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Optimization of an Imported Bracket Geometry — with Measuring Dimensions
Introduction
In some application fields, there is a strong focus on weight reduction. For example, this is the case in the automotive industry, where every gram has a distinct price tag.
The bracket in this tutorial is used for mounting a heavy component on a vibrating foundation. It is thus important to keep the natural frequency well above the excitation frequency to avoid resonances. The bracket is also subjected to shock loads, which can be treated as a static acceleration load. This gives an optimization problem, where results from two different study types must be considered simultaneously. In this example, the weight of a mounting bracket is reduced, given an upper bound on the stresses and a lower bound on the first natural frequency, by changing the size and location of three holes on the bracket.
In this tutorial, you will import the bracket geometry from a CAD file and use the Offset Faces and Transform Faces operations to reparameterize the already existing holes in the bracket to move and resize them.
An integral part of setting up the optimization is defining the geometrical, or design, constraints, which in this case will ensure that the holes will not come close to each other or the edges of the bracket. You can achieve this by adding dimension features to the geometry sequence. These measure the modified geometry and create parameters with the values. Using the parameters, you can then formulate conditions in parameter check features to abort the building of the geometry in case a measured value is outside the acceptable interval. When a parameter check is triggered during the optimization, the solver will not solve the forward problem; instead, it will continue with the next iteration.
The tutorial Optimization of an Imported Bracket Geometry — with Parameter Expressions demonstrates an alternative way of setting up the design constraints for the optimization by defining parameter expressions that you can use directly in the constraints section of the optimization study settings.
Note: This application requires the Optimization Module, the Structural Mechanics Module, and the Design Module.
Model Definition
The original bracket together with a sketched mounted component are shown in Figure 1. The bracket is made of steel.
The component, which can be considered as rigid when compared with the bracket, has its center of gravity at the center of the circular cutout in the bracket. The mass is 4.4 kg, the moment of inertia around its longitudinal axis is 7.1·10−4 kg·m2, and the moment of inertia around the two transverse axes is 9.3·10−4 kg·m2.
Figure 1: Bracket supporting a heavy component.
We can reduce the weight of the bracket by drilling holes in the bracket’s vertical surface and optimizing the holes’ size and placement. This is done by adding the Offset Faces and Transform Faces geometry operations to reparameterize the holes on the imported bracket geometry.
Optimization Parameters
Five geometrical parameters are used in the optimization. They are summarized in Table 1 and shown in Figure 2.
Table 1: Geometrical parameters.
Parameter
Description
Lower limit (mm)
Upper limit (mm)
RC
Radius of the central hole
3
15
ZCO
Vertical distance from the bend to the edge of the central hole
1
23
RO
Radius of the outer hole
3
15
ZOO
Vertical distance from the bend to the edge of the outer hole
8
30
YOO
Horizontal distance from the edge of the bracket to outer hole
3
29
Figure 2: Geometrical parameters that are optimized in the study.
There are two possible paths to take when considering how to set up the control parameters for the optimization. Here we optimize the actual radii and positions of the holes. An alternative would be to define the control parameters that are the displacements relative to the original geometry that is imported from the CAD file
You will measure all the original geometrical parameters that will be optimized in the study, by projecting the geometry onto a plane and using Dimensions and Constraints operations to create new parameters.
In the geometry sequence, one Transform Faces node is added to change both the position and the radius, defined by a scaling factor, of the central hole. By using the Transform Faces operation, we can move and change the size of the holes by applying a linear transformation (displacement, rotation, and isotropic scaling) to the faces of the holes. For the outer holes, we add an Offset Faces node to change the radius of the holes and we add two Transform Faces operations to displace the holes. Using the Offset Faces operation, we change the size of the holes on the bracket by offsetting the hole faces in the normal direction.
Constraints
Constraints can be divided into physical constraints and geometrical constraints.
The physical constraints are defined as follows:
•
The lowest natural frequency must be at least 60 Hz.
•
When exposed to a peak acceleration of 4g in all three global directions simultaneously, the effective stress is not allowed to exceed 80 MPa anywhere. This criterion is nondifferentiable because the location of the peak stress can jump from one place to another. A gradient-free optimization algorithm must thus be used.
Geometrical constraints have to ensure that there must be at least 3 mm of material between two holes or between a hole and an edge. This criterion is enforced through limits on the control parameters and as constraints.
The COBYLA solver uses sampling in the control variable space to approximate the objective function, the constraints, and the control variable bounds. Individual samples may be computed outside the bounds and in violation of the constraints. Therefore, it is important to parameterize the geometry in such a way that it is robust with respect to (small) constraints and bound violations.
Bounds and linear constraints are generally satisfied to high precision at the optimum point returned by the solver, but nonlinear constraints are often slightly violated. The reason is that the solver tends to converge from the outside of the feasible domain and terminates before the constraints are completely satisfied. Tightening the solver tolerances will decrease the constraint violation but is often not worth the computational effort; it is better to specify constraints with a safety margin.
Geometrical constraints
The geometrical constraints are shown in Figure 3 and summarized in Table 2.
Figure 3: Geometrical constraints.
As shown in Figure 3, geometrical constraints are set up so that spaces between different geometrical entities are bounded in specific ranges. That is important because we do not want to have very fragile parts of the geometry that would break when loaded, and we want to preserve the original shape of the geometry. Here, we consider four different geometrical constraints, which are summarized in Table 2.
Table 2: Geometrical constraints.
Parameter
Description
Limit (mm)
d_O_Cmp
Distance between the edge of outer hole and component
3
d_O_O
Distance between the edges of two outer holes
3
d_O_C
Distance between the edges of outer and central holes
3
d_C_Cmp
Distance between the edges of central hole and component
3
In this tutorial, the geometrical constraints are enforced by Parameter Check nodes in the geometry sequence. A Parameter Check node is useful to ensure that a parameter value stays within a certain range. The check gives an error if the condition that you provide is false. If so, the geometry with this set of parameters is skipped in the optimization study. In this case, the checked parameters are created by dimension nodes that measure the modified geometry.
Results and Discussion
The initial geometry used in the optimization is shown in Figure 4. Three rather small holes have been introduced.
Figure 4: Initial geometry.
The optimal values of the geometrical parameters are shown in Table 3.
Table 3: Optimal values.
Parameter
Optimal value (mm)
Lower limit (mm)
Upper limit (mm)
RC
6.10
 3
 15
ZCO
7.19
 1
 23
RO
9.93
 3
 15
YOO
3.08
 3
 29
ZOO
22.20
 8
 30
The weight of the optimized bracket is about 187 g, a reduction of 17 g from the original 204.10 g. The stresses from the shock load on the optimized geometry are shown in Figure 5.
Figure 5: Stresses at peak load in the optimized design.
The optimal solution gives three fairly large holes.
There are several possible arrangements of the holes that give the same weight reduction within a small tolerance. It is therefore possible that the design variables are not always the same at convergence.
Notes About the COMSOL Implementation
The bracket geometry is imported from a CAD file saved in STEP format.
The component mounted on the bracket is not modeled in detail. It is replaced by a Rigid Connector having the equivalent inertial properties.
Application Library path: Design_Module/Tutorial_Examples/bracket_import_optimization_dimensions
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Structural Mechanics > Solid Mechanics (solid).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Eigenfrequency.
6
Click  Done.
Global Definitions
Parameters 1
In the parameters table, you will load the parameters needed to define the physics and initial values of optimization parameters used in the optimization study.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file bracket_import_optimization_dimensions_parameters.txt.
5
In the Home toolbar, click  Parameter Case.
Creating a parameter case for the original design parameters will come in handy when comparing the optimized and unmodified bracket geometry.
6
In the Settings window for Case, type Original Values in the Label text field.
Geometry 1
The bracket geometry for this tutorial has been saved in the STEP format. Make sure that the CAD Import Module geometry kernel is used.
1
In the Model Builder window, under Component 1 click Geometry 1.
2
In the Settings window for Geometry, locate the Advanced section.
3
From the Geometry representation list, choose CAD kernel.
Import 1
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file bracket_import_optimization_geom.step.
5
Locate the Length Unit section. From the Length unit list, choose From CAD document.
With this option, the length unit of the geometry is automatically set to the unit found in the imported file, in this case, mm. Otherwise, you can manually specify the length unit in the Settings window for the Geometry node.
6
Locate the Source section. Click  Import.
Original Geometry
In the next section, you will add measuring dimensions to get all the necessary measurements of the imported geometry. You will first project the geometry onto a plane and measure the radius of the holes, distances between the holes and the edges of the geometry, which will define your optimization parameters, as well as all other dimensions of the geometry that will be useful for future analysis.
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Original Geometry in the Label text field.
3
Locate the Plane Definition section. From the Plane list, choose yz-plane.
Original Geometry > Plane Geometry
In the Model Builder window, click Plane Geometry.
Original Geometry > Projection 1
1
In the Work Plane toolbar, click  Projection.
2
In the Settings window for Projection, locate the Projection section.
3
From the Projection type list, choose Edges and vertices.
Since you are only interested in the values of the measurements and you will not create geometry on this work plane, this projection can be set as construction geometry.
4
Locate the Assigned Attributes section. From the Construction geometry list, choose On.
5
Click  Build Selected.
Original Geometry > Distance 1
1
In the Sketch toolbar, click  Distance.
2
Select the two outer boundaries of the bracket, following the figure below, and apply Distance by clicking on the Graphics window.
Dimensions are created by default as constraining dimensions that control the geometry. The dimension is displayed in magenta in the Graphics window, indicating that the geometry is overdefined. This is expected as it cannot change the projection. Next, you will change this dimension into a measuring dimension that does not drive the measurements of the entities it is applied to, instead, it is updated according to the other features in the geometry sequence.
3
In the Settings window for Distance, locate the Dimension Value section. Click to clear the associated Constrain Distance toggle button (lock symbol).
4
Select the Create measuring parameter checkbox.
This creates a measuring parameter you can use in other geometry and nongeometry features.
5
In the Parameter name text field, type LY.
6
Click  Build Selected.
Original Geometry > Radius 1
1
In the Sketch toolbar, click  Radius.
2
Select the edge of the mounting component hole, following the figure below, and apply Radius by clicking in the Graphics window.
3
In the Settings window for Radius, locate the Dimension Value section. Click to clear the associated Constrain Radius toggle button (lock symbol).
4
Select the Create measuring parameter checkbox.
5
In the Parameter name text field, type RCmp.
6
Click  Build Selected.
Original Geometry > Distance 2-9, Radius 2-3
1
Proceed to create the measuring parameters listed in the table and shown in the image below by defining measuring dimensions according to the previous sections.
Parameter name
Description
ROorig
Original radius of outer holes
RCorig
Original radius of central hole
LZ
Bracket height
ZCOorig
Original distance from bend to bottom of central hole
THK
Plate thickness
YInd
Y position of the indentation
WInd
Width of indentation
HInd
Bend height
YOOorig
Original distance from edge to outer hole
ZOOorig
Original distance from bend to bottom of outer hole
Coordinate System - Central Hole
Next, set up work planes for the two outer and the central hole. These define the coordinate systems you will use to change the hole size and position using the Transform Faces operation.
1
In the Model Builder window, right-click Geometry 1 and choose Work Plane.
2
In the Settings window for Work Plane, type Coordinate System - Central Hole in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Circle perpendicular.
4
On the object imp1, select Edge 91 only.
5
Click  Build Selected.
Coordinate System - Outer Hole 1
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Coordinate System - Outer Hole 1 in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Circle perpendicular.
4
On the object imp1, select Edge 99 only.
5
Select the Reverse normal direction checkbox.
6
Click  Build Selected.
Coordinate System - Outer Hole 2
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Coordinate System - Outer Hole 2 in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Circle perpendicular.
4
On the object imp1, select Edge 81 only.
5
Click  Build Selected.
Z Displacement and Scaling - Central Hole
For modifying the central hole, you will add a Transform Faces operation to the geometry sequence. Using this feature, you can both displace the hole and change its radius by scaling it.
1
In the Geometry toolbar, click  Editing and choose Transform Faces.
2
In the Settings window for Transform Faces, type Z Displacement and Scaling - Central Hole in the Label text field.
3
On the object imp1, select Boundaries 31 and 32 only.
4
Locate the Coordinate System section. From the Work plane list, choose Coordinate System - Central Hole.
5
Locate the Scaling section. In the Factor text field, type RC/RCorig.
6
Locate the Displacement section. In the xw text field, type ZCO+RC-RCorig-ZCOorig.
7
Click  Build Selected.
Offset - Outer Holes
For the two outer holes, use the Offset Faces operation to change the radius. Note that only one Offset Faces operation is needed since the holes have the same radius.
1
In the Geometry toolbar, click  Editing and choose Offset Faces.
2
In the Settings window for Offset Faces, type Offset - Outer Holes in the Label text field.
3
On the object tf1, select Boundaries 28, 29, 34, and 35 only.
4
Locate the Offset section. In the Distance text field, type RO-ROorig.
5
Select the Reverse side checkbox.
6
Click  Build Selected.
Displacement - Outer Hole 1
Next, you can add the Transform Faces operations to displace the outer holes.
1
In the Geometry toolbar, click  Editing and choose Transform Faces.
2
In the Settings window for Transform Faces, type Displacement - Outer Hole 1 in the Label text field.
3
On the object off1, select Boundaries 34 and 35 only.
4
Locate the Coordinate System section. From the Work plane list, choose Coordinate System - Outer Hole 1.
5
Locate the Displacement section. In the xw text field, type
 -(ZOO-ZOOorig+RO-ROorig).
6
In the zw text field, type YOO-YOOorig+RO-ROorig.
7
Click  Build Selected.
Displacement - Outer Hole 2
1
In the Geometry toolbar, click  Editing and choose Transform Faces.
2
In the Settings window for Transform Faces, type Displacement - Outer Hole 2 in the Label text field.
3
On the object tf2, select Boundaries 28 and 29 only.
4
Locate the Coordinate System section. From the Work plane list, choose Coordinate System - Outer Hole 2.
5
Locate the Displacement section. In the xw text field, type
 -(ZOO-ZOOorig+RO-ROorig).
6
In the zw text field, type YOO-YOOorig+RO-ROorig.
7
Click  Build Selected.
Work Plane 5
You will define a new work plane that you will use for partitioning the bracket into two domains since the evaluation of the maximum stress is needed only for the upper part of the bracket.
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose yz-plane.
4
From the Offset type list, choose Through vertex.
5
On the object tf3, select Point 18 only.
6
Click  Build Selected.
Partition Objects 1
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object tf3 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
Click  Build Selected.
Geometrical Constraints
Next, add a work plane that you will use for projecting the modified geometry. Here you will set up measuring dimensions that define the parameters for the geometrical constraints used for the optimization.
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Geometrical Constraints in the Label text field.
3
Locate the Plane Definition section. From the Plane list, choose yz-plane.
Geometrical Constraints > Plane Geometry
1
In the Model Builder window, click Plane Geometry.
For a cleaner look in the Graphics window, disable the in-plane visualization of the outline of the 3D geometry.
2
In the Settings window for Plane Geometry, locate the Visualization section.
3
Find the In-plane visualization of 3D geometry subsection. Clear the Coincident entities (blue) checkbox.
4
Clear the Intersection (green) checkbox.
5
Clear the Projection (teal) checkbox.
Geometrical Constraints > Projection 1
1
In the Work Plane toolbar, click  Projection.
2
In the Settings window for Projection, locate the Assigned Attributes section.
3
From the Construction geometry list, choose On.
4
Click  Build Selected.
Geometrical Constraints > Point 1
Next, create points that you will constrain to the center of the projected circles. Later, you will use these points for setting up the measuring dimensions for the geometrical constraints.
1
In the Sketch toolbar, click  Point.
2
Apply the Point by clicking in the Graphics window near the left outer hole.
3
Locate the Assigned Attributes section. Select the Construction geometry checkbox.
Geometrical Constraints > Concentric 1
The Concentric constraint ensures that the point and the circle always stay concentric as the geometry is updated during the optimization study.
1
In the Sketch toolbar, click  Concentric.
2
Select one edge of the left outer hole and Point 1 and apply the Concentric constraint by clicking in the Graphics window.
Geometrical Constraints > Point 2-4, Concentric 2-4
Proceed to create three more points and apply the Concentric constraints. When done, you should have points at the center of the circles that are the projections of the optimized holes and at the center of the half circle at the top, as shown in the figure below.
Geometrical Constraints > Distance 1
1
In the Sketch toolbar, click  Distance.
2
Select the point in the left outer hole and the mounting component and apply Distance by clicking in the Graphics window.
3
In the Settings window for Distance, locate the Dimension Value section. Click to clear the associated Constrain Distance toggle button (lock symbol).
4
Select the Create measuring parameter checkbox.
5
In the Parameter name text field, type d_O_Cmp.
Geometrical Constraints > Distance 2-4
1
Add three more Distance constraints by following the steps in the section above to create the following dimension parameters.
Parameter name
Description
d_O_O
Distance between outer holes
d_O_C
Distance between outer and central holes
d_C_Cmp
Distance between central hole and component
Parameter Check 1
To ensure that the modified bracket geometry fulfills the geometrical constraints, add Parameter Check features where you can enter conditions using the previously defined dimension parameters.
1
In the Model Builder window, right-click Geometry 1 and choose Programming > Parameter Check.
2
In the Settings window for Parameter Check, locate the Parameter Check section.
3
In the Condition for message text field, type d_O_Cmp<3[mm]+RO+RCmp.
4
In the Message text field, type Distance between outer hole and component smaller than 3 mm.
5
Click  Build Selected.
Parameter Check 2
1
In the Geometry toolbar, click  Programming and choose Parameter Check.
2
In the Settings window for Parameter Check, locate the Parameter Check section.
3
In the Condition for message text field, type d_O_C<3[mm]+RO+RC.
4
In the Message text field, type Distance between outer hole and central hole smaller than 3 mm.
5
Click  Build Selected.
Parameter Check 3
1
In the Geometry toolbar, click  Programming and choose Parameter Check.
2
In the Settings window for Parameter Check, locate the Parameter Check section.
3
In the Condition for message text field, type d_O_O<3[mm]+2*RO.
4
In the Message text field, type Distance between outer holes smaller than 3 mm.
5
Click  Build Selected.
Parameter Check 4
1
In the Geometry toolbar, click  Programming and choose Parameter Check.
2
In the Settings window for Parameter Check, locate the Parameter Check section.
3
In the Condition for message text field, type d_C_Cmp<3[mm]+RC+RCmp.
4
In the Message text field, type Distance between central hole and component less than 3 mm.
5
Click  Build Selected.
Left Edges Indent
At this stage, the geometry setup is ready except for adding a safeguard in case the edges of the central hole and the edges of the faces defining the indentation intersect or come close to each other. This may result in very short edges or narrow face regions, which could become difficult to mesh. Since the indentation edges connect continuous faces, we can use the Ignore Edges virtual operation to hide them from the mesher and avoid potential meshing issues. First, create coordinate-based selections that contain the edges for the indentation faces.
1
In the Geometry toolbar, click  Selections and choose Box Selection.
2
In the Settings window for Box Selection, type Left Edges Indent in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Edge.
4
Locate the Box Limits section. In the x minimum text field, type LX-THK.
5
In the x maximum text field, type LX.
6
In the y minimum text field, type YInd.
7
In the y maximum text field, type WInd.
8
In the z minimum text field, type HInd+1[mm].
9
In the z maximum text field, type HInd+10[mm].
10
Locate the Resulting Selection section. Find the Cumulative selection subsection. Click New.
11
In the New Cumulative Selection dialog, type Ignore Edges Selection in the Name text field.
12
Click OK.
13
In the Settings window for Box Selection, click  Build Selected.
Right Edges Indent
1
In the Geometry toolbar, click  Selections and choose Box Selection.
2
In the Settings window for Box Selection, type Right Edges Indent in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Edge.
4
Locate the Box Limits section. In the x minimum text field, type LX-THK.
5
In the x maximum text field, type LX.
6
In the y minimum text field, type LY-WInd.
7
In the y maximum text field, type LY-YInd.
8
In the z minimum text field, type HInd+1[mm].
9
In the z maximum text field, type HInd+10[mm].
10
Locate the Resulting Selection section. Find the Cumulative selection subsection. From the Contribute to list, choose Ignore Edges Selection.
11
Click  Build Selected.
Central Edges Indent
1
In the Geometry toolbar, click  Selections and choose Box Selection.
2
In the Settings window for Box Selection, type Central Edges Indent in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Edge.
4
Locate the Box Limits section. In the x minimum text field, type LX-2*THK.
5
In the x maximum text field, type LX.
6
In the y minimum text field, type LY/2-YInd.
7
In the y maximum text field, type LY/2+YInd.
8
In the z minimum text field, type 0.
9
In the z maximum text field, type HInd+1[mm].
10
Locate the Resulting Selection section. Find the Cumulative selection subsection. From the Contribute to list, choose Ignore Edges Selection.
11
Click  Build Selected.
Ignore Edges 1
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Edges.
2
In the Settings window for Ignore Edges, locate the Input section.
3
From the Edges to ignore list, choose Ignore Edges Selection.
Change the model view to see the edges on the bottom side of the bracket.
4
Click the  Go to Default View button in the Graphics toolbar.
5
In the Geometry toolbar, click  Build All.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Structural steel.
4
Click the Add to Component button in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Solid Mechanics
Fixed (Bolts)
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
The exact way the bolts clamp the bracket to the foundation is not important for the results in the part being optimized.
2
In the Settings window for Fixed Constraint, type Fixed (Bolts) in the Label text field.
3
Select Boundaries 6–11 only.
Rigid Connector (Mounted component)
1
In the Physics toolbar, click  Boundaries and choose Rigid Connector.
The attached component has a high stiffness, and is bolted to the two upper bolt holes. It is modeled as being rigid, with only mass properties.
2
In the Settings window for Rigid Connector, type Rigid Connector (Mounted component) in the Label text field.
3
Select Boundaries 28, 29, 38, and 39 only.
4
Locate the Center of Rotation section. From the list, choose User defined.
5
Specify the Xc vector as
LX-THK/2
x
LY/2
y
LZ
z
Mass and Moment of Inertia 1
1
In the Physics toolbar, click  Attributes and choose Mass and Moment of Inertia.
2
In the Settings window for Mass and Moment of Inertia, locate the Mass and Moment of Inertia section.
3
In the m text field, type mCmp.
4
From the list, choose Diagonal.
5
Specify the I matrix as
IXCmp
0
0
0
IYZCmp
0
0
0
IYZCmp
Mesh 1
Free Triangular 1
1
In the Mesh toolbar, click  More Generators and choose Free Triangular.
2
Select Boundaries 4, 15, 17, 19, 21, 24, and 26 only.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Select Boundary 4 only.
5
Locate the Element Size section. From the Predefined list, choose Finer.
Size 2
1
In the Model Builder window, right-click Free Triangular 1 and choose Size.
2
Select Boundaries 15, 17, 19, 21, 24, and 26 only.
3
In the Settings window for Size, locate the Element Size section.
4
From the Predefined list, choose Extra fine.
5
Click  Build Selected.
Swept 1
In the Mesh toolbar, click  Swept.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 3.
4
Click  Build All.
Eigenfrequency Study
Run an eigenfrequency study on the initial geometry.
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Eigenfrequency Study in the Label text field.
3
In the Study toolbar, click  Compute.
Solid Mechanics
Add the peak loads, and perform a stationary study.
Body load 4g on bracket
1
In the Physics toolbar, click  Domains and choose Body Load.
2
In the Settings window for Body Load, type Body load 4g on bracket in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose All domains.
4
Locate the Force section. Specify the fV vector as
4*g_const*solid.rho
x
4*g_const*solid.rho
y
4*g_const*solid.rho
z
Rigid Connector (Mounted component)
In the Model Builder window, click Rigid Connector (Mounted component).
Force 4g on mounted component
1
In the Physics toolbar, click  Attributes and choose Applied Force.
2
In the Settings window for Applied Force, type Force 4g on mounted component in the Label text field.
3
Locate the Applied Force section. Specify the F vector as
4*g_const*mCmp
x
4*g_const*mCmp
y
4*g_const*mCmp
z
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Stationary.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Stationary Study
1
In the Settings window for Study, type Stationary Study in the Label text field.
2
In the Study toolbar, click  Compute.
Definitions
Prepare for the optimization by adding variables for the bracket mass and the maximum stress.
Stress Optimization Domain
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Stress Optimization Domain in the Label text field.
3
Select Domain 2 only.
Domain Probe 1
1
In the Definitions toolbar, click  Probes and choose Domain Probe.
2
In the Settings window for Domain Probe, type mass in the Variable name text field.
3
Locate the Probe Type section. From the Type list, choose Integral.
4
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 > Solid Mechanics > Material properties > solid.rho - Density - kg/m³.
Domain Probe 2
1
In the Definitions toolbar, click  Probes and choose Domain Probe.
2
In the Settings window for Domain Probe, type maxStress in the Variable name text field.
3
Locate the Probe Type section. From the Type list, choose Maximum.
4
Locate the Source Selection section. From the Selection list, choose Stress Optimization Domain.
5
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 > Solid Mechanics > Stress > solid.mises - von Mises stress - N/m².
Results
Modify the default stress plot to monitor the geometry and stresses in the optimized region.
Stress in Optimized Region
1
In the Model Builder window, under Results click Stress (solid).
2
In the Settings window for 3D Plot Group, type Stress in Optimized Region in the Label text field.
Volume 1
1
In the Model Builder window, expand the Stress in Optimized Region node, then click Volume 1.
2
In the Settings window for Volume, locate the Expression section.
3
From the Unit list, choose MPa.
Deformation
1
In the Model Builder window, expand the Volume 1 node.
2
Right-click Deformation and choose Delete.
Selection 1
1
Right-click Volume 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose Stress Optimization Domain.
Marker 1
1
Right-click Volume 1 and choose Marker.
2
Click the  Zoom Extents button in the Graphics toolbar.
Root
Set up the optimization study.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Empty Study.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Optimization Study
In the Settings window for Study, type Optimization Study in the Label text field.
General Optimization
In the Study toolbar, click  Optimization and choose General Optimization.
Eigenfrequency
1
In the Study toolbar, click  More Study Extensions and choose Study Reference.
2
In the Settings window for Study Reference, type Eigenfrequency in the Label text field.
3
Locate the Study Reference section. From the Study reference list, choose Eigenfrequency Study.
Stationary
1
In the Study toolbar, click  More Study Extensions and choose Study Reference.
2
In the Settings window for Study Reference, type Stationary in the Label text field.
3
Locate the Study Reference section. From the Study reference list, choose Stationary Study.
General Optimization
1
In the Model Builder window, click General Optimization.
2
In the Settings window for General Optimization, locate the Optimization Solver section.
3
From the Method list, choose COBYLA.
4
Click to expand the Solver Settings section. Clear the Stop if error checkbox.
5
Find the Constraint settings subsection. Select the Enforce design constraints strictly checkbox.
6
Click Replace Expression in the upper-right corner of the Objective Function section. From the menu, choose Component 1 > Definitions > comp1.mass - Domain Probe 1 - kg.
7
Locate the Objective Function section. In the table, enter the following settings:
Expression
Description
Evaluate for
comp1.mass
Bracket mass
Stationary
The first eigenfrequency is to be used in the optimization.
8
From the Solution list, choose Use first.
9
Locate the Control Variables and Parameters section. Click  Load from File.
10
Browse to the model’s Application Libraries folder and double-click the file bracket_import_optimization_ctrlvars.txt.
11
Locate the Constraints section. In the table, enter the following settings:
Expression
Lower bound
Upper bound
Evaluate for
real(freq)
minFreq
 
Eigenfrequency
comp1.maxStress/maxStressLimit
 
1
Stationary
12
Click to expand the Output section. Select the Plot checkbox.
13
In the table, enter the following settings:
Plot group
Plot window
Stress in Optimized Region
Graphics
14
Select the Copy optimized parameters to a parameter case checkbox.
This last step automatically creates a new parameter case with the optimum parameter values.
If some configurations are not valid, the optimization procedure should still continue. The default is to stop if an error occurs.
Solution 3
1
In the Study toolbar, click  Show Default Solver.
Run the optimization.
2
Click  Compute.
Results
Stress in Optimized Region
Examine the stress distribution in the optimized configuration.
1
In the Model Builder window, under Results click Stress in Optimized Region.
2
In the Stress in Optimized Region toolbar, click  Plot.
On the last line of Global Constraints Table 7, you will find the values of the natural frequency and maximum stress in the optimized configuration, as well as the values of the other constraints.
Objective Table 3
On the last line of Objective Table 3, you will find the optimal set of parameters and the minimum weight. Note that the value in the Objective column can be colored orange if the solution violates a constraint slightly but is still accepted within the tolerances.
Next, you can rebuild the geometry using the optimized parameters.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions > Parameters 1 click Case 2.
2
In the Settings window for Case, type Optimized Values in the Label text field.
3
In the Model Builder window, click Parameters 1.
4
In the Settings window for Parameters, click  Optimized Values.
Geometry 1
1
In the Geometry toolbar, click  Build All.
2
In the Model Builder window, under Component 1 click Geometry 1.