Topology Optimization of a Loaded Knee Structure

Imagine that you are designing a bracket that must carry a specified load offset down and to the side of its anchoring. As a first attempt, you try a fully solid construction like the one below.

Figure 1: Geometry of the structure with loads and constraints.

This design is clearly overdesigned and does not make efficient use of material. As a direct simulation reveals, stresses are unevenly distributed, meaning that some areas carry significantly less load than others. Provided that the stiffness of the part can be reduced to, for example, 40% of the value computed from the fully solid design without violating the specification, it is tempting to remove some material — saving both weight and costs. But what is the minimum amount of material necessary, and how should it be distributed?

This model demonstrates how you can apply the SIMP model (Solid Isotropic Material with Penalization) for structural topology optimization with COMSOL Multiphysics. Note that, in contrast to the way SIMP is usually applied, this example minimizes weight for a specified minimum stiffness, instead of maximizing stiffness for a specified maximum weight.

Model Definition

The area that may contain solid material is bounded by the initial L-shaped design; see Figure 1. The structure’s uppermost boundary is fixed while the load is carried evenly distributed over the outermost 5 cm of the boundary indicated in the figure. The material from which the structure is made is a structural steel with Young’s modulus, E0 = 200 GPa, and Poisson’s ratio, ν = 0.33. The part is to be cut from a sheet of 2 cm thickness.

In the SIMP model (see Ref. 1), the stress tensor is considered to be a function of the true Young’s modulus E0 and the material volume factor, θp, which is a function of the control variable, 0 ≤ θc ≤ 1:

Because the stiffness must for numerical reasons not vanish completely anywhere in the model, the penalized material volume factor, θp, is constrained such that 10-3 = θmin ≤ θp ≤ 1. The exponent p ≥ 1 is a penalty factor that makes intermediate densities provide less stiffness compared to what they cost in weight. The material volume factor is related to the control variable via a Helmholtz filter and a projection function based on the hyperbolic tangent function:

Here θf, β, and θβ are the filtered material volume factor, projection slope and projection point. The solution can be made independent of the mesh resolution by using a fixed value of the filter radius, Rmin, but in this example we will not change the mesh and thus apply the default filter radius based on the local mesh element size.

When there is a lower limit on the stiffness of any acceptable design, expressed equivalently as an upper bound on the total strain energy

(1)

the SIMP penalization forces θc toward either of its bounds. Increasing p therefore leads to a sharper solution. The maximum strain energy Wsmax is conveniently expressed as a factor times the lowest possible strain energy, as computed for the fully solid design in Figure 1.

Conceptually, the optimization problem is simply minimizing the amount of material used, measured as the solid fraction of the initial design domain

(2)

without violating the stiffness constraint in Equation 1. Here, A is the area of the design domain.

The Finite Element Mesh

Without regularization, problems of this type are intrinsically mesh-size dependent. For this model, generate an initial free triangular mesh using a maximum element size of 15 mm. This mesh, shown in Figure 2, is rather coarse and effectively limits the smallest possible size of details in the topology solution. When the mesh is later refined and the optimization solver restarted on the finer mesh, proper scaling of the regularization term ensures that the updated solution retains the same topology, only sharpening the details.

Figure 2: Finite element mesh of the structure.

Results and Discussion

Figure 3 shows one possible material distribution, optimal for the chosen regularization strategy. Black areas represent material and white areas represent void. Unless composite materials are considered, gray areas are unphysical so a useful optimized design should be essentially black and white.

Figure 3: Distribution of the control variable after optimization.

As can be seen in Figure 4, below, the stress is relatively evenly distributed over the solid material. Each individual member in the truss-like structure is subjected to roughly the same maximum stress, indicating that the design makes full use of the material’s stiffness. At the same time, the peak stress is not excessive: only about 3 times the mean stress in the solid areas.

Figure 4: The von Mises stress of the optimal distribution of the control variable.

Reference

1. M.P. Bendsøe and O. Sigmund, Topology Optimization Theory, Methods, and Applications, Springer, 2004.

Notes About the COMSOL Implementation

A Solid Mechanics interface is combined with a to represent the structural properties of the bracket, while the objective and constraint are defined directly in a study step. The internal strain energy is a predefined variable, solid.Ws, readily available to use as the objective function for the optimization problem.

The optimization solver is selected and controlled from the study step. For topology optimization, either the MMA, IPOPT or the SNOPT solver can be used. This example demonstrates a strategy based on using MMA with a limited number of iterations on successively finer meshes. It quickly and reliably produces trustworthy topologies showing good improvement of the objective function value. These solutions are not necessarily globally optimal, which may, however, be of less importance in practice.

Optimization_Module/Topology_Optimization/loaded_knee

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.

In the table, enter the following settings:


Name	Expression	Value	Description
F_load	10[kN]	10000 N	Applied load
h0	7.5[mm]	0.0075 m	Mesh size
WsRef	1	1	Reference strain energy
WsMaxFactor	1/0.4	2.5	Maximum allowed relative strain energy
d	2[cm]	0.02 m	Bracket thickness
beta	1	1	Projection slope

Note that the reference strain energy is initially set equal to 1 and must be updated before optimization is started.

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 loaded_knee_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.

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 .

Add a feature, which can be used to distinguish between void and solid regions. This variable is coupled back to the interface via the Young’s modulus. The filter radius should not be smaller than the mesh element size, so the default will work, but a fixed value has to be chosen to make the result of the optimization mesh independent.

Definitions

In the toolbar, click

and choose .

Topology Optimization

Density Model 1 (dtopo1)

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

In the β text field, type beta.

Add a node to compute total mass and area.

Definitions

Mass Properties 1 (mass1)

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

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

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 .

In the next steps, you fix the structure for rigid body translation and rotation by prescribing zero displacements along one of the sides, then apply the load.

Solid Mechanics (solid)

Fixed Constraint 1

In the toolbar, click

and choose .

Select Boundary 3 only.

Boundary Load 1

In the toolbar, click

and choose .

Select Boundary 6 only.

In the window for , locate the section.

From the list, choose .

Specify the Ftot vector as


0	x
-F_load	y

Mesh 1

Free Triangular 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type h0.

Click

The resulting mesh is shown in Figure 2.

Study 1

First compute only a stationary solution on which you can evaluate a reference value for the objective, as well as set up suitable plots.

In the toolbar, click

Results

Stress (solid)

Click the

button in the toolbar.

Penalized Material Volume Factor

In the window, expand the node, then click .

In the window for , type Penalized Material Volume Factor in the text field.

Surface 1

Plotting the Penalized Material Volume Factor gives a sharper and fairer picture of the topology, as seen by the interface.

In the window, expand the node, then click .

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

Locate the section. From the list, choose .

From the list, choose .

In the toolbar, click

Global Evaluation 1

In the window, expand the node.

Right-click and choose .

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

Click

Use the value from the window to update the value of the parameter WsRef. This will normalize the compliance constraint in such a way that a maximum allowed relative compliance increase can be easily prescribed.

Global Definitions

Parameters 1