anisompm: animating anisotropic damage...

AnisoMPM: Animating Anisotropic Damage Mechanics

Copyright of figures and other materials in the paper belongs original authors.

Presented by Ki-hoon Kim

2020.09. 01

Computer Graphics @ Korea University

J. Wolper et al.[University of Pennsylvania]SIGGRAPH 2020

Ki-hoon Kim | 2020-09-01 | # 2Computer Graphics @ Korea University

• Introduction

• Related Work

• Paper Summary

• Anisotropic Damage(CDM)

• AnisoMPM Spatial Discretization

• Anisotropic Elasticity

• Inextensibility

• Results&Discussion

Contents

Introduction


• Fracture follows us through our everyday lives

▪ But not all materials break in the same way

• Fracture is a notoriously difficult phenomenon to model

▪ There already exist many approach to animate isotropic fracture

▪ Most existing approaches to fracture focus on fracture mechanics

• We propose to focus on continuum damage mechanics

Introduction


• Core: the development of AnisoMPM

▪ Robust and general MPM for animating the dynamic fracture

▪ AnisoMPM is easy to implement in existing solver

• We use hyper-elasticity formulation based on QR-decomposition

▪ Much more efficient than SVD models

▪ Simple to implement

▪ Robust under extreme deformations

▪ Lacks robust treatments for materials with extremely stiff fibers

• Novel discretization for embedded directional inextensibility

Contribution

Related Work


• Modeling inelastic deformation: viscoelasticity, plasticity, fracture[D. Terzopoulos and K. Fleischer/SIGGRAPH 1988]

• Graphical modeling and animation of ductile fracture[J. O’Brien et al./SIGGRAPH 2002]

Physically Modeling Fracture


• A phase-field model for fracture in biological tissues[A. Raina and C. Miehe/Biomechanics and modeling in mechanobiology 2016]

Anisotropic Damage&Elasticity


• Finite element analysis of composite sheet-forming process[O’Bradaigh and Pipes/Composites Manufacturing 1991]

• Fast Simulation of Inextensible Hair and Fur[Müller et al./ VRIPHYS 2012]

Inextensibility


• Application of a particle-in-cell method to solid mechanics[Sulsky et al./ Computer Physics Communications 1995]

• A Material Point Method for Snow Simulation[Stomakhin et al./SIGGRAPH 2013]

Material Point Method


• CD-MPM: Continuum damage material point methods for dynamic fracture animation[J. Wolper et al./SIGGRAPH 2019]

Material Point Method(Cont.)

Paper Summary


Anisotropic Damage(CDM)

• A phase-field model for fracture in biological tissues[A. Raina and C. Miehe/Biomechanics and modeling in mechanobiology 2016]

▪ Derive local damage evolution

• From

• To

▪ Anisotropic Damage

𝛾𝑙0 𝑑, 𝛻𝑑 =𝑑2

2+

𝑙0

2𝛻𝑑 2𝑑V Eq (2)

ሶ𝑑 =1

𝜂< 1 − 𝑑 ෩𝐷 − 𝑑 − 𝑙0

2Δ𝑑 > Eq (6)


• Traditional MPM + Anisotropic Damage(CDM)

▪ Explicit and Implicit Damage Integration(Evolution)

AnisoMPM Spatial Discretization


• Make anisotropic Elasticity with QR-Elasticity

Anisotropic Elasticity


• With High-stiffness & Our Implicit Methods, suffer Locking Effect

• We present an embedded directed inextensibility solver

Inextensibility


Results

Anisotropic Damage(CDM)

A phase-field model for fracture in biological tissues[A. Raina and C. Miehe/Biomechanics and modeling in mechanobiology 2016]


• We define the crack phase-field as a set of damage 𝑑 ∈ [0,1]

▪ 𝑑 = 0, for unbroken material

▪ 𝑑 = 1, for fully broken material

• Spatially regularized crack surface function Γ𝑙0(𝑑)

▪ 𝑙0 is a length scale parameter

▪ 𝛾𝑙0 is a crack surface density function

▪ Ω0 is initial material space

Geometric Crack Modeling

Regularized crack surface function

Γ𝑙0 𝑑 = Ω0 𝛾𝑙0 𝑑, 𝛻𝑑 𝑑𝑉 Eq (1)


• Crack density per unit volume is

▪ This function is the key to modeling crack propagation

• The minimization principle of diffuse crack topology gives an expression for the regularized crack phase field as:

▪ Where 𝑊Γ = {𝑑|𝑑 𝑋, 𝑡 = 1 𝑎𝑡 𝑋 ∈ Γ 𝑡 }


Crack Density and Crack condition

𝛾𝑙0 𝑑, 𝛻𝑑 =𝑑2

2+

𝑙0

2𝛻𝑑 2𝑑V Eq (2)

𝑑 𝑋, 𝑡 = arg{ inf𝑑∈𝑊Γ

Γ𝑙0 𝑑 } Eq (3)


• The evolution of the damage field can be formulated as a function of the damage

▪ 𝜂 is a mobility constant that physically controls the viscosity of the crack evolution

▪ 1 − 𝑑 ෩ℋ is local crack driving force

▪ 𝐷𝑐 = 𝑑 − 𝑙02Δ𝑑 is a geometric resistance.

Δ is Laplacian(𝛻2)


Evolution of damage field

𝜂 ሶ𝑑 = 1 − 𝑑 ෩ℋ − 𝐷𝑐 Eq (4)


• We must ensure that the cracks are irreversible

▪ ሶΓ𝑙0 𝑑 ≥ 0

• Bounding the phase field: 𝑑 ∈ [0,1]

• Ensure a positive local crack growth: ሶ𝑑 ≥ 0

• Enforcing a positive driving force: ෩ℋ > 0▪ For 𝑑 = 0, ෩ℋ = 0

▪ For 𝑑 = 1, ෩ℋ = ∞

• 𝑠 is time interval [0, 𝑡]

• 𝑡 is current time

• ෩ℋ is simply defined to be the maximum value of ෩𝐷

▪ Over all material states during the simulation history


Local crack driving force modeling

ℋ 𝑋, 𝑡 = max𝑠∈[0,𝑡]

෩𝐷 𝑠𝑡𝑎𝑡𝑒 𝑋, 𝑠 ≥ 0 Eq (5)


• We can design crack driving force ෩𝐷

▪ The 3rd constraint is already fulfilled by taking the maximum

▪ 1st and 2nd constraints also require that

• ෩𝐷 = 0, for an unbroken state

• ෩𝐷 = ∞, for a broken state

• Our local damage evolution equation is

▪ < 𝑥 >≔𝑥+ 𝑥

2as the Macauley bracket


Local damage evolution equation

ሶ𝑑 =1

𝜂< 1 − 𝑑 ෩𝐷 − 𝑑 − 𝑙0

2Δ𝑑 > Eq (6)


• Add anisotropy to our damage model

▪ 𝜎+ is tensile portion of the Cauchy stress

• 𝜎+ = σ < 𝜎𝑖 > 𝐧𝑖 ⊗𝐧𝑖

• 𝐧𝑖 is eigenvector of Cauchy stress tensor

• 𝜁 controls the slope of the driving force

▪ We keep it equal to 1 for all explorations

▪ 𝜎𝑐 critical stress threshold

▪ 𝐴 is defined tensor for anisotropic fiber direction

Anisotropic Crack Driving Force

Crack Driving State Function

෩𝐷 = 𝜁 < Φ 𝜎+ − 1 > Eq (7)

Φ 𝜎+ =1

𝜎𝑐2 (𝐴𝜎

+: 𝜎+𝐴) Eq (8)


• Let 𝐚1 be the principal fiber direction, and 𝐚2 be the secondary fiber direction.

• We define 𝐴

▪ 𝛼1 and 𝛼2 are variables controlling the strength of the fiber direction

• If 𝛼1 = 𝛼2 = 0, isotropic materials

• If 𝛼1 ≠ 0, 𝛼2 = 0, transversely isotropic materials

• If 𝛼1 ≠ 0, 𝛼2 ≠ 0, orthotropic materials

▪ 𝛼1 and 𝛼2 ∈ {−1, 0}

Anisotropic Crack Driving Force

Fiber Direction

𝐴 = 𝐼 + 𝛼1 𝐚1 ⊗𝐚1 + 𝛼2(𝐚2 ⊗𝐚2) Eq (9)


• We couple our anisotropic damage with anisotropic elasticity through degrading relevant terms in the anisotropic energy density.

▪ Energy density: Ψ(𝐹)

▪ Deformation Gradient: 𝐹 =𝜕𝐱

𝜕𝐗

▪ A monotonically decreasing function of the damage, 𝑔 𝑑

• 𝑔 𝑑 = 1 − 𝑑 2 1 − 𝑟 + 𝑟

▪ If 𝑑 = 0, 𝑔 𝑑 = 1, (1 + 𝑟) in the paper.

▪ If 𝑑 = 1, 𝑔 𝑑 = 𝑟

• 𝑟 is residual stress▪ Ensure a small tensile stress even in regions of full damage

▪ Prevent degenerate deformation gradients.

▪ We degrade only the tensile portion of the elasticity Ψ+ 𝐹

▪ Ψ− 𝐹 : Compressive portion

▪ Ψ𝑓 𝐹 : Fiber energy

Elasticity Degradation

Damage to Energy Density

Ψ 𝐹 = 𝑔 𝑑 Ψ+ 𝐹 + Ψ− 𝐹 + Ψ𝑓 𝐹 Eq (10)

AnisoMPM Spatial Discretization


• Particle(material point) 𝑝 holds

▪ Position 𝐱𝑝

▪ Velocity 𝐯𝑝

▪ Mass 𝑚𝑝

▪ Deformation gradient 𝐹𝑝

Material point method- Particles(Lagrangian)


• Background grid node 𝑖 holds

▪ Constant Position 𝐱𝑖▪ Velocity 𝐯𝑖▪ Mass 𝑚𝑖

• Grid is needed for the computation of stress-based force

▪ Using cubic B-splines kernel

▪ 𝑤𝑖𝑝(𝐱𝑖 , 𝐱𝑝, ℎ) = 𝑁(1

ℎ(𝑥𝑝 − 𝑥𝑖))𝑁(

1

ℎ(𝑦𝑝 − 𝑦𝑖))𝑁(

1

ℎ(𝑧𝑝 − 𝑧𝑖))

• ℎ is Kernel radius

Material point method- Background grid(Eulerian)


Material point method- Full method overview


• 1. Rasterize particle data to the grid

▪ Mass 𝑚𝑖𝑛 = Σ𝑝𝑚𝑝𝑤𝑖𝑝

𝑛

▪ Velocity 𝐯𝑖𝑛 = Σ𝑝𝐯𝑝

𝑛𝑚𝑝𝑤𝑖𝑝𝑛 /𝑚𝑖

𝑛



• 2. Compute particle volumes and densities

▪ Only once



• 3. Compute grid forces

▪ Stress-based forces



• 4. Update velocities on grid



• 5. Grid-based body collisions



• 6. Solve the linear system

▪ Semi-implicit velocities update



• 7. Update deformation gradient



• 8~10. Update particle velocities and positions

▪ With particle-based body collision



• Rasterize particle data to the grid

▪ Make velocity field

• Compute grid forces

▪ MPM: Stress-based forces

▪ FLIP: Pressure forces

• Update velocity field

▪ 𝐯𝑛+1 = 𝐯𝑛 + Δ𝑡𝑚−1𝐟

• Update particle velocity

Material point method- Method flow comparison with FLIP









• Update particle



• Additional Lagrangian physical quantities

▪ Damage 𝑑

▪ Fiber direction {𝐚}

▪ Fiber Magnitudes {𝛼}

Aniso-MPM


• Remind Damage Evolution:

• We track our phase-field damage variables, 𝑑

▪ Similarly to momentum

▪ Update Rotation of fibers

• 𝑅𝑝𝑛 is rotation matrix of deformation gradient 𝐹𝑝

𝑛 = 𝑅𝑝𝑛𝑆𝑝

𝑛

▪ Polar Decomposition

Damage Evolution

ሶ𝑑 =1

𝜂< 1 − 𝑑 ෩𝐷 − 𝑑 − 𝑙0

2Δ𝑑 > Eq (6)

෩𝐷 = 𝜁 < Φ 𝜎+ − 1 > Eq (7)

Φ 𝜎+ =1

𝜎𝑐2 (𝐴𝜎

+: 𝜎+𝐴) Eq (8)

𝐴 = 𝐼 + 𝛼1 𝐚1 ⊗𝐚1 + 𝛼2(𝐚2 ⊗𝐚2) Eq (9)

𝐴𝑝𝑛 = 𝐼 + 𝛼1 𝑅𝑝

𝑛𝐚1 ⊗𝑅𝑝𝑛𝐚1 + 𝛼2(𝑅𝑝

𝑛𝐚2 ⊗𝑅𝑝𝑛𝐚2) Eq (9.1)


• From Eq.6

▪ From (Eq.7) ෩𝐷𝑝𝑛 = max(෩𝐷𝑝

𝐻, 𝜁 < Φ 𝜎+ − 1 >)

• ෩𝐷𝑝𝐻 is the maximum value in the history of particle 𝑝

▪ Δ𝑑𝑝𝑛=σ𝑖 𝑑𝑖

𝑛Δ𝑁𝑖𝑛 with 𝑑𝑖

𝑛 =Σ𝑝𝑤𝑖𝑝

𝑛 𝑑𝑝𝑛

Σ𝑝𝑤𝑖𝑝𝑛 [𝑤𝑖𝑝

𝑛 is weight function]

• Δ𝑁𝑖𝑛 = 𝑁𝑖

′′ 𝑥 𝑁𝑖 𝑦 𝑁𝑖 𝑧 + 𝑁𝑖 𝑥 𝑁𝑖′′ 𝑦 𝑁𝑖(𝑧)+𝑁𝑖 𝑥 𝑁𝑖 𝑦 𝑁𝑖

′′(𝑧)

• 𝑁𝑖(𝑥𝑝) is the interpolation function used for transfer

𝑑𝑝𝑛+1 = min 1, 𝑑𝑝

𝑛 + ሶ𝑑𝑝𝑛Δ𝑡

= min(1, 𝑑𝑝𝑛 +

Δ𝑡

𝜂< 1 − 𝑑𝑝

𝑛 ෩𝐷𝑝𝑛 − 𝑑𝑝

𝑛 − 𝑙02Δ𝑑𝑝

𝑛 >

Eq (11)

Explicit Integration of Damage

ሶ𝑑 =1

𝜂< 1 − 𝑑 ෩𝐷 − 𝑑 − 𝑙0

2Δ𝑑 > Eq (6)


• From Eq.6

and ignore bracket

• Make 𝐴𝐱 = 𝐛 form,

▪ 𝐱 is unknown 𝑑𝑛+1

Implicit Integration of Damage Modeling

ሶ𝑑 =1

𝜂< 1 − 𝑑 ෩𝐷 − 𝑑 − 𝑙0

2Δ𝑑 > Eq (6)

𝑑𝑝𝑛+1 = 𝑑𝑝

𝑛 + ሶ𝑑𝑝𝑛+1Δ𝑡

= 𝑑𝑝𝑛 +

Δ𝑡

𝜂{ 1 − 𝑑𝑝

𝑛+1 ෩𝐷𝑝𝑛 − 𝑑𝑝

𝑛+1 − 𝑙02Δ𝑑𝑝

𝑛+1 }

1 +Δ𝑡

𝜂෩𝐷𝑛 + 1 𝑑𝑛+1 −

Δ𝑡

𝜂𝑙02 Δ𝑑𝑛+1 =

Δ𝑡

𝜂෩𝐷𝑛 + 𝑑𝑛 Eq (12)

𝐴 + 𝐵 𝐝 = 𝐜 Eq (13)

𝐴 𝐵 𝐜


• Discretize Eq. (12)

▪ By writing a weak form of the PDE and then using the MLS(Moving Least Squares) shape function Θ𝑖 𝑥 , and its gradient

▪ To get a positive semi definite linear system for the unknown 𝑑𝑖

𝑛+1

Linear System for Implicit Integration

𝛻Θ𝑖 𝑥 = 𝑀𝑝−1𝑤𝑖𝑝

𝑛 (𝑥𝑖 − 𝑥𝑝𝑛)

𝐴 = 𝐴𝑖𝑖 = Σ𝑝𝑉𝑝𝑛 1 +

Δ𝑡

𝜂෩𝐷𝑝𝑛 + 1 𝑤𝑖𝑝

𝑛

𝐵 = 𝐵𝑖𝑗 = Σ𝑝𝑉𝑝𝑛 Δ𝑡

𝜂𝑙02 𝛻Θ𝑖 𝑥𝑝

𝑛𝑃

𝛻Θ𝑗 𝑥𝑝𝑛

𝐜 = 𝑐𝑖 = Σ𝑝𝑉𝑝𝑛 Δ𝑡

𝜂෩𝐷𝑝𝑛 + 𝑑𝑖

𝑛 𝑤𝑖𝑝𝑛


Explicit & Implicit Damage Comparison

Anisotropic Elasticity


• Through QR-decomposition of deformation gradient 𝐹 = 𝑄𝑅

▪ 𝑄 is rotation and 𝑅 is upper-triangular

• We write our additively-decomposed anisotropic energy as:

▪ Ψ𝜇 𝑅 =𝜇

2Σ𝑖𝑗𝑟𝑖𝑗

2 − 3 − 𝜇(𝐽 − 1) is shearing term

▪ Ψ𝜆 𝑅 =𝜆

2𝐽 − 1 2 is volumetric term

▪ 𝐽 = det(𝐹)

QR-Elasticity

𝑅 =

𝑟11 𝑟12 𝑟13𝑟22 𝑟23

𝑟33

Ψ 𝑅 = Ψ𝜇 𝑅 + Ψ𝜆 𝑅 + Ψ𝑓(𝑅)


• We add two terms to penalize stretching in the fiber direction

▪ Primary fiber direction stiffness 𝑘𝑥▪ Optionally secondary fiber direction stiffness 𝑘𝑦

QR-Elasticity-Fiber Term

(a) Isotropic: 𝑘𝑥 = 𝑘𝑦 = 0

(b) Transverse Isotropic: 𝑘𝑥 = 15𝜇, 𝑘𝑦 = 0

(c) Orthotropic: 𝑘𝑥 = 15𝜇, 𝑘𝑦 = 3𝜇

Ψ𝑓 𝑅 =𝑘𝑥2

𝑟11 − 1 2 +𝑘𝑦

2𝑟122 + 𝑟22

2 − 1

2


• We additively decompose our QR-elasticity into degradation

Elasticity Degradation

Degradation: Ψ 𝑅 = Ψ+ 𝑅 + Ψ− 𝑅 + Ψ𝑓 𝑅

QR-Elasticity: Ψ 𝑅 = Ψ𝜇 𝑅 + Ψ𝜆 𝑅 + Ψ𝑓(𝑅)

Ψ+ 𝑅 = ൝Ψ𝜇 𝑅 + Ψ𝜆 𝑅

Ψ𝜇 𝑅

𝐽 ≥ 1𝐽 < 1

, Ψ− 𝑅 = ቊ0

Ψ𝜆 𝑅𝐽 ≥ 1𝐽 < 1


Energy Visualization


Anisotropic Damage and Elasticity


• We struggle to model high stiffness material

▪ Without decreasing Δt for explicit MPM

• Even if the system were solvable using implicit MPM, it may still suffer from the classic locking effect

High Stiffness Material

Fibers are parallel to one another (a, c)Fibers are slightly perturbed (b, d)

Inextensibility


• We present an embedded directed inextensibility solver

▪ In which extreme stiffness is modeled with hard constraints

• we assume the inextensible fibers embedded in the continuum to only transform under rigid kinematics

Inextensibility


• Enforcing constraint equation in Eulerian space as follows:

▪ 𝐝 =1

2(𝜕𝐯

𝜕𝐱+

𝜕𝐯

𝜕𝐱

𝑇) is the Eulerian rate-of-strain tensor

▪ 𝐚 is the current unit fiber direction in Eulerian space

• Our constrained conservation of momentum equation is

▪ 𝜎𝑓𝑢𝑙𝑙 = 𝜎 + 𝜆 𝐚⊗ 𝐚 is the full stress

▪ 𝜆= 𝜆(𝐱, 𝑡; 𝐚) represents the unknown tension along the fiber direction enforcing the inextensibility constraint

Theory and Equations

𝐚⊗ 𝐚 :𝐝 = 0 Eq (14)

𝜌 𝐱, 𝑡𝐷𝐯

𝐷𝑡= 𝛻𝐱 ∙ 𝜎𝑓𝑢𝑙𝑙 + 𝜌 𝐱, 𝑡 𝐠 Eq (15)


• We follow [Jiang et al. 2016] to derive the weak form

• 𝛼, 𝛽 as the dimension indices

• 𝑖 as the traditional MPM grid index

▪ 𝑁𝑛 the number of MPM grid nodes

• 𝑧 as the grid cell index

▪ 𝑁𝑐 the number of MPM grid cells

Weak Form-Denotation


• Multiplying Eq.(15) by our test function 𝐪(𝐱, 𝑡)

▪ Integrating in world space

▪ Applying integration by parts and divergence theorem

▪ For simplicity,

• Ignoring gravity and assuming a zero-traction boundary

▪ 𝑞𝛼,𝛽 =𝜕

𝜕𝑥𝛽𝑞𝛼

Weak Form-Momentum Equation[Test Function]

Ω𝑡 𝑞𝛼𝜌𝐷𝑣𝜶

𝐷𝑡𝑑𝐱 = Ω𝑡− 𝑞𝛼,𝛽𝜎𝛼𝛽𝑑𝐱 − Ω𝑡 𝑞𝛼,𝛽𝜆𝑎𝛼𝑎𝛽𝑑𝐱 Eq (16)

𝜌 𝐱, 𝑡𝐷𝐯

𝐷𝑡= 𝛻𝐱 ∙ 𝜎𝑓𝑢𝑙𝑙 + 𝜌 𝐱, 𝑡 𝐠 Eq (15)


• Expand velocity and q at nodes

▪ 𝑣𝛼 𝐱 = 𝑣𝑖𝛼𝑁𝑖 𝐱

▪ 𝑞𝛼 𝐱 = 𝑞𝑖𝛼𝑁𝑖(𝐱)

• 𝑞𝛼,𝛽 𝐱 = 𝑞𝑖𝛼𝑁𝑖,𝛽(𝐱)

▪ 𝑁𝑖(𝐱) is weight function from nodes

• Expand 𝜆 at cell centers

▪ 𝜆 𝐱 = 𝜆𝑧Γ𝑧 𝐱

▪ Γ𝑧(𝐱) is weight function from cell centers

• We pick 𝑞𝑖𝛼 = 𝛿 𝑖,𝛼 = 𝑖∗,𝛼∗ [if 𝑖, 𝛼 = 𝑖∗, 𝛼∗ then 1, else 0]

• We get the final form of our momentum equation

▪ 𝐵(𝑧,𝑖𝛼) = Ω𝑡𝑛 𝑎𝛼 𝐚 ∙ 𝛻𝑁𝑖 Γ𝒛 𝐱 d𝐱

▪ 𝑏𝑖𝛼 =𝑚𝑖

Δ𝑡𝑣𝑖𝛼𝑛 − Ω𝑡𝑛𝑁𝑖,𝛽𝜎𝛼𝛽𝑑𝐱

Weak Form-Momentum Equations

𝑚𝑖

Δ𝑡𝑣𝑖𝛼𝑛+1 + 𝐵(𝑧,𝑖𝛼)𝜆𝑧 = 𝑏𝑖𝛼 Eq (17)


• Similarly manipulate Eq. (14)

▪ By the introducing the test function, ℎ(𝐱, 𝑡)

• Interpolate our equation over grid nodes and cells

▪ 𝑣𝛼 𝐱 = 𝑣𝑖𝛼𝑁𝑖 𝐱

▪ ℎ 𝐱 = ℎ𝑧Γ𝑧(𝐱)

• And let ℎ𝑧 = 𝛿𝑧=𝑧∗[if 𝑧 = 𝑧∗ then 1, else 0]

Weak Form-Constraints[Manipulate & Interpolate]

𝐚⊗ 𝐚 :𝐝 = 0 Eq (14)

Ωt ℎ(𝐱, 𝑡) 𝐚 ⊗ 𝐚 : 𝐝𝑑𝐱 = 0 Eq (18)

Ωt Γ𝑐𝑎𝛼𝑎𝛽𝑣𝑖𝛼𝜕𝑁𝑖 𝑥

𝜕𝑥𝛽𝑑𝐱 = 0 Eq (19)

𝐝 =1

2(𝜕𝐯

𝜕𝐱+

𝜕𝐯

𝜕𝐱

𝑇) is the Eulerian rate-of-strain tensor

𝐚 is the current unit fiber direction in Eulerian space


• Extracting the 𝑣𝑖𝛼 term, we derive our linear constraint equations

▪ Giving us 𝑁𝑐 equations of 𝑣𝑖𝛼𝑛+1 and 𝜆𝑧

Weak Form-Constraints

𝐵(𝑧,𝑖𝛼)𝑣𝑖𝛼 = 0 Eq (20)


• From 𝑏𝑖𝛼 =𝑚𝑖

Δ𝑡𝑣𝑖𝛼𝑛 − ,Ω𝑡𝑛𝑁𝑖,𝛽𝜎𝛼𝛽𝑑𝐱

• And

Weak Form-Discretization

Ω𝑡𝑛𝑁𝑖,𝛽𝜎𝛼𝛽𝑑𝐱 ≈ σ𝑝𝑉𝑝𝑛𝜎𝑝𝛼𝛽

𝑛 𝑁𝑖,𝛽(𝐱𝑝) Eq (21)

𝐵(𝑧,𝑖𝛼) = Ω𝑡𝑛 𝑎𝛼 𝐚 ∙ 𝛻𝑁𝑖 Γ𝒛 𝐱 d𝐱 ≈ σ𝑝𝑉𝑝𝑛Γ𝑧 𝐱𝑝 𝑎𝛼 𝐚 ∙ 𝛻𝑁𝑗(𝐱𝑝) Eq (22)


• Make a linear system as

▪ 𝑀 is 𝑑𝑖𝑚 × 𝑁𝑛 × 𝑑𝑖𝑚 × 𝑁𝑛 diagonal matrix

• 𝑀(𝑖𝛼,𝑖𝛼) = 𝑚𝑖

▪ 𝐛 is 𝑑𝑖𝑚 × 𝑁𝑛 vector

▪ 𝐵 is 𝑁𝑐 × 𝑑𝑖𝑚 × 𝑁𝑛 sparse matrix

Weak Form-Implementation

𝑀 𝐵𝑇

𝐵 0

𝐯𝛌

=𝐛𝟎

Eq (23)

𝑏𝑖𝛼 =𝑚𝑖

Δ𝑡𝑣𝑖𝛼𝑛 −න

Ω𝑡𝑛𝑁𝑖,𝛽𝜎𝛼𝛽𝑑𝐱

𝐵(𝑧,𝑖𝛼) = Ω𝑡𝑛 𝑎𝛼 𝐚 ∙ 𝛻𝑁𝑖 Γ𝒛 𝐱 d𝐱 ≈ σ𝑝𝑉𝑝𝑛Γ𝑧 𝐱𝑝 𝑎𝛼 𝐚 ∙ 𝛻𝑁𝑗(𝐱𝑝) Eq (22)

𝑚𝑖

Δ𝑡𝑣𝑖𝛼𝑛+1 + 𝐵(𝑧,𝑖𝛼)𝜆𝑧 = 𝑏𝑖𝛼 Eq (17)


Inextensibility Result

Results&Discussion


Results


Performance


• Some make more desirable results without anisotropic elasticity

▪ Bone and Orange

• Many parameters and options make difficult to tune

▪ This breadth enhance the flexibilityand increase the potential for artistic control

• Did not thoroughly explore brittle fracture

▪ Requires the development of new interpolation functions and treatments

• AnisoMPM is not designed to produce large-scale debris effect

▪ Yet another promising target for future work

Limitations and Future work

anisompm: animating anisotropic damage...

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