Conor McCune

ME 125 LP

Topology Optimization Final Report

Abstract

The purpose of this project is to determine how to use topology optimization in order to minimize the weight of a design while constraining the stiffness. This is useful to a wide variety of industries, as avoiding resonance is a fundamental principle of design. The basis of this project is the “Loaded Knee” optimization in COMSOL which has been adapted to fit our needs.  

Introduction

The value of this project ultimately lies in the ability to use this optimization in industry. As such, the actual “part” that we are optimizing holds little importance. That being said, in order to provide clarity to the project, I altered the geometry of the loaded knee to resemble a 2 dimensional side view of the hook on an industrial mixer. The hook my optimization is very loosely based around is shown here.I designed my hook to be made of structural steel and have a total height of 0.8 meters, width of 0.7 meters and a thickness of 2 inches. As the hook rotates, it is loaded at a specific frequency determined by the rotational speed. If this frequency matches one of the natural frequencies of the hook, then the deflection of the hook could increase until the part breaks. If I want to decrease the weight of the hook I need to remove material. I can either keep the outer boundary/shape the same and remove material inside of the boundary (internal optimization) or I can remove material by changing the shape of the hook(full optimization). This project will explore both methods of reducing weight while constrained by stiffness. Additionally I will comment on the impact that the optimizations have on the stress and deflection of the part.

Procedure

Using Comsol’s Loaded Knee example problem, I first needed to change the geometry. I increased the width of the middle horizontal bar, and added an additional rectangle on the bottom right to get the desired geometry of my hook. I then combined both domains into a single domain in order to apply a 0.025 meter inward offset to the entire geometry. This offset puts a boundary on the outside of the shape and will allow me to optimize either the entire geometry(full optimization) or only the internal structure(internal optimization) by changing the domains that the density model acts on. I then put all of the forces and boundary conditions on the geometry. The top of the hook is a fixed boundary condition because it is being held in place by the mixer. The lower arm of the hook has forces on it both vertically and horizontally from whatever is being mixed. For simplicity I put the value for both the horizontal and vertical force to be 50 lbs or 222.41 N. The weight of the arm is also a concern. The arm is made of structural steel and weighs 62.97 kg or 138.8 lbs before the optimization. I applied a gravitational volume force in order to accommodate this force. The picture below shows all of the forces on the geometry. The 50 pound boundary loads are in blue, and the gravitational force is in red. The top left horizontal edge is the fixed constraint that is not shown. You can imagine how this geometry could act as a hook for a mixer as the fixed edge follows a circular path out of and back into the page.

Figure 1: Geometry and Loading

  Now that I have the physical set-up completed, I need to set up the optimization. This optimization has the goal of minimizing the material needed, with a constraint on the strain energy. The smaller the strain energy, the larger the stiffness of the part, and therefore by constraining the strain energy we are constraining the stiffness. The strain energy constraint is a non-dimensional value comparing the strain energy of the optimized geometry to the strain energy of the initial unchanged geometry. This ratio has an upper bound I titled WsMax. If WsMax is 2 then the optimized geometry can have 2 times the strain energy of the initial system( WsInitial). Therefore, both WsMax and WsInitial are global parameters that must be defined by the user. However, the initial strain energy is unknown. In order to find that value, I added a stationary study and a probe to the entire domain, and found an initial strain energy of 0.01224 Joules. The reason this value is so small is because of the small displacement undergone by the hook. With only the weight of gravity and 50 lbs in both the horizontal and vertical directions, the hook has a very small displacement. For the actual topology module, I used the method of moving asymptotes (MMA) and constrained the maximum iterations to 50 due to the time it takes to solve. I added a parametric sweep that swept over different values of WsMax in order to see how the mass, stress, and displacement varied when I allowed the strain energy to increase. The values I chose for WsMax were 2,4, and 6. These values are the scalar multiples of the initial strain energy. I then ran this optimization two separate times. Once with the density model applied only to the inside (internal optimization), and once with the density model applied to the entire domain(full optimization) and found the mass, maximum stress, and maximum displacement for each case. The figures below show a color coded stress map on the deformed geometries. The scale of the deformation is vastly increased in order to be perceptible. The initial geometry is translucent behind the optimized geometry.

Results

Internal Optimization

Figure 2: Initial Geometry ( Strain Energy Ratio =1)

Figure 3: Internal Optimization: Strain Energy Ratio=2 

Figure 4: Internal Optimization: Strain Energy Ratio=4

Figure 5: Internal Optimization: Strain Energy Ratio=6

Table 1: Initial Geometry vs. Internal Optimization

Full Optimization

Figure 7: Full Optimization: Strain Energy Ratio =2

Figure 8: Full Optimization: Strain Energy Ratio =4

Figure 9: Full Optimization: Strain Energy Ratio=6

Table 2. Initial Geometry vs. Full Optimization

Table 3: Internal Optimization vs. Full optimization

Discussion

From Tables 1 and 2 we can see the comparison between the initial geometry and the final optimized geometries. The internal optimization(Table 1) is the most useful data in a practical sense, as the overall shape of the hook is the same as the initial design, but the mass is much lower. This sort of topology optimization is useful for any problem in which both the weight of the object, and any possible resonant frequencies is a concern. The initial hook design is made to be stiff enough such that the mixer could never excite any of the hooks' natural frequencies. However, the initial design of this hook weighed 138 lbs. In order to reduce the weight of the hook, while keeping the same shape, we turn to internal topology optimization, and remove material from the internal structure. The more material we remove, the lower the stiffness will be, and the more likely the hook is to be excited to one of its natural frequencies. However, by constraining the strain energy of the hook, we can limit how much we decrease the stiffness and therefore the natural frequency. The results in table 1 show the optimized mass for differing allowable strain energies. The maximum stress is also included to ensure the yield stress is never surpassed as well as the maximum deflection to ensure a hook would never deflect more than allowed. The results follow what we would expect. As we allow the strain energy to increase, the mass decreases, but the deflection and stress increase.

The full optimization solves a more niche problem. In this problem we do not care about the final shape of the hook, as long as it does not exceed the size and shape of the initial geometry. We want the lightest possible hook design that is still constrained by stiffness. The results follow what we would expect. Because the external geometry is no longer constrained, the optimization can output a shape with a much smaller mass for the same allowable strain energy when compared to the internal optimization(Table 3). With this small mass comes a larger deflection, and I would have expected a higher stress. However, the internal optimization experiences more stress than the full optimization. This is due to the stress concentrations that occur at the sharp corners of the design and because by decreasing the mass we are also decreasing the force of gravity on the hook. Because the full optimization can change the entire shape of the hook, it no longer has such drastic stress concentrations, and as such has a lower maximum stress.  

Conclusion

This project shows the relationship between mass, stress, displacement and strain energy. The optimization done here can be adapted to other 2D geometries in order to determine the minimum weight possible for a set shape, while maintaining the stiffness needed to avoid resonance(Internal Optimization). It can also be adapted to determine the ideal shape of an object in order to minimize its mass while constrained by stiffness(Full Optimization). Avoiding resonance is important to the design of many different parts, which makes this optimization extremely valuable to a wide range of products.