Topology optimization is attracting a lot of fans! This month I review what topology optimization is and some of the practical issues it raises.

Topology optimization is not new, the origins are usually attributed to Bendsoe and Kikuchi, reference 1, from 1988. By 2000 several FEA solvers had early versions. There is rejuvenated interest now that implementations are more accessible to engineers. There is also a natural alignment sensed between topology optimization and additive manufacturing processes.

## Comparisons

Topology optimization is unique within structural optimization methods: it can literally take empty design space and produce a configuration.

Other main forms of optimization have to start from an initial configuration; either mesh or geometry based.

- Sizing optimization varies the physical properties of 1D elements (cross-sectional properties of rods/beams) and 2D elements (thickness of shells). It works directly on the FEA mesh, not geometry.
- Shape optimization comes in two flavors. The first distorts the boundary shape of a 2-D or 3-D structure by distorting the mesh. The second allows parametric variation of underlying geometry dimensions.

These methods are best suited to improving fairly mature designs. Although, parametric shape optimization can give a wider range of candidate solutions – ideal for use at an intermediate design stage. Finally, topology optimization, with it’s completely radical free-form approach is ideal for concept design studies.

## The ‘free spirit’

So topology optimization is the ‘free spirit’ of the optimization world! But what is it actually doing, and does it really deliver optimum solutions?

Mathematically, the method fills design space with continuous material, then punches voids. The FEA implementations ‘knock out’ individual elements from a fixed mesh. However, it is not practical to literally delete elements, so density and Young’s modulus of each element parent material are lowered to form a ‘chewing gum’ type material. Any element can be modified.

The objective of the optimization is usually to minimize the compliance or strain energy of the overall structure. This in effect drives up its stiffness. However, a constraint is needed to give any meaningful solution. Commonly this is the target volume. There are some practical issues here:

- It is difficult to evolve chewing gum, and steel regions, with no transition. Fake materials between these two extremes are used. There is a tradeoff between an abruptness of transition, and convergence.
- Gentle transitions are difficult to interpret. Large amounts of ‘shades of gray’ transition material will defy physical interpretation of the structural boundaries. A visually well-defined configuration with a relative density boundary of say, 0.85 is a useful. A fuzzy outline shape evolving above a value of 0.3, is nebulous.
- Without special controls the solution may ‘checkerboard’; adjacent elements flipping hard between steel and chewing gum. Hence, foam-like regions are generated which are not physically realizable.
- As the mesh is refined, more root-like or branch-like configurations occur. In the limit, the stiffest optimum solution will look very fibrous.
- Traditionally, these trends are unacceptable, but rapid prototyping may make this a useful area to explore!

## Tough on strength constraints

Strength constraint in topology optimization is a tough challenge for program developers and methods are only recently evolving to include this in a rigorous way. Issues include:

- As element stiffness reduces toward zero, local stress levels may approach a singularity.
- Intact elements will form a Lego brick-like surface. If un-smoothed stresses are used, many artificial local stress concentrations will result and require additional intact material.
- Contours of density or Young’s modulus can be plotted and smoothed to give a clear indication of the evolving configuration. However, it is very difficult to assess what stresses would be present in the surfaces of such a notional structure.

The best that can probably be achieved is to use a stress constraint as a global ‘guiding’ constraint for the optimizer. It is not a rigorous limit, as found in more traditional methods.

## The ideas machine

So what does this leave us with? Well, topology optimization is an incredibly powerful tool which can give great insights into potential configurations. It opens up ideas which are just not intuitive. However, it must be clearly seen as an ideas machine! The configuration is not particularly ‘optimum’ from a traditional strength or even stiffness perspective. Transitioning the somewhat fuzzy mathematical material boundary into a true geometric configuration will mean moving off the mathematical optimum. The transitioned geometry will need to be formally checked for adequate strength and stiffness – this may rule out the configuration. Alternatively shape and/or sizing optimization studies can flow from this.

In summary, topology optimization is always a great starting point. That is the way the tool should be used, (and heavily re-used) as we explore design space. In many ways the term light-weighting is ideal. Not optimum, but better!

## References

[1] Bendsoe and Kikuchi. Generating optimal topologies in structural design using a homogenization method. Comp. Meth. Appl. Mech. Eng. **71** (1988)