meshless animation of fracturing solids mark pauly leonidas j. guibas richard keiser markus gross...
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Meshless Animation of Meshless Animation of Fracturing SolidsFracturing Solids
Mark PaulyLeonidas J. Guibas
Richard KeiserMarkus Gross
Bart AdamsPhilip Dutré
MotivationMotivation
Simulation of fracturing materials in many different applications.
MotivationMotivation
Simulation of fracturing materials in many different applications.
Requirements on fracturing algorithm:
MotivationMotivation
Simulation of fracturing materials in many different applications.
Requirements on fracturing algorithm:brittle or ductile fracture
MotivationMotivation
Simulation of fracturing materials in many different applications.
Requirements on fracturing algorithm:brittle or ductile fracture
arbitrary cracks
MotivationMotivation
Simulation of fracturing materials in many different applications.
Requirements on fracturing algorithm:brittle or ductile fracture
arbitrary cracks
control of fracture paths
MotivationMotivation
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
Related WorkRelated Work
O’Brien & Hodgins [99, 02]dynamic remeshing
element cutting difficult to avoid ill-
shaped elements
Related WorkRelated Work
O’Brien & 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
Meshless MethodsMeshless Methods
Advantagessampling of the volume
handling of large deformation
(re-)sampling of the domain
handling of discontinuities
Drawbacksboundary conditions
overhead for computing interpolation functions
ContributionsContributions
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
OverviewOverview
Part 1: Physics AnimationMeshless Continuum Mechanics
Modeling Discontinuities
Spatial Re-sampling
Part 2: Surface HandlingSurface Model
Crack Initiation & Propagation
Topological Events
Elasticity ModelElasticity Model
Meshless elasticity model derived from continuum mechanics.1
x x+u
displacementfield u
Müller et al.: Point Based Animation of Elastic, Plastic and Melting Objects, SCA 2004
1
t tu t tu
t tεt tσt tU
Simulation loop:
extf tf
Time integrationGradient of displacement fieldStrainStressBody forceAdd external forcesStrain energy
DiscretizationDiscretizationDiscrete set of nodes {xi}
Approximation of displacement field u:
x
ui
xi
u(x) i i(x) ui
evaluation point
summation overneighboring nodes i
displacement vectorof node i
shape functionof node i
Derivation of shape functions
using Moving Least Squares (MLS)
DiscretizationDiscretization
Shape functions i:
i(x) = i(x,xi) pT(x) [M(x)]-1 p(xi)
weight function
linear basis p(x) = [1 x]T
moment matrixM(x) = ii(x,xi) p(xi) pT(xi)
Weight function i(x,y):
i(x,y) = i(r) = 1-6r2+8r3-3r4 r10 r>1
r = ||x-y||/hi
with hi the support radius of node i0 1
1
0 r
i(r)
by construction they build a first order partition of unity (PU)
DiscontinuitiesDiscontinuities
Only visible nodes should interact
collect nearest neighbors
perform visibility test crack
DiscontinuitiesDiscontinuities
Only visible nodes should interact
collect nearest neighbors
perform visibility test crack
DiscontinuitiesDiscontinuities
Problem: undesirable discontinuities of the shape functions
not only along the crack
but also within the domain
crack
DiscontinuitiesDiscontinuities
Weight function Shape function
Visibility Criterion
DiscontinuitiesDiscontinuities
Solution: transparency method1
nodes in vicinity of crack partially interact
by modifying the weight function:
i’(xi,xj) = i(||xi-xj||/hi + (2ds/κ)2)
crack ds
crack becomes transparent near the crack tip
Organ et al.: Continuous Meshless Approximations for Nonconvex Bodies by Diffraction and Transparency, Comp. Mechanics, 1996
1
DiscontinuitiesDiscontinuities
Weight function
Shape function
Visibility Criterion Transparency Method
Re-samplingRe-sampling
xi
crack
Add simulation nodes when number of neighbors too small
Shape functions adapt automatically!
Local resampling of the domain of a node
distribute mass
adapt support radius
interpolate attributes
Re-sampling: ExampleRe-sampling: Example
Part 2Part 2Surface HandlingSurface Handling
Surface AnimationSurface 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:
surfels {si}
simulationnodes {pj}
xi xi + ji’(xi,xj)(uj+ujT(xj-xi))
same transparency weight
Crack PropagationCrack Propagation
Crack initiationwhere stress above threshold
crack created by inserting 3 crack nodes each carrying 2 opposing surfels connection is crack front
external force
external force
one fracturesurface
crack front
Crack PropagationCrack Propagation
Crack propagationpropagate crack nodes along propagation direction
re-project first and last node
up-sample if necessary
external force
external force
one fracturesurface
Crack Propagation: ExampleCrack Propagation: Example
Crack EventsCrack Events
Splittingwhen crack propagates through the material
split front in two new fronts
each one propagates independently
block of material
Crack EventsCrack Events
Mergingwhen two fronts propagate close to each other
merge fronts and associated fracture surfaces
block of material
Crack Events: ExampleCrack Events: Example
Brittle FractureBrittle Fracture
Initial statistics:4.3k nodes
249k surfels
Final statistics:6.5k nodes
310k surfels
Simulation time:22 sec/frame
Controlled FractureControlled Fracture
Initial statistics:4.6k nodes
49k surfels
Final statistics:5.8k nodes
72k surfels
Simulation time:6 sec/frame
Ductile FractureDuctile Fracture
Initial statistics:2.2k nodes
134k surfels
Final statistics:3.3k nodes
144k surfels
Simulation time:23 sec/frame
ConclusionConclusion
Advantagesdecoupling of physics and surface representationdynamic adaptation of shape functions
during crack propagation when re-sampling of spatial domain
Drawbacksexcessive fracturing simulation nodes visibility testing is still costly
each test = ray-surface intersection test
Future WorkFuture Work
Real-time simulationsimplification of algorithms
efficient data structures
efficient caching schemes
Solve excessive up-sampling issuevariant of the virtual node algorithm
Thank you!Thank you!
Contact informationMark Pauly [email protected]
Richard Keiser [email protected]
Bart Adams [email protected]
Phil Dutré [email protected]
Markus Gross [email protected]
Leonidas J. Guibas [email protected]