The promise of computer aided manufacturing is to make materializable structures that could not be fabricated using traditional methods. An example is 3D printed lattices, where variation in the lattice geometry and print media can define a vast spectrum of resulting material behaviour, ranging from fully flexible forms to completely stiff examples with high strength. While these “architected materials” offer huge promise for industrial applications, in practice they are difficult to generate and explore digitally, and even harder to simulate for mechanical testing. In this talk I will outline a range of approaches to the study of architected materials using machine learning. I will describe several projects using graph neural networks (GNNs) to model lattice geometry, and report on a few recent works that construct inverse models. These approaches are progress toward better methods for approximation of the material behaviour of the space of all lattice geometries, offering potential for real-time material feedback at the design stage, and a streamlined selection process for architected materials.

1. Exploring Architected Materials Using
Machine Learning

2. SIMULATION
SOFTWARE & ALGORITHMS
GEOMETRY CONSULTING
RESEARCH & DEVELOPMENT
Solving Critical
Geometry Problems
In Industry

3. Design Software for
Industrial 3D Printing

4. Introduction to architected materials/ metamaterials
Lattice compression prediction via GNNs
Inverse models for metamaterials
Industrial metamaterials & Metafold 3D
OUTLINE

5. AR C HI T EC T ED
MAT ER IALS

6. MICRO S T RU CT U RES
1. Small-scale architectures that modify
the macro-scale behaviour of an object.
Two defining principles
FCI
T

7. 2. Separation of scales. The mechanical
behaviour of microstructures is the
average behaviour of a sufficiently large
volume filled with those microstructures.
(Gibson & Ashby, 1999)
MICRO S T RU CT U RES
Two defining principles

8. MET AMAT ERIALS
Architected
Materials
Control macroscopic behaviour
through microstructure engineering

9. HO MO GENIZAT IO N
Separation of Scales
Lattices as Materials
D. Kochmann, ETH

10. Cellular/ Random
• Open cell foams
• Closed cell foams
• Spinodal topologies
• Stochastic composites
Slender Beam
• Reticular
• Truss-based
• beam-based lattices
• "Lattice"
Smooth Topologies
• Triply Periodic Minimal
Surfaces (TPMS)
• Triply Periodic Level
Surfaces (TPLS)
• Plate-based lattices
• Surface-based lattices
• "Shelllular" lattices
Lattices
Gaulon et al, 2018

11. MET AMAT ERIALS
Lattice-like structures
that go beyond nature
1” cube of 2mm unit cell
metamaterial gyroid has same
surface area as an 8.5” x 11”
sheet of paper

12. E N E RGY ME D ICA L A UTO MO TIV E
A E RO SP A CE
CO NSUME R
APPLIC AT IONS
Applications of metamaterials across industries

13. Clean Tech: Carbon Capture & Filtration
APPLIC AT IONS
More surface area: More impact
Cellular Fluidics, Dudukovic et al,
2021
Biofiltration media, Elliott et al., 2017
Intensified Carbon Capture Device,
Miramontes et al., 2020

14. CHALLENGES OF
MET AMAT ERIALS
Complexity & Fabrication
Exploration & Simulation

15. MACHINE LEAR NING F O R
AR C HI T EC T ED MAT ER I ALS

16. Using graph neural networks to approximate
mechanical response on 3D lattice structures
Elissa Ross & Daniel Hambleton
Advances in Architectural Geometry, 2020
Lattice GNNs

17. Learning the structure-property
relationship for truss lattices
THE LA T T ICE F EA
S URROGA TE PROBLEM
LATTICE
Current approach:
Finite Element Analysis (FEA)
expensive, slow
STIFFNESS
Proposed approach:
Machine Learning
fast, no cost (once trained)

18. LA T T ICE D A T A IS
HET EROGENEOUS
Typical ml models ingest only "flat" (euclidean) data
LA T T ICE D A T A IS
GEOM ETRIC, 3D
Typical ml models require data that is invariant to
transformations
CHALLENGES

19. Graph Neural Networks generalize the methods
of deep learning to graph-structured data
e.g. graphs with different
numbers of edges
GNNS A CCEPT
HET EROGENEOUS
D A T A
To learn a highly
nonlinear function
on some dataset
GOA L OF A
GNN • Graph classification/
regression
• node classification
• link prediction
GNN TA S KS
source
EUCLIDEAN
DATA
GRAPH-STRUCTURED
DATA

20. Related Work: Neural Message
Passing for Quantum Chemistry
Justin Gilmer, Samuel Schoenholz, Patrick Riley, Oriol Vinyals, George
Dahl, 2017
• Nodes have features. These features ("messages") are aggregated
("passed") according to the nodes in the 1-ring neighbourhood of a
particular node
• The k-th layer of the NN aggregates features from nodes that are k-
hops away
• Implemented in PyTorch-Geometric, a GNN library for PyTorch,
Matthias Fey and Jan Eric Lenssen, 2019: Fast graph representation
learning with PyTorch Geometric

21. Related Work: Elastic Textures for Additive
Fabrication
• Julian Panetta, Qingnan Zhou, Laigi Malomo, Nico
Pietroni, Paolo Cignoni, Denis Zorin, 2015
• parametric, tileable, printable cubic patterns with
a range of elastic material properties.

22. Lattices are built from unit cells
• Unit cell: a ‘recipe’ for a lattice
• Lattice pattern: more concise ‘recipe’
for the nodes and beams of a cubic
lattice. Divide cube in 48 equal
tetrahedra.

23. Lattice data is described by combinatorial and
geometric information
edges & nodes
What is the
graph of the
lattice pattern?
vertex nodes -- 0 DOF
edge nodes -- 1 DOF
face nodes -- 2 DOF
tet centre node -- 3 DOF
Nodes have
degrees of
freedom
Offsets
determine node
position
COMBINATORIAL GEOMETRIC

24. Using graph neural networks to approximate
mechanical response on 3D lattice structures
Elissa Ross & Daniel Hambleton
Advances in Architectural Geometry, 2020
GRA PH NEURA L NETWORK M OD EL

25. Data Representation for GNN
COMBINATORIAL
• Node features are either:
1. Offsets (these are independent of
embedding)
2. Geometric features to capture
“local stiffness”: valence, node
type, average edge length of
adjacent edges, bias toward
vertical, etc.
• Edge features: edge length, dot product
with each of the unit direction vectors
GEOMETRIC
UNIT CELL
LATTICE PATTERN
MERGED BOUNDARY
• Adjacency matrix:
What graph?

26. Datasets
One Lattice Topology
Single lattice combinatorial
type.
25K different offset positions.
92% accuracy
One Lattice Topology
with Morphing
25K morphed versions of
the One Type dataset.
86% accuracy
All Lattice Topologies
~6K different combinatorial
lattice types.
4 offset positions per type
~24K lattices.
No meaningful learning

27. RES U LTS

28. Trained model can predict compression stiffness to an
accuracy of over 92%
RESULTS
Using graph neural networks to approximate
mechanical response on 3D lattice structures
Elissa Ross & Daniel Hambleton
Advances in Architectural Geometry, 2020

29. Performance aware design
RESULTS
Using graph neural networks to approximate
mechanical response on 3D lattice structures
Elissa Ross & Daniel Hambleton
Advances in Architectural Geometry, 2020

30. Performance aware design
RESULTS
Using graph neural networks to approximate
mechanical response on 3D lattice structures
Elissa Ross & Daniel Hambleton
Advances in Architectural Geometry, 2020

31. Spinodoids
Inverse-designed spinodoid metamaterials
Siddhant Kumar, Stephanie Tan, Li Zheng and
Dennis M. Kochmann
Nature NPJ Computational Materials, 2020

32. Spinodoids
Inverse-designed spinodoid metamaterials
Siddhant Kumar, Stephanie Tan, Li Zheng and
Dennis M. Kochmann
Nature NPJ Computational Materials, 2020

33. Spinodoids
Inverse-designed spinodoid metamaterials
Siddhant Kumar, Stephanie Tan, Li Zheng and
Dennis M. Kochmann
Nature NPJ Computational Materials, 2020

34. Spinodoids
Inverse-designed spinodoid metamaterials
Siddhant Kumar, Stephanie Tan, Li Zheng and
Dennis M. Kochmann
Nature NPJ Computational Materials, 2020
Printing

35. Spinodoids
Inverse-designed spinodoid metamaterials
Siddhant Kumar, Stephanie Tan, Li Zheng and
Dennis M. Kochmann
Nature NPJ Computational Materials, 2020

36. Lattice
Dataset
Exploring the property space of periodic cellular
structures based on crystal networks
Lumpe & Stankovic
PNAS 2021
• Systematic investigation of publicly available crystallographic networks from a
structural point of view
• Unit cell catalogue, with properties based on numerical homogenization:
• Effective Young’s moduli, effective shear moduli, average connectivity,
scaling exponent indicating stretching vs. bending dominated

37. Lattice
Dataset
Exploring the property space of periodic cellular
structures based on crystal networks
Lumpe & Stankovic
PNAS 2021

38. INVERSE TRUSS
METAMATERIALS
• Bastek, Kumar, Telgen, Glaesener & Kochmann. Inverting the structure-
property map of truss metamaterials by deep learning, PNAS, 2021.
• Generated a data set with 3,000,000 samples of anisotropic unit cells
based on 262 elementary lattice topologies and affine transformations
• Inverse model to produce a family of truss unit cells that match a given
anisotropic stiffness tensor

39. INVERSE TRUSS
METAMATERIALS
• Bastek, Kumar, Telgen, Glaesener & Kochmann. Inverting the structure-
property map of truss metamaterials by deep learning, PNAS, 2021.
• Generated a data set with 3,000,000 samples of anisotropic unit cells
based on 262 elementary lattice topologies and affine transformations
• Inverse model to produce a family of truss unit cells that match a given
anisotropic stiffness tensor

40. INVERSE TRUSS
METAMATERIALS
• Indurkar, Karlapati, Shaikeea & Deshpande. Predicting deformation
mechanisms in architected metamaterials using GNN, arXiv preprint, 2022.
• Classified 17,201 diverse lattices into bending-dominated, stretching-
dominated or combined classes
• Accuracy over 90% on stretching vs. non-stretching, but only 82% for the
full classification intro three classes

41. ML O N MET AMAT ERIALS :
S U MMARY
Additional questions
Common themes

42. CHALLENG ES OF
IND U S TR IAL METAMATER IALS

43. 3D PRINTED
MET AMAT ERIALS
1. Traditional engineering software was not made
for the geometric freedoms of 3D printing
2. Developments in 3D printing hardware have
outpaced engineers’ capabilities to design,
iterate and bring products to market using AM
RESULT: Slow, frustrating, painfully inefficient
workflows that throttle the industrial adoption of
lattices & metamaterials
Stuck at the Gate

44. Detailed Parts
Large Parts
Formlabs Form 3L (SLA)
Nervous Systems
Autodesk Ember (DLP)
CU RRENT
CAPABILITIES

45. RES U LT
• Research has focused on a handful of
representatives of different cellular
materials and lattice types.
• Lattices used in industry are really lattices
as structures, not lattices as materials.
Limited use of metamaterials in industry
EOS +
Under Armour

46. MET AF O LD ’S
APPROAC H Guiding Objectives
1. Make working with lattices accessible to
facilitate reductions in global energy use
through light-weighting and other high surface
area clean technologies
2. Handle complexity needed to print lattices as
materials (high length scale separation)
3. Offer tools for both design and engineering of
metamaterial products

47. FE ATU R E S
ü Volumetric: Represent geometry using equations,
not surfaces
ü Cloud-native: query based model enables
analytics, collaboration, security
ü Hardware-integrated: patent pending technology
ü Easy: ML-powered metamaterials selection
speed up metamaterials discovery
LIG HT CYCLE S O F T W AR E

48. OU TC O MES
1. Meshless 3D printing software
eliminates computational bottleneck
2. Resolution and build volume are
decoupled through patent-pending
software-hardware integration
3. Printing more surface area opens new
possibilities in clean tech and beyond
LIG HT CYCLE S O F T W AR E

49. MET AF O LD VIS IO N
S UPPORT F RONTIER INNOV A TION
Help engineers develop remarkable new products using
metamaterials and bring them to market faster.
REDUCE GLOBA L ENERGY US E
Realize the promise of 3D printing as a transformative
manufacturing methodology for a lower carbon future.

50. T HANK YOU elissa@metafold3d.com
www.metafold3d.com