COMSOL 6.3 - Design Optimization of a Beam

When applying design optimization, one has to choose between different techniques with different strengths and weaknesses regarding robustness, learning curve, design freedom, and associated potential improvements. This example illustrates how to apply parameter, shape, and topology optimization to the problem of minimizing the weight of a beam subject to a displacement constraint. In this case the parameter and shape optimization happen to give similar designs with similar performance, while the topology optimization gives a design with significantly better performance. Although it is not unusual that the superior design freedom of topology optimization results in superior performance, it is in no way guaranteed. The optimization can get stuck in a local minimum and it is often possible to improve on the result by using it as an initial design for shape or parameter optimization as shown in the model Shape Optimization of a Tesla Microvalve.

Model Definition

The model geometry (Figure 1) consists of two regions: A fixed domain on which a distributed load is applied and a domain to be designed for optimal performance. For the parameter optimization it is the lower right corner that is moved, while the shape optimization changes the bottom right boundary. Finally, topology optimization is able to distribute material freely in the design domain.

Figure 1: The model geometry.

The beam is made of aluminum, and the displacement field is calculated under the assumption of linear elasticity. The displacement of the upper right corner is constrained to be less than 0.2 mm.

The parameter optimization moves the lower right point and constructs a new mesh in every iteration. This means that the gradient cannot be computed analytically using the adjoint method, so the use of a derivative free optimization solver (such as Nelder–Mead) is appropriate.

The shape optimization uses two 2nd order Bernstein polynomials to move the bottom right boundary. This is implemented using the and features. The Maximum displacement is set such that the lower boundary cannot move to the point where the topology changes. The lower left point is fixed, but the right point is allowed to slide along a Roller boundary.

For a detailed introduction to the use of structural topology optimization and how to use a Helmholtz filter for regularization, see the model Topology Optimization of an MBB Beam. The main points are that Young’s modulus varies spatially to reflect the material distribution. It is not possible to set zero void stiffness, because this causes the void displacement field to become undefined.

Results and Discussion

Figure 2 displays the result of the parameter optimization together with the distributed load and the mesh. The displacement field is shown in colors and the maximum value is located near the end of the beam,

Figure 2: The lower right corner is moved to the center of the beam, when its position is used as a control variable. The color scale show reveals that the displacement at the upper right corner satisfies the 0.5 mm constraint.

The shape optimization results in a somewhat similar design and the mass is only 1% lower, see Figure 3. However, the computation time is similar to the parameter optimization despite having more control variables. This is due to the fact that the mesh topology is fixed, which allows for the use of analytic computation of the gradient via the adjoint method. It is thus possible to use IPOPT, a quadratic gradient based optimizer. Moreover, the addition of more control variables is not expected to affect the computational time adversely.

Figure 3: The shape of the beam is optimized by moving the lower boundary vertically. The movement is large toward the right of the beam, which causes a concentration of elements in this area.

Figure 4 plots the design found with topology optimization. Four triangles have been cut out and the end of the beam is left unsupported, which allows for a seesaw effect because most of the load is to the left of the unsupported region. You can thus expect that the performance of the design is quite sensitive to variations in the load distribution. The model Bracket — Topology Optimization shows how multiple load cases can be used to find design that are robust to load variations.

Figure 4: Topology optimization has significant design freedom, so it is able to find a design, which is 41% lighter than the result of the shape optimization.

Notes About the COMSOL Implementation

This model combines the Optimization and Solid Mechanics interfaces. First, you set up and solve the parameter optimization. Then you define the shape optimization problem using the features available from the component node. Both the shape and parameter optimization work with a coarser mesh, but the topology soptimization needs a rather fine mesh if a structure with slender components is desired. This is particularly true, if the constraint on the displacement is relaxed to a higher value.

The topology optimization is implemented by adding a on the component. This is used to define a custom Young’s modulus for a new Linear Elasticity model under the Solid Mechanics interface. This new Linear Elasticity model is disabled in the studies for the shape and parameter optimization. The interface cannot be justified for the implicit geometry description associated with topology optimization, but the extra computational cost is manageable for this simple 2D problem.

Optimization_Module/Design_Optimization/beam_optimization

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

Add parameters for the geometry, including the y-coordinates for two points to be controlled by the parameter optimization.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L0	1[m]	1 m	Beam length
T0	20[cm]	0.2 m	Beam height
M0	L0T02700[kg/m^3]	540 kg/m	Beam weight
Yopt1	0[m]	0 m	Y position, point 1
Yopt2	0[m]	0 m	Y position, point 2
maxDisp	0.2[mm]	2E-4 m	Maximum displacement

Geometry 1

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


x (m)	y (m)
0	0
L0/2	Yopt1
L0	Yopt2
L0	0.9*T0
L0	T0
L0	T0
0	T0

Polygon 2 (pol2)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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


x (m)	y (m)
0	0.9*T0
L0	0.9*T0

Bottom Boundaries

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Locate the section. In the text field, type eps.

Locate the section. From the list, choose .

Right Boundaries

Right-click and choose .

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

Locate the section. In the text field, type L0-eps.

In the text field, type Inf.

In the toolbar, click

Click the

button in the toolbar.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Material Link 1 (matlnk1)

In the window, under right-click and choose > .

Solid Mechanics (solid)

Fix the left boundary. Then impose a nonuniform load on the upper boundary.

Fixed Constraint 1

In the window, under right-click and choose .

Select Boundaries 1 and 3 only.

Boundary Load 1

In the toolbar, click

and choose .

Select Boundary 5 only.

In the window for , locate the section.

From the list, choose .

Specify the fL vector as


0	x
-1e6[N/m]((X/L0)^4(1-X/L0))	y

Enable and use it do define a feature.

Definitions

Mass Properties 1 (mass1)

In the window, under right-click and choose > .

In the window for , locate the section.

From the list, choose .

Tip Displacement

In the toolbar, click

and choose .

In the window for , type Tip Displacement in the text field.

In the text field, type pnt_disp.

Select Point 7 only.

Study 1

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Solve the model without minimization to generate a plot to update while optimizing.

In the window, click .

In the window for , type Study 1: Parameter Optimization in the text field.

In the toolbar, click

Results

Parameter Optimization

In the window for , type Parameter Optimization in the text field.

Click to expand the section. From the list, choose .

In the text area, type Parameter Optimization - eval(mass1.mass) kg (eval(mass1.mass/M0*100,,%d) %).

From the list, choose .

In the text field, type 3.

Locate the section. Clear the checkbox.

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type solid.disp.

Click to expand the section. Select the checkbox.

In the text field, type 0.

In the text field, type 1.05*maxDisp.

Locate the section. From the list, choose .

Deformation

In the window, expand the node, then click .

In the window for , locate the section.

Select the checkbox. In the associated text field, type 100.

Mesh 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Deformation 1

Right-click and choose .

In the window for , locate the section.

Select the checkbox. In the associated text field, type 100.

Line 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type 1.

Locate the section. From the list, choose .

From the list, choose .

Deformation 1

Right-click and choose .

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

Locate the section. In the text field, type u.

In the text field, type v-1e-8*solid.FperLengthy.

Locate the section.

Select the checkbox. In the associated text field, type 100.

Arrow Line 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type solid.FperLengthx.

In the text field, type solid.FperLengthy.

Locate the section. From the list, choose .

Select the checkbox. In the associated text field, type 1e-6.

Locate the section. In the text field, type 160.

Locate the section. From the list, choose .

Deformation 1

Right-click and choose .

In the window for , locate the section.

Select the checkbox. In the associated text field, type 100.

In the toolbar, click

Study 1: Parameter Optimization

Set up and compute the solution to the parameter optimization problem.

Optimization

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

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

Locate the section. From the list, choose .

Locate the section. Click

twice.

In the table, enter the following settings:


Parameter name	Initial value	Scale	Lower bound	Upper bound
Yopt1 (Y position, point 1)	0[m]	1	0	0.9*T0
Yopt2 (Y position, point 2)	0[m]	1	0	0.9*T0

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	Evaluate for
comp1.pnt_disp/maxDisp		1	Stationary

Locate the section. Select the checkbox.

From the list, choose .

Locate the section. Find the subsection. From the list, choose .

In the window, click .

In the window for , locate the section.

Clear the checkbox.

In the toolbar, click

Results

Parameter Optimization

Click the

button in the toolbar.

Next solve the same problem using shape optimization with a and a .

Component 1 (comp1)

Next solve the same problem using shape optimization with a and a .

Free Shape Domain 1

In the toolbar, click

and choose .

Polynomial Boundary 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Click to expand the section. From the list, choose .

From the list, choose .

Locate the section. In the text field, type 0.9*T0.

Symmetry/Roller 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Root

Add a study for the shape optimization and run it before adding the optimization, so that a plot can be generated for visualizing the results while optimizing.

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 toolbar.

In the toolbar, click

to close the window.

Study 2

Step 1: Stationary

In the window for , locate the section.

From the list, choose .

In the window, click .

In the window for , locate the section.

Clear the checkbox.

In the text field, type Study 2: Shape Optimization.

In the toolbar, click

Results

Shape Optimization

In the window, right-click and choose .

In the window for , type Shape Optimization in the text field.

Locate the section. In the text area, type Shape Optimization - eval(mass1.mass) kg (eval(mass1.mass/M0*100,,%d) %).

From the list, choose .

In the text field, type 3.

Locate the section. From the list, choose .

Study 2: Shape Optimization

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 20.

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

Locate the section. 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.pnt_disp/maxDisp		1

Locate the section. From the list, choose .

Select the checkbox.

From the list, choose .

In the toolbar, click

Results

In the toolbar, click

Click the

button in the toolbar.

Component 1 (comp1)

Finally, set up a topology optimization problem using the density method. Start by adding the feature, use it to define a penalized Young’s modulus in an alternative .

Density Model 1 (dtopo1)

In the window, right-click and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

Locate the section. In the θ0 text field, type 1.

Prescribed Material 1

In the toolbar, click

Select Domain 2 only.

Materials

Topology Link 1 (toplnk1)

In the window, under right-click and choose > .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Mesh 1

Topology optimization uses an implicit geometry description, which requires a finer mesh

In the window, under right-click and choose .

Mesh 2

In the window for , locate the section.

From the list, choose .

Click

The quadratic discretization used in the interface cannot be justified for the implicit geometry description associated with topology optimization, but the extra computational cost is manageable for this simple 2D problem.

Root

Add a study for the optimization. Follow the (now) usual procedure for creating a plot to be used in the optimization.

Add Study

In the toolbar, click