approximation methods and applications in physics...
TRANSCRIPT
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Meshless
Approximation Methods and Applications in Physics Based Modeling and Animation
Bart Adams Martin Wicke
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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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Part I: Meshless Approximation Methods
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Meshless
Approximations
Approximate a function from discrete samples
Use only neighborhood information
1D 2D, 3D
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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
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Meshless
Approximation Methods
Fluid simulation using SPH Elastic solid simulation using MLS
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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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Smoothed Particle Hydrodynamics
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Smoothed Particle Hydrodynamics (SPH)
Integral representation of a scalar function f
Dirac
delta function
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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)
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Smoothed Particle Hydrodynamics (SPH)
Example: designing a smoothing kernel in 2D
For simplicity set
We pick
Satisfy the unity constraint (2D)
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Particle approximation by discretization
Smoothed Particle Hydrodynamics (SPH)
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Example: density evaluation
Smoothed Particle Hydrodynamics (SPH)
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Smoothed Particle Hydrodynamics (SPH)
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Derivatives
Smoothed Particle Hydrodynamics (SPH)
replace by
∇,Z
linear, product rule
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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)
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Smoothed Particle Hydrodynamics (SPH)
Example: gradient of our smoothing kernel
We have
with
Gradient using product rule:
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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.
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Similarly, starting from
This gradient is symmetric:
Smoothed Particle Hydrodynamics (SPH)
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Other differential operators
Divergence
Laplacian
Smoothed Particle Hydrodynamics (SPH)
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Smoothed Particle Hydrodynamics (SPH)
Problem: Operator inconsistency
Theorems derived in continuous setting don’t hold
Solution: Derive operators for specific guarantees
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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)
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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
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SPH Summary (1)
A scalar function f
satisfies
Replace Dirac
by a smooth function w
Discretize
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SPH Summary (2)
Function evaluation:
Gradient evaluation:
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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
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Preview: Particle Fluid Simulation
Solve the Navier‐Stokes momentum equation
pressureforce
viscosityforce gravityLagrangian
derivative
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Preview: Particle Fluid Simulation
Discretized
and solved at particles using SPH
density estimation
pressure force
viscosity force
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Preview: Particle Fluid Simulation
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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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Moving Least Squares
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Meshless
Approximations
Same problem statement:Approximate a function from discrete samples
1D 2D, 3D
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Moving Least Squares (MLS)
Moving least squares approach
Locally fit a polynomial
By minimizing
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with
Moving Least Squares (MLS)
Solution:
Approximation:
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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
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Demo (demo‐shapefunctions)
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Moving Least Squares (MLS)
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Moving Least Squares (MLS)
Derivatives
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Moving Least Squares (MLS)
Consistency
have to prove:
or:
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Problem: moment matrix can become singular
Example:
particles in a plane in 3D
Linear basis
Moving Least Squares (MLS)
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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.
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MLS Summary
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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
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Preview: Elastic Solid Simulation
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Preview: Elastic Solid Simulation
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Preview: Elastic Solid Simulation
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Part I: Conclusion
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SPH – MLS Comparison
SPH
localfast
simple weightingnot consistent
MLS
localslower
matrix inversion (can fail)consistent up to chosen order
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Lagrangian
vs
Eulerian
Kernels
Lagrangian
kernelsneighbors remain constant
Eulerian
kernelsneighbors change
[Fries & Matthies
2004]
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Lagrangian
vs
Eulerian
Kernels
Lagrangian
kernels are OK for elastic solid simulations, but not for fluid simulations
[Fries & Matthies
2004]
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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)
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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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Search Data Structures
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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
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Search Data Structures
Spatial hashing
limited adaptivity
cheap construction
and maintenance
kd‐trees
more adaptive, flexible
more expensive to
build and maintain
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Spatial Hashing: Construction
i,j
H(i,j)
…
…
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Spatial Hashing: Query…
…
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Spatial Hashing: Query
…
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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
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kd‐Trees: Construction
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kd‐Trees: Query
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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
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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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Application 1: Particle Fluid Simulation
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Tutorial Overview
Meshless
Methods
smoothed particle hydrodynamics
moving least squares
Applications
particle fluid simulation
elastic solid simulation
shape & motion modeling
Conclusions
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Fluid Simulation
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Eulerian
vs. Lagrangian
Eulerian
Simulation
Discretization
of space
Simulation mesh required
Better guarantees / operator consistency
Conservation of mass problematic
Arbitrary boundary conditions hard
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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)
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Navier‐Stokes Equations
Momentum equation:
Continuity equation:
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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)
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Momentum Equation
Left‐hand side is material derivative
“How does the velocity of this piece of fluid change?”
Useful in Lagrangian
setting
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Momentum Equation
Instance of Newton’s Law
Right‐hand side consists of
Pressure forces
Viscosity forces
External forces
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Density Estimate
SPH has concept of density built in
Particles carry mass
Density computed from particle density
ρi =Xj
wijmj
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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)
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Pressure Forces
Discretize
Use symmetric SPH gradient approximation
Preserves linear and angular momentum
ap =−∇pρ
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Pressure Forces
Symmetric pairwise
forces: all forces cancel out
Preserves linear momentum
Pairwise
forces act along
Preserves angular momentum
xi − xj
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Viscosity
Discretize
using SPH Laplace
approximation
Momentum‐preserving
Very unstable
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XSPH (artificial viscosity)
Viscosity an artifact, not simulation goal
Viscosity needed for stability
Smoothes velocity field
Artificial viscosity: stable smoothing
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Integration
Update velocities
Artificial viscosity
Update positions
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Apply to individual particles
Reflect off boundaries
2‐way coupling
Apply inverse impulse to object
Boundary Conditions
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Surface Effects
Density estimate breaks down at boundaries
Leads to higher particle density
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Surface Extraction
Extract iso‐surface of density field
Marching cubes
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Demo (sph)
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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]
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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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Application 2: Elastic Solid Simulation
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Goal
Simulate elastically deformable objects
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Goal
Simulate elastically deformable objects
efficient and stable algorithms~
different materialselastic, plastic, fracturing
~highly detailed surfaces
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Elasticity Model
What are the strains and stressesfor a deformed elastic material?
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Elasticity Model
Displacement field
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Elasticity Model
Gradient of displacement field
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Elasticity Model
Green‐Saint‐Venantnon‐linear strain tensor
symmetric 3x3 matrix
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Elasticity Model
Stress from Hooke’s
law
symmetric 3x3 matrix
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Elasticity Model
For isotropic materials
Young’s modulus E
Poisson’s ratio v
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Elasticity Model
Strain energy density
Elastic force
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Elasticity Model
Volume conservationforce
prevents undesirable shape inversions
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Elasticity Model
Final PDE
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Particle Discretization
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Simulation Loop
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Surface Animation
Two alternatives
Using MLS approximation of
displacement field
Using local first‐order approximation of
displacement field
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Surface Animation – Alternative 1Simply use MLS approximation
of deformation field
Can use whatever representation: triangle meshes, point clouds, …
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Surface Animation – Alternative 1Vertex position update
Approximate normal update
first‐order Taylor for displacement field at normal tip
tip is transformed to
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Surface Animation – Alternative 1Easy GPU Implementation
scalarsremain constant
only have to send particle deformations to the GPU
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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
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Demo (demo‐elasticity)
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Plasticity
Include plasticity effects
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Plasticity
Store some amount of the strain and subtract it from the actual strainin the elastic force computations
strain state variable
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Plasticity
Strain state variables updated byabsorbing some of the elastic strain
Absorb some of the elastic strain:
Limit amount of plastic strain:
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Plasticity
Update the reference shape andstore the plastic strain state variables
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Ductile Fracture
Initial statistics:2.2k nodes134k surfels
Final statistics:3.3k nodes144k surfels
Simulation time:23 sec/frame
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Modeling Discontinuities
Only visible
nodesshould interact
crack
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Modeling Discontinuities
Only visible
nodesshould interact
collect nearest neighbors
crack
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Modeling Discontinuities
Only visible
nodesshould interact
collect nearest neighbors
perform visibility test
crack
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Modeling Discontinuities
Only visible
nodesshould interact
collect nearest neighbors
perform visibility test
crack
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Modeling Discontinuities
Only visible
nodesshould interact
Discontinuity along the crack surfaces
crack
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Modeling Discontinuities
Only visible
nodesshould interact
Discontinuity along the crack surfaces
But also within the domain
undesirable!
crack
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Modeling Discontinuities
Weight function Shape function
Visibility Criterion
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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
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Modeling Discontinuities
Weight
function
Shape
function
Visibility Criterion Transparency Method
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Demo (demo‐shapefunctions)
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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
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Re‐sampling: Example
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Brittle Fracture
Initial statistics:4.3k nodes249k surfels
Final statistics:6.5k nodes310k surfels
Simulation time:22 sec/frame
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Summary
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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
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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
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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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Application 3: Shape & Motion Modeling
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Shape Deformations
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Shape Deformations: Objective
Find a realistic shape deformationgiven the user’s input constraints.
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Shape Deformations
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Shape Deformations
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Shape Deformations
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Deformation Field RepresentationUse meshless
shape functions to define
a continuous deformation field.
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Deformation Field Representation
Complete linear basis in 3D
Precompute
for every node and neighbor
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Deformation Field Optimization
We are optimizing the displacement field
nodal deformationsunknowns to solve for
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Deformation Field Optimization
The displacement field shouldsatisfy the input constraints.
Position constraint
quadratic in the unknowns
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Deformation Field Optimization
The displacement should be realistic.
Locally rigid (minimal strain)
Volume preserving
degree 6 in the unknowns non‐linear problem
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Deformation Field Optimization
The total
energy
to minimize
Solve using LBFGS
unknowns: nodal displacements
need to compute derivativeswith respect to unknowns
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Nodal Sampling & Coupling
Keep number of unknowns as low as possible.
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Nodal Sampling & Coupling
Ensure proper coupling by using material distance in weight functions.
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Nodal Sampling & Coupling
Set of candidate points:vertices and interior set of dense grid points
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Nodal Sampling & Coupling
Grid‐based fast marching tocompute material distances.
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Nodal Sampling & Coupling
Resulting sampling
is roughly uniform
over the material.Resulting coupling
respects the topology
of the shape.
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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
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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!