solid Faces

Defining glue regions and pairs on composite solid faces which are perpendicular to the stackdirection (edge faces) may produce poor stress continuity. If the glue definition is between edge facesbelonging to different PCOMPS definitions, and if the number of plies on each PCOMPS definition issmall and the same, and the ply thicknesses are similar, the stress continuity should be fairly smooth.This also applies to the results requested with the BGRESULTS case control command.

Additional Recommendations

When defining glue regions and pairs on geometry which are not tangent continuous, creating singleglue regions which cross corner transitions can result in non-uniform stress results around thecorners. It is recommended to break these areas into multiple regions and pairs as shown below.

When defining glue regions and pairs, it is recommended to not include the same element face inmultiple regions. In “A” below, an element is repeated in regions 2 and 3. In “B”, the same elementonly exists in region 2. “B” is recommended. Repeating element faces multiple times in the same ordifferent regions can significantly increase memory requirements and degrade performance.



Glue conditions

9.4 Glue Control ParametersThe glue control parameters on the BGPARM bulk entry can help you adjust the glue algorithm. Formost solutions, the default settings are appropriate, thus the BGPARM entry is not required.

You can optionally define multiple BGSET/BGPARM bulk entry pairs, each pair with unique glue setIDs (GSID), and then combine them with a single BGADD bulk entry.

The multiple BGSET/BGPARM bulk entry pairs are created to adjust certain glue parameters locally.Glue parameters can also be adjusted globally with a BGPARM bulk entry having the same GSID asthe BGSET case control command.

Global and local glue parameters have the following definition and rules:

• Global Glue Parameters

The BGPARM bulk entry, which uses the same GSID entered on the BGSET case controlcommand, defines global parameters.

Any of the parameters on the BGPARM bulk entry can be defined globally. A parameter’s defaultvalue is used if it is not defined globally or locally.

• Local Glue Parameters

The BGPARM bulk entries associated to individual BGSET bulk entries, which are then combinedwith a BGADD bulk entry, define local parameters.

The parameters GLUETYPE, PENN, PENT, PENTYP, and PENGLUE can be defined locally. Alocal parameter definition overrides a global definition.

Glue Parameter Descriptions for SOL 401

•


Glue conditions


o SOl 401 only supports GLUETYPE=2. As a result, the glue stiffness is determined by thePENTYP and PENGLUE parameters.

PENTYP – Changes the meaning of the PENGLUE parameter as described in the tablebelow. (Default=1)

PENGLUE - A scale factor used to adjust the glue stiffness. (Default=1.0)

PENTYP=1 (Default) PENGLUE is a unitless value (glue stiffness scale factor).PENTYP=2 PENGLUE has the units of F/L2.

Note: For glued coincident faces, there is little flexibility between the faces with default penaltyfactors. The glue condition created between non-coincident faces will not usually produce a localstiffness as accurate as using a conventional finite element for the connection. The flexibility inthe glue condition will depend on the value of the penalty factors. If you have non-coincidentfaces and the glue joint flexibility is important, then it is recommended that you model thisconnection with conventional finite elements.

Average modulus computation

The averaged modulus of the elements associated with the source side region is also used whencomputing the glue stiffness. The averaged modulus is computed as follows.

o The modulus is averaged for the elements associated with the source side region.

o For elements using orthotropic and anisotropic materials, the element modulus value used inthe average modulus computation is computed as follows.

MAT9: E=(C11+C22+C33)/3

MAT11: E= (E1+E2+E3)/3

o When a solid composite defined with the PCOMPS entry is included in a source glue region,the average modulus is computed by averaging all modulus values defined on all materialsdefined in the input file.

• SLIDE - Requests the sliding glue. Sliding glue includes normal stiffness but no tangentialstiffness. Gaps between the glued surfaces are preserved as sliding occurs.

0 - Sliding glue is off (default).

1 - Sliding glue is on.

• INTORD and REFINE help to improve the accuracy of the glue solution. The number oflocations where normals are projected (glue points) from the source region is dependent on thevalue assigned to the INTORD parameter, and on the element face type. The following tablesummarizes how the INTORD value adjusts the number of glue points for a particular elementface:

Number of Glue Points Used in Glue Element EvaluationFace Type INTORD=1 INTORD=2 (default) INTORD=3Linear Triangle 1 3 7Parabolic Triangle 3 7 12Linear Quad 1 4 9



Glue conditions

Parabolic Quad 4 9 16

REFINE will increase the number of glue points by refining the mesh on the source region. Partof the refinement process is to project element edges and grids from the associated target regionback to the source region. The resulting refinement on the source region is then more consistentwith the target side, which then gives a better distribution of glue elements. The refined grids andelements are only used during the solution. The glue results are transferred back to the originalmesh for post processing results.

Refinement occurs when set to 2 (default). REFINE=0 turns the refinement off.

• You can use the PREVIEW parameter to request the export of a bulk data representation of theelement edges and faces where glue elements are created. See Glue Preview.

9.5 Glue previewYou can optionally export a bulk data representation of the element edges and faces where glueconditions are created. To do so, set the PREVIEW parameter on the BGPARM bulk entry to “1”.For example:

$*BGSET 100 1 2 0.001000BGPARM 100 PREVIEW 1$*

The software writes a bulk data file containing dummy shell element entries for face locations, anddummy PLOTEL entries for edge locations. Dummy GRID, property, and material entries are alsowritten. You can import the file into a preprocessor to display both source and target glue locations.

The preview file has the naming convention:

<input_file_name>_glue_preview_<subcaseid>_<gluesetid>.dat

For example, if an input file named test.dat includes a subcase numbered 101 and a glue setnumbered 201, the resulting preview file name is:

test_glue_preview_101_201.dat

In the following simple example, the red mesh is glued to the green.


Glue conditions


Before applying any loading, the software creates the glue elements. This is the point in the solutionin which the preview output is written.

The following illustration shows how the preview appears after you import the preview file into theSimcenter preprocessor. Note that the colors were manually modified in Simcenter after importingthe file.



Chapter 10: Considerations for nonlinear analysis

10.1 Discrete system for a nonlinear continuum modelTheories in solid mechanics are dictated by three governing relationships:

• The state equilibrium that

Equation 10-1.

requires where σij are stress tensor components, bi are body forces, and xj are space coordinates.

• The constitutive relations represented by stress-strain relations, e.g., for linear elasticity,

Equation 10-2.

where εkl are strain tensor components and Dijkl are elastic constants.

• The compatibility represented by strain-displacement relations, e.g., for a small deformation,

Equation 10-3.

where ui are displacements.

These systems of governing differential equations must be satisfied for every infinitesimal elementthroughout the domain of the continuum. The complete set of state variables, namely displacements,may be determined by solving these systems of equations supplemented by boundary conditions,and in dynamic situations by initial conditions as well. For the nonlinear problems, the governingequations should be satisfied throughout the history of load application. The material nonlinearityis manifested in the constitutive relations. The geometric nonlinearity is pronounced in thestrain-displacement relations, but it also affects the equilibrium equation by changing applied loads.Changes in constraints affect the boundary conditions, which constitute contact problems.

Most of the known solutions for the solid mechanics problems are based on ideal geometry and linearapproximations. However, the real nature is more complicated and inherently nonlinear. The linearsystem is a very particular case of a general problem. Even the nonlinear solutions that we seek dealwith only a small subset of special cases in a general category of nonlinear problems. When the



nonlinear system is confronted, no general mathematical solutions exist and superposition no longerapplies. The system may even be non-conservative.

The first phase of the structural analysis is the idealization of a physical system into a simpler andmore manageable engineering problem. The idealization process involves simplifications of thegeometry, boundary and joint conditions, and loading conditions, etc. using engineering intuitions,experimental data, empirical observations, and classical solutions. If the idealized structural systemrenders a problem that cannot be resorted to a classical method of analysis, further idealization isrequired, namely discretization, for numerical analysis.

Finite elements represent spatial discretization of a continuum. As such, however, they donot immediately impose nonlinearity. When nonlinearity has to be taken into account for largedisplacements and/or stresses, a numerical model poses new dimensions to the discretization inaddition to the n-dimensional Euclidean space. That is, the discretization is applied to time, load,and material properties by using piece wise linear curves. While discretization allows approximatesolutions by numerical methods, it introduces numerous mathematical singularities which maycomplicate computational processes. Fortunately, the efficiency of modern digital computers makesit feasible to apply complicated computational procedures to the complex systems of engineeringproblems.

For the discrete system, governing differential equations are converted to algebraic equations. Thefinite element model represents a structure by an assemblage of finite elements interconnected atnodal points. State variables are the displacements (displacement method or stiffness approach.) ofthe nodal points which carry fictitious forces representing distributed stresses actually acting on theelement boundaries. The equilibrium requirements are satisfied at nodal points by the nodal forcebalance. The material constitutive laws are satisfied at the integration points of the element. Thecompatibility is ensured by the displacement continuity between elements. It is noted, however, thatthe compatibility of the nonconforming elements is ensured by a patch test.

10.2 Finite element formulation for equilibrium equationsThe variational principle renders the system governing equilibrium equations when applied to afunctional ll, representing a total potential of a continuum, i.e.,

Equation 10-4.

where U is the strain energy of the system and W is the potential energy of the external loads. Theequilibrium equations can be obtained by invoking the principle of virtual work or the Ritz method, i.e.,

Equation 10-5.

which implies that the total potential of the system must be stationary with respect to the statevariables (displacement) for equilibrium to be ensured. The functional IT is so called because itinvolves the integral of implicit functions of the state variables, {u}.



Considerations for nonlinear analysis

Considering a three-dimensional continuum for a nonlinear problem, the stationarity condition resultsin

Equation 10-6.

where the dots and δ denote infinitesimal increments and arbitrary variations, respectively. Theleft-hand side represents variations in the strain energy increment and the right-hand side representsvariations in the external work which consists of body forces bi (such as a gravity load), traction forcesti at the boundary surface (such as pressure loads), and concentrated forces pi. Now it remainsto determine admissible functions expressing the arguments of the functional II in terms of statevariables {u}, which are valid throughout the whole region and satisfy the boundary conditions.

The finite element method can be characterized by the following features distinguished from theconventional Ritz methods or the matrix method for frame structures:

• The whole region of the system is divided into numerous subdomains, called finite elements,which have simple geometrical shapes.

• The variational process is limited to each finite element, which aggregates into a whole regionwhen assembled.

• The admissible displacement field within each element, , can be expressed in terms of nodaldisplacements using interpolation functions known as shape functions, N, i.e.,

Equation 10-7.

where {u} is a displacement vector consisting of all nodal points of the element.

The strain-displacement relations for the element can then be established in terms of nodaldisplacements using the shape functions in Equation 10-7, i.e.,

Equation 10-8.

where

Equation 10-9.

and the element matrix [B] consists of derivatives of the shape functions, evaluated at the currentdeformed geometry. Notice that the geometric linear problem requires that the element matrix beevaluated only at the initial geometry. The software employs an approximate updated Lagrangianapproach for geometric nonlinear problems, by which linear strains are computed in the updatedelement coordinate system in order to eliminate the effects of the rigid body rotation but the




equilibrium is established at the final position in the stationary coordinate system. This method doesnot require reevaluation of the element matrix [B] (constant in the absence of large strains) while theelement coordinates are reevaluated continuously.

Equilibrium equations for an element may be obtained by reducing Equation 10-6 after the substitutionof Equations 10-7 and 10-8, based on the small deformation theory. Then the element boundarystresses are statically equivalent to the nodal forces which balance the applied external loads, i.e.,

Equation 10-10.

with

Equation 10-11.

and

Equation 10-12.

where [Ns] is an appropriate interpolation function for the traction force. Notice that the equilibriumequation for an incremental load may be expressed as

Equation 10-13.

where should be components of co-rotational stress which is independent of a rigid body rotation.

The element stiffness matrix can be obtained by substituting the constitutive relations into Equation10-13, i.e.,

Equation 10-14.

where

Equation 10-15.

and [D] is the material tangent matrix. The nodal forces of an element can then be expressed as




Equation 10-16.

where the element stiffness is

Equation 10-17.

Notice that this expression represents an element stiffness due to the material stiffness withoutgeometric nonlinear effects. As will be shown later, an additional stiffness [J(d)] due to initial stressesshould be included for an incremental process because the initial stresses exist from the secondincrement.

The equilibrium must be satisfied in the whole region throughout the complete history of loadapplication. Equilibrium equations for the global discrete system are obtained when all the elementsare assembled, i.e.,

Equation 10-18.

where Σ over m denotes a summation over all elements. For the incremental process, the equilibriumequation may be rewritten as

Equation 10-19.

with

Equation 10-20.

where {σ0} represents an initial stress or the stress state at the preceding load step.

Because of the approximations involved in the interpolation functions, the finite element modelprovides an approximate solution even if the equilibrium Equation 10-18 is satisfied exactly.Consequently, the differential equations of equilibrium are not satisfied exactly even for linearproblems, but the error decreases as the finite element mesh is refined. This convergence conditionis required and ensured by element formulations with regard to the element convergence criteria.




10.3 Coordinate transformationsThe coordinate transformation is one of the most frequent operations in the finite element method.Vectors and matrices defined in a particular coordinate system can be transformed into anothersystem. Coordinate systems involved are:

• Basic coordinate system: a Cartesian coordinate system on which local coordinate systemsare defined.

• Local coordinate system: defined by the user in the Bulk Data, which may include specialcoordinates such as cylindrical and spherical coordinate systems.

• Global coordinate system: a collective coordinate system which comprises all the local coordinatesystems specified for output quantities.

• Element coordinate system: a Cartesian coordinate system unique to each element.

• Displaced element coordinate system: similar to element coordinate system but defined in thedisplaced position.

• Material coordinate system: a Cartesian coordinate system used to orient anisotropic materialproperties.

• Modal coordinate system: a generalized coordinate system defined for each eigenmode.

It is noted that the global system is a Cartesian coordinate system, although non-Cartesian coordinatesystems are adopted to orient the local Cartesian coordinates for output quantities. In the software,all the displacements and forces, hence the system matrices, such as the stiffness matrix, areexpressed in the global coordinates. This implies that all the major computations involved in theanalysis are processed in Cartesian coordinates. Element and material coordinate systems aredefined in the element connectivity description. Now we only have to consider linear transformationsbetween Cartesian coordinate systems.

Let us consider a coordinate transformation between the primed and unprimed systems which areright-handed Cartesian coordinates. The transformation matrix T consists of direction cosines of unitvectors of the unprimed coordinate system, i.e.,

Equation 10-21.

where

Equation 10-22.

Notice that T is an orthogonal matrix and thus




Equation 10-23.

Because the work and energy are invariants with respect to coordinate transformation, i.e.,

Equation 10-24.

it follows that

Equation 10-25.

Then the equilibrium equation,

Equation 10-26.

may be expressed in the unprimed coordinate system by

Equation 10-27.

with

Equation 10-28.

It is noted that the modal matrix Φ is used as a transformation matrix for a modal transformationwhich is not elaborated here.

Equation 10-29.

The forces and displacements are transformed from element to global coordinates and vice versa, i.e.

where Tbe transforms from element to basic coordinates and Tbg transforms from global to basiccoordinates.

It is noted that Tbe is identical for all the nodes of an element but Tbg may vary from node to node inthe same element. The element stiffness matrix is transformed into global coordinates by




Equation 10-30.

for which the building blocks of Tbe and Tbg are (3x3) matrices formed for each nodal point and haveto be assembled for an entire element, e.g. for a three-noded triangular shell element

Equation 10-31.

where the superscript is used to associate each (3x3) matrix with the nodal point and is repeatedfor the rotational degrees of freedom.

10.4 Displacement sets and reduction of system equationsThe equilibrium equations, and thus system matrices, are reduced in size using the displacement set,which is a unique feature of this software. Mutually exclusive subsets of the global displacement set,{ug}, are defined as follows:

um Degrees-of-freedom eliminated by multipoint constraints

us Degrees-of-freedom eliminated by single-point constraints

uo Degrees-of-freedom omitted by static condensation

ur Degrees-of-freedom eliminated by a bulk data SUPORT to suppress rigidbody motion

ui Degrees-of-freedom which remain for solution after reductions

For convenience, complementary sets are defined as follows:

un = ug - umuf = un - usua = uf - uoul = ua - urThe subsets are defined by the user with a possible exception in the s-set if PARAM,AUTOSPC,YESis used. Notice that the rigid elements are equivalent to the multipoint constraints internally in theprogram, but they are not selectable in the subcases as for MPCs. Because the set-reductionoperations involve many basic modules and DMAP blocks, the mathematics for elimination ofconstraints and static condensation is reviewed here.

The multipoint constraint equations are formed in the module GP4 as follows:




Equation 10-32.

where

The module MCE1 partitions [Rmg] and solves for a transformation matrix [Gmn], i.e.,

Equation 10-33.

where

Then the module MCE2 partitions the global stiffness matrix, [Kgg], and reduce it to the n-set, i.e.,

Equation 10-34.

from which the system is reduced to

Equation 10-35.

where

and

The primes are used in K’nn, P’n, and Q’n to distinguish from Knn, Pn, and Qn, which are resultingmatrices after the reduction.

Equations in the n-set can be further reduced by eliminating single-point constraints, i.e.,

Equation 10-36.




which is reduced to

Equation 10-37.

where

with

Notice that the effects of constraint forces (Qs and Qm) are not visible in Equation 10-37. Thesingle-point constraint forces are recovered by

Equation 10-38.

Further reduction of equations in the f-set is performed by an elimination of the o-set, known as staticcondensation. The f-set is partitioned by the UPARTN module as follows:

Equation 10-39.

from which

Equation 10-40.

where

and

Then the reduced system of equations in the a-set is obtained as

Equation 10-41.

where

and




There are some rules to remember regarding the displacement sets in SOL 401. They are:

• The r-set is not supported. Do not use the SUPORT bulk entry.

• PARAM, AUTOSPC is not supported.

• PARAM, AUTOSPCR is not supported.

• Rigid elements are formulated with linear multipoint constraint equations and do not have largedisplacement capability. Consequently, erroneous results will be obtained if the rigid elementundergoes a large rotation. To avoid this, stiff elements should be used in place of rigid elementsfor large displacement analysis.

10.5 Nonlinear solution procedureThe general-purpose program developer faces the task of providing the best workable solutionmethod for a wide spectrum of problems, while maintaining flexibility by allowing the user to specifyoptional parameters. Based on the extensive numerical experiments, an attempt was made toestablish a general strategy suitable for most problems without requiring insight or experience.Variations in combining theories, algorithms, criteria and parameter values with numerous testproblems resulted in a succinct implementation.

The major feature of the nonlinear analysis is the requirement for the incremental and iterativeprocesses to obtain a solution. The main issue is how to choose the most efficient method from theoptions available for the incremental and iterative processes in the solution of nonlinear equilibriumequations. The increment size for time steps has the most significant effect on the efficiency andthe accuracy of the computation, particularly in the path-dependent problems. The incremental anditerative processes are complementary to each other because the larger the increment size themore iterations the solution requires. While an excessively small increment reduces the computingefficiency without any significant improvement in accuracy, a large increment may deteriorate theefficiency as well as the accuracy; it may even cause divergence.

It is impossible to optimize the incremental step size in the absence of prior knowledge of thestructural response. The best engineering judgement should be exercised to determine the incrementsize based on the severity of the nonlinearity. Needless to say, no incremental load steps are requiredwhen the response is linear. In principle, the size of the load increment (or time increment) should bechosen to yield a uniform rate of change in strains or stresses for the material nonlinear problems anda uniform rate of change in displacements for geometric nonlinear problems.

User specifications for solution methods in nonlinear analyses are allowed via:

• The NLCNTL bulk entry for the static analysis. It is selected by the NLCNTL case controlcommand.

• The TSTEP1 bulk entry for the load increment (time based). It is selected by the TSTEPNLcase control command.

• The EIGRL bulk entry for the modal analysis. It is selected by the METHOD case controlcommand.




The increment size can vary from subcase to subcase by specifying different TSTEPNL. It isrecommended to define separate TSTEPNL for every subcase even if the same values are specified,so that changes can be accommodated in the subcase level as needed.



Chapter 11: Geometric nonlinearity

11.1 Overview and user interfaceGeometric nonlinearities are manifested in problems involving large rotations and large deformation.The characteristics are follower forces due to large rotations, geometric stiffening due to initial stresseffect (as a result of large rotations), and large strains due to large deformation.

Geometric nonlinear effects should be significant if the deformed shape of the structure appearsdistinctive from the original geometry by a visual inspection. A more rigorous and quantitativedefinition for the large displacements can be derived from the plate theory of Kirchhoff and Love: thesmall deflection theory is valid for a maximum deflection of less than 20% of the plate thickness or2% of the small span length. However, this definition seems to be a little conservative for numericalanalysis, and there is no distinct limit for large displacements because geometric nonlinear effects arerelated to the boundary conditions as well as the dimensions of the structure. If the load-deflectioncurve of the critical point can be estimated, the loading point should be in the nonlinear portion ofthe curve.

Geometric nonlinear effects in the structure involving large rotations, whether rigid body rotations ordeformation induced rotations, are self-evident. Stiffening of a membrane, stiffness in a pendulumor snap-through of an arch belong to this category. The motion of a pendulum under gravity iscaused by geometric (differential) stiffness. Follower forces are manifested when the applied loadsare displacement dependent, such as pressure load and thermal load applied on the surface thatrotates. Centrifugal force is another example of follower forces. Large strain effects are pronouncedin metal forming problems which could have strains exceeding 100%. Finite strain formulation isrequired to treat the problems in this category. The software does not currently support the largestrain capability. In most structural applications, however, moderately large strains (20 to 30%)appear in local areas if there is any large deformation. The software can be used for that category ofproblems. Other geometric nonlinear effects are treated by updated element coordinates, gimbalangles (or rotation vector), and the differential stiffness [Kd].

The geometric nonlinearity is controlled by the parameter LGDISP with the following values inSOL 401:

• = 0 for geometrically linear analysis

• = 1 for geometrically nonlinear analysis

With values of 1 or 2 for LGDISP, all the potentially nonlinear elements become actively nonlinearelements unlike the material nonlinear model.

This solver has a distinct approach to the large rotation, for which the element coordinates arecontinuously updated to the current configuration during the iteration. The equilibrium is sought in thedeformed position. Consider the internal force computation as follows:



Equation 11-1.

The element matrix is defined from the strain definition as

Equation 11-2.

in which could be divided into two parts (linear and nonlinear), i.e.,

Equation 11-3.

Upon differentiation of Equation 11-1, we have

Equation 11-4.

where {σ} represents stresses with reference to the original coordinates. Substituting Equation11-3 and

Equation 11-5.

Equation 11-4 becomes

Equation 11-6.

with

Equation 11-7.

and



Geometric nonlinearity

Equation 11-8.

in which KL represents the usual linear stiffness matrix, KR a stiffness due to large rotation, and Kg ageometric stiffness dependent on the initial stress level.

Now it remains to define the nonlinear part of the element matrix (BN). The definition of finite strainsbased on the Lagrangian formulation (referred to the initial configuration) is as follows:

Equation 11-9.

Equation 11-10.

with other components obtained similarly. In matrix notation

where {εL} is the usual infinitesimal strain vector and {εN} is the nonlinear strain vector consisting ofthe second order terms, i.e.,

Equation 11-11.

where

Equation 11-12.

and




Equation 11-13.

Introducing shape functions (Ni) and nodal displacements {u} (using an example of a 10-nodedtetrahedron), displacement derivatives are expressed by

Equation 11-14.

and

Equation 11-15.

where

Equation 11-16.

and

Equation 11-17.

From the properties of matrices A and θ, it can be shown that




from which

Equation 11-18.

The initial stress stiffness [Kσ] can be derived as follows:

in which

where

Equation 11-19.

with I being the (3x3) identity matrix. Finally the geometric stiffness is

Equation 11-20.

It has been found that stiffness matrices caused by geometric nonlinearity (KR and Kσ) can becomputed from the matrices [A], [G], and [M] with the following observations:

• [G] is dependent upon the initial geometry, hence stays constant unless the geometry is updated.This matrix is used in forming [KR] and [Kσ].

• [A] is used in forming [KR]. [A] is dependent on the rotations and should be updated continuously.

• [M] is used in forming [Kσ]. [M] is dependent on the stresses and should be updated continuously.




The primary functions of nonlinear stiffness matrices can be interpreted as follows:

• The matrix [KR] takes into account the effects of large rotations. The large displacement effects,due to rigid body translation and rotation, are treated effectively in the absence of large strains byupdating element coordinates in the software.

• Geometric stiffness matrix [Kσ] takes into account the effects of the initial stresses. This effectbecomes important with geometric stiffening, and is used for instability analysis. The geometricstiffness matrix [Kσ] is equivalent to the differential stiffness [Kd] in the software.

11.2 Updated element coordinatesWhen the large displacement effect is included in the nonlinear analysis, the solver employs a methodof displaced element coordinate system. This method allows large rotations by updating elementcoordinates to the deformed geometry, and the equilibrium is computed in the deformed configuration.

11.2.1 Concept of convective coordinates

The concept is based on the fact that the rigid body motion does not contribute to the strain energyand is eliminated from the internal force computation. Consider a rod which underwent rigid bodymotion as well as deformation as shown below:

Figure 11-1. Net Deformation of a Rod

The net displacement ud is measured in the displaced element coordinate system by overlaying theoriginal element on top of the deformed element. The element force can simply be computed by

Equation 11-21.

where the superscript e denotes an elemental operation and the subscript d denotes the vectors inthe displaced element coordinate system. Then the element forces should be transformed into thecommon coordinate system (namely global coordinate system denoted by a subscript g) beforeassembly for global operations, i.e.,




Equation 11-22.

where the summation sign implies an assembly operation, and Tbd and Tbg are transformationmatrices from displaced to basic and from global to basic coordinate systems, respectively.

11.2.2 Updated coordinates and net deformation

Referring to Figure 11-2, a quadrilateral element is shown in its original and deformed positions(denoted by subscripts e and d, respectively) with reference to the basic coordinate system (denotedby a subscript b).

Figure 11-2. Element Coordinates vs. Displaced Coordinates

The element coordinate system is established by bisecting the diagonals of the quadrilateral.Transformation from the element coordinate system to basic coordinates is simply

Equation 11-23.

where the position vector (Xeb in Figure 11-2) of the element coordinate system with respect tothe basic coordinate system is denoted by < xe, ye, ze > Tbasic and transformation matrix [Tbe] is




composed of direction cosines of unit vectors of the element coordinate system with respect to thebasic coordinate system, i.e.,

Equation 11-24.

As the element deforms or displaces, the element coordinate system moves and this is defined as adisplaced coordinate system. The displaced coordinate system is established in the same manner asthe element coordinate system. Again the transformation should be performed similarly, i.e.,

Equation 11-25.

where < xd, yd, zd > Tbasic is the position vector of the displaced element coordinate system withrespect to the basic coordinate system (Xdb in Figure 11-2) and [Tbd] is formed simliarly to [Tbe].

In order to isolate the deformation from the rigid body displacements, nodal displacements arecomputed in the displaced element coordinate system by overlaying the original element as shownin Figure 11-3.

Figure 11-3. Computation of Net Deformation




The net displacements can be computed by subtracting the original nodal coordinates in the elementcoordinate system from the displaced nodal coordinates in the displaced element coordinate system,i.e.,

Equation 11-26.

in which the nodal coordinates in the element and displaced element coordinate systems can becomputed by the following transformations:

Equation 11-27.

and

Equation 11-28.

Substitution of Equations 11-27 and 11-28 into Equation 11-26 results in

Equation 11-29.

where {ug} is a total displacement (translational components only) in the global coordinates. In theabsence of the large displacement effect, the net displacement ud in Equation 11-29 is reduced to:

11.2.3 Provisions for global operation

It is noted that the net rotations (θ x, θ y, and θ z of each node associated with the shell andbeam elements) are computed by a gimbal angle approach (or rotation vector approach) beforecomputing element forces. Subsequently, the element forces have to be transformed to the globalcoordinate system before assembly for equilibrium check. The internal forces are computed using netdisplacements and rotations, ud, i.e.,




Equation 11-30.

if the material is linear or

Equation 11-31.

if nonlinear material is involved. Consequently, the tangent stiffness matrix is formed in the globalsystem by assembling the element stiffness matrices transformed into the global coordinate systemfrom the displaced coordinate system, i.e.,

Equation 11-32.

The update process is performed at every iteration and the updated nodal displacements ud are usedwhenever strains and stresses are computed. Effectively, the second order effect due to large rigidbody motion is eliminated. However, the displacement output shows the total displacements inthe global coordinates, i.e.,

Equation 11-33.

where the subscript i. denotes operations on each nodal point. The transformation matrix [Tbd] iscomputed for each element after each iteration and stored in the ESTNL data block for stiffness matrixupdate when requiredby the stiffness matrix update strategy. On the other hand, the transformation[Tbg] is computed for each nodal point and it is not stored but recomputed whenever it is needed. Thenodal coordinates in the undeformed geometry, Xb, are available from the data block BGPDT.

This approach can be interpreted as approximate updated Lagrangian method, since the motion ofthe body follows Lagrangian description. Stresses are computed in the deformed geometry just likeCauchy stress. However, this method of displaced coordinate system is a unique and salient featurein the software. The referential geometry in the updated Lagrangian method is brought up-to-date atevery incremental step upon convergence but fixed during the iterative process, which is inherentlydifferent from the current method of updating the coordinate system.

11.3 Follower forcesThe term "follower force" usually refers to the applied loads that change direction and magnitudewith structural displacements and rotations, e.g.,

Equation 11-34.




where p is the magnitude of the pressure on the surface A, interpolated by a shape function N, andñdA changes as a function of u. They generally occur with fluid pressures such as the pressurizedballoon, inflated tire, or the lift load on the airplane wing. Other physical applications involvekinematics such as the classical fire hose instability problem or inertia loads on spinning bodies. Inthe software, the term applies to specific load inputs as defined below.

11.3.1 Basic definition

For geometrical nonlinear analysis, static loads belong to one of two categories, namely:

• Loads defined by fixed vector inputs, which may be calculated once per run and cannot changedirection or magnitude.

• Loads defined by the location of one or more GRID points.

The first category includes simple forces, and enforced displacements. The second, follower forcecategory, includes the following Bulk Data inputs:

FORCE1, FORCE2 The direction changes with displacements of the referenced GRID points. Themagnitudes of these concentrated loads are constant.

PLOAD, PLOAD4 The pressure loads follow the surface of the solid elements (HEXA, PENTA,PYRAM and TETRA).

RFORCE Centrifugal loads change in magnitude and direction with motion of the massesattached to the GRID points. The effect may be destabilizing if large motionsoccur. It is recommended that lumped masses be used with these loads.

Also note that upstream superelements are assumed to be linear and therefore the upstream loadswill remain fixed in magnitude and direction. In addition, forces on omitted degrees of freedom (whenASET or OMIT data are present) should not be follower forces.

11.3.2 Implementation

The follower forces depend on the GRID displacements and therefore must be recalculated for eachnonlinear iteration and line search. The basic equation for residual error, as defined in Newton’smethod of iteration, becomes:

Equation 11-35.

where the applied load vector {Pal is now a variable. Corrective Loads are computed based on theupdated geometry and added to the initially applied loads to account for the follower forces, i.e.,

Equation 11-36.

where




Equation 11-37.

Note that thermal effects are included in the vector {F}.

Using the following equation presented in Newton’s method of iteration,

[KT] {Δui} = {Ri-1}

the tangent matrix can be calculated using derivatives of the loads, which is termed follower matrix.However, the nonlinear solution process ignores the stiffness effects of the changing loads anduse the approximation:

Equation 11-38.

The effect of the approximation is minor in most cases. However, it could become a major concern inthin shell models with pressure loads causing large rotations, where the converged solutions will becorrect but the rate of convergence may be slow or cause divergence. Also the buckling solutions ormodal analysis on preloaded structure with pressure load may not be correct due to the approximatetangent matrix if the effect of the follower matrix is significant.

The follower force effects in the analysis can be controlled by the parameter LGDISP. Three optionsare available in PARAM LGDISP:

• = 0 for no geometric nonlinearity

• = 1 for full geometric nonlinearity (including follower forces)



Chapter 12: Solution methods

12.1 Solution AlgorithmLet n represent the current time step, i the current iteration, (n-1) will be the last converged time step.If n=1, the previous converged time step will be the initial conditions.

U represents the displacement vector.

P represents the external force vector.

F represents the internal force vector.

R represents the residual (R= P-F) vector.

t represents time.

∆t represents time step size.

1. Assemble stiffness matrix for the structure if this is the first step, or if a stiffness update isrequested any time during the solution. Decompose the stiffness matrix.

2. Save last converged displacements (Un-1) and external forces (Pn-1)

3. tn=tn-1+∆t

4. Obtain the external force vector Pn1 for the first iteration of current time step.

a. If i>1: Include follower force effects to obtain Pn1.

5. Compute internal forces Fn1(Uni–1)

6. Compute residual: R = Pni - Fni

7. If i>1:

a. Compute: E2=∆Uni-1. R

b. Compute Er=E2/E1

c. If Er>1 or Er<-100, ndiv=ndiv+1

d. Write quasi Newton vectors for BFGS.

8. Compute displacement increment: ∆Uni=K-1 R

9. Compute E1=∆Uni.R

10. Compute Uni=Uni-1+ ∆Uni

11. Compute current stiffness parameter



12. Check for convergence.

a. If solution has converged and stiffness update is needed, go to 1.

b. If solution has converged and stiffness update isn’t necessary, go to 2.

c. If solution hasn’t converged and stiffness update is needed, go to 1 (skip 3).

d. If solution hasn’t converged and stiffness update isn’t needed, go to 4.

12.2 Adaptive Solution StrategiesNonlinear finite element computations comprise material processes, element force computations,and various global solution strategies. The computational procedure involves incremental anditerative processes ranging from local subincrements to global solution processes. Performanceof the finite element program can be scrutinized from three different perspectives: computationalefficiency, solution accuracy, and effectiveness. All of these attributes of the nonlinear program canbe improved by adaptive algorithms.

12.3 Newton’s method of iterationThe equilibrium equations in the g-set may be written as

{Pg} + {Qg} - {Fg} = {0} where {Pg}, {Qg}, and {Fg} represent vectors of applied loads, constraint forces,and element nodal forces, respectively. Element nodal forces are nonlinear functions of displacementsfor nonlinear elements. Since the equilibrium condition is not immediately attained in the presence ofnonlinear elements, an iterative scheme such as the Newton-Raphson method is required. Since theerror vanishes at constrained points and the constraint forces vanish at free points, the unbalancedforces acting at nodal points at any iteration step are conveniently defined as an error vector by

{Ra} = {Pa} - {Fa}.

Equation 12-1.

Notice that the a-set is equivalent to the I-set in the nonlinear analysis because the r-set does notexist. The subscript a will be dropped for simplicity in the following discussion.

Based on Newton's method, a linearized system of equations is solved for incremental displacementsby Gaussian elimination in succession. The Jacobian of the error vector emerges as the tangentialstiffness matrix. The equation to solve at the i-th iteration is

[KT] {Δui} = {Ri-1}

Equation 12-2.

where

and



Solution methods

{Δui} = {ui} – {ui-1},

{Ri} = {P} – {F(ui)}.

The iteration continues until the residual error {R} and the incremental displacements {Δu} becomenegligible, which is signified by the convergence criteria.

The tangential stiffness consists of the geometric stiffness in addition to the material stiffness, i.e.,without regard to the coordinate transformation,

Equation 12-3.

where [Km] and [Kd] refer to the material and the differential stiffness, respectively. The materialstiffness is given in Equation 4-17 with a material tangential matrix for [D]. The differential stiffness,which is caused by the initial stress, is defined as follows:

Equation 12-4.

where [BN] represents the second order effects in the strain-displacement relations, [G] consists ofderivatives of shape functions and [M] is a function of stresses. Notice that the initial displacementstiffness is not included in [KT] because its effects are already eliminated in the element formulation.

Newton's procedure was previously implemented using a corrective force. Recalling that the elementforces for linear elements are expressed as

{F} = [K] {u},

a corrective force vector may be defined as

{C} = {F} – [K] {u}

Equation 12-5.

The iteration starts with initial values

{u0} = last converged displacement;

{R0} = P – F(u0) = {ΔP} + {Re};

where {ΔP} is an incremental load vector and {Re} is a residual load error carried over from the lastconverged solution. Then the successive error vectors can be evaluated by

Ri+1 = Ri – K(ui+1 – ui) – (Ci+1 – Ci).

Equation 12-6.

It is noted that the corrective force vector vanishes for linear elements. The corrective forcecalculation has been removed, and the error vector is computed directly from the internal forces, i.e.,


Solution methods


{Ri+1} = {P} – {Fi}

Equation 12-7.

Then the residual load error is automatically carried over to the next incremental process.

The merit of the Newton-Raphson method is the quadratic rate of convergence, i.e.,

||u* – ui+1|| ≤ q||u* – ui||2

Equation 12-8.

where u* is a true value of {u}, q is a constant, and 1111 represents a vector norm. From a practicalstandpoint, however, determination of the tangential stiffness and its inverse at each iteration entailsa considerable amount of computation. As Figure 12-1 suggests, one may resort to the modifiedNewton's method which requires the tangential stiffness to be evaluated just once at the initialposition, {uo}, and used thereafter to solve for {Δui}. However, more iterations are required for a givenaccuracy by the modified Newton's method. The Gaussian elimination method is better suited for thisapproach than the iterative descent method because the decomposition is performed only once.



Solution methods

Figure 12-1. Newton’s Methods for Iteration


Solution methods


12.4 Stiffness update strategiesAmong other features of the solution algorithm, the stiffness matrix update has probably the mostprofound effect on the success of the nonlinear solution. In spite of its significance, however, it isvery difficult to implement a robust algorithm for update strategy due to the lack of a prior informationregarding the right timing for an update. In this section, stiffness update strategies are reviewed withrespect to the static analysis. Variations of the modified Newton’s method are adopted in the software.

However, the modified Newton’s method could lead to divergence when the stiffness changesdrastically, as demonstrated in the figure Newton’s Methods for Iteration, unless the tangentialstiffness is reevaluated at the critical point. To this end, an adaptive matrix update method isunavoidable. Stiffness update strategies are established to update the stiffness matrix on an asneeded bases such as probable divergence.

12.4.1 Update principles

Newton’s method could be trapped in an infinite loop, oscillating about the local maximum asillustrated in the figure Newton’s Methods for Iteration (a). This difficulty is overcome during theNewton’s iteration by discarding the differential stiffness, [Kd], when the tangential stiffness is notpositive definite as shown in (b) of that figure.

Control over the stiffness update method is achieved with the KUPDATE parameter on the NLCNTLentry:

1. If KUPDATE=1 (Full Newton method), stiffness is updated after each iteration.

2. If KUPDATE=0 (Auto method, default), stiffness is updated based on the change in the valueof current stiffness parameter since the last stiffness update.

3. If KUPDATE=-1 (Initial stiffness method), no stiffness update is made during the solution.

4. If KUPDATE=N (N>1,Quasi Newton method), if solution time step doesn’t converge in Niterations, stiffness is updated at the iterated configuration. If the parameter TSTEPK is setto YES (Default=NO) on the NLCNTL entry, then stiffness update is also performed beforebeginning a new time step.

Current stiffness parameter approach for the automatic stiffness update method (KUPDATE=0)

The current stiffness parameter approach proposed by Bergan and Crisfield [*] is used for automaticstiffness update method. The current stiffness parameter gives a scalar measure for the stiffness ofthe structure at the current loading condition.

Where, n is the time step index,

I is the teration index,

∆Pni = Pni-Pn-1i is the incremental applied load between time step n-1 and n,

∆P11 = P11 - P01 is the initial applied load (for the first iteration of the first time step),

P11 is the applied load in the first iteration of the first time step of the first subcase,



Solution methods

∆Uni = Uni - Un-1i is the incremental displacement between time step n-1 and n,

∆U11 = U11U01 is the initial displacement increment (after the first iteration of first step).

After the first iteration of the first step, S11=1, that is, the initial value of the current stiffness parameteris one. After each iteration, the value of current stiffness parameter is recomputed. The change incurrent stiffness parameter is computed for each iteration as:

Where, Sref is the reference value for current stiffness parameter. At the start of the solution, Sref isset to 1.0. Stiffness is updated if ∆S≥α, where α=5.0 for a problem with structural loading only, α=20.0for a problem with pure thermal load, and α=10.0 for a problem with a combination of structural andthermal load. The value for α can be defined by the user on the NLCNTL card through parameterCSTFPAR. Valid input for CSTFPAR is a real number.

Sref is updated upon updating stiffness to correspond to the value of Sni.

1. Bergan. P, Horrigmoe. G, Krakeland . B, and Soreide T., SOLUTION TECHNIQUES FORNON-LINEAR FINITE ELEMENT PROBLEMS, International Journal for Numerical Methodsin Engineering, Vol. 12, 1677-1696 (1978)

2. Crisfield M. A. , Non-linear Finite Element Analysis of Solids and Structures, Volume 1:Essentials, John Wiley & Sons, Chichester, 1991

12.4.2 Divergence criteria

The MAXDIV parameter in the NLCNTL entry requires an integer to specify a limit on the probabledivergence conditions allowed for each time step to continue. In each iteration of a time step, thefollowing three quantities are computed:

Equation 12-9.

Equation 12-10.

Equation 12-11.

where,

Rni-1=Pni-1-Fni-1 (Uni-2) is the residual for iteration i-1 for time step n,


Solution methods


ΔUni=K-1 Rni-1 is the displacement increment computed in iteration i-1 for time step n,

Rni=Pni-Fni (Uni-1) is the residual for iteration i for time step n,

εpi is the computed error for the force norm for iteration i for time step n,

εpi-1 is the computed error for the force norm for iteration i-1 for time step n,

εwi is the computed error for the energy norm for iteration i for time step n,

εwi-1 is the computed error for the energy norm for iteration i-1 for time step n.

For the first iteration of every new time step, the value for λpi and λwi is set to 0.9999. The variableNDIV is initiated to 0 in the first iteration of each time step. NDIV is incremented if:

1. If Er>1 or Er<-100, NDIV=NDIV+1

2. If λwi>1.00 and λpi>1.00, NDIV=NDIV+1

NDIV is reset to 0 if neither of the two conditions above are met in any given iteration. The solutionis considered to have diverged if NDIV>MAXDIV. MAXDIV is a NLCNTL parameter which controlsmaximum allowable divergences in a time step (Default=3).

Upon divergence, the software will:

1. Revert the solution to the last converged point.

2. Reform stiffness at the last converged configuration (unless KUPDATE=-1, in that case nostiffness update is performed).

3. Perform bisection, and attempt to solve the time step with a reduced size from the last convergedpoint.

For a timestep of size Δt, on bisection the time step is reduced to:

Δt1=1/2 Δt

Bisection continues until the solution converges, that is,

Δtk = 1/2 Δtk-1 = 1/2k Δt

where k is a bisection count. Once the bisection is successful (rendering a converged solution),the integration proceeds to the next time step. If k = 1, the same time step size is used for thenext time step.

If k > 1, an effort is made to accelerate the solution process by increasing the time step size forthe next step. The time step size for the next time step is influenced by all of the following factors:

a. The number of iterations (i) that were required to reach convergence with time step size of Δtk.

b. Number of bisections (k) performed.

c. Remaining time left (δt) to complete the original time step of size Δt.

The time step size used for the next time step is m Δtk, where 1 ≤ m ≤ k – 1. The largestvalue of m that satisfied the following conditions is used:

A. I ≤ 3 or I ≤ MAXITR/2m, where MAXITR is the maximum permissible iterations for a giventimestep.



Solution methods

B. δt/m Δtk is an integer greater than 0.

Although the goal of increasing time step size is to reduce the number of time steps tillcompletion of solution, for some problems, further bisection may be required in subsequentsteps.

The maximum number of bisections is limited by the parameter MAXBIS (default=5). Thebisection process is activated on an as-needed basis. You also have an option to suppressbisection by specifying MAXBIS=0.

d. If the solution fails to converge even after attempting maximum permissible bisections, thesolution is terminated with a fatal message. Results corresponding to the last converged userrequested output time are printed.

12.5 Convergence criteriaThe convergence test is an important factor that affects accuracy and overall efficiency in nonlinearfinite element analysis. Out-of-balance forces and changes in displacements should vanish uponconvergence in an iterative process. The energy error accommodates both quantities and is usuallyadequate for most problems. However, the displacements could be in gross error while the residualload error is negligible, or vice versa.

In order to ensure accurate and consistent convergence, multiple criteria with errors measured interms of displacements, loads, and energy should be combined. It is the error function and theconvergence tolerance that characterize the criteria. Error functions are formulated using theweighted normalization so that the error measures are dimensionless. Tolerances should be realisticfor the solution scheme to be efficient. In this context, variations are considered in search of the bestworkable combinations of error functions and tolerances for a wide class of structural problems.

12.5.1 Rudimentary considerations

The convergence test is a decision-making process, on which termination of the iterative process isbased, while the true solution is not known. The convergence criteria are extremely important for theincremental/iterative solution strategy to be effective and efficient, because improper criteria couldcause inefficiency as well as inaccuracy. It is rather astounding to:find a scarcity of publications onthis subject, considering the significant impact of the convergence criteria on the accuracy and theefficiency of the computation. Two distinct aspects are involved in the convergence criteria:

• Error functions to be minimized by the iteration.

• Tolerances of error functions within which errors are acceptable.

Both aspects must be defined properly for the criteria to be effective, for the solution scheme tobe efficient, and for the solution to be accurate.

There are no universally accepted convergence criteria to date in the field of finite element analysis.Conditions to be met by ideal convergence criteria for a general-purpose finite element analysis havebeen contemplated. The convergence criteria should:

• be satisfied for linear cases at all times.

• be independent of structural units.


Solution methods


• be reliable (cancellation of errors are not acceptable).

• render consistent accuracy.

• be independent of structural characteristics (stiffening or softening).

• be able to handle all the loading cases including constant loading, unloading, and no externalloading (applicable to creep analysis).

• have smooth transitions after the stiffness updates and loading changes.

These conditions dictate the formulation of error functions to be discussed.

12.5.2 Convergence conditions

The iteration continues until the convergence is attained by satisfying the convergence criteria andthe residual error vector at convergence is carried over to the next incremental step. When theconvergence criteria are satisfied, the out-of-balance forces and the changes in displacements shouldbe sufficiently small so that the remaining error is not physically significant nor will it cause anydetrimental effects, numerically or physically, on the succeeding incremental steps. Convergencetolerances have the following effects:

• excessively tight tolerances cause a waste of computing resources for unnecessary accuracy.

• excessively loose tolerances cause not only inaccuracy but convergence difficulties in thesubsequent steps due to cumulative errors.

The fundamental difficulty of the convergence tests for a structural analysis lies in the fact thatthe base vectors (forces and displacements) involve inconsistent units, namely, combinations offorces and moments or translations and rotations. Indiscriminate use of these vectors will causeunit-dependent convergence criteria. For example, while an error in forces is dominant when themodel is expressed in newton-meter, the error would be dominated by moments if the same model isdescribed in newton-millimeter.

The most natural and reasonable criterion for the convergence test is formulated in terms of anenergy error. The energy error is the logical choice because both the out-of-balance forces {R}and the change in displacements {Δu} should be minimized by the iteration process. Furthermore,energy quantities do not pose problems of inconsistent units due to mixed units associated withtranslations and rotations.

Although the convergence test in terms of energy errors is usually adequate, some distinct errorsare not detected with this criterion; i.e., displacements are in gross error while the residual loaderror is negligible, or vice versa. This would be the case if the degrees-of-freedom in error have avery small or a very large stiffness. Such cases compel the need for criteria in terms of loads anddisplacements. Nominally, by visualizing the load-deflection curve for a one-dimensional case, it canbe noticed that the convergence criterion in terms of loads governs the stiffening structure and thecriterion in terms of displacements governs the softening structure. Scalar error functions for thesecriteria are formulated to be dimensionless by introducing the weighted normalization.



Solution methods

12.5.3 Error functions and weighted normalization

Error functions are defined for errors in displacement, residual force and energy criterion. Fordisplacement and residual force error criterion, L2 vector norms are used. The L2 norm is indicatedby ||V||2, where V represents a vector (for example, ΔU). In solution 401, error criterion correspondingto displacement, force and energy (work) are computed in every iteration. Error in energy (work) isused as the default convergence criterion. These criteria are computed as follows:

• EPSU (displacement criterion)

Equation 12-12.where,

Uni is the displacement vector for iteration i of time step n,

Uni-1 is the displacement vector for iteration i-1 of time step n, and

Un-1 is the displacement vector at convergence for time step n-1.

The displacement error computation can be summarized as the ratio of L2 norm of theincremental displacement in the current iteration and the L2 norm of the incremental displacementin the current time step. The error in displacement in the first iteration for every time step shouldbe 1.000.

• EPSP (force criterion)

where,

Pni is the external force vector for iteration i of time step n,

Fni is the internal force vector for iteration i of time step n,

Rni is the residual force vector for iteration i of time step n, and

Rn1 is the residual force vector for iteration 1 of time step n.

Equation 12-13.

• EPSW (work criterion)

Equation 12-14.Where,


Solution methods


ΔUniis the incremental displacement vector computed for iteration i of time step n,

Rni is the residual force vector for iteration i of time step n,

Uni is the total displacement vector for iteration i of time step n,

Pni is the external force vector for iteration i of time step n,

Fni is the internal force vector for iteration i of time step n,

Rn1 is the residual force vector for iteration 1 of time step n, and

Un1 is the total displacement vector for iteration 1 of time step n.

Default tolerances for these error criterion are:

EPSU (Displacement): –1.0E-2

EPSP (Force): –1.0E-2

EPSW (Work): –1.0E-6

You can specify custom convergence tolerances on the NLCNTL bulk entry. The solution isconsidered to have converged if the computed error criteria are less than the tolerance specified.

12.5.4 Implementation

The convergence tolerance determines the efficiency of the solution scheme as well as the accuracyof the solution. The tolerance should be realistic, not too tight nor too loose. It is difficult to chooseoptimal default values for the convergence tolerances. However, efforts have been made to set thedefault values to provide reliable solutions to the general class of problems. Thus, default tolerancesshould be adhered to until good reasons are found to change them.

The following three error functions (in terms of displacements, loads, and energy) are computedand compared to tolerances.

Eu < EPSU (=10-2 by default)

Ep < EPSP (=10-2 by default)

Ew < EPSW (=10-6 by default)

where EPSU, EPSP, and EPSW are tolerances specified in the NLCNTL entry. However, onlythose criteria chosen by the user (combinations of U,P, and/or W) are designed to be satisfied forconvergence.

It is noted that divergence conditions are established independent of convergence criteria.



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