cd-mpm: continuum damage material point methods for
TRANSCRIPT
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
Changyong Song| 6/17/2021| # 3Computer Graphics @ Korea University
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.
Changyong Song| 6/17/2021| # 4Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 5Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 6Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 7Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 8Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 9Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 10Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 11Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 12Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 13Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 14Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 15Computer Graphics @ Korea University
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.
Changyong Song| 6/17/2021| # 16Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 17Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 18Computer Graphics @ Korea University
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..
Changyong Song| 6/17/2021| # 19Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 20Computer Graphics @ Korea University
PFF-MPM
Data Flow and Discretization
• Traditional MPM vs PFF - MPM
Changyong Song| 6/17/2021| # 21Computer Graphics @ Korea University
PFF-MPM
Data Flow and Discretization
Changyong Song| 6/17/2021| # 22Computer Graphics @ Korea University
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”)
Changyong Song| 6/17/2021| # 23Computer Graphics @ Korea University
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.
Changyong Song| 6/17/2021| # 24Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 25Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 26Computer Graphics @ Korea University
Elastic Degradation
Decomposition of 𝛙𝐸
• For simple decomposition of the tensile & compressive part
• Visualization
Changyong Song| 6/17/2021| # 27Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 28Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 29Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 30Computer Graphics @ Korea University
PFF-MPM
Data Flow and Discretization
Changyong Song| 6/17/2021| # 31Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 32Computer Graphics @ Korea University
PFF-MPM Spatial Discretization
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.
Changyong Song| 6/17/2021| # 33Computer Graphics @ Korea University
PFF-MPM Spatial Discretization
Dynamics : Grid to Particle & Strain Update
• Particle velocities
Updated by
• Velocity gradients
Updated by
• Strain
Update by
Changyong Song| 6/17/2021| # 34Computer Graphics @ Korea University
PFF-MPM Spatial Discretization
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.
Changyong Song| 6/17/2021| # 35Computer Graphics @ Korea University
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)
Changyong Song| 6/17/2021| # 36Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 37Computer Graphics @ Korea University
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)
Changyong Song| 6/17/2021| # 38Computer Graphics @ Korea University
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.
Changyong Song| 6/17/2021| # 39Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 40Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 41Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 42Computer Graphics @ Korea University
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
Changyong Song| 6/17/2021| # 43Computer Graphics @ Korea University
Result
CCC vs NACC
Changyong Song| 6/17/2021| # 44Computer Graphics @ Korea University
Result
Cohesion coefficient & Hardening
Changyong Song| 6/17/2021| # 45Computer Graphics @ Korea University
Result
Compare with Drucker-Prager
Changyong Song| 6/17/2021| # 46Computer Graphics @ Korea University
Result
More 3D Examples
• All demos were run on an Intel Core i7-8700K CPU
12 Threads at 3.70 GHz.
Changyong Song| 6/17/2021| # 47Computer Graphics @ Korea University
• 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
Changyong Song| 6/17/2021| # 48Computer Graphics @ Korea University
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