COMSOL 6.3 - 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 that 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, two load cases are used, but this actually corresponds to four load cases, because symmetry is imposed. This results in a 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 two cases are considered and symmetry is imposed, the optimization instead results in the design displayed in Figure 3.

Figure 3: When two cases are considered and symmetry is imposed, 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 dataset.

Figure 4: The displacement field for the optimized design after it has been transferred to another component using a Filter dataset, a mesh part, and an 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.

Continuation is used in the SIMP exponent as well as the projection slope. This is achieved with a , where the option has been enabled.

This model demonstrates how to export all domains together. This simplifies the import, but the curved boundaries will be described 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 feature, while the design free restricted is restricted using the feature. A Solid Mechanics interface represents the structural properties of the bracket, 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.

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

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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.

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

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The GMRES solver is used rather than the MUMPS solver, because it is faster.

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 and choose .

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.

In the window, under click .

The geometry should now look like that in Figure 1.

In the window, collapse the node.

Global Definitions

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