The Secret to Design Super-Lightweight Components

In mathematics, an optimization problem consists on minimizing or maximizing a function taking into account some constraints.

I want to show you an example of how to apply mathematics in order to solve a problem in the real life: the design of super-lightweight components. That's possible by means of a topology optimization.

In a topology optimization, the function to be maximized is the stiffness of a component. The constraint to be satisfied is a limit on the mass.

Therefore, the result of the optimization is a component that not only is lighter but also is the stiffest possible for its weight.

That's the secret to design super-lightweight components. And they will be very strong!

Let me show you a practical example.

The first thing to do in a topology optimization is to determine the space to be optimized. That's called the design space.

Then, it's necessary to mesh the component and apply the boundary conditions (loads and constraints).

During the optimization process, it's decided which elements are really working and which are being a little bit lazy. The working elements are kept, while the lazy ones are removed. That's the way to keep the really important elements. That's Intelligent Use of Material!

Result of the topology optimization

In this case, the optimization has been carried out with a weight reduction of 70%.

Final Design

As you can see in the Finite Element Analysis of the optimized design, low-stressed areas (dark blue) are minimal, which means good use of material. This is the objective of a topology optimization, to use material only where it is necessary. There are no lazy elements. And there are no excessive working ones.

If you would like to learn more about topology optimization or want to find out how you can do it too, I invite you to visit my website:

www.topologyoptimizationforyou.jimdo.com 

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Learning From Trees - How to Reduce Notch Stresses

Bionics is the science that applies systems and methods found in nature to engineering and technology.

Trees, for example, can show us how to design optimized mechanical components by reducing notch stresses. Let's take a look at the base of a tree:

As you can see, the shape of the base is not a quarter-circle. It can be approximated using the Method of Tensile Triangles, developed by Claus Mattheck and his co-workers from Karlsruhe Institute of Technology in Germany.

The procedure to draw the tensile triangles is quite simple. The first triangle starts at the bottom with a 45º angle. This creates a new notch on top of the triangle, which can be bridged symmetrically with a second triangle. This second triangle has to start from the middle of the hypotenuse of the first triangle. The third triangle is drawn in the same way. Three tensile triangles are usually enough. The corners have to be rounded, except the lower one.

As you can see in the picture below, the method is quite accurate:

This is a biological solution to the serious problem of reduction of notch stresses. However, engineers still utilize quarter-circle transitions in order to do the same.

Quarter-circle transition

In the following pictures we can see the two options. On the left, the engineering solution: the quarter-circle transition. On the right, nature's solution: tensile triangles. Which one is the best?

Using the Finite Element Method to calculate the stresses, we can figure it out.

The quarter-circle transition shows red spots, which indicate high stressed areas. On the contrary, the transition using tensile triangles is less stressed (maximum von Mises stresses are about 20% lower than using quarter-circles).

This means that by imitating the optimized shape from the base of trees we could reduce 20% the maximum stresses in a notch, making our mechanical components not only stronger but also more durable.

We still have a lot to learn from nature!

So please treat it with care!