Bracket — Stress Optimization with Fatigue Evaluation

Bracket — Stress Optimization
with Fatigue Evaluation

This model demonstrates how to improve fatigue properties using shape optimization. The fatigue properties are not optimized directly. Instead a heuristic methodology is applied where an approximate value of the maximum stress is minimized subject to constraints on the mass and stiffness.

Model Definition

The model is based on the model Bracket — Fatigue Evaluation in the Fatigue Module Application Library. The initial stiffness as well as fatigue properties can thus be evaluated immediately. The fatigue analysis indicates that failure is likely to occur near some small fillets as illustrated in Figure 1.

Figure 1: The initial fatigue properties.

The, , and shape optimization features are used to allow modification of the fillet details, where the maximum stress occurs. COMSOL does not support optimization of the fatigue properties directly, so a heuristic approach is applied. The method is not guaranteed to improve the fatigue properties and therefore it is critical that these are evaluated before and after optimization. The heuristic approach involves using the ap-norm of the von Mises stress as the objective function,

For large values of p the objective is a good approximation of the maximum stress, but too large values can cause numerical problems for the optimization solver, so the model uses p = 50.The objective is scaled with the initial value, so (strictly speaking) the maximum stress, σmax, does not play a role; it only serves to prevent unit warnings. COMSOL Multiphysics comes with built-in variables for the volume and the stiffness, so it is easy to specify these constraints. The number of optimization iterations is limited to 20 and an iterative solver is used for the structural mechanics to save computation time.

Results and Discussion

As one might expect the optimization increases the fillet to reduce the maximum stress. This can be seen in Figure 2.

Figure 2: The shape optimization increases the fillet radius at the point of maximum stress.

To verify that the optimization has indeed improved the fatigue properties, a fatigue analysis is performed on the optimized design, and the result can be seen in Figure 3.

Figure 3: The shape optimization has removed the stress concentration at the fillet as compared to the initial design in Figure 2.

Notes About the COMSOL Implementation

The MMA optimization solver is used, but note that the (default) globally convergent behavior is disabled. The study step zeros the shape optimization variables, so the model uses the withsol operator to plot the fatigue usage in the deformed frame.

Optimization_Module/Shape_Optimization/bracket_fatigue_optimization

Modeling Instructions

Root

In this example you will start from an existing model from the Fatigue Module.

Application Libraries

From the menu, choose .

In the window, select in the tree.

Click

Start by computing the stiffness of the initial design in a new .

Results

Initial Compliance

In the toolbar, click

In the window for , type Initial Compliance in the text field.

Locate the section. From the list, choose .

Global Evaluation 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
pExp	50	50	P-norm exponent
sigmaMax	100[MPa]	1E8 Pa	Maximum stress
Ws0	1.07[J]	1.07 J	Characteristic stiffness

Component 1 (comp1)

In the window, expand the node.

Definitions

In the window, expand the node.

Free Boundaries

In the window, expand the node.

Right-click and choose .

In the window for , type Free Boundaries in the text field.

Locate the section. From the list, choose .

Click

In the dialog box, type 24, 25, 28, 29, 63, 64, 70, 71 in the text field.

Click .

Symmetry Boundaries

Right-click and choose .

In the window for , type Symmetry Boundaries in the text field.

Locate the section. Click

Click

In the dialog box, type 15, 65 in the text field.

Click .

Free Domains

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type Free Domains.

Locate the section. From the list, choose .

Locate the section. Under , click

In the dialog box, select in the list.

Click .

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Objective Function

In the window, right-click and choose .

In the window for , type Objective Function in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
obj	aveop1((solid.mises/sigmaMax)^pExp)^(1/pExp)		Objective function

Component 1 (comp1)

Free Shape Domain 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Free Shape Boundary 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

The bracket thickness is 8 mm, so we set a maximum displacement that is smaller to avoid inverted elements.

Locate the section. In the dmax text field, type 3[mm], which corresponds to twice the maximum displacement.

Symmetry/Roller 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbartwice.

Initial Design

In the window, click .

In the window for , type Initial Design in the text field.

Step 1: Stationary

In the window, expand the node.

Right-click and choose .

Optimized Design

In the window, click .

In the window for , type Optimized Design in the text field.

Right-click and choose .

Shape Optimization

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 15.

In the text field, type 0.2.

Click in the upper-right corner of the section. From the menu, choose .

Use an if-statement to evaluate the objective for the last solution.

Locate the section. In the table, enter the following settings:


Expression	Description
if(para==1, comp1.obj, 0)	Objective Function

From the list, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. In the table, enter the following settings:


Expression	Lower bound	Upper bound
comp1.fsd1.relVolume		1
comp1.solid.Ws_tot/Ws0		1

This prevents the optimized design from being heavier or less stiff.

In the toolbar, click

Results

Applied Loads (solid) 1, Stress (solid) 1

In the window, under , Ctrl-click to select and .

Right-click and choose .

Optimized Design

Solver Configurations

In the window, expand the node.

Solution 3 (sol3)

In the window, expand the node, then click .

In the window for , locate the section.

Clear the check boxto save computational time.

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose to reduce the computational time further.

Shape Optimization

In the window, under click .

In the window for , locate the section.

Select the check box.

From the list, choose .

In the toolbar, click

Initial Fatigue

In the window, click .

In the window for , type Initial Fatigue in the text field.

In the window, expand the node.

Parametric Sweep, Step 1: Fatigue

In the window, under , Ctrl-click to select and .

Right-click and choose .

Optimized Fatigue

In the window, click .

In the window for , type Optimized Fatigue in the text field.

Parametric Sweep

Right-click and choose .

Step 1: Fatigue

In the window for , locate the section.

Find the subsection. From the list, choose .

In the toolbar, click

Results

Shape Optimization 1

In the window, under right-click and choose .

Cycles to Failure, Optimized

In the window, under click .

In the window for , type Cycles to Failure, Optimized in the text field.

The study step returns zero for the dependent variables not solved for, so plotting the fatigue usage in the deformed configuration requires the use of the withsol operator.