CD-MPM: Continuum Damage Material Point Methods for

CD-MPM: Continuum Damage Material Point Methods for Dynamic Fracture Animation

Joshuah Wolpher et al.SIGGRAPH 2019

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

Presented by Changyong Song

6/17/2021

Computer Graphics @ Korea University

Changyong Song| 6/17/2021| # 2Computer Graphics @ Korea University

Introduction

Numerical Approach for Fracture Animation

• Fracture Animation :

Animation of breaking solid objects, like glass or pottery etc..

• In Computer Graphics, Fracture Animation requires :

Methods for large-scale topological changes at varying rates.

Robust procedures for tracking the evolving crack fronts.

• Physics-based Approach

FEM : discretization of the governing equation.

BEM : Numerical solution of boundary integral equation.

MPM : Meshless method by combining Lagragian material particles with Eulerian grids.

FEM BEM MPM


Motivation

• In this Paper, Why they use MPM for Fracture Animation? If extreme deformation is occurred, Mesh based method(e.g. FEM) is required

re-meshing routine. MPM need only track the unmeshed particles. Can natural multi-material coupling Effortless collision handling

• Limits of MPM Sharp material discontinuity -> Make difficult to predict crack evolution

• Properties of Continuum damage mechanics Another approach to fracture.

• FEM is paired with Fracture mechanics.

CDM has the advantage of predicting the crack tip more accurately. Benefit of capturing fracture while crack propagation.

• Plasticity Use Cohesive Cam Clay Model for volume preserving when fracture is

occured.


Contributions

• Phase Field Fracture MPM (PFF – MPM)

an augmented MPM solver

High visual fidelity (For sharp material discontinues)

• A more efficient analytic return mapping scheme

• Non Associated Cam Clay (NACC)

• a plasticity scheme


Related Work

Fracture Simulation

• Mass Spring Model

The first and represented continuum materials as point masses connected by springs with stress based yields thresholds.

• “Physically Based Simulation of Cracks on Drying 3D Solids”, Kimiya Aoki(Toyohashi University of Technology) et al./ COMPUTER GRAPHICS INTERNATIONAL, 2004


Related Work

Fracture Simulation

• Finite element methods(FEM) The most successful for brittle, ductile, and thin shell materials. But difficult geometric and computational challenges. Numerous re-meshing

• XFEM(extended FEM) “Robust Extended Finite Element Method for complex cutting of

deformables”, Dan Koschier(RWTH Aachen University) et al. / ACM Transactions on Graphics, 2017

A re-meshing free cutting algorithm

Better conserve mass and preserve material stiffness of simulated materials.

Limit : Additional Complexities• Floating point arithmetic • Self Collision on embedded meshes


Related Work

Fracture Approach

• Fracture Theory for numerical modeling approaches :

Discontinuous method

• Allow fracture surfaces to be represented in the displacement field as discontinuities.

Typically Composed Griffith theory & CDM

Continuous method

• Model the displacement as being continuous everywhere.

PD (peridynamics)

∙ Avoiding derivative computation by replacing partial differential equations to integral ones

Combining PD & PFF (Phase Field Fracture)

∙ “Phase field based peridynamics damage model for delamination of composite structures”, Pranesh Roy(Indian Institute of Science) et al. / Composite Structures, 2017

∙ For brittle fracture


Related Work

Material Point Method

• MPM has proven to be a promising discretization choice for simulating many solid and fluid materials.

Snow

• “A material point method for snow simulation”, Alexey Stomakhin(University of California Los Angeles) et al. / ACM Transactions on Graphics, 2013

Foam

• “Continuum foam : a material point method for shear-dependent flows”, Yonghao Yue(Columbia University) el al. / ACM transactions on Graphics, 2015

Cloth

• “A material point method for thin shells with frictional contact”, Qi Guo(University of California) et al, / ACM transactions on Graphics, 2018


• Think about I’m in the bus stop.

What If I want to express that the bus keeps passing based on where I am?

• In this time we use “Eulerian Coordinates”

• A coordinate system that prioritizes bus stops over buses.

What if I want to express that I’m riding a bus passing a bus stop?

• In this time we use “Lagrangian Coordinates”

• A coordinate system that prioritizes me(bus) over bus stops.

Background

Lagrangian/Eulerian Coordinates


• Sometimes, We need to express more than one direction at the same time to express situation.

e.g. stress

• By using Tensor, we can express this.

𝜎12 : Acting on the plane in the 1 direction with the stress of 2 direction.

In Eulerian Coordinates, we call this “Cauchy’s stress tensor”

Background

Stress Tensor


• Consider the deformation below

Basis Vector መ𝐈(𝑋 coordinate) : Initial Position (Reference)

Basis Vector ො𝐞(𝑥 coordinate) : Deformed Position (Current)

• The coordinate 𝑥 is determined by mapping function

Background

Deformation Mapping Function


• Also, we can differentiate this mapping function.

This mean that It is possible to differentiate the position after deformation into the initial position.

We called this “Deformation Gradient Tensor”

Deformation Gradient can be decomposed by Rotation & stretch

• U : Right stretch tensor

• V : Left stretch tensor

Left Cauchy deformation tensor

Background

Deformation Gradient Tensor


• In general, it is natural to define stress and surface area based on the state after fraction/deformation.

• But Sometimes, When defining the deformation of an object based on the state before deformation, it is necessary to define the corresponding stress based on the state before deformation as well

e.g. Tensor in Lagrangian coordinate

• A method for defining the stress based on the state before deformation is the 1st Piola Kichhoff stress tensor (Lagrange Method)

Background

1st Piola Kirchoff Stress Tensor


• Represent continuum material as :

Particles

Grid of nodes

• “A material point method for snow simulation”, Alexey Stomakhin(University of California Los Angeles) et al. / ACM Transactions on Graphics, 2013

• Loop for n steps of simulation :

P2G transfer

Compute forces

Update Velocities

G2P transfer

Update Positions

Background

Material Point Method


Background

Griffith’s Theory

• Previous Fracture theories required the curvature of the crack to find the force for fracture. But if curvature equals to 0?

• e.g. A crack stabbed with a knife(sharp discontinuity)

• Griffith Crack(Griffith’s Crack Theory) “The phenomena of rupture and flow in solids”, Alan Arnold Griffith

et al. / transactions of the royal society of london, 1921 Assume the object is hyper-elastic. Derived a criterion for crack growth using an energy approach. Initially, all materials are assumed to have subtle damage. If the crack developed, the total energy is minimized. Can calculate as the sum of Strain Energy & Surface Released Energy.


Background

CDM & Phase Field Model

• Continuum Damage Mechanics (CDM)

Combining continuum & fracture mechanics.

“smeared” cracks represented as continuum

Local CDM : compare with defined maximal stress

Non-Local CDM : track an evolving field of damage variables.

• Phase Field Model

Modeling method for find adjacent surface of two different surface.

• “An introduction to phase-field modeling of microstructure evolution”, NeleMoelans(Katholieke Universiteit Leuven) et al./ Elsevier, 2008

Using Phase Field variable

Length Parameter

• Smooth change of interface


• Elastic materials are linear materials

The stress changes in proportion to the strain rate.

• Hyper Elasticity

The elastic deformation is very Large

• e.g. rubber

In hyper elastic materials, the relationship between stress and strain rate is obtained by stress - strain energy density function.

There are various models for express Hyper elasticity

• Neo – Hookean

• Mooney-Rivlin

• Arruda-Boyce

• Odgen

Background

Hyper Elasticity


Background

Plasticity & Yield Surface

• Plasticity

Property of a material that is permanently deformed when it receives external force.

Deformation gradient can be decomposed by Plasticity & Elasticity.

𝑭 = 𝑭𝐸𝑭𝑃

• Yield Surface (Yield Function)

3D surface in principal stress space

Boundary of elastic and plastic deformation

Condition :

• e.g. : Drucker-Prager, Von mises, Cam-Clay..


• What is return mapping algorithm? Use 𝑭𝐸 , to see where particle in stress space

• “An Enhanced Void-Crack based Rousselier Damage Model for Ductile Fracture with the XFEM“, Meor Iqram BIN Meor Ahmad et al. / International Journal of Damage Mechanics, 2018

Project to yield surface points back

Hardening state update per particle

• Associated vs Non – Associated Associated

• Projection direction is enforced to Yield function’s Gradient.

Non – Associated• Projection direction is not enforced to Yield function’s Gradient.

Background

Plasticity : Return Mapping


PFF-MPM

Data Flow and Discretization

• Traditional MPM vs PFF - MPM


PFF-MPM



Governing Equation

Griffith’s Theory

• Material :

With material space Ω0 and deformed space Ω𝑡

Under deformation map 𝒙 = Φ 𝑿, 𝑡

• 𝒙, 𝑿 are world and material coordinates respectively.

• Total Free Energy (Griffith’s theory of fracture)

𝐹 =𝜕Φ

𝜕𝑋is the deformation gradient.

Solving the minimization of 휀 predicts the crack propagation.

: damaged finite-strain hyper-elastic energy density function

: energy released

: discontinuous boundary due to fracture

: critical energy release rate (called “fracture toughness”)


Governing Equation

Released Energy with Phase Field

• In Computer Graphics Use a level set method to separate the continuum into a

heathy/damaged region (with interface )• “A Level Set Method for Ductile Fracture”, Jan Hegemann(University

of Munster) et al. / ACM SIGGRAPH, 2013

This approach requires frequent reinitialization of SDF(Signed Distance Function) & cannot resolve non – manifold topology.

• Use Phase Field Approximation to the surface integral

𝑙0 : discretization dependent length scale parameter• 𝑙0 -> 0 : the volume integral to converge to the surface integral

Phase field variable 𝑐 𝑿, 𝑡 ∈ 0,1 ∶• 𝑐 = 1 corresponds to healthy material and 𝑐 = 0 to fully damaged

material.


Governing Equation

Elastic Degradation

• The traditional hyper elastic energy density Ψ𝐸 can be additively decomposed into a tensile/compressive contribution.

: tensile(degrade) part

: compressive part

• Material separation is permitted along cracked regions with 𝑔(𝑐)

• Combining the above, the free energy functional ca be written as

: Split Energy Density

: Phase Field Approximation


Elastic Degradation

Variation of the Neo-Hookean model

• For simple decomposition of the tensile & compressive contribution.

“Hybrid Grains: Adaptive Coupling of Discrete and continuum Simulations of Granular Media”, Yonghao Yue(The university of Tokyo, Columbia University) et al. / ACM Trans. Graph, 2018

: the energy of shearing change

: the energy of volume change

: Kirchhoff stress

• 𝜅 : bulk modulus, 𝜇 : shearing modulus

• 𝒃 = 𝑭𝑭𝑻 : Left Cauchy-Green strain

• dev 𝐴 = 𝐴 − 𝑇𝑟(𝐴)

𝑑𝐼 : deviatoric part of any stress tensor A

• 𝑑= problem dimension


Elastic Degradation

Decomposition of 𝛙𝐸

• For simple decomposition of the tensile & compressive part

• Visualization


Phase-Field Evolution

Local Damage Mechanics

• Local damage mechanics assumes the damage state of each material point is only locally dependent on its own stress history.

Material’s damage is linearly related to the maximum eigenvalue of the Cauchy Stress.

Simple and efficient but tend to cause undesired artifact.

Crack propagation has strong mesh direction and resolution dependency

• “On mesh bias of local damage models for concrete”, Peter Grasseland Milan Jirasek(LSC) et al./ Fracture Mechanics of concrete structures Vol.5


Phase-Field Evolution

Parabolic Phase Field Evolution

• To avoid the problems with local damage mechanics :

Non – Local CDM

Target a phase field evolution rule that is constructed using Ginzburg – Landau theory.

• Euler Lagrangian equation for 𝑐


Governing Equation

Momentum Conservation

• Focus on the response of hyper-elastoplastic solids.

Using backward Euler, minimizing an incremental potential.

• Lagrangian momentum conservation

𝑅 is density

is first Piola-Kirchhoff stress

This formula equals to


PFF-MPM



PFF-MPM Spatial Discretization

Dynamics : Particle to Grid

• Notation

𝑝, 𝑞 : particle quantities

𝑖, 𝑗, 𝑘 : grid node quantities

𝑛, 𝑛 + 1 : quantities at discrete time 𝑡𝑛, 𝑡𝑛+1

∆𝑡 = 𝑡𝑛+1 - 𝑡𝑛

• Particles to Grid : APIC

Mass & momentum are transferred to the grid with APIC

• “The Affine Particle-In-Cell Method”, Chenfanfu Jiang (University of California Los Angeles) et al. / SIGGRAPH 2015

• 𝐶𝑝𝑛 is APIC velocity gradient, 𝑤𝑖𝑝

𝑛 is the quadratic B-spline interpolation



Dynamics : Grid Update

• Grid Velocity is updated with MLS-MPM forces

“A Moving Least Squares Material Point Method with Displacement Discontinuity and Two-Way Rigid Body Coupling”, Yuanming Hu (MIT CSAIL) et al. / ACM Transactions on Graphics 2018

• 𝑉𝑝0 is the original volume of particle 𝑝

• 𝑀𝑝−1 = 4

∆𝑥2for a quadratic particle-grid kernel

• * = 𝑛 𝑜𝑟 𝑛 + 1 for symplectic Euler/Implicit Euler respectively.



Dynamics : Grid to Particle & Strain Update

• Particle velocities

Updated by

• Velocity gradients

Updated by

• Strain

Update by



Phase-Field Transfer

• Phase Field transfer

After solving for 𝑐𝑝𝑛+1 on the grid, we transfer the Phase Field

back to particles

•

• Compare the new Phase Field against its current value to prevent any material healing.


Generalized Non-associated Return Mapping

Plastic Flow

• Return mapping by using Plastic flow theory :

The Material’s behavior can be simulated after reaching the yield surface (yield condition).

The Plastic Flow equation can be formulated through evolution of the

elastic part of the Left Cauchy – Green strain Tensor 𝒃𝐸 = 𝑭𝐸𝑭𝐸𝑇.

Split Equation 2 parts.

• Elastic Get “trial” step going from 𝒃𝐸,𝑛 to 𝒃𝐸,𝑡𝑟

Only consider “Elastic” part.

• Plastic Integrate from 𝒃𝐸,𝑡𝑟 to 𝒃𝐸,𝑛+1 by choosing proper projection direction.

Direction 𝑮 : 𝜕𝑦

𝜕𝜏(for associated plasticity)

Direction 𝑮 : 𝒅𝐞𝐯(𝝏𝒚

𝝏𝝉) (for non-associated plasticity)


Plastic Flow

Implicit Euler Integration

• The ODE to be integrated from 𝑏𝐸,𝑡𝑟 to 𝑏𝐸,𝑛+1 is then :

By Lie derivative (Tensor to Vector field)

• In CD – MPM Tech Doc, equation (23)

Solving Initial Value Problem

𝛾 is unknown scalar.

• Reformulate above equation by using Implicit Euler Integration :

For efficiency and ease of implementation

Geometric correctness

𝛿𝛾 = 𝛾Δ𝑡


Plastic Flow

Isotropic materials

• For Isotropic materials :

Much more convenient to solve flow Rule equation.

• Equation is in the diagonal space (obtained by SVD of Deformation Gradient).

• Can rewrite quadratic function.

For rewrite quadratic function, decompose stress tensor two part.

• We only use pressure part.

Ƹ𝑠 = dev( Ƹ𝜏) : deviatoric portion

𝑝 = −1

𝑑tr( Ƹ𝜏) : pressure portion

𝑞 =6 −𝑑

2Ƹ𝑠 : stress magnitude of yield point

𝑑 : problem dimension(2 or 3)


Plastic Flow

Isotropic materials (con’t)

• Stiff materials (like metal) :

Ƹ𝑠𝑛+1 & Ƹ𝑠𝑡𝑟 are approximately in the same direction.

𝑝𝑛+1 = 𝑝𝑡𝑟

• 𝑝𝑡𝑟 is computed from 𝑏𝐸,𝑡𝑟

The only essential unknown for plastic projection is 𝑞𝑛 +1.

• This makes greatly simplify return mapping.


NACC

MCC & CCC

• Use Associated-Flow Rule

Modified Cam – Clay Yield Surface (MCC)• “On the generalized stress-strain behavior of wet clay”, K. H. Roscoe

(Cambridge University) et al./ 1968

• Modeling clay and soil plasticity

• Lacks cohesion and thus has no stress under tension.

Useful for dry, granular materials

Cohesive Cam – Clay (CCC)• “Dynamic Anti crack propagation in snow”, J. Gaume (Swiss Federal Institute of

Technology) et al. / Nature Communications, 2018

• Augmented MCC model.

• Physically accurate anti-crack propagation in snow.

• Volume gain problem


NACC

Yield Surface Criteria

• Expand CCC by adopting a Non – Associated flow rule

Preserve volume during plasticity propagation.

• Yield Surface(Function)

𝛽 : cohesion coefficient

𝑝0 = 𝐾sinh(ξ𝑚𝑎𝑥(−𝛼, 0)) : related to hardening behavior

𝐾 =2

3𝜇 + 𝜆 : bulk modulus

ξ : hardening factor

𝛼 : track hardening


NACC

Return Mapping

• In Yield Surface

If Second term is negative, this equation does not have solution for 𝑞.

Discontinuity requires that return mapping broken down into 3 cases.

• Set 𝑝𝑚𝑎𝑥 = 𝑝0 & 𝑝𝑚𝑖𝑛 = −𝛽𝑝0

3 Cases

• 𝑝𝑡𝑟 > 𝑝0

• 𝑝𝑡𝑟 <−𝛽𝑝0

• −𝛽𝑝0 < 𝑝𝑡𝑟 < 𝑝0 Valid Range

Project to the tips of the ellipsoid


NACC

Von Mises & Drucker-Prager

• Applying return mapping approach by reformulation

VM(Von Mises) yield criteria

• 𝜏𝑦 : yield strength

DP(Drucker-Prager) yield criteria

• There was no visually significant difference from previous studies.

“Drucker-Prager Elastoplasticity for Sand Animation” , Gergely Klar(University of California, Los Angeles) et al. / ACM Trans, 2016


Result

CCC vs NACC


Result

Cohesion coefficient & Hardening


Result

Compare with Drucker-Prager


Result

More 3D Examples

• All demos were run on an Intel Core i7-8700K CPU

12 Threads at 3.70 GHz.


• Phase Field Fracture MPM (PFF – MPM)

an augmented MPM solver

• Griffith’s Theory

• Phase Field Model

• A more efficient analytic return mapping scheme

• Non Associated Cam Clay (NACC)• a plasticity scheme

• Use Flow Rule

• Faster than other method e.g. x2 than CCC Model

• Pair Together

Easy to pair

Work well

Conclusion


Limitation

• CD-MPM has limitations :

Rendering can be challenging

• Rendering debris is difficult due to the delicate balance between surface smoothness & sharpness.

• Smoothing too much removes the intricate patterns.

• Smoothing too little leaves surface undesirably textured.

• Future Work :

Anisotropic fracture

Documents

CD-MPM: Continuum Damage Material Point Methods for