Chapter 13 Nonlinear Simulations
1
Chapter 13
Nonlinear Simulations
13.1
Basics of Nonlinear Simulations
13.2
Step-by-Step: Translational Joint
13.3
Step-by-Step: Microgripper
13.4
More Exercise: Snap Lock
13.5
Review
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
2
Section 13.1
Basics of Nonlinear Simulations
Key Concepts
• Nonlinearities
• Causes of Structural Nonlinearities
• Steps, Substeps, and Iterations
• Newton-Raphson Method
• Force/Displacement Convergence
• Solution Information
• Line Search
• Contact Types
• Contact versus Target
• Contact Formulations
• Additional Contact Settings
• Pinball Region
• Interface Treatment
• Time Step Controls
• Update Stiffness
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
3
Nonlinearities
Fo
rc
e
{F
}
Displacement {D}
Fo
rc
e
{F
}
Displacement {D}
[1] In a linear
simulation, [K]
(slope of the line)
is constant.
[2] In a nonlinear
simulation, [K] (slope
of the curve) is
changing with {D}.
• In a nonlinear simulation, the
relation between nodal force {F} and
nodal displacement {D} is nonlinear.
• we may write
K(D)⎡⎣ ⎤⎦ D{ } = F{ }
• Challenges of nonlinear simulations
come from the difficulties of solving
the above equation.
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Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
4
Causes of Structural Nonlinearities
• Geometry Nonlinearity
• Due to Large Deflection
• Topology Nonlinearity
• Contact Nonlinearity
• Etc.
• Material Nonlinearity
• Due to Nonlinear Stress-Strain
Relations
To include geometry
nonlinearity, simply
turn on
.
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
5
Steps, Substeps, and Iterations
• Steps (Load Steps)
• Each step can have its own analysis settings.
• Substeps (Time Steps)
• In dynamic simulations, time step is used
for integration over time domain.
• In static simulation, dividing into substeps is
to achieve or enhance convergence.
• Iterations (Equilibrium Iterations)
• Each iteration involves solving a linearized
equilibrium equation.
[1] Number of
steps can be
specified here.
[3] Each step
can have its
own analysis
settings.
[2] To switch
between steps,
type a step number
here.
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
6
Displacement D{ }
Fo
rc
e
F {}
D
0
D
1
D
2
D
3
D
4
F
0
F
1
F
2
F
3
F
0
+ ΔF
P
0
P
1
P
2
P
3
P
4
P
1
′
P
2
′
P
3
′
P
4
′
Newton-Raphson Method
[1] Actual response
curve, governed by
K(D)⎡⎣ ⎤⎦ D{ } = F{ }
[2] Displacements at
current time step
(known).
[5] Displacements at next
time step (unknown).
[3] External
force at
current time
step (known).
[4] External
force at next
time step
(known).
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
7
Suppose we are now at
P
0
and the time is increased one substep further so that
the external force is increased to
F
0
+ ΔF , and we want to find the displacement
at next time step
D
4
.
Starting from point
P
0
, calculates a tangent stiffness [K], the
linearized stiffness, and solves the following equation
K⎡⎣ ⎤⎦ ΔD{ } = ΔF{ }
The displacement
D
0
is increased by ΔD to become D1 . Now, in the D-F space,
we are at
(D
1
,F
0
+ ΔF ) , the point
P
1
′ , far from our goal
P
4
. To proceed, we need to
"drive" the point
P
1
′ back to the actual response curve.
Substituting the displacement
D
1
into the governing equation, we can
calculate the internal force
F
1
,
K(D
1
)⎡⎣ ⎤⎦ D1{ } = F1{ }
Now we can locate the point
(D
1
,F
1
) , which is on the actual response curve. The
difference between the external force (here,
F
0
+ ΔF ) and the internal force (here,
F
1
) is called the residual force of that equilibrium iteration,
F
1
R = (F
0
+ ΔF )− F
1
If the residual force is smaller than a criterion, then the substep is said to be
converged, otherwise, another equilibrium iteration is initiated. The iterations
repeat until the convergence criterion satisfies.
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平衡迭代
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
8
[1] You can turn
on and
set the criterion.
[2] You can turn
on and
set the criterion.
[3] When shell
elements or beam
elements are used,
can be
activated.
[4] When shell
elements or beam
elements are used,
can be
activated.
Force/Displacement
Convergence
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
9
Solution Information
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
10
Line Search
D
0
D
1
F
0
F
0
+ ΔF
Calculated ΔD
Goal
Fo
rc
e
Displacement
[1] In some cases, when the F-
D curve is highly nonlinear or
concave up, the calculated ΔD
in a single iteration may
overshoot the goal.
[2] Line search can be
turned on to scale
down the incremental
displacement. By
default, it is .
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
11
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
12
Contact Types
• Bonded
• No Separation
• Frictionless
• Rough
• Frictional
• Linear versus Nonlinear Contacts
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
13
Contact versus Target [1] To specify a contact
region, you have to select a set
of faces (or edges),
and select a set of
faces (or edges).
[2] If is set to
, the roles of
and will
be symmetric.
• During the solution, will
check the contact status for each point
(typically a node or an integration
point) on the faces against
the faces.
• If is set to ,
the roles of and
will be symmetric.
• If is set to ,
the checking is only one-sided.
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
14
Contact Formulations
[1] Workbench
offers several
formulations to
enforce contact
compatibility.
[2] is input here.
The input value (default to 1.0) is
regarded as a scaling factor to multiply a
stiffness value calculated by the program.
• MPC (multi-point constraint)
• Pure Penalty
• Normal Lagrange
• Augmented Lagrange
Chapter 13 Nonlinear Simulations
Section 13.1 Basics of Nonlinear Simulations
15
Additional Contact
Settings
• Pinball Region
• Interface Treatment
• Time Step Controls
• Update Stiffness
Chapter 13 Nonlinear Simulations
Section 12.2 Translational Joint
16
60
20
20
40
Section 13.2
Translational Joint
Problem Description
[3] All connectors
have a cross section
of 10x10 mm.
[1] The
translational joint
is used to connect
two machine
components, so
that the relative
motion of the
components is
restricted in this
direction.
[2] All leaf springs
have a cross section
of 1x10 mm.
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Chapter 13 Nonlinear Simulations
Section 12.2 Translational Joint
17
Results
0
30
60
90
120
0 10 20 30 40
Fo
rc
e
(N
)
Displacement (mm)
[1] Nonlinear
Solution.
[2] Linear Solution.
101.73
74.67
Chapter 13 Nonlinear Simulations
Section 13.3 Microgripper
18
Section 13.3
Microgripper
Problem Description
The microgripper is made of PDMS and actuated by a SMA (shape memory alloy)
actuator; it is tested by gripping a glass bead in a lab. In this section, we want to
assess the gripping forces on the glass bead under an actuation force of 40 µN
exerted by the SMA device. More specifically, we will plot a gripping force-versus-
actuation-force chart.
Chapter 13 Nonlinear Simulations
Section 13.3 Microgripper
19
Results
[1] contact
status. [2] contact
pressure.
Chapter 13 Nonlinear Simulations
Section 13.4 Snap Lock
20
Section 13.4
Snap Lock
Problem Description
7
20
20
7
10
30
17
7
5
10
5
8
The purpose of this
simulation is to find out
the force required to push
the insert into the
position and the force
required to pull it out.
Chapter 13 Nonlinear Simulations
Section 13.4 Snap Lock
21
[2] It requires
236 N to pull
out.
[1] It requires 189 N
to snap in.
[3] The curve is
essentially symmetric.
Remember that we
didn't take the
friction into account.
Results (Without Friction)
Chapter 13 Nonlinear Simulations
Section 13.4 Snap Lock
22
Results (With Friction)
[1] It requires 328 N
to snap in.
[2] It requires 305 N to
pull out.
[3] Because of
friction, the curve is
not symmetric.
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