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 is chosen to be the total strain energy in this example. 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.

This 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 2nd 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 re-meshed.

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 topology optimization feature. A Solid Mechanics interface represents the structural properties of the beam, while an Optimization interface allows adding the objective and the constraints for the optimization problem. The elastic strain energy density is a predefined variable, solid.Ws, available to use as the objective function for the optimization problem.

Only half of the hook geometry must be modeled, because of the assumed symmetry, which is implemented as a Symmetry condition on the symmetry plane. Plots for postprocessing and inspection of the solution while solving are set up using a Mirror data set, 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 an Optimization study step. For topology optimization, either the MMA or the SNOPT solver can be used. They have different merits and weaknesses. MMA tends to be braver in the beginning, proceeding quickly toward an approximate optimum, while SNOPT is more cautious but also converges more efficiently close to the final solution thanks to its 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 of excessive stress concentrations.

References

1. B.S. Lazarov and O. Sigmund, “Filters in topology optimization based on Helmholtz-type differential equations,” International Journal for Numerical Methods in Engineering, 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,” Structural and Multidisciplinary Optimization, 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.

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

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.

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 , click to expand 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 in the window toolbar.

In the toolbar, click

to close the window.

Materials

Topology Link 1 (toplnk1)

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.

Definitions

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.

Locate the section. From the list, choose .

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

Prescribed Material 1

In the toolbar, click

and choose .

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