Parametric Growth Processes for Metamaterial Design

Jonàs Martínez

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Introduction

Introduction

Introduction

Mechanical Metamaterials

Auxetic Light and stiff Shape-morphing
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Jian et al. 2018 Lee et al. 2014 Coulais et al. 2016
Introduction

Mechanical Metamaterials

Functional grading Soft robotics Topology optimization
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Kumar et al. 2020 Vanneste et al. 2020 Zhu et al. 2017
Introduction

Additive Manufacturing

Fused filament fabrication Stereolithography apparatus Selective laser sintering
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Introduction

Periodic Metamaterials

  • Most widespread metamaterials

  • Efficient simulation, compact storage

  • Grading can be challenging

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Introduction

Periodic Homogenization

Heterogeneous microstructure Representative volume element Homogeneous equivalent structure
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Introduction

Stochastic Metamaterials

  • Less widespread metamaterials

  • Costly simulation

  • Implicit grading

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Introduction

Stochastic Homogenization

Approximate cut-off techniques, for side length D>0D > 0:

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D=1D=1 D=10D=10 D=100D=100

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Introduction

Parametric Growth Process

  • Each point is the origin of a cell.

  • Cells grow according to a growth law and are forbidden to overlap.

  • Growth law parameterized with distance functions.
Introduction

Parametric Growth Process: Euclidean Distance

Points Distance Growth Process
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Introduction

Parametric Growth Process: Star-shaped Distance

Points Distance Growth Process
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Introduction

Parametric Growth Process: Grading

Grading Field Result Close-up
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Introduction

Parametric Growth Process: Symmetries

One-axis reflection Three-fold rotation
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Contributions

Contributions

Contributions

Contributions

2D Periodic 2D Stochastic (auxetic) 3D Periodic
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2D Periodic

Contributions

2D Periodic 2D Stochastic (auxetic) 3D Periodic
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2D Periodic

Publication

Star-shaped metrics for mechanical metamaterial design

Jonàs Martínez, Mélina Skouras, Christian Schumacher, Samuel Hornus, Sylvain Lefebvre, and Bernhard Thomaszewski

SIGGRAPH 2019

2D Periodic

Star-shaped Set

  • Origin OO

  • Minimum radial spans aa

  • Maximum radial span bb

  • Star-shaped set S\mathcal{S}

2D Periodic

Star-shaped Distance dSd_{\mathcal{S}}

dS(p,q)=∥q−p∥f(∠(q−p))d_{\mathcal{S}}(p, q) = \frac{\left\| q - p\right\|}{f(\angle(q-p))}

where:

  • ∥⋅∥\left\| \cdot \right\| Euclidean norm

  • ∠(⋅)\angle( \cdot) Angle of vector

  • f:[0,2π]→[a,b]f: [0, 2 \pi] \rightarrow [a, b] Periodic function

2D Periodic

Star-shaped Parameterization

Interpolation over equally spaced angles

Polygonal Outline Smooth Outline
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2D Periodic

Distance vs. Growth Process

Distance based Growth Process
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2D Periodic

Distance vs. Growth Process

Distance based Growth Process
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2D Periodic

Distance vs. Growth Process

Distance based Growth Process
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2D Periodic

Discrete Growth Process Algorithm

QQ priority queue of grid cells ordered by distance dSd_{\mathcal{S}}.

  • Insert in QQ grid cells containing nuclei.

  • while QQ is not empty do

    • Pop grid cell gg from QQ
    • if gg is unlabeled then
      • Label with originating nuclei
    • Insert the 44-neighbourhood grid cells in QQ
2D Periodic

2D Point Lattices LL

Diagonal Triangular Honeycomb
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2D Periodic

Symmetries

TL\mathcal{T}_{L} Point group symmetries lattice LL
TS\mathcal{T}_{\mathcal{S}} Point group symmetries star-shaped set S\mathcal{S}
TL∩S=TL∩TS\mathcal{T}_{L \cap \mathcal{S}}=\mathcal{T}_{L} \cap \mathcal{T}_{\mathcal{S}} subgroup shared symmetries

  • The discrete growth process is (approximatively) invariant to any transformation of TL∩S\mathcal{T}_{L \cap \mathcal{S}}

  • Example:
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    TL∩S=\mathcal{T}_{L \cap \mathcal{S}}= Three-fold rotation
2D Periodic

Growth Process Algorithm: Robustness

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Threefold symmetry?
2D Periodic

Growth Process Algorithm: Robustness

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Threefold symmetry
2D Periodic

Grading

Discrete Continuous
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2D Periodic

Isotropic Materials: Three-fold Rotation Symmetry

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2D Periodic

Orthotropic Materials: One-axis Reflectional Symmetry

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2D Periodic

Printed Results

Auxetic Continuous grading Discrete grading
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2D Stochastic (auxetic)

Contributions

2D Periodic 2D Stochastic (auxetic) 3D Periodic
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2D Stochastic (auxetic)

Publication

Random Auxetic Porous Materials from Parametric Growth Processes

Jonàs Martínez

Computer-Aided Design 2021

2D Stochastic (auxetic)

Auxetic Materials

Negative Poisson ratio Applications
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Jian et al. 2018 Excellent shock absorption, fracture toughness, or vibrational absorption
2D Stochastic (auxetic)

Polymeric Auxetic Foams

Uncompressed polyurethane foam Tri-axially compressed
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Mardling et al. 2020
2D Stochastic (auxetic)

2D Random Auxetic Networks

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Reid et al. 2018 Liu et al. 2019
2D Stochastic (auxetic)

Random Materials

Seamless geometry gradation Isotropic elasticity Symmetry-breaking resilient
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Kumar et al. 2020 Tarantino et al. 2019 Portela et al. 2020
2D Stochastic (auxetic)

Our Approach

Pre-processing Porous Material
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Parametric optimization Discrete Growth Process
2D Stochastic (auxetic)

Parametric Growth Process

Points Distance S\mathcal{S} Thickness S∗\mathcal{S}^* Growth Process
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2D Stochastic (auxetic)

Cell Regularization

S\mathcal{S} S∗\mathcal{S}^* Cell Growth Non-regularized Regularized
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  • Numerical results show that cell regularization has little impact.
2D Stochastic (auxetic)

Stochastic Homogenization

  • Hooke's Law:

σ=Cϵ\sigma = C \epsilon

[σ1σ2σ12]=[c11c12c13c12c22c23c13c23c33][ϵ1ϵ22ϵ12]\begin{bmatrix} \sigma_{1} \\ \sigma_{2} \\ \sigma_{12}\end{bmatrix}=\begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{12} & c_{22} & c_{23} \\ c_{13} & c_{23} & c_{33} \end{bmatrix} \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ 2 \epsilon_{12}\end{bmatrix}

Stress σ\sigma
Strain ϵ\epsilon
Elasticity tensor CC

2D Stochastic (auxetic)

Isotropic Elasticity

  • Closest isotropic elasticity tensor CisoC_{iso} (closed-form):

min⁡c11iso,c12iso∥C−Ciso∥F\min_{c_{11}^{iso},c_{12}^{iso}} \left\lVert{C - C_{iso}}\right\rVert_{F}

  • Deviation from isotropy δiso\delta_{iso}:

δiso=∥C−Ciso∥F∥C∥F≥0\delta_{iso} = \frac{\left\lVert C - C_{iso}\right\rVert_{F}}{\left\lVert C \right\rVert_{F}} \ge 0

Homogenized elasticity tensor CC
Frobenius norm ∥⋅∥F\left\lVert \cdot \right\rVert_{F}

2D Stochastic (auxetic)

Three-fold Symmetry and Isotropy

Hypothesis
Let S\mathcal{S} and S∗\mathcal{S}^* be three-fold symmetric. The deviation from isotropy δiso\delta_{iso} tends toward 00 as the size ss of the growth process increases.

2D Stochastic (auxetic)

Three-fold Symmetry and Isotropy

Hypothesis
Let S\mathcal{S} and S∗\mathcal{S}^* be three-fold symmetric. The deviation from isotropy δiso\delta_{iso} tends toward 00 as the size ss of the growth process increases.

Intuition

  • Rotation of angle aa around origin R(a)R(a)
  • Let pp be a point (nucleus of a cell), for n∈Zn \in \mathbb{Z}:

R(2π3n)(p+S)=R(2π3n)p+R(2π3n)S=R(2π3n)p+SR(\frac{2\pi}{3}n) (p + \mathcal{S}) = R(\frac{2\pi}{3}n) p + R(\frac{2\pi}{3}n) \mathcal{S} = R(\frac{2\pi}{3}n) p + \mathcal{S}

which hints that the discrete growth process is approximatively three-fold symmetric.

In 2D, three-fold symmetry leads to linear isotropic elasticity.

2D Stochastic (auxetic)

Three-fold Symmetry and Isotropy

Hypothesis
Let S\mathcal{S} and S∗\mathcal{S}^* be three-fold symmetric. The deviation from isotropy δiso\delta_{iso} tends toward 00 as the size ss of the growth process increases.

Numerical Results

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2D Stochastic (auxetic)

Isotropic Poisson's Ratio Minimization

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2D Stochastic (auxetic)

Isotropic Poisson's Ratio Minimization

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2D Stochastic (auxetic)

Isotropic Poisson's Ratio Minimization

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2D Stochastic (auxetic)

Experimental Results

  • Laser-cut sheets made of Styrene-Butadiene Rubber (SBR).

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2D Stochastic (auxetic)

Grading Results

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2D Stochastic (auxetic)

Grading Results

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3D Periodic

Contributions: 3D Periodic

2D Periodic 2D Stochastic (auxetic) 3D Periodic
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3D Periodic

Publication

3D periodic Cellular Materials with Tailored Symmetry and Implicit Grading

Semyon Efremov, Jonàs Martínez, Sylvain Lefebvre

Symposium on Physical Modeling 2021

3D Periodic

3D Parametric Growth Process

3D star-shaped set Lattice Growth Process
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Closed Cell 3x3 Open Cell 3x3
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3D Periodic

Crystallographic Symmetries

  • Geometric transformations (e.g., rotation, reflection) that leave the crystal unchanged

  • 32 crystallographic point groups

  • Only (1,2,3,4,6)-fold rotational symmetries are possible

3D Periodic

3D Lattice Parameterization

  • Bravais lattices are periodic set of points defined as:

{ia⃗+jb⃗+kc⃗}\{i\vec{a} + j\vec{b} + k\vec{c} \}

(i,j,k)(i,j,k) any integers.
(a⃗,b⃗,c⃗)(\vec{a}, \vec{b}, \vec{c}) primitive vectors

  • Bravais lattices cover all crystal systems
3D Periodic

3D Star-shaped set parameterization

Spherical Harmonics Spherical Polyhedra
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3D Periodic

3D Symmetries

TL\mathcal{T}_{L} Point group symmetries lattice LL
TS\mathcal{T}_{\mathcal{S}} Point group symmetries star-shaped set S\mathcal{S}
TL∩S=TL∩TS\mathcal{T}_{L \cap \mathcal{S}}=\mathcal{T}_{L} \cap \mathcal{T}_{\mathcal{S}} subgroup shared symmetries

  • The discrete growth process is (approximatively) invariant to any transformation of TL∩S\mathcal{T}_{L \cap \mathcal{S}}


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    TL∩S=\mathcal{T}_{L \cap \mathcal{S}}= Hexagonal symmetry
3D Periodic

Geometry Gradation

S1\mathcal{S}_1 S1+S22\frac{\mathcal{S}_1 + \mathcal{S}_2}{2} S2\mathcal{S}_2
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3D Periodic

Numerical Homogenization

Structure Magnitude displacement Von Mises stress
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(1) Monchiet et al. A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast, 2012
(2) Willot, Fourier-based schemes for computing the mechanical response of composites with accurate local fields, 2015

3D Periodic

Geometric and Physical Symmetries

Closed cell Open cell
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3D Periodic

3D Printed Results

Triclinic material Monoclinic material
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3D Periodic

3D Printed Results: Grading

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3D Periodic

3D Printed Results: Grading

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3D Periodic

Surface Frame Structure

Rendered 3D printed
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Limitations and Future Work

Limitations and Future Work

Non-linear elasticity Growth process Beyond elasticity
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Thank You


2D Periodic code

https://github.com/mfx-inria/starshaped2d


2D Stochastic code

https://github.com/mfx-inria/auxeticgrowthprocess2d

References

Bourgeat et al. 2004, Approximations of effective coefficients in stochastic homogenization, Annales de l'I.H.P. Probabilités et statistiques
Coulais et al. 2016, Combinatorial design of textured mechanical metamaterials, Nature
Lee et al. 2014, Ultralight, Ultrastiff Mechanical Metamaterials, Science
Jian et al. 2018, Auxetic multiphase soft composite material design through instabilities with application for acoustic metamaterials, Soft Matter
Liu et al. 2019, Realizing negative Poisson's ratio in spring networks with close-packed lattice geometries, Physical Review Materials
Kumar et al. 2020, Inverse-designed spinodoid metamaterials, npj Computational Materials
Mardling et al. 2020, The use of auxetic materials in tissue engineering, Biomaterials science
Portela et al. 2020, Extreme mechanical resilience of self-assembled nanolabyrinthine materials, PNAS
Reid et al. 2018, Auxetic metamaterials from disordered networks, PNAS
Tarantino et al. 2019, Random 3D-printed isotropic composites with high volume fraction of pore-like polydisperse inclusions and near-optimal elastic stiffness, Acta Materialia
Vanneste et al. 2020, Anisotropic Soft Robots Based on 3D Printed Meso-Structured Materials: Design, Modeling by Homogenization and Simulation, IEEE Robotics and Automation Letters
Zhu et al. 2017, Two-Scale Topology Optimization with Microstructures, ACM Transactions on Graphics