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Stephanie Wang University of California Los Angeles May 6th, 2020 SIMULATION AND VISUALIZATION OF DUCTILE FRACTURE WITH THE MATERIAL POINT METHOD (MPM) Particle count: 77,000 Simulation time: 2 Mesh-process time: 5

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SLIDE 1

SIMULATION AND VISUALIZATION OF DUCTILE FRACTURE WITH THE MATERIAL POINT METHOD (MPM)

Stephanie Wang University of California — Los Angeles May 6th, 2020

Particle count: 77,000 Simulation time: 2 Mesh-process time: 5

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SLIDE 2

COLLABORATORS

▸ PhD Advisor: Joseph Teran, UCLA ▸ Xuchen Han, UCLA ▸ Qi Guo, UCLA ▸ Mengyuan Ding, UCLA ▸ Steven Gagniere, UCLA ▸ Leyi Zhu, University of Science and Technology of

China

▸ Theodore Gast, JIXIE EFFECTS (UCLA) ▸ Chenfanfu Jiang, University of Pennsylvania (UCLA)

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SLIDE 3

Particle count: 60,000 Simulation time: 11 Mesh-process time: —-

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SLIDE 4

Particle count: 60,000 Simulation time: 11 Mesh-process time: 5

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SLIDE 5

Particle count: 207,000 Simulation time: 16 Mesh-process time: 13

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SLIDE 6

Particle count: 207,000 Simulation time: 16 Mesh-process time: 13

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SLIDE 7

Particle count: 207,000 Simulation time: 16 Mesh-process time: 13

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SLIDE 8

OUTLINE

▸ Material Point Method (MPM) ▸ Grid-particle transfer ▸ Force computation ▸ Simulation and visualization of ductile fracture ▸ Yield surfaces ▸ Mesh-processing ▸ Discussion

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SLIDE 9

Particle count: 200,000 Simulation time: 35 Mesh-process time: 16

THE MATERIAL POINT METHOD

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SLIDE 10

▸ Particles for state ▸ Grid for computations ▸ Interpolation between particles and grid ▸ Similar to FEM: Vertices for state, Mesh for computations

ROUGH ALGORITHM

MATERIAL POINT METHOD (MPM)

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SLIDE 11

ROUGH ALGORITHM

mn

i = TransferP2G(mp)

vn

i = TransferP2G(vn p)

f n

i = ComputeForce()

˜ vn+1

i

= vn

i + ∆t

mn

i

f n

i

vn+1

p

= TransferG2P(˜ vn+1

i

) xn+1

p

= xn

p + ∆tvn+1 p

⇐ Beware!

MATERIAL POINT METHOD (MPM)

mp vn

i

xn+1

p

notation meaning when where position after forces particle velocity before forces grid mass never changes particle

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SLIDE 12

ROUGH ALGORITHM

mn

i = TransferP2G(mp)

vn

i = TransferP2G(vn p)

f n

i = ComputeForce()

˜ vn+1

i

= vn

i + ∆t

mn

i

f n

i

vn+1

p

= TransferG2P(˜ vn+1

i

) xn+1

p

= xn

p + ∆tvn+1 p

⇐ Beware!

MATERIAL POINT METHOD (MPM)

mp vn

i

xn+1

p

notation meaning when where position after forces particle velocity before forces grid mass never changes particle

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SLIDE 13

PARTICLE-GRID TRANSFER

INTERPOLATION SCHEME

▸ Compactly supported kernel function ▸ Spline: C1 (C2) piecewise-polynomial

ˆ N(x)

Quadratic Cubic

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SLIDE 14

INTERPOLATION SCHEME

▸ Tensor product: ▸ Compute weights: ▸ Partition of unity ▸ Barycentric embedding ▸ Conservation of momenta, non-increasing energy

N(x) = ˆ N(x) ˆ N(y) ˆ N(z)

PARTICLE-GRID TRANSFER

wn

ip = N(xn i xn p)

rwn

ip = rN(xn i xn p)

X

i

wn

ip = 1

X

i

wn

ipxn i = xn p

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SLIDE 15

INTERPOLATION SCHEME

TransferP2G

mn

i =

X

p

wn

ipmp

mn

i vn i =

X

p

wn

ipmpvn p

Mass Momentum

TransferG2P

vn+1

p

= X

i

wn

ip˜

vn+1

i

PARTICLE-GRID TRANSFER

Kernel at particle Kernel at node

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SLIDE 16

PIC, FLIP, APIC, RPIC, ……

mn

i =

X

p

wn

ipmp

vn

i =

1 mn

i

X

p

wn

ipmpvn p

f n

i = ComputeForce()

˜ vn+1

i

= vn

i + ∆t

mn

i

f n

i

vn+1

p

= X

i

wn

ip˜

vn+1

i

xn+1

p

= xn

p + ∆tvn+1 p

mn

i =

X

p

wn

ipmp

Dn

p =

X

i

wn

ip(xn i − xn p)(xn i − xn p)T

vn

i =

1 mn

i

X

p

wn

ipmp(vn p + Bn p(Dn p)−1(xn i − xn p))

f n

i = ComputeForce()

˜ vn+1

i

= vn

i + ∆t

mn

i

f n

i

vn+1

p

= X

i

wn

ip˜

vn+1

i

Bn+1

p

= X

i

wn

ipvn i (xn i − xn p)T

xn+1

p

= xn

p + ∆tvn p

Particle In Cell (PIC) Affine Particle In Cell (APIC)

PARTICLE-GRID TRANSFER

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SLIDE 17

▸ Particle In Cell (PIC): Harlow 1964 ▸ Fluid Implicit Particle (FLIP): Brackbill and Ruppel 1986 ▸ Affine Particle In Cell (APIC): Jiang et al. 2015 ▸ Rigid Particle In Cell (RPIC): Jiang et al. 2015 ▸ Polynomial Particle In Cell (PolyPIC): Fu et al. 2017 ▸ Extended Particle In Cell (XPIC): Hammerquist et al. 2017

PIC, FLIP, APIC, RPIC, ……

PARTICLE-GRID TRANSFER

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SLIDE 18

ROUGH ALGORITHM

mn

i = TransferP2G(mp)

vn

i = TransferP2G(vn p)

f n

i = ComputeForce()

˜ vn+1

i

= vn

i + ∆t

mn

i

f n

i

vn+1

p

= TransferG2P(˜ vn+1

i

) xn+1

p

= xn

p + ∆tvn+1 p

⇐ Beware!

MATERIAL POINT METHOD (MPM)

mp vn

i

xn+1

p

notation meaning when where velocity before forces grid position after forces particle mass never changes particle

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SLIDE 19

ROUGH ALGORITHM

mn

i = TransferP2G(mp)

vn

i = TransferP2G(vn p)

f n

i = ComputeForce()

˜ vn+1

i

= vn

i + ∆t

mn

i

f n

i

vn+1

p

= TransferG2P(˜ vn+1

i

) xn+1

p

= xn

p + ∆tvn+1 p

⇐ Beware!

MATERIAL POINT METHOD (MPM)

mp vn

i

xn+1

p

notation meaning when where velocity before forces grid position after forces particle mass never changes particle

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SLIDE 20

FORCE COMPUTATION

DEFORMATION GRADIENT

F

xa xb

Xb Xa

x = Φ(X, t) F(X, t) = ∂Φ ∂X(X, t)

Ω0 Ωt

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SLIDE 21

FORCE COMPUTATION

DEFORMATION GRADIENT

mesh-based forces: F per triangle particle-based forces: F per particle

Φ = X

e

V 0

e Ψ(Fe)

Φ = X

p

V 0

p Ψ(Fp)

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SLIDE 22

FORCE COMPUTATION

FORCE AS ENERGY GRADIENT

▸ First Piola-Kirchoff stress ▸ Total potential energy ▸ ``F is a function of x” ▸ Energy is a function of x ▸ Force can be computed from x

P(F) = ∂Ψ ∂F(F)

Φ = X

p

V 0

p Ψ(Fp)

Fn+1

p

= I + ∆t X

i

vi(rωn

ip)T

! Fn

p

fi = − ∂Φ ∂xi fi = ∂Φ ∂xi = X

p

V 0

p

✓∂Ψ ∂F(Fp(x)) ◆ (Fn

p)T rωn ip

Fn

e =

X

q

xn

q rNq(Xe)T

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SLIDE 23

FORCE COMPUTATION

HYPER-ELASTIC MODELS

▸ St. Venant Kirchhoff potential with Hencky strain ▸ (Easy for analytical plastic projection)

F = UΣVT ψ(F) = µtr((ln Σ)2) + λ 2 (tr(ln Σ))2 ∂ψ ∂F = U(2µΣ−1 ln Σ + λtr(ln Σ)Σ−1)VT

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SLIDE 24

FORCE COMPUTATION

FINITE ELEMENT ELEMENT

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SLIDE 25

FORCE COMPUTATION

FINITE ELEMENT ELEMENT

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SLIDE 26

FORCE COMPUTATION

FINITE ELEMENT ELEMENT

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SLIDE 27

FORCE COMPUTATION

FINITE ELEMENT ELEMENT

Φ = X

e

V 0

e Ψ(Fe)

Φ = X

e

V 0

e Ψ(Fe)

Fn

e =

X

q

xn

q rNq(Xe)T

Fn

e =

X

q

xn

q rNq(ξe)T

! X

q

XqrNq(ξe)T !−1

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SLIDE 28

PARTICLE MPM

FORCE COMPUTATION

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SLIDE 29

FORCE COMPUTATION

PARTICLE MPM

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SLIDE 30

FORCE COMPUTATION

PARTICLE MPM

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SLIDE 31

FORCE COMPUTATION

PARTICLE MPM

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SLIDE 32

FORCE COMPUTATION

PARTICLE MPM

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SLIDE 33

FORCE COMPUTATION

PARTICLE MPM

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SLIDE 34

FORCE COMPUTATION

PARTICLE MPM

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SLIDE 35

FORCE COMPUTATION

PARTICLE MPM

Φ = X

p

V 0

p Ψ(Fp)

Fn+1

p

= I + ∆t X

i

vi(rωn

ip)T

! Fn

p

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SLIDE 36

FORCE COMPUTATION

LAGRANGIAN MPM

Φ = X

e

V 0

e Ψ(Fe)

Fn

e =

X

q

xn

q rNq(Xe)T

f n

i =

X

q

ωn

iqf n q

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SLIDE 37

Particle count: 4,000 Simulation time: 5 Mesh-process time: 0.2

SIMULATION AND VISUALIZATION OF DUCTILE FRACTURE

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SLIDE 38

RANKINE YIELD SURFACE [MÜLLER ET AL. 2014]

▸ Constraining maximal principal stress ▸ Mode I yielding (tension) ▸ Softening rule

y(τ) = max

kuk=kvk=1 uT τv − τC ≤ 0

⌧ n+1

C

= ⌧ n

C + ↵

✓ max

kuk=kvk=1 uT ✏n+1v −

max

k˜ uk=k˜ vk=1 ˜

uT ✏tr˜ v ◆

YIELD SURFACES

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SLIDE 39

Particle count: 130,000 Simulation time: 15 Mesh-process time: 8

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SLIDE 40

VON MISES (J2) YIELD SURFACE

▸ Constraining shear stress ▸ Mode II and III yielding (shear) ▸ Softening can be added

y(τ) = kτ tr(τ)IkF τC  0

YIELD SURFACES

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SLIDE 41

τC/E = 0.5 τC/E = 0.7

τC/E = 1

Particle count: 60,000 Simulation time: 11 Mesh-process time: 4

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SLIDE 42

Particle count: 60,000 Simulation time: 11 Mesh-process time: 5

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SLIDE 43

MESH-PROCESSING

THREE STEPS OF CREATING FRACTURING MESH

▸ Fracturing topology (that evolves with time) ▸ Extrapolate positions for the added vertices ▸ Smoothing crack surface to reduce mesh-dependent noise ▸ Advantage: per-frame post-process instead of per-time-step

treatment

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SLIDE 44

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 45

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 46

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 47

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 48

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 49

MESH-PROCESSING

FRACTURING TOPOLOGY

duplicated vertices

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SLIDE 50

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 51

MESH-PROCESSING

FRACTURING TOPOLOGY

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SLIDE 52

MESH-PROCESSING

1 2 3 3 4 5 6

▸ Subdivided mesh ▸ Edge-stretching

cutting criterion

▸ Evolves with time

FRACTURING TOPOLOGY

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SLIDE 53

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 54

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 55

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 56

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 57

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 58

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 59

MESH-PROCESSING

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 60

MESH-PROCESSING

1 2 3

▸ Granular view ▸ Locally rigid motion ▸ Merging vertices

based on topology

3 4 5

EXTRAPOLATING POSITIONS FOR ADDED VERTICES

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SLIDE 61

MESH-PROCESSING

SMOOTHING CRACK SURFACE

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SLIDE 62
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SLIDE 63

MESH-PROCESSING

SMOOTHING CRACK SURFACE

1 2 3

▸ Collect all ever broken

edges

▸ Gauss-Siedel

smoothing

▸ Smooth only the

undeformed configuration

3 4 5

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SLIDE 64

DISCUSSION

LIMITATIONS AND FUTURE DIRECTIONS

▸ Crack patterns can be affected by particle sampling density, mesh

topology, grid resolution

▸ Finding appropriate parameters for edge-stretching threshold and

crack smoothing iterations

▸ Exploring different yield surfaces and flow rules

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SLIDE 65

NUMERICAL METHODS

MESH V.S. PARTICLE

Particle-based forces (grid velocity updated F) Mesh-based forces (mesh geometry updated F)

Delaunay mesh for visualization requires quality mesh for simulation has artificial fracture no artificial fracture 6-8 particles per cell 2 particles per cell automatic self-collision easy coupling with other MPM material

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SLIDE 66
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SLIDE 67

Small grid dx Large grid dx Lagrangian

Particle count: 8,000 Simulation time: 0.6 Post-process time: 0.5

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SLIDE 68

Particle count: 2,290,000 Element count: 930,000 Simulation time: 50 Mesh-process time: 5

ACKNOWLEDGEMENT

The work is supported by NSF CCF-1422795, ONR (N000141110719, N000141210834), DOD (W81XWH15-1-0147), Intel STC-Visual Computing Grant (20112360) as well as a gift from Adobe Inc.

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SLIDE 69

THANKS FOR LISTENING!

Particle count: 5,500 - 77,000 Simulation time: 0.2 - 2 Mesh-process time: 0.3 - 5