Freedom and Responsibility – Unit cells – Cell Size Distribution – Parameters – Integration


A quote I often use in my course on “Design for Additive Manufacturing”, comes from the novelist Toni Morrison:

Freedom is…not having no responsibilities, it’s choosing the ones you want.

Despite bringing with it some new constraints, Additive Manufacturing (AM) has undeniably opened up the design space of what is possible. But with that freedom, and this is particularly true of cellular materials design, comes a responsibility the designer must now bear – what IS the most optimal design solution? And how can you know for sure? And all this before you even discuss manufacturability, inspection and field performance.

In this post, I wish to state what I think are the 4 most important design questions that need answering when working with cellular materials for AM. In presenting these questions, I presuppose that a cellular material design is advantageous to begin with, in comparison to say, a topology optimized design – which is a question for another time. I also refrain from discussing answers to these questions in this post, but will address these in future posts. My goal is to provide a framework for thinking to those seeking to design with cellular materials.

The 4 Questions

From a practical standpoint, when it comes to incorporating cellular material designs into structures, there are really four main questions that the designer needs to address:

  1. What specific unit cell(s) should be selected?
  2. How best should the size of the cells vary spatially?
  3. How best should the cell parameters be optimized?
  4. How best should these cells be integrated with the larger form of the structure?

I elaborate on each of these 4 below, in turn.

1. Unit Cell(s) Selection

A designer can select from, quite literally, an infinitely large list of unit cells. Indeed, a designer could even construct a cell with basic elements such as a beam or a face, instead of selecting from an available library. There are different ways to classify cellular materials, but it helps to think of cellular materials at the following four levels (the first two follow from nTopology‘s approach in their software):

  1. Dimensionality: Whether the cells occupy a volume or a surface
  2. Tessellation: The compartmentalization of space into independent volumes of a certain shape
  3. Elements: The use of beams and/or shells/faces within the tessellated space, and
  4. Topology: The actual arrangement of the elements within the tessellated space

Thus, a hexagonal honeycomb has volumetric dimensionality with a hexagonal prism tessellation with shell elements arranged along the prism edge-walls. Other examples are shown in Figure 1 to demonstrate the range of these four levels.

Figure 1. A selection of cell shapes with a 4-level classification scheme borrowed from nTopology‘s “Rule Builder”

2. Cell Size Distribution

Once a cell shape has been selected, the larger structure needs to be populated with these cells. The main concern then is one of size, and its distribution, which may be termed Cell Size Distribution (CSD), following a similar rationale employed for particles in their size distribution (PSD). With cellular materials, to a first approximation, a designer has the option of defining a regular periodic structure or a stochastic structure with some governing sizing rule – both these examples are shown in Figure 2.

Figure 2. Different strategies to distributing cell size across a volume (left) specification of a regular cell size and (right) definition of a stochastic cell size distribution governed by a sizing law (all images from nTopology)

Of course, we are not constrained to using one cell shape alone. Figure 3 demonstrates how two or more shapes may be combined to form more complex patterns for 2D structures (a full catalog of such shapes can be found in Chavey, 1989).

Figure 3. Mathematical descriptions of tessellation, shown here for 2D polygons (Wikimedia Commons, Attr: Tomruen, following Chavey, 1989)

Different shapes taken together may be varied across space in their sizes. While this is harder to do for polygons with more than 4 sides, it is easy to realize for structures constructed with 90 degree corners, as shown in Figure 4 (Casanova et al., SFF Symposium 2018, under preparation).

Figure 4. 3D printed honeycomb shapes with 90 degree corners combined in different ways to form tessellated structures (Casanova et al., under preparation, 2018)

3. Cell Parameter Optimization

A selected cell shape, in addition to being prescribed by its topology as discussed in the first question, is also prescribed by parameters describing its features. Consider the hexagonal honeycomb of a bee – as shown in Figure 5, the thickness of the walls (beams in 2D) varies across space. A typical way to represent this structure parametrically is to describe it in terms of the length (l) and thickness (t) of the walls, and for natural honeycombs, the radius of curvature at the corners. The optimization of these parameters is a crucial driver for the overall performance of the structure that is comprised of the cellular material. Consider the equation in Figure 5, for the effective stiffness E* – which goes by the cube of the ratio (l/t) and is thus highly sensitive to variation in these parameters.

Figure 5. The hexagonal honeycomb can be described in terms of the lengths of its walls, their thicknesses and further, the radius at the corner (Ack: Derek Goss – ASU, Alex Grishin – PADT)

Software, such as nTopology, permit modulation of thickness parameters in three different ways: (1) as a stipulated value applied globally, or (2) one varying spatially per a prescribed function (modifier), or (3) the solver locally optimizes the thickness of cells in response to a global load case (see Figure 6: left, middle and right, respectively). This is a crucial aspect of optimizing with cellular materials, something which exploits Additive Manufacturing’s capability to attain local tunability of structure and is difficult to achieve otherwise.

Figure 6. Three methods of prescribing cell thickness – (left) global prescription of a thickness value, (middle) thickness specified in terms of a function and (right) thickness optimized by the solver in response to local stresses (Screen captures from nTopology)

4. Integration

A crucial question that designers may be tempted to sweep under the rug is how one should integrate these cellular materials into the actual engineering structure and its adjacent parts in an assembly where one or more components contain cellular materials.

One key aspect of integration involves termination of cells at skin boundaries. This is relatively straightforward when bridging between 2 surfaces with conformal lattices, but more challenging in complex structures such as the one shown in Figure 7. How thick should the skin be? What is the best way to blend lattices with the skin? Are there regions that don’t need a skin or a sharp boundary (for bone integration in implants, for example)? While software have capabilities to enable this computationally, it is not always clear what is the best solution from a part performance and manufacturability standpoint.

Figure 7. Infilling a geometry with a lattice material raises questions around how to terminate these structures at the boundary of a structure (Screen captures from nTopology)

In Conclusion

I hope this article helps establish a framework for thinking about designing cellular materials – each of these is a rather deep question, as are the interactions between them and their relative importance when working towards a specific optimization objective. In future posts, I will attempt to unravel each of these 4 questions with the best of the available information in the literature and conducted by our own research group. Also, as alluded to in the introduction, the separation between topology optimization and cellular material design is somewhat arbitrary – and optimal solutions may well lie at the intersection of the two – which is an exciting area ripe for exploration.

To discuss these ideas further, or point out something I may have missed, please contact me. I will gladly update this post with credit if it improves the content. Thank you for reading!



  • Special thanks to the wonderful folks at nTopology for making available their Beta Analysis tool to my students and me at Arizona State University. Most of the images in this post are screenshots from their software.

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