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Designing Mechanical Meta-Materials by Learning Equivariant Flows

Mehran Mirramezani, Anne S. Meeussen, Katia Bertoldi, Peter Orbanz, Ryan P. Adams

TL;DR

This work shows how to drastically expand the design space of a class of mechanical meta-materials known as cellular solids, by generalizing beyond translational symmetry, by transforming a reference geometry according to a divergence free flow that is parameterized by a neural network and equivariant under the relevant symmetry group.

Abstract

Mechanical meta-materials are solids whose geometric structure results in exotic nonlinear behaviors that are not typically achievable via homogeneous materials. We show how to drastically expand the design space of a class of mechanical meta-materials known as cellular solids, by generalizing beyond translational symmetry. This is made possible by transforming a reference geometry according to a divergence free flow that is parameterized by a neural network and equivariant under the relevant symmetry group. We show how to construct flows equivariant to the space groups, despite the fact that these groups are not compact. Coupling this flow with a differentiable nonlinear mechanics simulator allows us to represent a much richer set of cellular solids than was previously possible. These materials can be optimized to exhibit desirable mechanical properties such as negative Poisson's ratios or to match target stress-strain curves. We validate these new designs in simulation and by fabricating real-world prototypes. We find that designs with higher-order symmetries can exhibit a wider range of behaviors.

Designing Mechanical Meta-Materials by Learning Equivariant Flows

TL;DR

This work shows how to drastically expand the design space of a class of mechanical meta-materials known as cellular solids, by generalizing beyond translational symmetry, by transforming a reference geometry according to a divergence free flow that is parameterized by a neural network and equivariant under the relevant symmetry group.

Abstract

Mechanical meta-materials are solids whose geometric structure results in exotic nonlinear behaviors that are not typically achievable via homogeneous materials. We show how to drastically expand the design space of a class of mechanical meta-materials known as cellular solids, by generalizing beyond translational symmetry. This is made possible by transforming a reference geometry according to a divergence free flow that is parameterized by a neural network and equivariant under the relevant symmetry group. We show how to construct flows equivariant to the space groups, despite the fact that these groups are not compact. Coupling this flow with a differentiable nonlinear mechanics simulator allows us to represent a much richer set of cellular solids than was previously possible. These materials can be optimized to exhibit desirable mechanical properties such as negative Poisson's ratios or to match target stress-strain curves. We validate these new designs in simulation and by fabricating real-world prototypes. We find that designs with higher-order symmetries can exhibit a wider range of behaviors.
Paper Structure (21 sections, 5 theorems, 47 equations, 9 figures, 1 table)

This paper contains 21 sections, 5 theorems, 47 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Cellular Solids and Symmetric Shapes
  3. Symmetric Shapes from Symmetry-Preserving Flows
  4. Symmetry- and volume-preserving flows
  5. Volume preservation
  6. Symmetrization
  7. Flow construction
  8. Implementation
  9. Applications
  10. Designing under uniaxial tension
  11. Designing under uniaxial compression
  12. Related Work
  13. Limitations and Future Work
  14. Appendix
  15. The 17 Wallpaper Groups
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathbb{G}$ be a space group on $\mathbb{R}^n$. For each ${\phi\in\mathbb{G}}$, denote by $c_i(\phi)$ the linear coefficient of $b_\phi$ with respect to the shift basis vector $b_i$, that is, ${\phi(x)=A_{\phi}x+\sum_{i\leq n}c_i(\phi)b_i}$. Then is a finite subset of $\mathbb{G}$. For each ${\phi\in\mathbb{G}}$, there are unique elements ${\hat{\phi}\in\widehat{\mathbb{G}}}$ and ${\tau_\phi

Figures (9)

  • Figure 1: (a) Example of three shape functions ${s:\Omega\subset\mathbb{R}^2\to{\lbrace 0,1 \rbrace}}$, from overvelde2012compaction. (b) Real-world experiment demonstrating nonlinear behaviors of fabricated solids specified by the shape functions in (a), from overvelde2012compaction. (c) A vector field defined by a flow equivariant under the space group p4, constructed as in \ref{['sec:symmetric_shapes']}.
  • Figure 2: Examples of the reference shape $s_0$ for different space groups. The labels p1, pg, etc. refer to the crystallographic naming standard for such groups see \ref{['sec:wallpaper_groups']}.
  • Figure 3: An overview of our modeling framework implemented in JAX. An equivariant neural network parameterized by $\theta$ works in tandem with a differentiable mechanics solver to flow an initial geometry to learn cellular solids with desired functionalities from a rich set of volume- and symmetry-preserving geometries. The upper left shape is $s_0$ and the upper right shape is $s_\theta$.
  • Figure 4: Undeformed configurations (i.e., $\epsilon=0$) of three cellular solids designs with 50% volume fraction and pore shapes respecting p2gg symmetry group, and their linear stress-strain responses with the corresponding target curves during a uniaxial tension of the top edge up to $\epsilon=0.1$.
  • Figure 5: Undeformed configurations (i.e., $\epsilon=0$) of two cellular solids designs that have negative Poisson's ratios with 50% volume fraction and pore shapes respecting pg and p4 symmetry groups (first column), and their deformed configurations from simulations during a uniaxial pulling of the top edge up to $\epsilon=0.1$ (second column). Experimental realization of the same cellular solids under pulling confirms negative Poisson's ratios for both designs. The experimental measurements are also in strong numerical agreement with simulation results (third and fourth columns). Scale bars: 5cm.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: $\Gamma$ does not increase divergence
  • proof
  • ...and 1 more