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Virtual Materials Testing Karel Matous College of Engineering Collegiate Associate Professor of Computational Mechanics Director of Center for Shock-Wave Processing of Advanced Reactive Materials Shock W ave - processing of Advanced Reactive

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Karel Matous ̌

Virtual Materials Testing

College of Engineering Collegiate Associate Professor of Computational Mechanics Director of Center for Shock-Wave Processing of Advanced Reactive Materials

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Department of Aerospace and Mechanical Engineering

2

High Energy Ball Milling (HEBM)

Shock

C-SWARM Verification Prediction Validation/UQ Discovery

Truly multiscale in space, time, and constitutive equations Chemo-thermo-mechanical behavior Solid-solid state transformations

Shock W ave-processing of Advanced Reactive Materials

  • Demonstration Ni/Al and Discovery c-BN systems
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SLIDE 3

Department of Aerospace and Mechanical Engineering

3

Mesoscale Microscale Nanoscale Macroscale

▪ Targeted Characteristics:

– Layer Thickness – Layer Tortuosity – Reactant Surface Area Contact

▪ Targeted Characteristics:

– Crystal Size – Crystal Shape – Crystal Orientation

▪ Targeted Characteristics:

– Porosity – Pellet Size and Shape

Ni/Al demonstration system Shock W ave-processing of Advanced Reactive Materials

▪ Targeted Characteristics:

– Particle Size – Particle Shape

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SLIDE 4

Department of Aerospace and Mechanical Engineering

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Real material Surrogate medium macroscale microscale/mesoscale multiscale analysis statistical equivalence Model reduction

Macro-scale Meso-scale shock zone transition zone inert zone Micro-scale O(0.1 m) O(0.1 mm) O(0.1 μm) Macro-continuum Micro-continuum

Hierarchical multiscale modeling concept Microstructure-Statistics- Property-Relations Ensemble averaging Image-based (Data-Driven) Modeling

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SLIDE 5

Department of Aerospace and Mechanical Engineering

4

Real material Surrogate medium macroscale microscale/mesoscale multiscale analysis statistical equivalence Model reduction

Hierarchical multiscale modeling concept Microstructure-Statistics- Property-Relations Ensemble averaging

Macro-scale Micro-scale O(0.1 m) O(0.1 μm) Macro-continuum Micro-continuum O(0.1 mm) Meso-scale

Image-based (Data-Driven) Modeling

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50 100 150 200 0.1 0.2 0.3 0.4 0.5 radius (µm) Srs Smm Sm4 Sm5 Sm6 S55

10 20 30 40 50 60 70 80 0.05 0.1 0.15 0.2 0.25 diameter (µm) volume fraction voxel pack cell

Department of Aerospace and Mechanical Engineering

pack cell

100μm

5

scan - 19123 particles cell - 1082 particles scan - 1445x1288x798 cell - 400x400x400

µm µm

100μm

2048 CPUs cp = 53.91%

C

cp = 55.27%

T

cp = 54.20%

S

9- bins

Parallel Genetic Algorithm

N=500,000

Glass beads

Image-based (Data-Driven) Modeling

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SLIDE 7

Department of Aerospace and Mechanical Engineering

Rice & Mustard - cp=0.667

µm

Resolution 69.4

6

1.006 1.5777 2.1494 2.7211 3.2928 0.0201 0.1662 0.3124 0.4585 0.6046 5 10 15 20 25 30 d ε pdf 5 10 15 20 25 30

e D (mm)

Polydisperse Systems

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SLIDE 8

Department of Aerospace and Mechanical Engineering

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Ni-Al HEBM compact (blue=Ni)

raw data, processed data

Resolution ~5 nm

FIB/SEM (ND) Nano-tomography (Argonne) FIB/SEM - 5 nm Nano-tomography - 11.8 nm

Image-based (Data-Driven) Modeling

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Department of Aerospace and Mechanical Engineering

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1µm

Nickel crystals mean 61.40 nm, variance: 6.36 nm2 Aluminum crystals mean 45.68 nm, variance: 0.66 nm2

Image-based (Data-Driven) Modeling

25 15 5 Count [1010] 660 3300 Thickness [nm] 2000 2 1

Effective Stress

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SLIDE 10

Department of Aerospace and Mechanical Engineering

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Impact Simulations in Heterogeneous Materials

Ne = 6,690,165 Dofs = 3,431,023

fixed 1.0 mm 1.0 mm 1.0 mm v = 100 m/s periodic

  • Aluminum powder
  • Voce-Kocks hardening

80 ellipsoids 60% volume fraction

||σ||

10+3 10+1 100 [MPa] 10-3 Hardening 315 240 210 [MPa] 280

Δt = 0.1 ns

  • LANL Mustang, 4800 cores
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SLIDE 11

Department of Aerospace and Mechanical Engineering

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Third-order statistics

  • LANL Mustang, 7200 cores

HS bound TPA-WR spheres TPA-WR icosahedra TPA-WR dodecahedra TPA-WR octahedra TPA-WR hexahedra TPA-WR tetrahedra ke/km cp 1 0.2 0.4 0.6 4 7 10 13 16 19 icosahedra

  • ctahedra

tetrahedra d d 2d 2d 3d 4d r1 r2 d 2d 3d 4d r2 d 2d 3d 4d r2 3d 4d q = 0°

  • Morphology is important

Image-based (Data-Driven) Modeling

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Department of Aerospace and Mechanical Engineering

Heterogeneous Layers

  • S. Xu, D. Dillard and J. Dillard

Welds From Internet

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Department of Aerospace and Mechanical Engineering

  • Cohesive modeling

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t ⟦u⟧ P P

traction-separation law

  • Cohesive law based on lower scale physics

?

t ⟦u⟧ t ⟦u⟧ t ⟦u⟧

lRUC

Multiscale Modeling of Heterogeneous Layers

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SLIDE 14

0t

q0u y

Department of Aerospace and Mechanical Engineering

  • Critical assumption: lc ≪ ℴ(L)

0ϕ(X) = X + 0u(X)

∈ Ω±

0F = 1 + ⇥X 0u(X)

Ω±

Adherends

Multiscale Cohesive Model

Micro Interface

1ϕ(X, Y ) = 0F (X)Y + 1u(Y )

∈ Θ0 F =0F + ⇤Y

1u(Y )

=1 + 1 lc 0u(X) ⇥ 0N + ⇤Y

1u(Y )

⇥ Θ0

Macro Interface: Average Deformation Gradient

0ϕ(X) ⇥ =0ϕ+ − 0ϕ− = 0u(X) ⇥

  • n Γ0

0F = 1 + 1

lc 0u(X) ⇥ ⊗ 0N

  • n Γ0

13

Matous et al., 2008 Mosby and Matous, 2014, 2015

̌

̌

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SLIDE 15

⇥X · 0P + f = 0 Ω±

0P = ∂0W

∂0F Ω± ⇥Y · 1P = 0 Θ0

1P = ∂1W

∂F Θ0 F = 1 + 1 lc 0u(X) ⇥ 0N + ⇥Y

1u(Y )

Department of Aerospace and Mechanical Engineering

Strong and W eak Forms

Macroscale Strong Form Microscale Strong Form

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0P · N = tp

  • n ∂Ωt

0u = 0u p

  • n ∂Ωu

t+ + t− = 0

  • n Γ0

Boundary Conditions Hill-Mandel Lemma

  • Microscale weak form
  • Yields closure on 0t
  • Restrictions on BC

Macroscale Weak Form

0R =

  • Ω±

0P : ⇥X(δ0u) dV

  • Ω±

f · δ0u dV

  • ∂Ωt

tp · δ0u dA +

  • Γ0

0t ·

  • δ0u

⇥ dA = 0

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SLIDE 16

inf

0u⇥ 0W(

0u ⇥ ) = inf

0F inf 1u

lc |Θ0| ⇤

Θ0 1W

0F ( 0u ⇥ ) + Y

1u

⇥ dV

Department of Aerospace and Mechanical Engineering

Hill-Mandel Lemma

No assumption on form of 0t

0t = 0N ·

1 |Θ0|

  • Θ0

1P dV

At microscale equilibrium

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Microscale Boundary Condition Admissibility

    

1u = 0

  • n ∂Θ

1u+ =1 u− || ¯

t+ = −¯ t−

  • n ∂Θ

¯ t = 0

  • n ∂Θ

1 |Θ0|

  • Θ0

Y

1u dV =

1 |Θ0|

  • ∂Θ0

1u · N Θ dA = 0

1P = ∂1W

∂F

  • F =0F +Y 1u

0t = ∂0W

∂ 0u⇥

1R = lc

|Θ0| ⇤

Θ0 1P : ⇥Y (δ1u) dV = 0

0u⇥R =

  • 0N ·

1 |Θ0| ⇤

Θ0 1P dV 0t

⇥ · ⇤ δ(0u) ⌅ = 0

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SLIDE 17

1W(F , ω) = (1 − ω)1W(F )

Department of Aerospace and Mechanical Engineering

Constitutive Response of Adhesive Layer

  • Isotropic damage law
  • Damage surface

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g(¯ Y , χt) = G( ¯ Y ) − χt ≤ 0 G( ¯ Y ) = 1 − exp ⇤ − ¯ Y − Yin p1Yin ⇥p2⌅ , H = ∂G( ¯ Y ) ∂ ¯ Y

  • Irreversible dissipative evolution equations
  • Different constitutive laws can be used

˙ ω = ˙ κH → ˙ ω = µ

  • φ(g)

⇥ ˙ χt = ˙ κH → ˙ χt = µ

  • φ(g)

⇥ ⇧ ⌅⇤ ⌃

viscous regularization

1/µ ≈ 𝜐 [s] Epoxy 𝜐-ℴ(10-6-10-2)

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Department of Aerospace and Mechanical Engineering

  • Digital Cell - 1000x1000x200 𝜈m3

Np = 4774

  • 10 % volume fraction

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  • 20 micron particles

1/2 lRUC lRUC 2 lRUC 1/2 lRUC - 70x70x200 𝜈m3 lRUC - 140x140x200 𝜈m3 2 lRUC - 280x280x200 𝜈m3 1/2 lRUC - Np = 23 lRUC - Np = 93 2 lRUC - Np = 374 Srs r [𝜈m]

50 100 150 0.2 0.4 0.6 0.8 1 Spp Spm Smp Smm 1/2 lRUC 1 lRUC 2 lRUC

lc = 200 𝜈m

1/2 lRUC

Image-Based Modeling, Heterogeneous Layers

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SLIDE 19

JusK = p Jus1K2 + Jus2K2

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  • Mixed mode loading ⟦un⟧=⟦us1⟧=⟦us2⟧=1/√2⟦us⟧

Representative Unit Cell Study

  • Ne≃12,317,628
  • Nn≃2,103,957
  • Dofs≃6,280,495

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lRUC 2 lRUC •Ne≃48,537,975

  • Nn≃8,294,617
  • Dofs≃24,758,080
  • Mean element size 1.5 𝜈m
  • ˙

0u/lc

  • = 1.00 [s−1]
  • ˙

0u/lc

  • = 0.01 [s−1]
  • ˙

0u/lc

  • = 0.10 [s−1]

0tn [MPa]

0un

  • [µm]

5 10 15 18 36 54 72

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Department of Aerospace and Mechanical Engineering

1/2 lRUC

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2 lRUC

RUC Study

ω lRUC

lcell ≈ 2 lstat lcell ≈ lstat lcell ≈ 1/2 lstat

0tn [MPa]

0un

  • [µm]

1 2 3 4 5 6 9 18 27

Shear Normal

0tn [MPa]

lcell [µm]

50 100 150 200 250 300 8 9 10 23 24 25

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Department of Aerospace and Mechanical Engineering

Multiscale Cohesive Model - Mixed Mode Loading

  • Isocontours of ω ≥ 0.999
  • 512 CPUs

20

0.0 0.2 0.4 0.6 0.8

30.0 15.0 0.00

||𝝉||

  • 2 lRUC - Np = 374

[MPa]

  • 10 % volume fraction
  • 20 micron particles
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SLIDE 22

ω

Department of Aerospace and Mechanical Engineering

Particle Diameter Effect

  • 10 % volume fraction

21

5 𝜈m

  • Smaller particles - higher strength
  • Non-monotonic fracture toughness

20 𝜈m 10 𝜈m

d = 20 µm d = 10 µm d = 5 µm

0tn [MPa]

0un

  • [µm]

2 4 6 9 18 27

Effect of constraint on Gn versus Gs

  • H. Parvatareddy and D. Dillard, IJF 1999.
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SLIDE 23

ω

Department of Aerospace and Mechanical Engineering

Particle Diameter Effect

  • 10 % volume fraction

21

5 𝜈m

  • Smaller particles - higher strength
  • Non-monotonic fracture toughness

20 𝜈m 10 𝜈m

Shear Normal

Gs [MPa·µm] Gn [MPa·µm] d [µm]

5 10 15 20 25 30 35 40 45 50 55 55 60 65 70 75 80

Effect of constraint on Gn versus Gs

  • H. Parvatareddy and D. Dillard, IJF 1999.
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Department of Aerospace and Mechanical Engineering

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X2 X1 X3

t = 10 µs

High-Performance Computing

  • Highly scalable finite strains

PGFem3D solver

  • Multiscale FE2 capability
  • Quasi-steady or transient analysis
  • Complex constitutive Eq.

Ne = 123,168,768 Nn = 28,366,848 10 Nonlinear time steps 4 Linear iterations ∆t = 1 µs v = 1 m/s

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1R = lc

|Θ0| ⇤

Θ0 1P : ⇥Y (δ1u) dV = 0

0u⇥R =

  • 0N ·

1 |Θ0| ⇤

Θ0 1P dV 0t

⇥ · ⇤ δ(0u) ⌅ = 0

Department of Aerospace and Mechanical Engineering

Hierarchically Parallel Multiscale Solver

i = i+1 Initialize Send requests to microscale Check convergence No Yes Build macroscale Assemble from micro Done

Downscaling Upscaling Key Microscale Server

No Yes Check convergence j = j+1 Build Receive request from macroscale Send result to mactoscale

0R =

  • Ω±

0P : ⇥X(δ0u) dV

  • Ω±

f · δ0u dV

  • ∂Ωt

tp · δ0u dA +

  • Γ0

0t ·

  • δ0u

⇥ dA = 0

23

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SLIDE 26

Department of Aerospace and Mechanical Engineering

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Macro-scale

  • No-slip on top/bottom
  • h= 20 mm, d = 20 mm

E = 205 GPa, ν = 0.25 320K elements in Macro Micro-scale

  • 210 x 210 x 210 µm3
  • 98 voids, 30 µm diameter

E = 3 GPa, ν = 0.29 10.2M elements in cell

  • Nonlinear hyperelastic constitutive model

5296 RUCs

Generalized Computational Theory of Homogenization

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SLIDE 27

Department of Aerospace and Mechanical Engineering

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9.43B Node, 53.75B Elements, 28.08B DOF

0.0 200 400 600 0.0 208 416 625

1 2

1 2

e

0.00 0.30 0.60 0.15 0.45

Multi-scale Simulations, PGFem3D - GCTH

he(min)=191 nm 5296 RUCs

σeq [MPa] ||0t|| [MPa]

  • Full system on LLNL Vulcan — 393,216 cores, 786,432 threads
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SLIDE 28

Department of Aerospace and Mechanical Engineering

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micro macro micro micro MPI_COMM_WORLD mm_inter micro_all

migrate Load-balancing for microscale simulations

  • Time-based metrics for adaptation, load-

balancing heuristics

  • Non-blocking data migration among

servers/computing nodes

  • Overlay computations with data migration

Ideal 512 RUC 1k = 1024

Speedup

  • No. of cores (servers)

4k (8) 8k (16) 16k (32) 32k (64) 64k (128) 128k (256) 256k (512) 1 2 4 8 16 32 64

RUC (512 cores) 1.46M Elements 262K Nodes 773K DOF

  • LLNL Vulcan
  • 262,114 cores

Multi-scale Simulations, PGFem3D - GCTH

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SLIDE 29

Department of Aerospace and Mechanical Engineering

27

F i x e d

u

2 mm 40 mm 10 mm 5 mm

Macroscale

  • Mode I loading

E = 15 GPa, ν = 0.25 10K elements 322 cohesive elements Microscale

  • 250 x 250 x 125 µm3
  • 40 voids, 40 µm diameter

E = 5 GPa, ν = 0.34 249K elements in RUC

322 RUCs

  • 80M Elements, 42.5M DOFs

Multi-scale Simulations, PGFem3D - GCTH

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SLIDE 30

Department of Aerospace and Mechanical Engineering

28

Fully Coupled Multi-scale DCB Failure: Mode-I

||0t|| [MPa]

1

σeq [MPa]

2

  • Numerically resolve O(105) scales (1 cm to 100 nm)

1 2 25 50 75 100 15 30 45 60

ω

0.0 1.0

5 10 15 20 25 0.5 1 1.5 2

Opening Disp. [µ m] Force [kN]

Opening Displacement [µm] Force [kN]

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SLIDE 31

Department of Aerospace and Mechanical Engineering

29

Nonlinear Manifold-based Reduced Order Model

Digital Database

q

1 N

{ }

Isomap — Generalizes the Multidimensional Scaling Reproducing kernel map for reconstruction map Neural Network for input to reduce space map

Input parameters Manifold RUC simulations

Reduced space

  • Homogenization and Localization capabilities
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SLIDE 32

Department of Aerospace and Mechanical Engineering

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0U = 0λ1(0e1 ⊗ 0e1) + 0λ2(0e2 ⊗ 0e2) + 0λ3(0e3 ⊗ 0e3),

Generating parameters — 6 HEALPix grid

Loading Case Description

  • No. Simulations

Mode 1

0λ1 ≥ 0, 0λ2 ≥ 0, 0λ3 ≥ 0

4032 Mode 2

0λ1 ≤ 0, 0λ2 ≥ 0, 0λ3 ≥ 0

4032 Mode 3

0λ1 ≤ 0, 0λ2 ≤ 0, 0λ3 ≥ 0

4032 Mode 4

0λ1 ≤ 0, 0λ2 ≤ 0, 0λ3 ≤ 0

4032

Loading Modes

Ne = 486,051, DOFs = 250,035

cp = 0.4 95 particles

F = 0F (X) + rY 1u(Y )

Nonlinear Manifold-based Reduced Order Model Deformation gradient

  • D = 9 × Ne = 4,374,459
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SLIDE 33

Department of Aerospace and Mechanical Engineering

31 Mode − 4 Mode − 3 Mode − 2 Mode − 1 Residual Variance Dimension

1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

  • Reduced dimension

Pair-wise distance ξi and ξj Neighborhood graph G on M Geodesic distances Geodesic minimal spanning tree ➝ d - reduced dimension Low dimensional embedding, A ➝ ζi - reduced vectors Inverse map, f†N, link ζi and ξi Neural Network, link ηi and ζi

Digital Database

q

1 N

{ }

Nonlinear Manifold-based Reduced Order Model

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SLIDE 34

Department of Aerospace and Mechanical Engineering

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MNROM — V erification

ROM FEM Frequency [%] kF kF

1.42 1.67 1.92 2.17 2.42 2 4 6 8 10 12 14

403 leave-one-out cases for each Mode

Mode − 4 Mode − 3 Mode − 2 Mode − 1

Frequency [%]

0Em [%]

0.25 0.5 0.75 1 1.28 10 20 30 40 50 60

Er = kF ROM

r

F FEM

r

kF kF FEM

r

kF ⇥ 100 [%],

0Er =

1 Θ0 R

Θr ErdΘ

k0F rkF ,

  • Averaged error in matrix
  • Local distribution of F
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SLIDE 35

Department of Aerospace and Mechanical Engineering

33

MNROM — Localization Characteristics

  • Query point

Almansi strain

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SLIDE 36

Department of Aerospace and Mechanical Engineering

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MNROM — Homogenization Characteristics

FEM ROM k ˆ Ck

0WC( ˆ

C) [MPa] 1.735 1.745 1.755 1.765 1.775 0.2 0.4 0.6 0.8 1 1.2 1.4

FEM ROM J

0WJ(J) [MPa]

0.78 0.88 0.98 1.08 1.18 1.28 1 2 3 4

inf

0u

0W(0u) = inf

0u inf 1u

1 |Θ0| Z

Θ0 1W(0F + r

Y

1u) dΘ.

Source κ [MPa] µ10 [MPa] µ01 [MPa] MNROM 105 13.45 13.45 FEM 94 12.40 12.40

  • Median error 10%
  • 400 Query points

Effective properties Homogenized potentials

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SLIDE 37

Department of Aerospace and Mechanical Engineering

Real material Surrogate medium macroscale microscale/mesoscale multiscale analysis statistical equivalence Testing inside scanner Numerical analysis Macroscale Validation Model reduction Mesoscale Validation

3µm resolution

Modeling with Co-Designed Experiments

35

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SLIDE 38

Department of Aerospace and Mechanical Engineering

Microtomography In Situ Testing

Displacment [mm] Force [N] 1 2 3 4 5 6 10 20 30 40 50 60 70 80 Average Loading Average Unloading

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Material System

200-300 µm glass beads Silicone rubber matrix Pure rubber - Calibration Np,cylinder ≈ 35,000 Np,VO = 11,686 cp = 0.505

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SLIDE 39

5 mm

20 40 60 0.01 0.02 0.03 0.04 0.05 0.06 0.07 radius (µm) pdf

rmax 106 µm ravg ~25 µm

2 4 6 50 100 150 200 250 300 Displacment [mm] Force [N] Loading/Unloading 1 Loading/Unloading 2 Relaxation Point Image

Department of Aerospace and Mechanical Engineering

Microtomography In Situ Testing

voids after debonding

37

Nv = 17,008

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SLIDE 40

Department of Aerospace and Mechanical Engineering

“Virtual” FE2 Micro-computer Tomography

3µm resolution

38

1x1x1 cm3 = ℴ(1012) voxels detectability ~ 1 micron 1x1x1 mm3 = ℴ(109) elem. mean element size ~ 1 micron

  • 1000 RUCs
  • Trillion number of elements
  • Billion number of equations
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SLIDE 41

Department of Aerospace and Mechanical Engineering

39

Conclusions

  • Thanks to Colleagues, Research Staff, Students

3D is a must Microstructure should be respected Link with materials science is important Statistics and data learning is needed Computational homogenization is promising High-performance computing is called for Nonlinear model reduction can make an impact Co-designed simulations and experiments are required

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SLIDE 42

Karel Matous

College of Engineering Collegiate Associate Professor of Computational Mechanics Director of Center for Shock-Wave Processing of Advanced Reactive Materials Department of Aerospace & Mechanical Engineering University of Notre Dame

367 Fitzpatrick Hall of Engineering Notre Dame, IN 46556 Email: kmatous@nd.edu www.nd.edu/~kmatous www.cswarm.nd.edu

̌