Obtaining the Solution Chapter Three. Training Manual October 15, 2001 Inventory # 001565 3-2 3. Obtaining the Solution Characteristics of a nonlinear

Obtaining the Solution

Chapter Three

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Basic Structural Nonlinearities 6.0


Training Manual3. Obtaining the Solution

Characteristics of a nonlinear solution:

• The hard work and expense of performing a nonlinear analysis occurs primarily during solution.– During the solution setup, you make the decisions that most

significantly affect the convergence behavior.• Overcoming convergence difficulties can require

considerable amounts of engineering time.– Most of the expense (CPU time) required in a nonlinear

analysis is consumed in solution.

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Training Manual… Obtaining the Solution

• In this chapter, we will introduce the procedures for nonlinear solution via the following topics:A. Basic concepts B. Automatic solution controlC. Results file optionsD. Solution optionsE. Nonlinear optionsF. Advanced nonlinear optionsG. Transient options

• The purpose is to give you an understanding of how to obtain converged solutions through use of the options that are available in the Solution Controls dialogue box. Other, less commonly used options are discussed in the Advanced Structural Nonlinearities Training Manual.

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Obtaining the Solution A. Basic concepts

• Before discussing the details of how to obtain a nonlinear solution, we will present some basic concepts of how a nonlinear solution is organized and managed.– Basic organization: Load steps, substeps, and equilibrium

iterations.– Basic management: Time and time-step controls.

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Obtaining the Solution ... Basic concepts

• A nonlinear solution is organized into three levels of operation:– Load steps are top-level, user-defined load changes.

“Constant value” loads vary linearly within a load step.– Substeps are program-defined load increments within a load

step.– Equilibrium iterations are

corrective solutions to obtain convergence within a substep.

Two Load Step Solution“Time”

Load

Load Step 2

Load Step 1

Substeps

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• ANSYS linearly interpolates loads for all substeps within a load step.

• For simple “constant value” loads you must use multiple load steps to define the load history.

“Time”t1 t2

Load

t3 t4

L1L2

L3

L4

LS1LS2 LS3 LS4

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Obtaining the Solution ... Basic concepts• Understand how ANSYS manages load history for an analysis with

multiple load steps:

Load

t2 “Time”t1

Load

t1 t2 “Time”

Newly applied loads are ramped from zero at the start of the load step to their full value at the end of the load step.

Loads which are unchanged will retain their values for the next load step.

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Load

“Time”t1 t2

When a load is redefined, its value is ramped (by default) from the value it had at the end of the previous load step.

When a load is deleted, the effect is a step change to zero. This is generally not advised, because it usually causes a convergence failure. Better modeling practice is to ramp the load to zero over a small time increment.

Load

“Time”t1 t2

Deleted

Reapplied

Load history management (cont’d):

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• Every load step and substep is associated with a unique value of “time”.– Substeps are therefore also known as time steps.

• “Time” is used as a tracking parameter in all static and transient analyses, whether or not they are truly time-dependent.– Therefore, for rate-independent static analyses, “time” can be in

terms of any arbitrary units.

“Time” is a tracking parameter

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• “Time” has the following characteristics:– A “time” value is specified

for the end of each load step.Solution > Sol’n Control

– Each substep is associated with a unique value of “time”.

– “Time” must be positive and nonzero.

– “Time” is always monotonically increasing. (The clock never stops ticking!)

“Time”

Load

10.0 23.05.7 14.6 18.2

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Obtaining the Solution... Basic conceptsCharacteristics of “Time” (cont’d)

– For rate-dependent or transient analyses, “time” must have consistent, chronological units.• Typically seconds for transient dynamic; hours for creep.

– For a rate-independent static analysis, you can define “time” in terms of any arbitrary units.• Hint: You can set “time” equal to the applied load, for

convenience. (If load is negative, use the absolute value.)• For example, if you apply a force of -14,500 in load step 1,

specify the “time” at the end of load step 1 to be 14,500.• This can make output and results interpretation easier.

– If you do not specify any values for Time at end of loadstep, the “time” at the end of each load step will default to the load step number. • “Time” = 1.0 at end of load step 1, etc.

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• The time increment, t, controls the load increment, F.F = t*(F2 - F1)/(t2 - t1)

• The time increment can be specified by the user or automatically predicted and controlled by ANSYS.

• The automatic time stepping algorithm predicts and controls the time increment (load increment) for all substeps within a load step.

F

Time

Load

F1

F2

tt1 t2

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Obtaining the Solution ... Basic concepts• Recall that breaking the load into

increments improves convergence by bringing the start point within the radius of convergence. F

uustart

F1• Auto time stepping automatically adjusts the load increment sizes (up and down) throughout the solution.– Smaller increments when convergence

is difficult, larger increments when convergence is easy.

• You control the range of adjustment by specifying a starting, minimum, and maximum time increment.

Time

Load

tmaxtstart tmin

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• “Time” and time step controls are just two of the available nonlinear solution controls.

• Other controls do the following:– Account for geometric nonlinearities.– Manage the large volume of data that is generated in a

nonlinear solution. – Specify which equation solver to use.– Set up restart controls.– Define convergence tolerances.– Control the number of equilibrium equations.– Enhance solution convergence.– Control program behavior in the event of non-convergence.

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• The number of available nonlinear solution controls can seem overwhelming at first.

• Fortunately, for the majority of applications, a fairly simple procedure will suffice ….

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Obtaining the SolutionB. Automatic solution control

• Automatic solution control is active by default.– Provides comprehensive, automatic, and intelligent nonlinear

tool settings to obtain robust and efficient converged solutions.– The recommended procedure is first to attempt a solution using

automatic controls.– If it converges in reasonable time, no trial-and-error is

needed!• Eliminates trial-and-error approach for about 70% of all

user nonlinear applications.– If convergence is slow or fails, then you can adjust

various solution options to enhance convergence.

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Obtaining the Solution... Automatic solution control

Recommended procedure:

• Apply all loads and boundary conditions.

• Go to the Basic tab in the Solution Controls dialogue box: Solution > Sol’n Control (Basic tab will be on top.)

• First, specify Large Displacement analysis, with the appropriate analysis type (Static or Transient).

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Obtaining the Solution... Automatic solution control• Next, specify a starting, minimum, and maximum time

increment.

• This is often specified using the Number of substeps, which is inversely related to the time increment:tstart = (tend - tbegin)/Nstart

tmin = (tend - tbegin)/Nmax

tmax = (tend - tbegin)/Nmin

Nstart

Nmax

Nmin

• Remember that for a rate-independent analysis, Time at end of loadstep can default to the load step number, or you can specify it to equal the load.

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• What values should you specify for Nstart, Nmax, and Nmin?– The “best” values are highly problem-dependent. – If you have no idea, try the default settings. Note that relying on

all-default settings will give you a warning:

– The default values vary with the physics of the model. The actual values used are echoed to the output:

– The defaults tend to favor robustness (ease of convergence) over efficiency. In the case shown above, the time increment can become as small as 1/200,000 of the total time of the load step.

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• Then Solve. Let automatic solution control define all the other tool settings.

• As the solution proceeds, you will see a graphical display of the iteration history.– The residual (force

imbalance) appears as a purple line.

– The force convergence criterion appears as a blue line.

– Whenever the residual dips below the criterion, a substep has converged, and the next load increment is applied. Convergence

F}

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• Three important “feedback” files are created during solution.– The error file (jobname.err).– The monitor file (jobname.mntr).– The output file (jobname.out).

• In an interactive run on a PC, the output is not written to a file. Instead it simply scrolls by in the Output Window.

• On a PC, you can capture the output on a file using Utility Menu>File>Switch Output to>File

• You should always review the contents of these files.– As part of verification.– To learn how to improve convergence behavior.

• We will introduce you to the contents of these files in the next workshop.

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• You might sometimes decide, based on the graphical solution tracking plot or the feedback files, that you want to stop the solution prematurely.

• There are two ways to terminate a solution:– If running interactively, hit the STOP button.– This will terminate the solution at the end of the current

iteration.

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Terminating a solution (cont’d):– If running in batch mode, create an “abort” file, jobname.abt.

• This must be an ASCII file in your working directory.• Contains the word nonlinear on the first line.• The program searches for this file at the start of every

iteration.– If it exists, the solution aborts before starting the next

iteration.

• You will be able to restart following termination due to STOP or jobname.abt.

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Obtaining the Solution B. Automatic solution control …WorkshopPlease refer to your Workshop Supplement for instructions on:

• W2. Automatic Solution Control - Fishing Rod

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What if automatic solution control doesn’t work for your model?

• Although automatic solution control is expected to work well for about 70% of all user applications, there will still be models for which it does not give a converged, efficient solution.

• In these cases, hands-on control is needed. The most commonly used controls are all accessible under the various tabs of the Solution Control dialogue box.

• The remainder of this chapter will discuss most of the solution controls available in the Solution Control dialogue box: Solution > Sol’n Control

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• The time step size is the most important solution parameter affecting robustness, accuracy, and efficiency.– In general, a smaller step size has the following effects:

• Improves convergence.• Improves accuracy.• Reduces the number of equilibrium iterations per substep.• But – often degrades the overall solution efficiency.

• In most cases, modify the step size controls before changing any other controls.– Remember, increasing the number of substeps is the same as

decreasing the time step size:

t = (tend - tbegin)/N

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Obtaining the SolutionC. Results file options

• By default, All solution items will be written to the .rst file for only the last substep.

• You can change the type of results data, and the substeps for which the data is written.– Write less data by choosing

a subset of results items.– Write more substeps to

allow animation or results history plots.

• These controls are found on the Basic tab of the Solution Controls dialogue box.

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Obtaining the Solution C. Results file options … Workshop

Please refer to your Workshop Supplement for instructions on:

• W3. Results File Options - Fishing Rod (part 2)

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Obtaining the Solution D. Solution options

• The Sol’n Options tab of the Solution Controls dialogue box allows you to:– Change the equation solver.– Set up the restart options.

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Obtaining the Solution ... Solution options – Solvers

• For a nonlinear analysis, there are three choices for an equation solver– Sparse

• Relatively robust and efficient– Preconditioned Conjugate Gradient

• Most efficient, but less robust– Frontal

• Most robust, but least efficient

• By default, the program will choose a solver appropriate for your model’s characteristics. The choice is reported in the output.

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Obtaining the Solution ... Solution options – Solvers

Other factors affecting solver choice• Sparse solver:

– Default choice for most problems– Superior to PCG for ill-conditioned matrices (number of iterations

to convergence in file.PCS over 1,000)– If unsymmetric matrices (contact with friction) are present.– Uses parallel processing (for all major platforms in 6.0 and later)

• PCG solver:– Works best with fine meshes, 3D models and solid elements –

SOLID185,186,187,95,92, and 45s– Superior to sparse solver if convergence is fast (number of

iterations to convergence in file.PCS in hundreds)• Frontal solver:

– Only for small problems (< 50,000 DOF)• Original direct solver in ANSYS

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Obtaining the Solution ... Solution options – Restart

Restarting an analysis:

• A restart allows you to modify and continue your analysis, without having to rerun it from the beginning.

• Reasons why you might restart your analysis:– Add more load steps to a completed analysis.– Recover from a convergence failure.– Investigate a different loading sequence.– Write results for any particular substep.

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• The program uses the following files in a restart:– Restart database file: jobname.rdb

• Written at the start of the 1st substep of the 1st load step.– Load history file: jobname.ldhi

• Updated for every load step.– Restart history variables files: jobname.rNNN (.r001, .r002, etc.)

• Contains element saved records, solution commands, and status for a single substep.

• By default, one .rNNN file is written for the last substep of the last load step, or for the last converged substep.

• You can specify the quantity and frequency of writing .rNNN files (will also write the last converged substep).– Frequency options:

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• For example, write .rNNN every 3rd substep of each load step, with no limit on the total number of .rNNN files:

No. of files (0 = no limit)Every 3rd substep (of each load step)

SubstepsRestart points

r001

Load

Time

r002

r004 r005 (last converged)

LS1 LS2

r003

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• To restart: Solution > Restart– Brings up a list of available restart points, and the Restart dialog

box.

Specify the desired restart point

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• At this point, the program automatically:– Resumes the .rdb file.

• Jobname.rdb written at the start of 1st substep of 1st load step.

• You will need to re-create any data such as components, APDL parameters, etc. that were created after the 1st substep of the 1st load step.

– Reads the .ldhi file, and gets the correct boundary conditions, time, etc. corresponding to the requested load step and substep.

– Rebuilds the solution commands and status.

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• Then, issue Solve. The solution will restart from the specified load step and substep.– All existing .rNNN files for substeps after the specified restart

point will be deleted.– All existing results sets for substeps after the specified restart

point will be deleted.

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• You can also restart to write out complete results to the .rst file at any available restart point.– Writing All solution items for every substep can create a

huge .rst file!– Selectively writing results only as needed can save on disk

space.


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• Procedure to write results set at restart point:– Solution > Restart– Specify desired load step and substep; use Create .rst file.

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Procedure (cont’d):– Also, change the results file options as

needed.– Then Solve.

• Creates results set at the specified restart point.– Results become available even if

not previously written due to prior results file options settings.

• Does not affect any existing .rNNN files.

• All items stored in the .RST file at and beyond the restart point will be deleted (items before the restart point are not affected).

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Obtaining the Solution D. Solution options ... Workshop


• W4. Solution Options - Tensile Bar

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Obtaining the Solution E. Nonlinear options

• The options on the Nonlinear tab allow you to control various tools that directly affect the convergence behavior:– Convergence tolerance

criteria.– Maximum number of

equilibrium iterations.– Convergence-

enhancement tools.– Bisection (cutback)

criteria.– Creep strain rate option.

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Obtaining the Solution ... Nonlinear opt’ns – Convergence criteria

• Recall that the Newton-Raphson method iterates to a converged solution using the equation [KT]{u} = {F} - {Fnr}– The program solves this equation repeatedly until the residual

(force imbalance), {F} - {Fnr}, becomes acceptably small.

• The largest acceptable value for the residual is called the force convergence criterion.

F

u

The solution is converged when Residual < Criterion

Crit

erio

n

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• Expressed mathematically:If: ||{R}|| < (R Rref)Then: The solution is converged.

• Where – ||{R}|| is a vector norm of the residual (a norm is an operator that

reduces a vector to a single scalar value).• L1 norm of the residual: ||{R}||1 = |Ri|• L2 (SRSS) norm of the residual: ||{R}||2 = (R2

i)1/2

• Infinite norm of the residual: ||{R}|| = max(|Ri|)– (R Rref) is the Force convergence criterion

• R is a tolerance factor and Rref is a reference force value– Rref can be the norm of all applied forces and reactions, ||{F}||

(automatically scales the criterion to the magnitude of load)

Default

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• The default convergence criterion works very well for most engineering applications:

||{R}||2 < (0.5% * ||{F}||2)• For special situations, you can change the criteria.

– You can tighten or loosen the convergence criterion.• Tighter criterion gives better accuracy, but more difficult

convergence.– You can also choose other item(s) to check for convergence.

• Force, Moment, Displacement, and Rotation criteria are available.

– In addition, you can change the norm that is used to measure the convergence items.• L1, L2, or Infinite norms.

• Note: If you change any convergence criteria, the program deletes all the default criteria!

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• To change the convergence criteria:

• The first time that you change criteria, a menu list of default values appears. (The defaults shown are based on auto solution controls being turned off.)

• Note: The action button is Replace. Remember, if you change any convergence criteria, the program deletes all the default criteria!

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• Clicking Replace brings up this dialogue box:

• You can add a displacement checking criterion:||{ui}|| < (u uref)

• You must also re-establish a force checking criterion.

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Why must you re-establish a force convergence criterion?

Relying on displacement convergence alone can in some cases lead to erroneous results.

Big

Res

idua

l

• Because displacement-based checking is a relative measure of convergence, it should only be used as a supplement to force-based convergence.

• Force-based convergence provides an absolute measure of convergence, as it is a measure of equilibrium between the internal and external forces.

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• If you do not specify a value for Minimum reference value, you will get a warning:

• The Minimum reference value (MINREF) is a safety feature that prevents your solution from trying to converge to a zero tolerance.– If free-body (unconstrained) systems or mechanisms have no external

forces, the criterion (R * ||{F}||2) will be zero.• If the criterion is zero, the solution will never converge!

– In such cases, the program redefines the criterion to be (R * MINREF)

– The default value of MINREF is 0.01. If this is nonsensical for your model, specify a more realistic value.

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Convergence criteria guidelines:• Default convergence criteria work well most of the time.

– You should rarely need to change the criteria.• To tighten or loosen a criterion, don’t change the default

reference value, but instead change the tolerance factor by one or two orders of magnitude.

• Do not use a “loose” criterion to eliminate convergence difficulties.– This simply allows the solution to “converge” to an incorrect

result!• Tightening the criterion requires more equilibrium iterations.• Review any MINREF warning messages during solution.

Make sure the minimum reference value used makes sense for the problem being solved.

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Obtaining the Solution E. Convergence criteria … Workshop


• W5. Convergence Criteria - Truss Structure

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Obtaining the Solution ... Nonlinear opt’ns – No. of eq. Iters.

• If a solution doesn’t converge in a reasonable number of equilibrium iterations, the program will bisect the load increment and solve again.

• The default (SOLCONTROL ,ON) is to let the program choose between 15 and 26 iterations depending on the physics of the problem. With SOLCONTROL,Off; default is 25 for all cases.

• You can specify the number of equilibrium iterations.– Useful if the analysis shows a convergent trend but needs more

iterations to achieve convergence.

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Obtaining the Solution ... Nonlinear opt’ns – Convergence tools

• Two convergence enhancement tools are available:– Line search– Predictor

• These tools enlarge the radius of convergence, making convergence behavior more robust and efficient.

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Obtaining the Solution ... Nonlinear opt’ns – Line search

• By default, the program turns line search on when contact elements are present.

• You can override the default to turn it on or off explicitly.

• When active, line search multiplies the displacement increment by a program-calculated scale factor between 0 and 1, whenever a stiffening response is detected.

• Line search will not destabilize a solution if activated, and in many cases will enhance a slowly converging solution.

• Line search does, however, require some additional CPU to calculate the line search parameter.

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Obtaining the Solution ... Nonlinear opt’ns – Line search

• Models that include hyperelasticity, contact, large-deflection trusses, or softening-stiffening response can often benefit from line search.

• Line search is particularly effective in overcoming oscillating convergence.

Oscillating convergence

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Obtaining the Solution E. Line search … Workshop


• W6. Line Search - Spar

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Obtaining the Solution ... Nonlinear opt’ns – Predictor

• The predictor attempts to accelerate convergence by predicting the degree of freedom solution for the first equilibrium iteration of every substep. The predictor will extrapolate the results of the last substep to obtain a starting point for the next solution.

Substep 2

Displacement

Load

Substep 1Newton-Raphson Iterations

Predictor Calculations

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Obtaining the Solution ... Nonlinear opt’ns – Predictor

• By default, the program turns the predictor on in all cases, unless rotational DOFs are present or the model contains SOLID65 elements (concrete cracking/crushing elements).

• The predictor is useful when the problem has a smooth nonlinear response. Plasticity is a good example.

• If the nonlinear response is not smooth, or large rotations are incorporated in the analysis, the predictor can cause divergence.– Turn the predictor off in such cases.

• Default predictor settings are good for most cases.

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Obtaining the Solution ... Nonlinear opt’ns – Cutback control

• If a substep fails to converge, the program will bisect the load increment and solve again.– Upon bisection, the current substep is discarded, the time step

size is reduced, and the program automatically restarts the solution.• Bisection is expensive! It is usually more efficient to set up

auto timestep controls to avoid bisection.– If the bisected load increment still fails to converge, the program

will bisect again.– If the time step size bisects down to its minimum, the solution

stops.

• The program also uses other criteria to trigger bisection. You can adjust some of these criteria using Cutback Control.

• The default cutback controls are adequate for most cases.

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Obtaining the Solution ... Nonlinear opt’ns – Cutback control• The Plastic strain ratio limits how

much equivalent plastic strain is acceptable within a single substep.

– Too much plastic strain in one substep can lead to an inaccurate solution.

– The default limit is 0.15 (15%).– A smaller limit can improve

accuracy in some cases.

• Explicit and Implicit Creep ratio will similarly limit the amount of creep strain in a single substep.

– The defaults are 0.1 (10%) for explicit and 0 (no limit) for implicit, as shown.

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Obtaining the Solution ... Nonlinear opt’ns – Cutback control• If the displacement increment

u at any DOF exceeds the Incremental displacement limit, the solution will bisect.– The default of 1e7 will usually

have no effect, because the solution terminates if any DOF response exceeds 1e6.

– A smaller value can be useful in some contact analyses, to control over-penetration and loss of contact.

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Obtaining the Solution ... Nonlinear opt’ns – Cutback control• For a transient dynamic analysis,

the program will bisect if the number of substeps in the highest-mode cycle is less than that specified for Points per cycle.

– The default is 13, using too few substeps per cycle can lead to artificial period elongation. 13 substeps has, in one test case, given about 2% cycle elongation (problem dependent).

– Specify more points per cycle for better accuracy, particularly if acceleration results are needed.

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Obtaining the Solution ... Nonlinear opt’ns – Cutback control• If the trend of iterations

indicates that the number of iterations required for convergence exceeds the maximum number allowed, the program will bisect before using all iterations.

– Can significantly improve overall efficiency.

• You can force the program to use all iterations before bisecting.

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Obtaining the Solution ... Nonlinear opt’ns – Creep option

• You can specify the implicit creep option to be on or off for a given load step.– It is off by default (Prog chosen is the same as Off ).– You might turn it off for establishing initial load over a small

starting time increment.– Then turn it on to proceed with an implicit creep analysis for

remaining load steps.

• We will study this option again in the Advanced Structural Nonlinearities training manual.

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• The options on the Advanced NL tab allow you to control what the program does if there is a convergence failure.

Obtaining the Solution F. Advanced nonlinear options

• The options are:– Do not terminate analysis.

This can be dangerous – it means that you will continue the next load step, despite nonconvergence.

– Terminate analysis and Exit. This is the default. Both the solution and your ANSYS session will stop.

– Terminate but Do Not Exit. This can be useful if you need to postprocess after a convergence failure.

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• The program can terminate for reasons other than convergence failure.

• Other options on the Advanced NL tab control other termination criteria.– Nodal DOF sol’n. By default, if any DOF response exceeds 1e6,

the solution will terminate with a DOF limit exceeded error. You can specify a different limit (typically much smaller).

– Cumulative Iter. The cumulative number of equilibrium iterations. By default, there is no limit.

Obtaining the Solution … Advanced nonlinear options

– Elapsed time and CPU time. These criteria originated in the mainframe era, when system administrators monitored account usage by the second. They may find renewed use in the new era of e-CAE ASPs. By default, there are no limits.

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• Another set of options on the Advanced NL tab, the Arc-length options, pertain to advanced postbuckling analysis, and are beyond the scope of this seminar. The arc-length method is discussed in the Advanced Structural Nonlinearities Training Manual.

Obtaining the Solution … Advanced nonlinear options

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Obtaining the Solution G. Transient options

• In a nonlinear analysis, both static and transient analyses use “time”.

• Time integration effects (transient inertial and damping effects) distinguish one analysis type from the other.

• The equations of equilibrium have these forms:Transient Dynamic: [M]{ü} + [C]{ů} + [K]{u} = {F}Static: … + … + [K]{u} = {F}

– Note that inertia and damping terms are not computed in the static equation of equilibrium.

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Obtaining the Solution ... Transient options

• These terms are activated when you choose a transient analysis:

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Obtaining the Solution … Transient options

• Once you have specified a transient analysis, the options on the Transient tab allow you to manage various tools that control transient effects.

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• You can turn transient effects off and on within a transient analysis. (Useful for establishing non-zero initial conditions.)

• You can also switch between stepped (default for transient) and ramped (default for static) loads.

“Time”

Load Ramped

Substeps Substeps

“Time”

Load Stepped

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• Rayleigh damping coefficients are specified as transient options.– Alpha damping is viscous damping, applied to the mass matrix:

• [C] = [M]• Alpha damping can cause unintended effects if used with an

artificially large mass (used sometimes for applying a known base acceleration history as an equivalent force history).

– Beta damping is applied to the stiffness matrix:• [C] = [KT]• Beta damping can cause unintended effects due to the

changing nature of [KT].

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• The amplitude decay integration parameter, GAMMA, can be useful for introducing a small amount of numerical damping to a transient analysis.– Useful in suppressing high-frequency “ringing” response.

• The other integration parameters are rarely used.

Documents

Obtaining the Solution Chapter Three. Training Manual October 15, 2001 Inventory # 001565 3-2 3. Obtaining the Solution Characteristics of a nonlinear