Bracket — Topology Optimization

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.

The various examples based on a bracket geometry form a suite of tutorials which summarizes the fundamentals when modeling structural mechanics problems in COMSOL Multiphysics and the Structural Mechanics Module. This example also requires the Optimization Module.

Model Definition

The model shows how to determine the optimal material distribution in a bracket geometry. The bracket is symmetric about the plane x = 0 and is made of a linear elastic material, structural steel.

It is optimized in two load configurations: First a single load case, which gives rise to an asymmetric design with two poorly connected halves. In a second analysis, eight load cases are used (two load groups times four constraint groups). This results in a symmetric design with a stronger connection between the two halves.

The original geometry, which can be considered as the design space, is imported as a geometry sequence (Figure 1).

Figure 1: The computational domain of the bracket is partitioned, so that the circles around the holes can be excluded from the design space.

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. 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. To reduce the probability of this happening, the model is solved using continuation in the SIMP exponent and the projection slope, that is, a sequence of optimization problems are solved. One optimization result is thus used as the initial guess of the next. This is achieved by combining a and an study step.

Results

The result of the optimization with one load case is shown in Figure 2.

Figure 2: A filtered volume plot, colored with the displacement magnitude. Only a single load case is considered, so the design becomes asymmetric and the two halves are poorly connected.

When eight load cases are considered, the optimization instead results in the design displayed in Figure 3.

Figure 3: When eight load cases are considered, the design becomes symmetric, and the connection between the two halves is stronger when compared to the single load case.

The following plot shows the displacement field for the optimized solution after the geometry has been transferred to another component via a .

Figure 4: A plot of the displacement field for the optimized design after it has been transferred to another component using a Filter dataset, a mesh part and a Import geometry feature. The hole furthest away from the camera is unconstrained for Load case 1.

Notes About the COMSOL Implementation

This example demonstrates a strategy based on using the MMA solver with a limited number of iterations. The filtered variable is solved on a finer mesh to get a smooth design for postprocessing.

The example makes use of box selections for imposing the boundary conditions on the imported geometry. This can be a significant time saver in the context of topology optimization, because many designs with similar boundary conditions can quickly be generated and tested against each other.

This model demonstrates how to export all domains together. This simplifies the import, but the Mesh part describes curved boundaries with flat triangles, so the hole geometry will be affected in this process, which might affect a contact analysis.

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 bracket, while an Optimization study step 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.

In this example the solver settings are changed slightly from their default values, in order to speed up the solution:

•

Since the solid mechanics problem is linear, it is enough to use one pass through the segregated solver, so it is set to terminate after one iteration, rather than on a tolerance.

•

When the constraints are changed, the structure of the stiffness matrix will change. For this reason, the caching of sparsity patterns is switched off.

•

The PARDISO solver is used rather than the MUMPS solver. The former is in general faster, but can be less accurate for ill-conditioned problems. For a case like this, there is no need for the additional potential accuracy.

Finally, it is worth noting that the problem of finding a stiff design can be quite different from finding a design that does not fail, 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/bracket_topology_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

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file bracket_topology_optimization_stl_parameters.txt.

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 bracket_topology_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.

In the window, collapse the node.

Global Definitions

Geometrical Parameters

Add a new parameter group and a function for imposing the forces on the two large holes. The forces correspond to a torque on the bracket.

In the window, under click .

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

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
Lmin	5 [mm]	0.005 m	Filter radius
meshsz	Lmin	0.005 m	Mesh size
meshsz2	Lmin/2	0.0025 m	Fine mesh size)
P0	2.5 [MPa]	2.5E6 Pa	Peak load intensity

Definitions

Analytic 1 (an1)

In the toolbar, click

and choose .

In the window for , type load in the text field.

Locate the section. In the text field, type F*cos(atan2(py,abs(px))).

In the text field, type F,px,py.

Locate the section. In the text field, type Pa, m, m.

In the text field, type Pa.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click

In the toolbar, click

to close the window.

Materials

Topology Link 1 (toplnk1)

In the window for , locate the section.

From the list, choose .

Solid Mechanics (solid)

Fixed Constraint 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

and choose .

Fixed Constraint 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

and choose .

Fixed Constraint 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

and choose .

Fixed Constraint 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

and choose .

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 .

Locate the section. Specify the FA vector as


0	t1
0	t2
load(-P0,Y-YC,Z)	n

In the toolbar, click

and choose .

Mesh 1

Mesh one side of the domain using swept and tetrahedral meshes.

Free Triangular 1

In the toolbar, click

and choose .

Select Boundaries 1, 4, 33, 37, 43, and 47 only.

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/2.

Swept 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Domains 1, 2, and 5–8 only.

Free Tetrahedral 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Domains 3 and 4 only.

Copy the mesh from one side to the other.

Copy Domain 1

In the window, right-click and choose .

Select Domains 1–8 only.

In the window for , locate the section.

Select the

toggle button.

Select Domains 9–16 only.

Compute a characteristic elastic strain energy.

Study 1

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Click

In the table, enter the following settings:


Load case	lg1	Weight	cg1	cg2	cg3	cg4
Load case 1	√	1.0	√	√	√	√

Global Definitions

Parameters 2

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Vfrac	0.5	0.5	Volume fraction
simpP	1	1	SIMP exponent
proj_beta	1	1	Projection slope

The characteristic strain energy is taken as double the computed value to compensate for the fact that half of the material will be removed.

Set up the density topology feature and let it control Young’s modulus.

Definitions

Density Model 1 (dtopo1)

In the window, under click .

In the window for , locate the section.

From the Rmin list, choose .

In the text field, type meshsz.

The filter radius must be larger or equal to the mesh size.

Locate the section. From the list, choose .

In the β text field, type proj_beta.

Locate the section. From the pSIMP list, choose .

In the text field, type simpP.

Locate the section. From the list, choose .

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

Prescribed Material 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Study 1

Topology Optimization

In the window, under click .

In the window for , locate the section.

In the text field, type 50.

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


Expression	Lower bound	Upper bound
comp1.dtopo1.theta_avg		Vfrac

In the window, click .

In the window for , type One Load Case in the text field.

In the toolbar, click

Use the plot to show the design during the optimization.

In the window, click .

In the window for , locate the section.

Select the check box.

From the list, choose .

In general it is best to scale the objective function with the initial value, but in this case the scale is close enough to 1 that it is unnecessary.

In the window, click .

In the window for , locate the section.

Clear the check box.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

twice.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
simpP (SIMP exponent)	1 2 3 4
proj_beta (Projection slope)	2 4 6 8

Click to expand the section. Select the check box.

In the toolbar, click

Results

One Load Case

In the window, under click .

In the window for , type One Load Case in the text field.

Threshold

In the window, expand the node, then click .

In the toolbar, click

Click the

button in the toolbar.

The resulting design is asymmetric due to the load and the two halves being poorly connected. If one of the bolt breaks, significant stiffness will be lost. Add a second load case together with a study using two load groups and four constraints giving a total of eight load cases. Run the optimization again to see what effect this has on the optimal design.

Solid Mechanics (solid)

Boundary Load 2

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

and 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 toolbar.

In the toolbar, click

to close the window.

Study 2

Step 1: Stationary

Add the eight load cases with the load groups changing most frequently. The purpose is to avoid unnecessary updates of the stiffness matrix by first changing the right hand side.

In the window for , locate the section.

Select the check box.

Click

eight times.

In the table, enter the following settings:


Load case	lg1	Weight	lg2	Weight	cg1	cg2	cg3	cg4
Load case 1	√	1.0		1.0		√	√	√
Load case 2		1.0	√	1.0		√	√	√
Load case 3	√	1.0		1.0	√		√	√
Load case 4		1.0	√	1.0	√		√	√
Load case 5	√	1.0		1.0	√	√		√
Load case 6		1.0	√	1.0	√	√		√
Load case 7	√	1.0		1.0	√	√	√
Load case 8		1.0	√	1.0	√	√	√