mechanical-nonlin 13.0 ch01 overview

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Customer Training Material

Lecture 1

Overview

Structural Nonlinearities

L1-1 ANSYS, Inc. Proprietary

© 2010 ANSYS, Inc. All rights reserved.Release 13.0

December 2010



ANSYS Mechanical Structural Nonlinearities

Customer Training Material Chapter Overview

• In this chapter, an overview of the basics of nonlinear finite-element

analysis (FEA) is presented:

A. What is “Nonlinear” Behavior?

B. Types of Nonlinearities

C. Nonlinear solution using linear solvers

D. Nonlinear FEA issues

• The purpose is to give you an understanding of the fundamental

nature of nonlinear FEA.

• The capabilities described in this section are generally applicable to

Structural licenses and above.



December 2010




Customer Training Material A. What is nonlinear behavior?

• Recall, in the 1600s, Robert Hooke discovered a simple linear

relationship between force (F) and displacement (u), known as

Hooke’s Law:

F = Ku

– .

• A linear structure obeys this linear relationship. A common

example is a simple spring:

F

u K

F

u



December 2010

- - ,

which is based on linear matrix algebra.




Customer Training Material ... What is nonlinear behavior?

• Significant classes of structures do not have a linear relationship

between force and displacement.

• Because a plot of F versus u for such structures is not a straight line,

such structures are said to be nonlinear.

– ,

load, KT (the tangent stiffness).

FKT



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u




Customer Training Material ... What is nonlinear behavior?

• A structure is nonlinear if the loading causes significant changes in

stiffness. Typical reasons for stiffness change are:

–

– Large deflections, such as a loaded fishing rod

– ,



December 2010




Customer Training Material B. Types of Nonlinearities

• There are three main sources of nonlinearities:

– Geometric nonlinearities: If a structure

experiences large deformations, its

changing geometric configuration can

cause nonlinear behavior.

– Material nonlinearities: A nonlinear stress-strain

relationship, such as metal plasticity shown on

the ri ht, is another source of nonlinearities.

– Contact: A “changing status” nonlinearity, where

bodies come into or out of contact with each

other.



December 2010




Customer Training Material … Types of Nonlinearities

• Of course, all three types of nonlinearities are commonly

encountered in combination.

– Workbench Mechanical can readil handle combined nonlinear effects.

Rubber Boot Seal

An example of

nonlinear geometry

lar e strain and lar e

deformation),

nonlinear material

(rubber), and

changing statusnonlinearities

(contact).



December 2010




Customer Training Material B. Nonlinear solution using linear solvers

• How does Workbench Mechanical solve for a changing stiffness?

– In a nonlinear analysis, the response cannot be predicted directly with a

set of linear equations.

– However, a nonlinear structure can be analyzed using an iterative series

of linear approximations, with corrections.

– ANSYS uses an iterative process called the Newton-Raphson Method .

Each iteration is known as an equilibrium iteration.

Load

F

34 A full Newton-Raphson

iterative analysis for one

1

increment of load. (Fouriterations are shown.)



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u Displacement




Customer Training Material … Nonlinear solution using linear solvers

• The actual relationship between load and displacement (shown

with a blue dotted line) is not known beforehand.

• onsequen y, a ser es o near approx ma ons w correc ons s

performed. This is a simplified explanation of the Newton-Raphsonmethod (shown as solid red lines)

– In the Newton-Raphson Method, the total

load F a is applied in iteration 1. The result

is x 1. From the displacements, the internalF

a4

Newton-Raphson Method

orces 1 can e ca cu a e . a 1, en

the system is not in equilibrium. Hence,

a new stiffness matrix (slope of red line) is

calculated based on the current conditions. 1

2F1

The difference of Fa - F1 is the out-of-balanceor residual forces. The residual forces must

be ‘small’ enough for the solution to converge. xx1



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– This process is repeated until Fa = Fi. In this example, after iteration 4, the

system achieves equilibrium and the solution is said to be converged.




Customer Training Material … Nonlinear solution using linear solvers

• The difference between external and internal loads, {Fa} - {Fnr }, is

called the residual . It is a measure of the force imbalance in the

structure.

• The goal is to iterate until the residual becomes acceptablysmall; that is, until the solution is converged .

• When convergence is achieved, the solution is in equilibrium,

within an acceptable tolerance.

Fa

{Fa} {Fnr }Fnr



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u




Customer Training Material ... Nonlinear solution using linear solvers

The Newton-Raphson method:

• Is not guaranteed to converge in all cases!

• converge on y e s ar ng con gura on s ns e e

radius of convergence.

Load

Diverging!

Load

Converged

F F

u Displacementustartu Displacementustart



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radius of convergencear ng ns e e ra us

of convergence





• Two techniques can help you obtain a converged solution:

F

F1

uustart

Apply load incrementally to move

the tar et closer to the start

uustart

Use convergence-enhancement

convergence

• Workbench Mechanical combines both strategies to obtain



December 2010

convergence.





• As a general rule, sudden changes to any aspect of a system will cause

convergence difficulties.

• With this in mind, it is useful to understand how loads are managed

– Load steps differentiate changes in general loading.

• In the Figure at the bottom right, F a and F b are loadsteps.

– Subste s a l the loads in an incremental fashion

• Because of the complex response, it

may be necessary to apply the load

incrementally. For example, F a1 may be

Fb

Fb2

near o e a oa . er e oa

for F a1 is converged, then the full F a load

is applied. F a has 2 substeps while F bhas 3 substeps in this example

Fa

Fa1

– Equilibrium iterations are the corrective

solutions to obtain a converged substep

• In the example on right, the iterations between the dotted lines indicate equilibrium

xa xb



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iterations.




Customer Training Material C. Nonlinear FEA Issues

• Three main issues arise whenever you do a nonlinear finite element

analysis:

– Obtaining convergence

– Balancing expense versus accuracy

– Verification

It takes care and skill

to juggle these three

issues successfully!



December 2010




Customer Training Material … Nonlinear FEA Issues

Obtaining convergence…

• Usually your biggest challenge.

• Solution must start within the radius of convergence.

– The radius of convergence is unknown!

• so u on converges, e s ar was w n e ra us.

• If solution fails to converge, the start was outside the radius.

– Trial-and-error is sometimes required.

– Experience and training reduce your trial-and-error effort.

• Difficult problems might require many load increments, and many

iterations at each load increment, to reach convergence.

– When many iterations are required, the overall solution time increases.



December 2010





Balancing expense versus accuracy…

• All FEA involves a trade-off between expense (elapsed time, disk and

memor re uirements and accurac .

• More detail and a finer mesh generally lead to a more accuratesolution, but require more time and system resources.

• on near ana yses a an ex ra ac or, e num er o oa

increments, which affects both accuracy and expense.

– More load increments will often improve the accuracy, but will alsogenerally increase the expense.

• Other nonlinear parameters, such as contact stiffness (discussed

later), can also affect both accuracy and expense.

• Use your own engineering judgment to determine how much

accuracy you need, how much expense you can afford.



December 2010





Verification…

• In a nonlinear analysis, as in any finite-element analysis, you must

verify your results.

• Due to the increased complexity of nonlinear behavior, nonlinear

results are generally more difficult to verify.

• Sensitivit studies increasin mesh densit decreasin load

increment, varying other model parameters) become more expensive.

Stress

Mesh Density

Typical Sensitivity Study



December 2010

• Later chapters will provide case-by-case details of modeling tips for

different nonlinear situations.

mechanical-nonlin 13.0 ch01 overview

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