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Meshless Approximation Methods and Applications in Physics Based Modeling and Animation Bart Adams Martin Wicke

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Page 1: Meshless PPT

Meshless

Approximation Methods  and Applications in Physics Based  Modeling and Animation

Bart Adams Martin Wicke

Page 2: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 3: Meshless PPT

Part I: Meshless Approximation  Methods

Page 4: Meshless PPT

Meshless

Approximations

Approximate a function from discrete samples

Use only neighborhood information

1D 2D, 3D

Page 5: Meshless PPT

Meshless

Approximation Methods

Smoothed Particle Hydrodynamics (SPH)

simple, efficient, no consistency guarantee

popular in CG for fluid simulation

Meshfree

Moving Least Squares (MLS)

a little more involved, consistency guarantees

popular in CG for elasto‐plastic solid simulation

Page 6: Meshless PPT

Meshless

Approximation Methods

Fluid simulation using SPH Elastic solid simulation using MLS

Page 7: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 8: Meshless PPT

Smoothed Particle  Hydrodynamics

Page 9: Meshless PPT

Smoothed Particle Hydrodynamics (SPH)

Integral representation of a scalar function f

Dirac

delta function

Page 10: Meshless PPT

Replace Dirac

by a smooth function w

Desirable properties of w

1.

compactness:

2.

delta function property:

3.

unity condition (set f

to 1):

4.

smoothness

Smoothed Particle Hydrodynamics (SPH)

Page 11: Meshless PPT

Smoothed Particle Hydrodynamics (SPH)

Example: designing a smoothing kernel in 2D

For simplicity set

We pick

Satisfy the unity constraint (2D)

Page 12: Meshless PPT

Particle approximation by discretization

Smoothed Particle Hydrodynamics (SPH)

Page 13: Meshless PPT

Example: density evaluation

Smoothed Particle Hydrodynamics (SPH)

Page 14: Meshless PPT

Smoothed Particle Hydrodynamics (SPH)

Page 15: Meshless PPT

Derivatives

Smoothed Particle Hydrodynamics (SPH)

replace    by 

∇,Z

linear, product rule

Page 16: Meshless PPT

Particle approximation for the derivative

Some properties:

simple averaging of function values

only need to be able to differentiate w

gradient of constant function not necessarily 0

will fix this later

Smoothed Particle Hydrodynamics (SPH)

Page 17: Meshless PPT

Smoothed Particle Hydrodynamics (SPH)

Example: gradient of our smoothing kernel

We have

with

Gradient using product rule:

Page 18: Meshless PPT

Alternative derivative formulation

Smoothed Particle Hydrodynamics (SPH)

Old gradient formula:

Product rule:

Use (1) in (2):

(1)

(2)

Gradient of constant function now always 0.

Page 19: Meshless PPT

Similarly, starting from

This gradient is symmetric:  

Smoothed Particle Hydrodynamics (SPH)

Page 20: Meshless PPT

Other differential operators

Divergence

Laplacian

Smoothed Particle Hydrodynamics (SPH)

Page 21: Meshless PPT

Smoothed Particle Hydrodynamics (SPH)

Problem: Operator inconsistency

Theorems derived in continuous setting don’t hold

Solution: Derive operators for specific guarantees

Page 22: Meshless PPT

Problem: particle inconsistency

constant consistency in continuous setting

does not necessarily give constant consistency in 

discrete setting (irregular sampling, boundaries)

Solution: see MLS approximation

Smoothed Particle Hydrodynamics (SPH)

Page 23: Meshless PPT

Problem: particle deficiencies near boundaries

integral/summation truncated by the boundary

example: wrong density estimation

Solution: ghost particles

Smoothed Particle Hydrodynamics (SPH)

realparticles

ghostparticles

boundary

Page 24: Meshless PPT

SPH Summary (1)

A scalar function f

satisfies

Replace Dirac

by a smooth function w

Discretize

Page 25: Meshless PPT

SPH Summary (2)

Function evaluation:

Gradient evaluation:

Page 26: Meshless PPT

SPH Summary (3)

Further literature

Smoothed Particle Hydrodynamics, Monaghan, 1992

Smoothed Particles: A new paradigm for animating highly deformable bodies, 

Desbrun

& Cani, 1996

Smoothed Particle Hydrodynamics, A Meshfree

Particle Method, Liu & Liu, 2003

Particle‐Based Fluid Simulation for Interactive Applications, Müller

et al., 2003

Smoothed Particle Hydrodynamics, Monaghan, 2005

Adaptively Sampled Particle Fluids, Adams et al., 2007

Fluid Simulation, Chapter 7.3 in Point Based Graphics, Wicke

et al., 2007

Many more

Page 27: Meshless PPT

Preview: Particle Fluid Simulation

Solve the Navier‐Stokes momentum equation

pressureforce

viscosityforce gravityLagrangian

derivative

Page 28: Meshless PPT

Preview: Particle Fluid Simulation

Discretized

and solved at particles using SPH

density estimation

pressure force

viscosity force

Page 29: Meshless PPT

Preview: Particle Fluid Simulation

Page 30: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 31: Meshless PPT

Moving Least Squares

Page 32: Meshless PPT

Meshless

Approximations

Same problem statement:Approximate a function from discrete samples

1D 2D, 3D

Page 33: Meshless PPT

Moving Least Squares (MLS)

Moving least squares approach

Locally fit a polynomial

By minimizing

Page 34: Meshless PPT

with

Moving Least Squares (MLS)

Solution:

Approximation:

Page 35: Meshless PPT

by construction they are consistent

up to the order of the basis by construction they build a partition of unity

Moving Least Squares (MLS)

Approximation:

with shape functions

weight functioncomplete polynomial basis

moment matrix

Page 36: Meshless PPT

Demo (demo‐shapefunctions)

Page 37: Meshless PPT

Moving Least Squares (MLS)

Page 38: Meshless PPT

Moving Least Squares (MLS)

Derivatives

Page 39: Meshless PPT

Moving Least Squares (MLS)

Consistency

have to prove:

or: 

Page 40: Meshless PPT

Problem: moment matrix can become singular

Example: 

particles in a plane             in 3D

Linear basis

Moving Least Squares (MLS)

Page 41: Meshless PPT

Moving Least Squares (MLS)

Stable computation of shape functions

translate basis byscale by 

It can be shown that this moment matrix has  a lower condition number.

Page 42: Meshless PPT

MLS Summary

Page 43: Meshless PPT

MLS Summary (2)

Literature

Moving Least Square Reproducing Kernel Methods (I) Methododology

and 

Convergence, Liu et al., 1997

Moving‐Least‐Squares‐Particle Hydrodynamics –I. Consistency and Stability, 

Dilts, 1999

Classification and Overview of Meshfree

Methods, Fries & Matthies, 2004

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

et al., 2004

Meshless

Animation of Fracturing Solids, Pauly

et al., 2005

Meshless

Modeling of Deformable Shapes and their Motion, Adams et al., 2008

Page 44: Meshless PPT

Preview: Elastic Solid Simulation

Page 45: Meshless PPT

Preview: Elastic Solid Simulation

Page 46: Meshless PPT

Preview: Elastic Solid Simulation

Page 47: Meshless PPT

Part I: Conclusion

Page 48: Meshless PPT

SPH – MLS Comparison

SPH

localfast

simple weightingnot consistent

MLS

localslower

matrix inversion (can fail)consistent up to chosen order

Page 49: Meshless PPT

Lagrangian

vs

Eulerian

Kernels

Lagrangian

kernelsneighbors remain constant

Eulerian

kernelsneighbors change

[Fries & Matthies

2004]

Page 50: Meshless PPT

Lagrangian

vs

Eulerian

Kernels

Lagrangian

kernels are OK for elastic solid  simulations, but not for fluid simulations

[Fries & Matthies

2004]

Page 51: Meshless PPT

Moving Least Squares Particle  Hydrodynamics (MLSPH)

Use idea of variable rank MLS

start for each particle with basis of highest rank

if inversion fails, lower rank

Consequence: shape functions are not smooth

(SPH)

(MLS)

Page 52: Meshless PPT

Tutorial Overview

Meshless Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 53: Meshless PPT

Search Data Structures

Page 54: Meshless PPT

Search for Neighbors

Approximate integrals using sums over samples

Brute force: O(n2)

Local kernels with limited support

Sum only over neighbors: O(n log

n)

Finding neighbors efficiently key

Page 55: Meshless PPT

Search Data Structures

Spatial hashing

limited adaptivity

cheap construction 

and maintenance

kd‐trees

more adaptive, flexible

more expensive to 

build and maintain

Page 56: Meshless PPT

Spatial Hashing: Construction

i,j

H(i,j)

Page 57: Meshless PPT

Spatial Hashing: Query…

Page 58: Meshless PPT

Spatial Hashing: Query

Page 59: Meshless PPT

Spatial Hashing

No explicit grid needed

Particularly useful for sparse sampling

Hash collisions lead to spurious tests

Grid spacing s

adapted to query radius r

d = 2r

d = r2d

cells searched

3d

cells searched

Page 60: Meshless PPT

kd‐Trees: Construction

Page 61: Meshless PPT

kd‐Trees: Query

Page 62: Meshless PPT

Comparison

Spatial hashing:

construct from n

points: O(n)

insert/move single point: O(1)

query: O(rρ) for average point density ρ

hash table size and cell size must be properly chosen

kd‐Trees:

construct from n

points: O(n

log n)

query: O(k

log n) for k

returned points

handles varying query types or irregular sampling

Page 63: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 64: Meshless PPT

Application 1: Particle Fluid Simulation

Page 65: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 66: Meshless PPT

Fluid Simulation

Page 67: Meshless PPT

Eulerian

vs. Lagrangian

Eulerian

Simulation

Discretization

of space

Simulation mesh required

Better guarantees / operator consistency

Conservation of mass problematic

Arbitrary boundary conditions hard

Page 68: Meshless PPT

Eulerian

vs. Lagrangian

Lagrangian

Simulation

Discretization

of the 

material

Meshless

simulation

No guarantees on 

consistency

Mass preserved 

automatically (particles)

Arbitrary boundary 

conditions easy (per 

particle)

Page 69: Meshless PPT

Navier‐Stokes Equations

Momentum equation:

Continuity equation:

Page 70: Meshless PPT

Continuity Equation

Continuum equation automatically fulfilled

Particles carry mass

No particles added/deleted No mass loss/gain

Compressible Flow

Often, incompressible flow is a better approximation

Divergence‐free flow (later)

Page 71: Meshless PPT

Momentum Equation

Left‐hand side is material derivative

“How does the velocity of this piece of fluid change?”

Useful in Lagrangian

setting

Page 72: Meshless PPT

Momentum Equation

Instance of Newton’s Law

Right‐hand side consists of

Pressure forces

Viscosity forces

External forces

Page 73: Meshless PPT

Density Estimate

SPH has concept of density built in

Particles carry mass 

Density computed from particle density

ρi =Xj

wijmj

Page 74: Meshless PPT

Pressure

Pressure acts to equalize density differences

CFD: γ

= 7, computer graphics: γ

= 1

large K and γ

require small time steps

p = K(Ãρ

ρ0

!γ− 1)

Page 75: Meshless PPT

Pressure Forces

Discretize

Use symmetric SPH gradient approximation

Preserves linear and angular momentum

ap =−∇pρ

Page 76: Meshless PPT

Pressure Forces

Symmetric pairwise

forces: all forces cancel out

Preserves linear momentum

Pairwise

forces act along                

Preserves angular momentum

xi − xj

Page 77: Meshless PPT

Viscosity

Discretize

using SPH Laplace

approximation

Momentum‐preserving

Very unstable

Page 78: Meshless PPT

XSPH (artificial viscosity)

Viscosity an artifact, not simulation goal

Viscosity needed for stability

Smoothes velocity field

Artificial viscosity: stable smoothing

Page 79: Meshless PPT

Integration

Update velocities

Artificial viscosity

Update positions

Page 80: Meshless PPT

Apply to individual particles

Reflect off boundaries

2‐way coupling

Apply inverse impulse to object

Boundary Conditions

Page 81: Meshless PPT

Surface Effects

Density estimate breaks down at boundaries

Leads to higher particle density

Page 82: Meshless PPT

Surface Extraction

Extract iso‐surface of density field

Marching cubes

Page 83: Meshless PPT

Demo (sph)

Page 84: Meshless PPT

Extensions

Adaptive Sampling [Adams et al 08]

Incompressible flow [Zhu et al 05]

Multiphase flow [Mueller et al 05]

Interaction with deformables

[Mueller et al 04]

Interaction with porous materials 

[Lenaerts

et al 08]

Page 85: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 86: Meshless PPT

Application 2: Elastic Solid Simulation

Page 87: Meshless PPT

Goal

Simulate elastically deformable objects

Page 88: Meshless PPT

Goal

Simulate elastically deformable objects

efficient and stable algorithms~

different materialselastic, plastic, fracturing

~highly detailed surfaces

Page 89: Meshless PPT

Elasticity Model

What are the strains and stressesfor a deformed elastic material?

Page 90: Meshless PPT

Elasticity Model

Displacement field

Page 91: Meshless PPT

Elasticity Model

Gradient of displacement field

Page 92: Meshless PPT

Elasticity Model

Green‐Saint‐Venantnon‐linear strain tensor

symmetric 3x3 matrix

Page 93: Meshless PPT

Elasticity Model

Stress from Hooke’s

law

symmetric 3x3 matrix

Page 94: Meshless PPT

Elasticity Model

For isotropic materials

Young’s modulus E

Poisson’s ratio v

Page 95: Meshless PPT

Elasticity Model

Strain energy density

Elastic force

Page 96: Meshless PPT

Elasticity Model

Volume conservationforce

prevents undesirable shape inversions

Page 97: Meshless PPT

Elasticity Model

Final PDE

Page 98: Meshless PPT

Particle Discretization

Page 99: Meshless PPT

Simulation Loop

Page 100: Meshless PPT

Surface Animation

Two alternatives

Using MLS approximation of 

displacement field

Using local first‐order approximation of 

displacement field

Page 101: Meshless PPT

Surface Animation – Alternative 1Simply use MLS approximation

of deformation field

Can use whatever representation: triangle meshes, point clouds, …

Page 102: Meshless PPT

Surface Animation – Alternative 1Vertex position update

Approximate normal update

first‐order Taylor for displacement field at normal tip

tip is transformed to

Page 103: Meshless PPT

Surface Animation – Alternative 1Easy GPU Implementation

scalarsremain constant

only have to send particle deformations to the GPU

Page 104: Meshless PPT

Surface Animation – Alternative 2

Use weighted first‐order Taylor approximationfor displacement field at vertex

Updated vertex position

avoid storing per‐vertex shape functions at the cost of more computations

Page 105: Meshless PPT

Demo (demo‐elasticity)

Page 106: Meshless PPT

Plasticity

Include plasticity effects

Page 107: Meshless PPT

Plasticity

Store some amount of the strain and subtract it from the actual strainin the elastic force computations

strain state variable

Page 108: Meshless PPT

Plasticity

Strain state variables updated byabsorbing some of the elastic strain

Absorb some of the elastic strain:

Limit amount of plastic strain:

Page 109: Meshless PPT

Plasticity

Update the reference shape andstore the plastic strain state variables

Page 110: Meshless PPT

Ductile Fracture

Initial statistics:2.2k nodes134k surfels

Final statistics:3.3k nodes144k surfels

Simulation time:23 sec/frame

Page 111: Meshless PPT

Modeling Discontinuities

Only visible

nodesshould interact

crack

Page 112: Meshless PPT

Modeling Discontinuities

Only visible

nodesshould interact

collect nearest neighbors

crack

Page 113: Meshless PPT

Modeling Discontinuities

Only visible

nodesshould interact

collect nearest neighbors

perform visibility test

crack

Page 114: Meshless PPT

Modeling Discontinuities

Only visible

nodesshould interact

collect nearest neighbors

perform visibility test

crack

Page 115: Meshless PPT

Modeling Discontinuities

Only visible

nodesshould interact

Discontinuity along the crack surfaces

crack

Page 116: Meshless PPT

Modeling Discontinuities

Only visible

nodesshould interact

Discontinuity along the crack surfaces

But also within the domain

undesirable!

crack

Page 117: Meshless PPT

Modeling Discontinuities

Weight function Shape function

Visibility Criterion

Page 118: Meshless PPT

Modeling Discontinuities

Solution: transparency method1

nodes in vicinity of crack 

partially interact

by modifying the weight 

function

crack becomes transparentnear the crack tip

Organ

et al.: Continuous Meshless

Approximations for Nonconvex

Bodies by Diffraction and Transparency, Comp. Mechanics, 1996

1

crack

Page 119: Meshless PPT

Modeling Discontinuities

Weight 

function

Shape 

function

Visibility Criterion Transparency Method

Page 120: Meshless PPT

Demo (demo‐shapefunctions)

Page 121: Meshless PPT

Re‐sampling

crack

Add simulation nodes when number of  neighbors too small

Shape functions adapt automatically!

Local

re‐sampling of the  domain of a node

distribute mass

adapt support radius

interpolate attributes

Page 122: Meshless PPT

Re‐sampling: Example

Page 123: Meshless PPT

Brittle Fracture

Initial statistics:4.3k nodes249k surfels

Final statistics:6.5k nodes310k surfels

Simulation time:22 sec/frame

Page 124: Meshless PPT

Summary

Page 125: Meshless PPT

Summary

Efficient algorithms

for elasticity: shape functions precomputed

for fracturing: local cutting of interactions

No bookkeeping for consistent mesh

simple re‐sampling

shape functions adapt automatically

High‐quality surfaces

representation decoupled from volume discretization

deformation on the GPU

Page 126: Meshless PPT

Limitations

Problem with moment matrix inversions

cannot handle shells (2D layers of particles)

cannot handle strings (1D layer of particles)

Plasticity simulation rather expensive

recomputing

neighbors

re‐evaluating shape functions

Fracturing in many small pieces expensive

excessive re‐sampling

Page 127: Meshless PPT

Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

Page 128: Meshless PPT

Application 3: Shape & Motion Modeling

Page 129: Meshless PPT

Shape Deformations

Page 130: Meshless PPT

Shape Deformations: Objective

Find a realistic shape deformationgiven the user’s input constraints.

Page 131: Meshless PPT

Shape Deformations

Page 132: Meshless PPT

Shape Deformations

Page 133: Meshless PPT

Shape Deformations

Page 134: Meshless PPT

Deformation Field RepresentationUse meshless

shape functions to define

a continuous deformation field.

Page 135: Meshless PPT

Deformation Field Representation

Complete linear basis in 3D

Precompute

for every node and neighbor

Page 136: Meshless PPT

Deformation Field Optimization

We are optimizing the displacement field

nodal deformationsunknowns to solve for

Page 137: Meshless PPT

Deformation Field Optimization

The displacement field shouldsatisfy the input constraints.

Position constraint

quadratic in the unknowns

Page 138: Meshless PPT

Deformation Field Optimization

The displacement should be realistic.

Locally rigid (minimal strain)

Volume preserving

degree 6 in the unknowns non‐linear problem

Page 139: Meshless PPT

Deformation Field Optimization

The total

energy

to minimize

Solve using LBFGS

unknowns: nodal displacements

need to compute derivativeswith respect to unknowns

Page 140: Meshless PPT

Nodal Sampling & Coupling

Keep number of unknowns as low as possible.

Page 141: Meshless PPT

Nodal Sampling & Coupling

Ensure proper coupling by using material distance in weight functions.

Page 142: Meshless PPT

Nodal Sampling & Coupling

Set of candidate points:vertices and interior set of dense grid points

Page 143: Meshless PPT

Nodal Sampling & Coupling

Grid‐based fast marching tocompute material distances.

Page 144: Meshless PPT

Nodal Sampling & Coupling

Resulting sampling

is roughly uniform

over the material.Resulting coupling

respects the topology

of the shape.

Page 145: Meshless PPT

Surface DeformationUse Alternative 1 of the surface

animation algorithms discussed before

Vertex positions and normals

updated on the GPU

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Shape Deformations

100k vertices, 60 nodes  55 fps

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Shape Deformations

500k vertices, 60 nodes  10 fps

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Demo (demo‐dragon)

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Deformable Motions

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Deformable Motions: ObjectiveFind a smooth path for a deformable object 

from given key frame poses.

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Deformation Field Representation

shape functions in spaceshape functions in time

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Deformation Field Representation

Frames: discrete samples in time

keyframe

1 keyframe

2 keyframe

3

frame 1 frame 2 frame 3 frame 4 frame 5

Solve only at discrete frames: nodal displacements 

Use meshless

approximation to define continuous displacement field

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Deformation Field Representation

Complete quadratic basis in 1D

Precompute

for each frame for every neighboring frame

keyframe

1 keyframe

2 keyframe

3

frame 1 frame 2 frame 3 frame 4 frame 5

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Deformation Field Optimization

We want a realistic motion interpolating the keyframes.

keyframe

1 keyframe

2 keyframe

3

frame 1 frame 2 frame 3 frame 4 frame 5

handle constraints

rigidity constraintsvolume preservation constraints

acceleration constraintsobstacle avoidance constraints

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Deformation Field Optimization

We want a smooth motion.

Acceleration constraint

for all nodes in all frames

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Deformation Field Optimization

We want a collision free motion.

Obstacle avoidance constraint

for all nodes in all frames

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Deformable Motions

solve time: 10 seconds, 25 frames

59 nodes

500k vertices

2 keyframes

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Adaptive Temporal Sampling

Number of unknowns to solve for: 3NT keep as low as possible!

Constraints only imposed at frameswhat at interpolated frames?

Adaptive temporal sampling algorithm

keyframe

1 keyframe

2 keyframe

3

frame 1 frame 2 frame 3 frame 4 frame 5

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Adaptive Temporal Sampling

Solve only at the key frames.

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Adaptive Temporal Sampling

Evaluate over whole time interval.

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Adaptive Temporal Sampling

Introduce new frame where energy highest and solve.

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Adaptive Temporal Sampling

Evaluate over whole time interval.

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Adaptive Temporal Sampling

Iterate until motion is satisfactory.

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Deformable Motions

interaction rate: 60 fps, modeling time: 2.5 min, solve time: 16

seconds, 28 frames

66 nodes

166k vertices

7 keyframes

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Demo (demo‐towers)

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Demo (demo‐animation‐physics)

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Summary

Realistic shape and motion modeling

constraints from physical principles

Interactive and high quality

MLS particle approximation

low number of particles

shape functions adapt to sampling and object’s shape

decoupled surface representation

adaptive temporal sampling

Rotations are however not interpolated exactly

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Tutorial Overview

Meshless

Methods

smoothed particle hydrodynamics

moving least squares

data structures

Applications

particle fluid simulation

elastic solid simulation

shape & motion modeling

Conclusions

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Conclusions

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Conclusions

Why use a meshless

method?

requires only a neighborhood graph

resamping

is easy

topological changes are easy

Why use a mesh‐based approach?

more mathematical structure to be exploited

consistency of differential operators

exact conservation of integral properties

Or maybe use a hybrid technique?

PIC/FLIP

particle level set

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Website

All material available at

http://graphics.stanford.edu/~wicke/eg09‐tutorial

Contact information

[email protected]

[email protected]

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Acknowledgements

Collaborators

Funding

Philip Dutré

Matthias Teschner

Matthias Müller

Markus Gross

Maks

Ovsjanikov

Richard Keiser

Mark Pauly

Michael Wand

Leonidas

J. Guibas

Hans‐Peter Seidel

Fund for Scientific Research, Flanders

Max Planck Center for Visual Computing 

and Communication

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Thank You!