meshless animation of fracturing solids mark pauly leonidas j. guibas richard keiser markus gross...

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 pauly@inf.ethz.ch

Richard Keiser keiser@inf.ethz.ch

Bart Adams barta@cs.kuleuven.ac.be

Phil Dutré phil@cs.kuleuven.ac.be

Markus Gross grossm@inf.ethz.ch

Leonidas J. Guibas guibas@cs.stanford.edu