Point Based Animation of Elastic, Point Based Animation of Elastic, Plastic and Melting ObjectsPlastic and Melting Objects

Mark PaulyMark PaulyAndrew NealenAndrew Nealen

Marc AlexaMarc Alexa

ETH ZürichETH Zürich TU DarmstadtTU Darmstadt StanfordStanford

Matthias MüllerMatthias MüllerRichard KeiserRichard KeiserMarkus GrossMarkus Gross

9555549李盈璁

Outline

Related Work Advantages & Disadvantages Elasticity Model Simulation Loop Time Integration Surface Animation (省略 ) Result

Related Work

Desbrun & Cani [95,96,99] Physics: Smoothed Particle Hydrodynamics (SPH) Surface: Implicit with suppressed distance blending

Tonnesen [98] Physics: Lennard-Jones based forces Surface: Particles with orientation

Advantages & Disadvantages

Advantages No volumetric mesh needed Natural adaptation to topological changes

Disadvantages Difficulty of getting sharp fracture lines Neighboring Phyxels are not explicitly given

Throughout this work we use Spatial Hashing [Teschner et al. 03] for fast neighbor search (when needed)

Elasticity Model

Continuum Elasticity Elastic Strain Estimation of Derivatives Discrete Energy Density Elastic Forces

Reference configuration

Continuum Elasticity

Deformed configuration

= Elastic Memory

Displacement (vector) field:

u(x) = [ u(x,y,z) v(x,y,z) w(x,y,z) ]T

u(x)x x+u(x)

x’u(x’)

x’+u(x’)

Elastic Strain

→ Strain depends on the spatial derivatives of u(x)no

strain

strain

u(x)

uuuuε TTGreen

zyx

zyx

zyx

www

vvv

uuu

u

Next: Compute spatial derivatives of the x component u

Estimation of Derivatives - 1

Computation of the unknown u,x, u,y and

u,z at xi by Linear approximation

,2

,

,

( ) ( )x

i i y

z

u

u u u O

u

x x x x

,

,

,

( )x

j i y j i

z

u

u u u

u

x x

ijj

jj wuu 2)~( Minimize

→ WLS/MLS approximation of derivatives

uixi

xj

x = xij

uj

Estimation of Derivatives - 2

i

Tj i iju u u xx

2( )j j ij

j

e u u w

, , ,i

T

x y zu u u u x

Set partial derivatives of e with respect to u,x, u,y and u,z to zero to obtain minimizer of e

Linear approximation of uj as seen from xiActual value of uj at point xj

1

i

Tij ij ij j i ij ij

j j

u w u u w

x x x x

Use SVD for the 3x3 Matrix inversion for stability

Vector of Unknown Partial Derivatives

uixi

xj

x = xij

uj

Discrete Energy Density -1

Strain from u

Cεσ

Stress via material law (Hooke)

3 3

1 1i ij ij

i j

U ε σ

Energy density (scalar)

T TT T T TGreen ε u u u u

Discrete Energy Density -2

Use Smoothed Particle Hydrodynamics (SPH) Method Mass of each Phyxel mi is fix during the simulation Distribute the mass around the Phyxel using a polynomial

weighting kernel wij with compact support

The density around Phyxel i is

From which we compute the volume vi as

i j ijj

m w

/i i iv m

Elastic Forces

Estimate volume vi represented by phyxel i via SPH

UvU ii

Elastic energy of phyxel i

Depends on ui and

uj of all neighbors j

Phyxel i and all neighbors j receive a force

j

ji ffj

ij

Uv

uf

Simulation Loop

,t extf

t tu t tu t tε t tσtf

Verlet Integration (= new displacements)

Estimation of Derivatives

External Forces (Gravity, Interaction)

Computation of Strains, Stresses,

Elastic Energy and per Phyxel

Body Forces

External Forces (Gravity, Interaction)

Verlet Integration (= new displacements)

Estimation of Derivatives

Computation of Strains, Stresses,

Elastic Energy and per Phyxel

Body Forces

Time Integration

Verlet (Explicit) Time Stepping

Newtons Second Law of Motion

2

2i i

iit m

u f

a

2 3 41 2 1 6t t t t t t t t t O t u u v a b

2 3 41 2 1 6t t t t t t t t t O t u u v a b

+

= 2 42t t t t t t t O t u u u a

Result

彈性物體1、彈性物體2

Thank You !