COMSOL 6.4 - Exporting and Importing a Topology-Optimized Hook

Exporting and Importing a
Topology-Optimized Hook

Topology optimization in a structural mechanics context can answer the question: Given that you know the loads on the structure, which distribution of the available material maximizes stiffness? Or, conversely, how much material is necessary to obtain a predefined stiffness, and how should it be distributed? Such investigations typically occur during the concept design stages.

The conflicting goals of stiffness maximization and mass minimization lead to a continuum of possible optimal solutions, depending on how you balance the goals against each other. This topology optimization example demonstrates how to use a penalization method (SIMP) to obtain an optimal distribution of a fixed amount of material such that stiffness is maximized. Changing the amount of material available leads to a different solution, which is also Pareto optimal, representing a different balance between the conflicting objectives.

Model Definition

The model studies optimal material distribution in a hook, which is symmetric about the plane z = 0 and consists of a linear elastic material, structural steel. It is subjected to two distributed load cases: One at the tip of the hook and one along its lower inner curve. The geometry is imported as a geometry sequence (Figure 1).

Figure 1: The computational domain of the hook is reduced by exploiting symmetry.

The optimality criterion is defined by the objective function, which in this example is chosen to be the total strain energy. Note that the strain energy exactly balances the work done by the applied load, so minimizing the strain energy minimizes the displacement induced at the points where loads are applied, effectively minimizing the compliance of the structure — maximizing its stiffness. The other, conflicting, objective is minimization of total mass, which is implemented as an upper bound on the mass of the optimized structure.

To regularize the problem we introduce a minimum length scale via a filter radius in a Helmholtz filter. See the Topology Optimization of an MBB Beam example for illustrations of this process. Ideally, the topology of the resulting designs should be mesh independent and for designs with moderate complexity this can be achieved, but if the optimal design is very complex, other designs with slightly different topologies perform only marginally worse, so the optimization problem has many local minima, and it is likely that a slightly suboptimal design is identified. The result of the optimization is shown in Figure 2.

Figure 2: A multislice plot colored with the displacement magnitude shows the topology optimized design before it is exported.

In this example the complexity of the design is constrained using milling constraints, see Ref. 4 or Topology Optimization of a Beam with Milling Constraints for details. The example demonstrates a strategy based on using MMA with a limited number of iterations. The filtered variable is solved on a finer mesh to get a smooth design for postprocessing. You can use this to export a STL file, but to transfer the geometry to a second component for verification purposes, it is best to use a dataset. This approach enables the recycling of selections from the other component, but the parameters for the geometry feature can still require some tweaking to get a valid mesh for computation of the displacements in the imported geometry.

Results

The following plot shows the displacement field for the optimized solution after the geometry has been transferred and remeshed.

Figure 3: A plot of the displacement field for the optimized design after it has been imported as an STL file.

Notes About the COMSOL Implementation

The control variable field, Helmholtz filter, and SIMP penalization are defined through the use of a feature. A Solid Mechanics interface represents the structural properties of the beam, while the objective and the constraints are defined directly in the study step. The elastic strain energy density is a predefined variable, solid.Ws, available to use as the objective function for the optimization problem.

It is sufficient to include only half of the hook geometry in the model because of the assumed symmetry, which is implemented as a Symmetry condition on the symmetry plane. Results plots and inspection of the solution while solving are set up using a Mirror dataset to show the complete solution. The geometry is parameterized, making it easy to experiment with different hook sizes, but keep in mind that the mesh size is taken as the filter radius.

The optimization solver is selected and controlled from the study step. For topology optimization, either of the MMA, IPOPT, or GCMMA solver can be used. They have different merits and weaknesses. MMA and GCMMA tend to be braver in the beginning, proceeding quickly toward an approximate optimum, while IPOPT is more cautious but also converge more efficiently close to the final solution thanks to their second-order approximation of the objective.

Finally, it is worth noting that the problem of finding a stiff design can be quite different from finding a design that does not break, and it is generally advised to use shape or parameter optimization to ensure that the design is free from excessive stress concentrations.

References

1. B.S. Lazarov and O. Sigmund, “Filters in topology optimization based on Helmholtz-type differential equations,” Int. J. Numer. Methods Eng., vol. 86, pp. 765–781, 2011.

2. F. Wang, B.S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscipl. Optim., vol. 43, pp. 767–784, 2011.

3. M.P. Bendsøe, “Optimal shape design as a material distribution problem,” Structural Optimization, vol. 1, pp. 193–202, 1989.

4. L Høghøj and E.A. Träff, “An advection–diffusion based filter for machinable designs in topology optimization,” Comp. Meth. App. Mech. & Eng., vol. 391, pp. 114488, 2022.

Optimization_Module/Topology_Optimization/hook_optimization_stl

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

Geometry 1

Create the geometry. To simplify this step, insert a prepared geometry sequence.

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file hook_optimization_stl_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

In the window, under click .

The geometry should now look like that in Figure 1. Note that the inserted geometry is parameterized and that the parameters used are automatically added to the list of global parameters in the model.

Global Definitions

Geometric Parameters

In the window, under click .

In the window for , type Geometric Parameters in the text field.

Now there is a group for geometric parameters. Add a second group for parameters related to the optimization.

Parameters 2

In the toolbar, click

and choose > .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
volfrac	0.5	0.5	Volume fraction
Lmin	2[mm]	0.002 m	Filter radius
meshsz	Lmin	0.002 m	Mesh size
meshsz2	Lmin/2	0.001 m	Fine mesh size

Mesh 1

Swept 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Size

In the window, click .

In the window for , locate the section.

Click the button.

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

In the text field, type meshsz/4.

Click

Use structural steel for the solid material.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Topology Link 1 (toplnk1)

In the window, under > click .

In the window for , locate the section.

From the list, choose .

The Young’s modulus is now coupled to the material. Next, set up a mirror dataset for seeing the full design, while optimizing.

Topology Optimization

Density Model 1 (dtopo1)

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the Rmin list, choose .

In the text field, type Lmin.

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

In the table, enter the following settings:


X	Y	Z
0	0	-1

From the Ldif list, choose .

In the text field, type 0.

Locate the section. From the list, choose .

Locate the section. In the θ0 text field, type 0.1.

Component 1 (comp1)

Prescribed Material 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Add a and feature to avoid 90 degree angles at domain boundaries. In this case, these features will also cause the optimization to find design without internal cavities.

Prescribed Material Boundary 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Prescribed Void Boundary 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Definitions

Variables 1

In the window, under right-click and choose .

In topology optimization the geometry is not sufficiently well defined to warrant 2nd order shape functions.

Solid Mechanics (solid)

In the window for , click to expand the section.

Fixed Constraint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Roller 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

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

From the list, choose .

This avoids problems for the verification study, if there are numerical inaccuracies in the normal vector orientation.

Take both loads into account using the load case functionality.

Boundary Load 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specify the Ftot vector as


0	x
-100[kN]	y
0	z

In the toolbar, click

and choose .

Boundary Load 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

and choose .

Study 1: Optimization

In the window, click .

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

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Select the checkbox.

Click

In the table, enter the following settings:


Load case	lg1	Weight	lg2	Weight
Load case 1	√	1.0		1.0
Load case 2		1.0	√	1.0

Initialize the study to generate plots to show while solving.

In the toolbar, click

Results

Mirror 3D 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Threshold

In the window, expand the > node, then click .

In the window for , locate the section.

From the list, choose .

Study 1: Optimization

Topology Optimization

In the window, under click .

In the window for , locate the section.

In the text field, type 25.

The control variable will not converge to the tolerance in so few iterations, but the actual design will not change noticeably by using more iterations.

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


Expression	Lower bound	Upper bound
comp1.dtopo1.theta_avg		volfrac

Click to expand the section. Select the checkbox.

From the list, choose .

In the toolbar, click

The optimization has removed material near the tip of the hook.

Results

Output material volume factor

In the window, under > click .

In the toolbar, click

Click the

button in the toolbar.

Add a finer mesh and solve for the filtered material volume factor on this.

Mesh 1

In the window, under right-click and choose .

Mesh 2

Size

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type meshsz2.

In the text field, type meshsz2/4.

Click

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Find the subsection. In the table, clear the checkbox for .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 2: Smooth Design (mesh2)

In the window for , type Study 2: Smooth Design (mesh2) in the text field.

Solution 2 (sol2)

In the toolbar, click

Step 1: Stationary

In the window, expand the node, then click > .

In the window for , click to expand the section.

Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

Results

Filter 1

Create a new component from the filter dataset.

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

Mesh-Based Geometry 1

Remesh Faces 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Size

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Click

Materials

Use structural steel for the solid material.

Material Link 1 (matlnk1)

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

Solid Mechanics (solid)

Copy/paste the physics from the first component and fix the selections.

In the window, under right-click and choose .

Solid Mechanics (solid2)

In the window, right-click and choose .

In the dialog, click .

We can use 2nd order displacements now that we have an explicit geometry representation.

In the window for , click to expand the section.

From the list, choose .

Fixed Constraint 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Roller 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Boundary Load 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Boundary Load 2

In the window, click .

In the window for , locate the section.

From the list, choose .

Component 1: Optimization

In the window, click .

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

Component 2: Verification

In the window, click .

In the window for , type Component 2: Verification in the text field.

Mesh 3

Free Tetrahedral 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

From the list, choose .

Click

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Find the subsection. In the table, clear the checkbox for .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 1: Optimization

Step 1: Stationary

In the window for , locate the section.

In the column of the table, under , clear the checkbox for .

Study 2: Smooth Design (mesh2)

In the window, under click .

In the window for , locate the section.

In the column of the table, under , clear the checkbox for .

Study 3: Verification

In the window, under click .

In the window for , locate the section.

In the column of the table, under , clear the checkbox for .

Click to expand the section. In the table, enter the following settings:


Component	Mesh
Component 1	Mesh 1

Locate the section. Select the checkbox.

Click

In the table, enter the following settings:


Load case	lg1	Weight	lg2	Weight
Load case 1	√	1.0		1.0
Load case 2		1.0	√	1.0

In the window, click .

In the window for , type Study 3: Verification in the text field.

Locate the section. Clear the checkbox.

In the toolbar, click

Add a new 3D plot group to plot the displacement.

Results

Displacement (solid2)

In the toolbar, click

In the window for , type Displacement (solid2) in the text field.

Locate the section. From the list, choose .

Volume 1

Right-click and choose .

Displacement (solid2)

Click the

button in the toolbar.

In the toolbar, click

Topology Optimization 1

In the window, under right-click and choose .

Geometry Modeling Instructions

If you want to create the geometry yourself, follow these steps.

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
r1	7.5[mm]	0.0075 m	Hole radius
r2	20[mm]	0.02 m	Max upper radius
r3	50[mm]	0.05 m	Max lower radius
r4	25[mm]	0.025 m	Min lower radius
Lc	90[mm]	0.09 m	Hole to hook center distance
w1	2.5[mm]	0.0025 m	Width of load surface
w2	12.5[mm]	0.0125 m	Total width