Four Questions when Designing Cellular Materials for Additive Manufacturing

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.

Size Effects in the Characterization of Cellular Materials

Key Takeaway: The number of cells in a test specimen composed of cellular structures greatly influences its mechanical response, and the nature of this relationship depends on the unit cell design and the imposed loading condition

Fig 1. A 3D-printed Voronoi honeycomb specimen under compression

Problem Statement:

The implementation of cellular (or lattice) materials in functional parts depends on our ability to predict their behavior under different loading and environmental conditions. This is typically studied by constructing test specimens filled with the unit cell under consideration and then subjecting that specimen to controlled tests (as shown in Figure 1) to determine an effective property that represents the behavior of that specific unit cell design, in combination with the specific Additive Manufacturing (AM) process and material used to construct it. A quick evaluation of testing the identical unit cell design with specimens of varying size quickly reveals a troubling size dependence – for example, under compression, test specimens with more cells produce a higher effective stiffness, all other variables being held constant. This represents a challenge to anyone who wants to characterize these materials – in this post, I attempt to answer two questions: Why is there a size effect? and What can we do about it? This is a short summary of a much more detailed peer-reviewed paper we published in the proceedings of the 2017 SFF Symposium.

Empirical Observations

First, let us see some evidence of this size effect. In figure 2, a strong increasing trend is seen for the effective modulus as a function of the number of cells in both axial and transverse directions for a hexagonal honeycomb structure. The effective modulus (calculated from measured load-displacement response normalized by the cross-sectional area and gauge length of the cellular specimen) is a homogenized property that is intended to represent the behavior of structures composed of these unit cells. However, the plot below clearly shows such a homogenized estimation is highly dependent on the number of cells needed to construct it. Similar effects have also been demonstrated in numerical and experimental studies conducted by Onck et al. (2001), Andrews et al. (2001)Tekoglu et al. (2011), and Yang (2016).

Fig 2. Size dependence in a hexagonal honeycomb specimen – increasing cells in both axial and transverse directions has the effect of increase the calculated effective modulus


Now let us move on to the first question posed at the top of this post: Why is there a size effect with regard to characterizing cellular materials? The answer to this lies in the gap between a theoretical model that assumes that a given cell is completely surrounded by other cells in all directions ad infinitum, and the reality that experiments involve finite sized specimens that introduce two edge effects as shown in Figure 3: free surfaces that have cells that are “partially constrained,” and the interface with the testing equipment that generates a frictional interaction. Neither of these is explicitly accounted for in the calculations of a homogeneous effective property.

Fig 3. The two primary sources of the size effect – free surfaces and frictional constraints

Generating Valid Data

Given this challenge, the question now is this: What can you do about this size effect? One option is to not use a homogenization approach at all – in other words, do not seek to use such testing procedures to extract any meaningful data. Alternatives to this approach would be to use bulk properties (if it can be shown that they are representative) or model the cellular material as an assemblage of beams and junctions – which are then characterized independent of the structure they constitute. However, the reality is that homogenization approaches have a powerful advantage in that they are far less expensive computationally to implement in analysis – as a result it is important that a method be developed to extract meaningful data that is ideally size-independent or at a minimum mitigates the effect of this dependence.

Brezny and Green (1990) proposed a method to compensate for the size effect by assuming that the material in the middle of the specimen equals the bulk value, but that the outer edges have a reduced modulus. It is more common however, to study this size effect to identify the size at which the property of interest (such as modulus or strength) stabilizes – and use this size for subsequent material modeling. Towards this, Onck et al. (2001), Andrews et al. (2001) and Tekoglu et al. (2011) have all plotted measured properties as a function of an L/d or H/d ratio, where L and H represent the length (or width) and height of the specimens, and d represents an average unit cell size.

An alternative way of representing the size effect, as we did in Le et al. (2017), is to count the number of constrained and unconstrained cells in the specimen, each term defined in Figure 4. This allows for the definition of a “percentage of constrained cells” in the specimen, which has the effect of combining the L/d and H/d terms into one value.

Fig 4. Defining cells as being either partially or fully constrained

In compression tests conducted on over two dozen different sizes of hexagonal, square and triangular honeycombs each (such as shown in Figure 5 for the square honeycomb), we extracted a material modulus by resolving the globally applied load and measured displacement at the level of each individual beam and then using beam theory to develop a relationship between this applied load and measured displacement.

Fig 5. Typical load-displacement response for a square honeycomb under compression

For the square honeycomb, the relationship between this material modulus and the percentage of constrained cells is shown in Figure 6. Using a percentage of constrained cells as a criterion, it appears that the material modulus approaches saturation beyond 70%. The letters A-E correspond to groupings of specimens with the number of cells increasing in either the axial or transverse cells: Group A contained all the specimens that had only 4 axial cells (but an increasing number of transverse cells), while group B contained all the specimens that had 4  transverse cells (but an increasing number of axial cells). Group C contained all of the specimens that had 7 axial cells, and group D contained all the specimens that had 7 transverse cells. Note how the two groups bifurcate – addition of more cells in different directions do not follow the same trends (hence the need to treat L/d and H/d separately in the previously reported studies). This bifurcation of modulii converges on a smaller range of values beyond the 70% constrained condition. These trends are also observed for hexagonal and triangular honeycombs with slight differences.

Fig 6. Predicted material modulus as a function of the percentage of constrained cells (material modulus is NOT the same as an effective modulus, though the above trend holds for either)

Whether one characterizes these modulii (or any other property) on the basis of the L/d or H/d ratio, or as we propose, using the percentage of constrained cells as a metric, the fact remains that such a study is necessary to ensure the measured quantity is an accurate representation of the fundamental behavior of the unit cell by doing a size effect study specific to the unit cell of interest.

Practical Implications

So what does this all mean in practice? Quite simply, if you are a designer or an analyst relying on experimental data for modeling cellular materials – be they honeycombs or foams, ensure that the data you are using is coming from specimens that are large enough to where edge effects can be ignored. Keep in mind that the size at which this effect can be ignored varies both by the design of the unit cell and the nature of the loading conditions (compression, shear or bending). This is similar to doing a mesh refinement study in Finite Element Analysis to show that you have arrived at a solution that is, for all practical purposes, mesh-insensitive – there is no single element size that works for all geometries and conditions.

If you are studying these materials experimentally, consider alternative ways of building up these models that do not involve homogenization – in addition to introducing the size-dependence discussed here – which itself is unit cell design and loading condition dependent, homogenization models are themselves only good enough for the specific unit cell they represent. In a future post I will discuss this point in more detail and review some alternative methods to homogenization techniques.


Thank you for reading! For comments on this post, or if you’d like to collaborate on similar ideas, please contact me.