Cellular Design Strategies in Nature: A Classification

What types of cellular designs do we find in nature?

Cellular structures are an important area of research in Additive Manufacturing (AM), including work we are doing here at PADT. As I described in a previous blog post, the research landscape can be broadly classified into four categories: application, design, modeling and manufacturing. In the context of design, most of the work today is primarily driven by software that represent complex cellular structures efficiently as well as analysis tools that enable optimization of these structures in response to environmental conditions and some desired objective. In most of these software, the designer is given a choice of selecting a specific unit cell to construct the entity being designed. However, it is not always apparent what the best unit cell choice is, and this is where I think a biomimetic approach can add much value. As with most biomimetic approaches, the first step is to frame a question and observe nature as a student. And the first question I asked is the one described at the start of this post: what types of cellular designs do we find in the natural world around us? In this post, I summarize my findings.

Design Strategies

In a previous post, I classified cellular structures into 4 categories. However, this only addressed “volumetric” structures where the objective of the cellular structure is to fill three-dimensional space. Since then, I have decided to frame things a bit differently based on my studies of cellular structures in nature and the mechanics around these structures. First is the need to allow for the discretization of surfaces as well: nature does this often (animal armor or the wings of a dragonfly, for example). Secondly, a simple but important distinction from a modeling standpoint is whether the cellular structure in question uses beam- or shell-type elements in its construction (or a combination of the two). This has led me to expand my 4 categories into 6, which I now present in Figure 1 below.

Figure 1. Classification of cellular structures in nature: Volumetric – Beam: Honeycomb in bee construction (Richard Bartz, Munich Makro Freak & Beemaster Hubert Seibring), Lattice structure in the Venus flower basket sea sponge (Neon); Volumetric – Shell: Foam structure in douglas fir wood (U.S. National Archives and Records Administration), Periodic Surface similar to what is seen in sea urchin skeletal plates (Anders Sandberg); Surface: Tessellation on glypotodon shell (Author’s image), Scales on a pangolin (Red Rocket Photography for The Children’s Museum of Indianapolis)

Setting aside the “why” of these structures for a future post, here I wish to only present these 6 strategies from a structural design standpoint.

  1. Volumetric – Beam: These are cellular structures that fill space predominantly with beam-like elements. Two sub-categories may be further defined:
    • Honeycomb: Honeycombs are prismatic, 2-dimensional cellular designs extruded in the 3rd dimension, like the well-known hexagonal honeycomb shown in Fig 1. All cross-sections through the 3rd dimension are thus identical. Though the hexagonal honeycomb is most well known, the term applies to all designs that have this prismatic property, including square and triangular honeycombs.
    • Lattice and Open Cell Foam: Freeing up the prismatic requirement on the honeycomb brings us to a fully 3-dimensional lattice or open-cell foam. Lattice designs tend to embody higher stiffness levels while open cell foams enable energy absorption, which is why these may be further separated, as I have argued before. Nature tends to employ both strategies at different levels. One example of a predominantly lattice based strategy is the Venus flower basket sea sponge shown in Fig 1, trabecular bone is another example.
  2. Volumetric – Shell:
    • Closed Cell Foam: Closed cell foams are open-cell foams with enclosed cells. This typically involves a membrane like structure that may be of varying thickness from the strut-like structures. Plant sections often reveal a closed cell foam, such as the douglas fir wood structure shown in Fig 1.
    • Periodic Surface: Periodic surfaces are fascinating mathematical structures that often have multiple orders of symmetry similar to crystalline groups (but on a macro-scale) that make them strong candidates for design of stiff engineering structures and for packing high surface areas in a given volume while promoting flow or exchange. In nature, these are less commonly observed, but seen for example in sea urchin skeletal plates.
  3. Surface:
    • Tessellation: Tessellation describes covering a surface with non-overlapping cells (as we do with tiles on a floor). Examples of tessellation in nature include the armored shells of several animals including the extinct glyptodon shown in Fig 1 and the pineapple and turtle shell shown in Fig 2 below.
    • Overlapping Surface: Overlapping surfaces are a variation on tessellation where the cells are allowed to overlap (as we do with tiles on a roof). The most obvious example of this in nature is scales – including those of the pangolin shown in Fig 1.
Figure 2. Tessellation design strategies on a pineapple and the map Turtle shell [Scans conducted at PADT by Ademola Falade]

What about Function then?

This separation into 6 categories is driven from a designer’s and an analyst’s perspective – designers tend to think in volumes and surfaces and the analyst investigates how these are modeled (beam- and shell-elements are at the first level of classification used here). However, this is not sufficient since it ignores the function of the cellular design, which both designer and analyst need to also consider. In the case of tessellation on the skin of an alligator for example as shown in Fig 3, was it selected for protection, easy of motion or for controlling temperature and fluid loss?

Figure 3. Varied tessellation on an alligator conceals a range of possible functions (CCO public domain)

In a future post, I will attempt to develop an approach to classifying cellular structures that derives not from its structure or mechanics as I have here, but from its function, with the ultimate goal of attempting to reconcile the two approaches. This is not a trivial undertaking since it involves de-confounding multiple functional requirements, accounting for growth (nature’s “design for manufacturing”) and unwrapping what is often termed as “evolutionary baggage,” where the optimum solution may have been sidestepped by natural selection in favor of other, more pressing needs. Despite these challenges, I believe some first-order themes can be discerned that can in turn be of use to the designer in selecting a particular design strategy for a specific application.


This is by no means the first attempt at a classification of cellular structures in nature and while the specific 6 part separation proposed in this post was developed by me, it combines ideas from a lot of previous work, and three of the best that I strongly recommend as further reading on this subject are listed below.

  1. Gibson, Ashby, Harley (2010), Cellular Materials in Nature and Medicine, Cambridge University Press; 1st edition
  2. Naleway, Porter, McKittrick, Meyers (2015), Structural Design Elements in Biological Materials: Application to Bioinspiration. Advanced Materials, 27(37), 5455-5476
  3. Pearce (1980), Structure in Nature is a Strategy for Design, The MIT Press; Reprint edition

As always, I welcome all inputs and comments – if you have an example that does not fit into any of the 6 categories mentioned above, please let me know by messaging me on LinkedIn and I shall include it in the discussion with due credit. Thanks!

Modeling 3D Printed Cellular Structures: Approaches

How can the mechanical behavior of cellular structures (honeycombs, foams and lattices) be modeled?

This is the second in a two-part post on the modeling aspects of 3D printed cellular structures. If you haven’t already, please read the first part here, where I detail the challenges associated with modeling 3D printed cellular structures.

The literature on the 3D printing of cellular structures is vast, and growing. While the majority of the focus in this field is on the design and process aspects, there is a significant body of work on characterizing behavior for the purposes of developing analytical material models. I have found that these approaches fall into 3 different categories depending on the level of discretization at which the property is modeled: at the level of each material point, or at the level of the connecting member or finally, at the level of the cell. At the end of this article I have compiled some of the best references I could find for each of the 3 broad approaches.

1. Continuum Modeling

The most straightforward approach is to use bulk material properties to represent what is happening to the material at the cellular level [1-4]. This approach does away with the need for any cellular level characterization and in so doing, we do not have to worry about size or contact effects described in the previous post that are artifacts of having to characterize behavior at the cellular level. However, the assumption that the connecting struts/walls in a cellular structure behave the same way the bulk material does can particularly be erroneous for AM processes that can introduce significant size specific behavior and large anisotropy. It is important to keep in mind that factors that may not be significant at a bulk level (such as surface roughness, local microstructure or dimensional tolerances) can be very significant when the connecting member is under 1 mm thick, as is often the case.

The level of error introduced by a continuum assumption is likely to vary by process: processes like Fused Deposition Modeling (FDM) are already strongly anisotropic with highly geometry-specific meso-structures and an assumption like this will generate large errors as shown in Figure 1. On the other hand, it is possible that better results may be had for powder based fusion processes used for metal alloys, especially when the connecting members are large enough and the key property being solved for is mechanical stiffness (as opposed to fracture toughness or fatigue life).

Fig 1. Load-displacement curves for ULTEM-9085 Honeycomb structures made with different FDM toolpath strategies

2. Cell Level Homogenization

The most common approach in the literature is the use of homogenization – representing the effective property of the cellular structure without regard to the cellular geometry itself. This approach has significantly lower computational expense associated with its implementation. Additionally, it is relatively straightforward to develop a model by fitting a power law to experimental data [5-8] as shown in the equation below, relating the effective modulus E* to the bulk material property Es and their respective densities (ρ and ρs), by solving for the constants C and n.


While a homogenization approach is useful in generating comparative, qualitative data, it has some difficulties in being used as a reliable material model in analysis & simulation. This is first and foremost since the majority of the experiments do not consider size and contact effects. Secondly, even if these were considered, the homogenization of the cells only works for the specific cell in question (e.g. octet truss or hexagonal honeycomb) – so every new cell type needs to be re-characterized. Finally, the homogenization of these cells can lose insight into how structures behave in the transition region between different volume fractions, even if each cell type is calibrated at a range of volume fractions – this is likely to be exacerbated for failure modeling.

3. Member Modeling

The third approach involves describing behavior not at each material point or at the level of the cell, but at a level in-between: the connecting member (also referred to as strut or beam). This approach has been used by researchers [9-11] including us at PADT [12] by invoking beam theory to first describe what is happening at the level of the member and then use that information to build up to the level of the cells.

Fig 2. Member modeling approach: represent cellular structure as a collection of members, use beam theory for example, to describe the member’s behavior through analytical equations. Note: the homogenization equations essentially derive from this approach.

This approach, while promising, is beset with some challenges as well: it requires experimental characterization at the cellular level, which brings in the previously mentioned challenges. Additionally, from a computational standpoint, the validation of these models typically requires a modeling of the full cellular geometry, which can be prohibitively expensive. Finally, the theory involved in representing member level detail is more complex, makes assumptions of its own (e.g. modeling the “fixed” ends) and it is not proven adequately at this point if this is justified by a significant improvement in the model’s predictability compared to the above two approaches. This approach does have one significant promise: if we are able to accurately describe behavior at the level of a member, it is a first step towards a truly shape and size independent model that can bridge with ease between say, an octet truss and an auxetic structure, or different sizes of cells, as well as the transitions between them – thus enabling true freedom to the designer and analyst. It is for this reason that we are focusing on this approach.


Continuum models are easy to implement and for relatively isotropic processes and materials such as metal fusion, may be a good approximation of stiffness and deformation behavior. We know through our own experience that these models perform very poorly when the process is anisotropic (such as FDM), even when the bulk constitutive model incorporates the anisotropy.

Homogenization at the level of the cell is an intuitive improvement and the experimental insights gained are invaluable – comparison between cell type performances, or dependencies on member thickness & cell size etc. are worthy data points. However, caution needs to be exercised when developing models from them for use in analysis (simulation), though the relative ease of their computational implementation is a very powerful argument for pursuing this line of work.

Finally, the member level approach, while beset with challenges of its own, is a promising direction forward since it attempts to address behavior at a level that incorporates process and geometric detail. The approach we have taken at PADT is in line with this approach, but specifically seeks to bridge the continuum and cell level models by using cellular structure response to extract a point-wise material property. Our preliminary work has shown promise for cells of similar sizes and ongoing work, funded by America Makes, is looking to expand this into a larger, non-empirical model that can span cell types. If this is an area of interest to you, please connect with me on LinkedIn for updates. If you have questions or comments, please email us at info@padtinc.com or drop me a message on LinkedIn.

References (by Approach)

Bulk Property Models

[1] C. Neff, N. Hopkinson, N.B. Crane, “Selective Laser Sintering of Diamond Lattice Structures: Experimental Results and FEA Model Comparison,” 2015 Solid Freeform Fabrication Symposium

[2] M. Jamshidinia, L. Wang, W. Tong, and R. Kovacevic. “The bio-compatible dental implant designed by using non-stochastic porosity produced by Electron Beam Melting®(EBM),” Journal of Materials Processing Technology214, no. 8 (2014): 1728-1739

[3] S. Park, D.W. Rosen, C.E. Duty, “Comparing Mechanical and Geometrical Properties of Lattice Structure Fabricated using Electron Beam Melting“, 2014 Solid Freeform Fabrication Symposium

[4] D.M. Correa, T. Klatt, S. Cortes, M. Haberman, D. Kovar, C. Seepersad, “Negative stiffness honeycombs for recoverable shock isolation,” Rapid Prototyping Journal, 2015, 21(2), pp.193-200.

Cell Homogenization Models

[5] C. Yan, L. Hao, A. Hussein, P. Young, and D. Raymont. “Advanced lightweight 316L stainless steel cellular lattice structures fabricated via selective laser melting,” Materials & Design 55 (2014): 533-541.

[6] S. Didam, B. Eidel, A. Ohrndorf, H.‐J. Christ. “Mechanical Analysis of Metallic SLM‐Lattices on Small Scales: Finite Element Simulations versus Experiments,” PAMM 15.1 (2015): 189-190.

[7] P. Zhang, J. Toman, Y. Yu, E. Biyikli, M. Kirca, M. Chmielus, and A.C. To. “Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing: theory and validation,” Journal of Manufacturing Science and Engineering 137, no. 2 (2015): 021004.

[8] M. Mazur, M. Leary, S. Sun, M. Vcelka, D. Shidid, M. Brandt. “Deformation and failure behaviour of Ti-6Al-4V lattice structures manufactured by selective laser melting (SLM),” The International Journal of Advanced Manufacturing Technology 84.5 (2016): 1391-1411.

Beam Theory Models

[9] R. Gümrük, R.A.W. Mines, “Compressive behaviour of stainless steel micro-lattice structures,” International Journal of Mechanical Sciences 68 (2013): 125-139

[10] S. Ahmadi, G. Campoli, S. Amin Yavari, B. Sajadi, R. Wauthle, J. Schrooten, H. Weinans, A. Zadpoor, A. (2014), “Mechanical behavior of regular open-cell porous biomaterials made of diamond lattice unit cells,” Journal of the Mechanical Behavior of Biomedical Materials, 34, 106-115.

[11] S. Zhang, S. Dilip, L. Yang, H. Miyanji, B. Stucker, “Property Evaluation of Metal Cellular Strut Structures via Powder Bed Fusion AM,” 2015 Solid Freeform Fabrication Symposium

[12] D. Bhate, J. Van Soest, J. Reeher, D. Patel, D. Gibson, J. Gerbasi, and M. Finfrock, “A Validated Methodology for Predicting the Mechanical Behavior of ULTEM-9085 Honeycomb Structures Manufactured by Fused Deposition Modeling,” Proceedings of the 26th Annual International Solid Freeform Fabrication, 2016, pp. 2095-2106

Modeling 3D Printed Cellular Structures: Challenges

In this post, I discuss six challenges that make the modeling of 3D printed cellular structures (such as honeycombs and lattices) a non-trivial matter. In a following post, I will present how some of these problems have been addressed with different approaches.

At the outset, I need to clarify that by modeling I mean the analytical representation of material behavior, primarily for use in predictive analysis (simulation). Here are some reasons why this is a challenging endeavor for 3D printed cellular solids – some of these reasons are unique to 3D printing, others are a result of aspects that are specific to cellular solids, independent of how they are manufactured. I show examples with honeycombs since that is the majority of the work we have data for, but I expect that these ideas apply to foams and lattices as well, just with varying degrees of sensitivity.

1. Complex Geometry with Non-Uniform Local Conditions

I state the most well-appreciated challenge with cellular structures first: they are NOT fully-dense solid materials that have relatively predictable responses governed by straightforward analytical expressions. Consider a dogbone-shaped specimen of solid material under tension: it’s stress-strain response can be described fairly well using continuum expressions that do not account for geometrical features beyond the size of the dogbone (area and length for stress and strain computations respectively). However, as shown in Figure 1, such is not the case for cellular structures, where local stress and strain distributions are non-uniform. Further, they may have variable distributions of bending, stretching and shear in the connecting members that constitute the structure. So the first question becomes: how does one represent such complex geometry – both analytically and numerically?

Fig 1. Honeycomb structure under compression showing non-uniform local elastic strains [Le & Bhate, under preparation]

2. Size Effects

A size effect is said to be significant when an observed behavior varies as a function of the size of the sample whose response is being characterized even after normalization (dividing force by area to get stress, for example). Here I limit myself to size effects that are purely a mathematical artifact of the cellular geometry itself, independent of the manufacturing process used to make them – in other words this effect would persist even if the material in the cellular structure was a mathematically precise, homogeneous and isotropic material.

It is common in the field of cellular structure modeling to extract an “effective” property – a property that represents a homogenized behavior without explicitly modeling the cellular detail. This is an elegant concept but introduces some practical challenges in implementation – inherent in the assumption is that this property, modulus for example, is equivalent to a continuum property valid at every material point. The reality is the extraction of this property is strongly dependent on the number of cells involved in the experimental characterization process. Consider experimental work done by us at PADT, and shown in Figure 2 below, where we varied both the number of axial and longitudinal cells (see inset for definition) when testing hexagonal honeycomb samples made of ULTEM-9085 with FDM. The predicted effective modulus increases with increasing number of cells in the axial direction, but reduces (at a lower rate) for increasing number of cells in the longitudinal direction.

This is a significant challenge and deserves a full form post to do justice (and is forthcoming), but the key to remember is that testing a particular cellular structure does not suffice in the extraction of effective properties. So the second question here becomes: what is the correct specimen design for characterizing cellular properties?

Fig 2. Effective modulus under compression showing a strong dependence on the number of cells in the structure [Le & Bhate, under preparation]

3. Contact Effects

In the compression test shown in the inset in Figure 2, there is physical contact between the platen and the specimen that creates a local effect at the top and bottom that is different from the experience of the cells closer the center. This is tied to the size effect discussed above – if you have large enough cells in the axial direction, the contribution of this effect should reduce – but I have called it out as a separate effect here for two reasons: Firstly, it raises the question of how best to design the interface for the specimen: should the top and bottom cells terminate in a flat plate, or should the cells extend to the surface of contact (the latter is the case in the above image). Secondly, it raises the question of how best to model the interface, especially if one is seeking to match simulation results to experimentally observed behavior. Both these ideas are shown in Figure 3 below. This also has implications for product design – how do we characterize and model the lattice-skin interface? As such, independent of addressing size effects, there is a need to account for contact behavior in characterization, modeling and analysis.

Fig 3. Two (of many possible) contact conditions for cellular structure compression – both in terms of specimen design as well as in terms of the nature of contact specified in the simulation (frictionless vs frictional, for example)

4. Macrostructure Effects

Another consideration related to specimen design is demonstrated in an exaggerated manner in the slowed down video below, showing a specimen flying off the platens under compression – the point being that for certain dimensions of the specimen being characterized (typically very tall aspect ratios), deformation in the macrostructure can influence what is perceived as cellular behavior. In the video below, there is some induced bending on a macro-level.

5. Dimensional Errors

While all manufacturing processes introduce some error in dimensional tolerances, the error can have a very significant effect for cellular structures – a typical industrial 3D printing process has tolerances within 75 microns (0.003″) – cellular structures (micro-lattices in particular) very often are 250-750 microns in thickness, meaning the tolerances on dimensional error can be in the 10% and higher error range for thickness of these members. This was our finding when working with Fused Deposition Modeling (FDM), where on a 0.006″ thick wall we saw about a 10% larger true measurement when we scanned the samples optically, as shown in Figure 4. Such large errors in thickness can yield a significant error in measured behavior such as elastic modulus, which often goes by some power to the thickness, amplifying the error. This drives the need for some independent measurement of the manufactured cellular structure – made challenging itself by the need to penetrate the structure for internal measurements. X-ray scanning is a popular, if expensive approach. But the modeler than has the challenge of devising an average thickness for analytical calculations and furthermore, the challenge of representation of geometry in simulation software for efficient analysis.

Fig 4. (Clockwise from top left): FDM ULTEM 9085 honeycomb sample, optical scan image, 12-sample data showing a mean of 0.064″ against a designed value of 0.060″ – a 7% error in thickness

6. Mesostructural Effects

The layerwise nature of Additive Manufacturing introduces a set of challenges that are somewhat unique to 3D Printed parts. Chief among these is the resulting sensitivity to orientation, as shown for the laser-based powder bed fusion process in Figure 5 with standard materials and parameter sets. Overhang surfaces (unsupported) tend to have down-facing surfaces with different morphology compared to up-facing ones. In the context of cellular structures, this is likely to result in different thickness effects depending on direction measured.

Fig 5. 3D Printed Stainless Steel Honeycomb structures showing orientation dependent morphology [PADT, 2016]
For the FDM process, in addition to orientation, the toolpaths that effectively determine the internal meso-structure of the part (discussed in a previous blog post in greater detail) have a very strong influence on observed stiffness behavior, as shown in Figure 6. Thus orientation and process parameters are variables that need to be comprehended in the modeling of cellular structures – or set as constants for the range of applicability of the model parameters that are derived from a certain set of process conditions.

Fig 6. Effects of different toolpath selections in Fused Deposition Modeling (FDM) for honeycomb structure tensile testing  [Bhate et al., RAPID 2016]


Modeling cellular structures has the above mentioned challenges – most have practical implications in determining what is the correct specimen design – it is our mission over the next 18 months to address some of these challenges to a satisfactory level through an America Makes grant we have been awarded. While these ideas have been explored in other manufacturing contexts,  much remains to be done for the AM community, where cellular structures have a singular potential in application.

In future posts, I will discuss some of these challenges in detail and also discuss different approaches to modeling 3D printed cellular structures – they do not always address all the challenges here satisfactorily but each has its pros and cons. Until then, feel free to send us an email at info@padtinc.com citing this blog post, or connect with me on LinkedIn so you get notified whenever I write a post on this, or similar subjects in Additive Manufacturing (1-2 times/month).