This neural network predicts material performance
One of the essential challenges of metamaterial design is being able to generate a material with specified material properties, like compression stiffness for example. It is reasonably straightforward to design new materials, then simulate or even physically test their properties. This is the forward problem. But the inverse problem — generating a material given a desired property — has remained elusive. A solution to this problem is exactly what is described in the recent paper Inverse-designed spinodoid metamaterials, by ETH researchers Siddhant Kumar, Stephanie Tan, Li Zheng and Dennis M. Kochmann (Nature NPJ Computational Materials, 2020).
After reading this paper I became a little obsessed with spinodoids. These are a class of materials that have a remarkable range of material expressions (both visually and in physical response), and a shockingly simple parametrization. It is a beautiful, clear and well-written paper that I highly recommend reading. I will only briefly summarize it here, then focus on a few of its fascinating themes.
First though, let’s look at some example geometries.
3D printing spinodoids
As soon as we saw this paper at MESH, we knew we wanted to print these geometries using a new 3D printing technology we’ve been developing. We created a pipeline that ingests the four parameters that control the spinodoid geometry, and that outputs a physical object: a little cube of spinodoid!
The results showed a clear range of material and visual effects. We focused on printing some of the benchmark materials in the paper, namely the lamellar, columnar and cubic topologies, together with the two realizations that resemble trabecular bone (scroll down for those).
The printing process identified some possible challenges with these geometries. First, printability is tricky, since the spinodoid topologies are full of overhanging geometry. As we were using a DLP-based methodology, we needed to take care when selecting our geometry.
A second challenge is that the materials also ended up being quite dense, and they lack the lines of sight through the print that make TPMS geometries so visually compelling. When we tried to print less dense materials, they become structurally less stable and fragile to print. One of the intriguing surprises from some early failures was just how organic these materials looked and felt, almost like tissue, which was frankly a bit creepy but gives some hints toward potential applications.
With the inverse model it is possible to specify exactly the density of the desired material. We are presently exploring ways to work with these materials to make them more amenable to applications like infill for 3D printed designs, for example by shifting from a solid material to a shell material.
What is a spinodoid?
Spinodal topologies are naturally occurring geometries that result from certain self-assembly processes such as phase separation in nanoporous metallic foams. Like triply periodic minimal surfaces, spinodal topologies consist of non-intersecting smooth surfaces that have nearly zero mean curvature. These topologies pose practical difficulties because they are obtained by simulating the Cahn-Hilliard phase separation process, which can take hours per simulation according to Kumar et al.
In the paper under discussion, spinodoids (“spinodal-like” materials) are proposed as an approximation to spinodal topologies that possess numerous benefits: they are succinctly parametrized by only four parameters, yet present a vast space of anisotropic materials with varying material properties (anisotropic materials are materials whose properties vary according to the direction in which force is applied).
Spinodoids are defined as a Gaussian random field (GRF), where the wave vectors are from a restricted sample on the unit sphere. The sampling of the wave vectors is biased through a non-uniform distribution function, which facilitates the creation of anisotropic materials. The resulting parameterization is given by four variables: a density, and three angles. Some example topologies are seen below.
The phase field generated through this process is then thresholded to yield either a solid or shell type topology.
The importance of non-periodicity & randomness
Over the last several years, a theme that has been emerging for me mathematically is the efficiency of randomness. When given a choice between a “human-made” discretization and a random model of any kind of a design space, the random one will probably perform better. Think a grid search will identify the best hyperparameters for your machine learning model? Random search will likely be better. Monte Carlo methods are proving to be tremendously useful in geometry processing. And the superiority of random samples can also partly account for why architectural CAD models — which are absolutely full of straight lines — are often the most disastrous meshes!
I’ve been thinking about periodic lattice geometries for many years now. At MESH we’ve worked on constructing machine learning models to predict the behaviour of lattice geometries, even after some morphing is introduced. It continues to seem to me that the answer to the structure-property riddle is just around the corner, if only we could understand a little more about the way that geometry contributes to performance.
Turns out the nonperiodic geometries are way ahead of that game! The design space of nonperiodic metamaterials like spinodoids is limitless. In addition, these nonperiodic geometries do not suffer from the same symmetry-induced sensitivity to fabrication defects that we see with periodic lattices. In this way, although it may seem counter-intuitive at first, the nonperiodicity of the spinodoid materials allows for easier creation of functionally graded materials (materials with properties that vary spatially), and creates a more robust output.
Nonperiodic for the win, yet again!
First, find a good forward model
On the machine learning side of the Inverse design paper, one of the key observations is that the hunt for an inverse model cannot start until a good forward model has been established. This is a useful principle to keep in mind, especially when the potential payoff of the inverse model is so alluring that it is tempting to jump right to that tough nut of a problem without first understanding the landscape.
Kumar et al. construct a relatively simple multi-layer perceptron model with six hidden layers, and achieve good success predicting stiffness (actually a vector of elastic moduli) from the four design parameters defining a spinodoid, on a dataset of 22K geometries. A key challenge in the inverse model is in defining an error function that can cope with non-uniqueness (there may be multiple sets of design parameters that generate materials with the same stiffness). In a clever and efficient solution, the authors use the forward model as an ingredient in this error function, which serves a dual purpose of avoiding computationally expensive finite element modelling, and providing some necessary gradients which are computed automatically by the back-propagation algorithm.
Further questions
A number of interesting opportunities are ahead in the study of spinodoids. As noted by the authors, there is no reason to limit the application of the machine learning model to stiffness, and indeed one could consider models that would predict a wide range of material properties.
From a printability perspective, I think it will be important to understand how printability might constrain the design space of materials, and what possibilities are there for enhancing printability without dramatically altering the target stiffness? For example, how might we apply nonperiodic or periodic support lattices to extend the space of printable spinodoids? This may be an especially important consideration for fragile shell-based realizations.
In a recent preprint, The local and global geometry of trabecular bone (Callens, né Betts, Müller & Zadpoor, 2020) the authors use tools from differential geometry to catalogue and deeply understand the forms of trabecular bone in great detail. I would love to see the ideas of that work inform the inverse model for spinodoids outlined in Kumar et al. to create a truly powerful candidate for approximating bone.
2021 outlook: MORE spinodoids!
This paper was a joy to read and explore, and I’ve only touched on a few highlights here. My colleagues are probably sick of hearing me talk about spinodoids, because that’s all I did for a few months of 2020. But I have no plans to stop being excited about these. At MESH we’ll continue to work with these ideas and materials, and to push their use in 3D printing, as a flexible and tuneable material.
