Meshless Animation of Fracturing Solids

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Meshless Animation of Fracturing Solids
Richard Keiser Markus Gross
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Motivation

• Simulation of fracturing materials in many
  different applications.

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Motivation

• Simulation of fracturing materials in many
  different applications.
• Requirements on fracturing algorithm

4
Motivation

• Simulation of fracturing materials in many
  different applications.
• Requirements on fracturing algorithm
• brittle or ductile fracture

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Motivation

• Simulation of fracturing materials in many
  different applications.
• Requirements on fracturing algorithm
• brittle or ductile fracture
• arbitrary cracks

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Motivation

• Simulation of fracturing materials in many
  different applications.
• Requirements on fracturing algorithm
• brittle or ductile fracture
• arbitrary cracks
• control of fracture paths

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Motivation

• Simulation of fracturing materials in many
  different applications.
• Requirements on fracturing algorithm
• brittle or ductile fracture
• arbitrary cracks
• control of fracture paths
• highly detailed surfaces

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Related Work

• OBrien Hodgins 99, 02
• dynamic remeshing
• element cutting
• difficult to avoid ill-shaped elements

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Related Work

• OBrien Hodgins 99, 02
• dynamic remeshing
• element cutting
• difficult to avoid ill-shaped elements
• Molino, Bao Fedkiw 04
• virtual node algorithm
• embedded surface in copied tetrahedra
• restricted decomposition of tetrahedras

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Meshless Methods

• Advantages
• sampling of the volume
• handling of large deformation
• (re-)sampling of the domain
• handling of discontinuities
• Drawbacks
• boundary conditions
• overhead for computing interpolation functions

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Contributions

• A meshless animation framework for stiff-elastic
  and plasto-elastic materials that fracture
• handling of brittle and ductile fracture
• allows arbitrary crack initiation and propagation
• allows for easy control
• highly detailed surfaces due to decoupling of
  physics and surface representation

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Overview

• Part 1 Physics Animation
• Meshless Continuum Mechanics
• Modeling Discontinuities
• Spatial Re-sampling

• Part 2 Surface Handling
• Surface Model
• Crack Initiation Propagation
• Topological Events

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Elasticity Model

• Meshless elasticity model derived from continuum
  mechanics.1

Simulation loop
Time integration
Gradient of displacement field
Strain
Stress
Body force
Add external forces
Strain energy
1
Müller et al. Point Based Animation of Elastic,
Plastic and Melting Objects, SCA 2004
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Discretization

• Discrete set of nodes xi
• Approximation of displacement field u

u(x) ? ?i ?i(x) ui

• Derivation of shape functions
• using Moving Least Squares (MLS)

x
xi
ui
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Discretization

• Shape functions ?i

?i(x) ?i(x,xi) pT(x) M(x)-1 p(xi)
? by construction they build a first order
partition of unity (PU)
r x-y/hi with hi the support radius of node
i
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Discontinuities

• Only visible nodes should interact
• collect nearest neighbors
• perform visibility test

crack
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Discontinuities

• Only visible nodes should interact
• collect nearest neighbors
• perform visibility test

crack
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Discontinuities

• Problem undesirable discontinuities of the shape
  functions
• not only along the crack
• but also within the domain

crack
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Discontinuities
Visibility Criterion
Weight function
Shape function
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Discontinuities

• Solution transparency method1
• nodes in vicinity of crack partially interact
• by modifying the weight function

crack
ds
?i(xi,xj) ?i(xi-xj/hi (2ds/?)2)

• crack becomes transparent
• near the crack tip

1
Organ et al. Continuous Meshless Approximations
for Nonconvex Bodies by Diffraction and
Transparency, Comp. Mechanics, 1996
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Discontinuities
Visibility Criterion
Transparency Method
Weight function
Shape function
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Re-sampling

• Add simulation nodes when number of neighbors too
  small

• Local resampling of the domain of a node
• distribute mass
• adapt support radius
• interpolate attributes

xi
Shape functions adapt automatically!
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Re-sampling Example
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Part 2Surface Handling
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Surface Animation

• All surfaces are represented using oriented point
  samples si wrapped around the simulation nodes
  pj
• Deformation of surfels is computed from
  neighboring simulation nodes

simulation nodes pj
surfels si
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Crack Propagation

• Crack initiation
• where stress above threshold
• crack created by inserting 3 crack nodes
• each carrying 2 opposing surfels
• connection is crack front

crack front
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Crack Propagation

• Crack propagation
• propagate crack nodes along propagation direction
• re-project first and last node
• up-sample if necessary

one fracture surface
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Crack Propagation Example
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Crack Events

• Splitting
• when crack propagates through the material
• split front in two new fronts
• each one propagates independently

block of material
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Crack Events

• Merging
• when two fronts propagate close to each other
• merge fronts and associated fracture surfaces

block of material
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Crack Events Example
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Brittle Fracture
Initial statistics 4.3k nodes 249k surfels
Final statistics 6.5k nodes 310k surfels
Simulation time 22 sec/frame
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Controlled Fracture
Initial statistics 4.6k nodes 49k surfels
Final statistics 5.8k nodes 72k surfels
Simulation time 6 sec/frame
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Ductile Fracture
Initial statistics 2.2k nodes 134k surfels
Final statistics 3.3k nodes 144k surfels
Simulation time 23 sec/frame
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Conclusion

• Advantages
• decoupling of physics and surface representation
• dynamic adaptation of shape functions
• during crack propagation
• when re-sampling of spatial domain
• Drawbacks
• excessive fracturing ? simulation nodes ??
• visibility testing is still costly
• each test ray-surface intersection test

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Future Work

• Real-time simulation
• simplification of algorithms
• efficient data structures
• efficient caching schemes
• Solve excessive up-sampling issue
• variant of the virtual node algorithm

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Thank you!

• Contact information
• Mark Pauly pauly_at_inf.ethz.ch
• Richard Keiser keiser_at_inf.ethz.ch
• Bart Adams barta_at_cs.kuleuven.ac.be
• Phil Dutré phil_at_cs.kuleuven.ac.be
• Markus Gross grossm_at_inf.ethz.ch
• Leonidas J. Guibas guibas_at_cs.stanford.edu