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8/12/2019 Mechanical-Nonlin 13.0 Ch01 Overview
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Customer Training Material
Lecture 1
Overview
Structural Nonlinearities
L1-1 ANSYS, Inc. Proprietary
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December 2010
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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.
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ANSYS Mechanical Structural Nonlinearities
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
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- - ,
which is based on linear matrix algebra.
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ANSYS Mechanical Structural Nonlinearities
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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ANSYS Mechanical Structural Nonlinearities
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
– ,
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ANSYS Mechanical Structural Nonlinearities
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.
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ANSYS Mechanical Structural Nonlinearities
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).
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ANSYS Mechanical Structural Nonlinearities
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
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ANSYS Mechanical Structural Nonlinearities
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.
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ANSYS Mechanical Structural Nonlinearities
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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ANSYS Mechanical Structural Nonlinearities
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
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ANSYS Mechanical Structural Nonlinearities
Customer Training Material ... Nonlinear solution using linear solvers
• 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
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convergence.
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ANSYS Mechanical Structural Nonlinearities
Customer Training Material ... Nonlinear solution using linear solvers
• 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.
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ANSYS Mechanical Structural Nonlinearities
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!
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ANSYS Mechanical Structural Nonlinearities
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.
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ANSYS Mechanical Structural Nonlinearities
Customer Training Material … Nonlinear FEA Issues
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.
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ANSYS Mechanical Structural Nonlinearities
Customer Training Material … Nonlinear FEA Issues
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
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• Later chapters will provide case-by-case details of modeling tips for
different nonlinear situations.