Chapter 13 Nonlinear Simulations 1

Chapter 13Nonlinear Simulations13.1 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 13.1Basics 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

Forc

e {F

}

Displacement {D}

Forc

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.

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 <Large Deflection>.

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 has 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{ }

Forc

e F {}

D

0 D

1 D

2 D

3 D

4

F

0

F1

F

2

F

3

F

0+ ΔF

P

0

P1

P

2 P

3 P

4

P1′

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, <Workbench> 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

D

1. Now, in the D-F space,

we are at (D

1,F

0+ ΔF ) , the point

P1′ , far from our goal

P

4. To proceed, we need to

"drive" the point P1′ back to the actual response curve.

Substituting the displacement D

1 into the governing equation, we can

calculate the internal force F1,

K(D

1)⎡⎣ ⎤⎦ D

1{ } = 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,

F1) is called the residual force of that equilibrium iteration,

F1R = (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.

Chapter 13 Nonlinear Simulations Section 13.1 Basics of Nonlinear Simulations 8

[1] You can turn on <Force

Convergence> and set the criterion.

[2] You can turn on <Displacement Convergence> and set the criterion.

[3] When shell elements or beam elements are used,

<Moment Convergence> can be

activated.

[4] When shell elements or beam elements are used,

<Rotation Convergence> 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

Forc

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 <Program Controlled>.

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 <Contact> faces (or edges), and select a set of <Target>

faces (or edges).

[2] If <Behavior> is set to <Symmetric>, the roles of

<Contact> and <Target> will be symmetric.

• During the solution, <Workbench> will

check the contact status for each point

(typically a node or an integration

point) on the <Contact> faces against

the <Target> faces.

• If <Behavior> is set to <Symmetric>,

the roles of <Contact> and <Target>

will be symmetric.

• If <Behavior> is set to <Asymmetric>,

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] <Normal Stiffness> 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 13.2Translational 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.

Chapter 13 Nonlinear Simulations Section 12.2 Translational Joint 17

Results

0

30

60

90

120

0 10 20 30 40

Forc

e (N

)

Displacement (mm)

[1] Nonlinear Solution.

[2] Linear Solution.

101.73

74.67

Chapter 13 Nonlinear Simulations Section 13.3 Microgripper 18

Section 13.3Microgripper

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 13.4Snap 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.