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Meshless Approximation Methods and Applications in Physics Based Modeling and Animation Bart Adams Martin Wicke Tutorial Overview Meshless Methods smoothed particle hydrodynamics moving least squares data structures

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SLIDE 1

Meshless Approximation Methods and Applications in Physics Based Modeling and Animation

Bart Adams Martin Wicke

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SLIDE 2

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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SLIDE 3

Part I: Meshless Approximation Methods

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SLIDE 4

Meshless Approximations

Approximate a function from discrete samples Use only neighborhood information

1D 2D, 3D

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SLIDE 5

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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SLIDE 6

Meshless Approximation Methods

Fluid simulation using SPH Elastic solid simulation using MLS

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SLIDE 7

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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SLIDE 8

Smoothed Particle Hydrodynamics

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SLIDE 9

Smoothed Particle Hydrodynamics (SPH)

Integral representation of a scalar function f Dirac delta function

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SLIDE 10

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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SLIDE 11

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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SLIDE 13

Example: density evaluation

Smoothed Particle Hydrodynamics (SPH)

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SLIDE 14

Smoothed Particle Hydrodynamics (SPH)

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Derivatives

Smoothed Particle Hydrodynamics (SPH)

replace by

∇,

Z linear, product rule

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SLIDE 16

Particle approximation for the derivative Some properties:

  • simple averaging of function values
  • nly 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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SLIDE 17

Smoothed Particle Hydrodynamics (SPH)

Example: gradient of our smoothing kernel

We have with Gradient using product rule:

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SLIDE 18

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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SLIDE 19

Similarly, starting from This gradient is symmetric:

Smoothed Particle Hydrodynamics (SPH)

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SLIDE 20
  • Other differential operators
  • Divergence
  • Laplacian

Smoothed Particle Hydrodynamics (SPH)

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SLIDE 21

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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SLIDE 22

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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SLIDE 23

Problem: particle deficiencies near boundaries

  • integral/summation truncated by the boundary
  • example: wrong density estimation

Solution: ghost particles

Smoothed Particle Hydrodynamics (SPH)

real particles ghost particles 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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SLIDE 27

Preview: Particle Fluid Simulation

Solve the Navier‐Stokes momentum equation

pressure force viscosity force gravity Lagrangian 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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SLIDE 29

Preview: Particle Fluid Simulation

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SLIDE 30

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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SLIDE 31

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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SLIDE 33

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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SLIDE 35

 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 function complete polynomial basis moment matrix

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SLIDE 36

Demo

(demo‐shapefunctions)

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SLIDE 37

Moving Least Squares (MLS)

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SLIDE 38

Moving Least Squares (MLS)

Derivatives

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SLIDE 39

Moving Least Squares (MLS)

Consistency

  • have to prove:
  • r:
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SLIDE 40

Problem: moment matrix can become singular

  • Example:
  • particles in a plane in 3D
  • Linear basis

Moving Least Squares (MLS)

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SLIDE 41

Moving Least Squares (MLS)

Stable computation of shape functions

translate basis by scale by

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

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SLIDE 42

MLS Summary

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SLIDE 43

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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SLIDE 44

Preview: Elastic Solid Simulation

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SLIDE 45

Preview: Elastic Solid Simulation

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SLIDE 46

Preview: Elastic Solid Simulation

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SLIDE 47

Part I: Conclusion

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SPH – MLS Comparison

SPH local fast simple weighting not consistent MLS local slower matrix inversion (can fail) consistent up to chosen order

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SLIDE 49

Lagrangian vs Eulerian Kernels

Lagrangian kernels

neighbors remain constant

Eulerian kernels

neighbors change

[Fries & Matthies 2004]

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SLIDE 50

Lagrangian vs Eulerian Kernels

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

[Fries & Matthies 2004]

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SLIDE 51

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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SLIDE 52

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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SLIDE 53

Search Data Structures

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SLIDE 54

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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SLIDE 55

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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SLIDE 56

Spatial Hashing: Construction

i,j

H(i,j)

… …

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SLIDE 57

Spatial Hashing: Query

… …

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SLIDE 58

Spatial Hashing: Query

…

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SLIDE 59

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 = r 2d cells searched 3d cells searched

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SLIDE 60

kd‐Trees: Construction

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SLIDE 61

kd‐Trees: Query

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SLIDE 62

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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SLIDE 63

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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SLIDE 64

Application 1:

Particle Fluid Simulation

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SLIDE 65

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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SLIDE 66

Fluid Simulation

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SLIDE 67

Eulerian

  • vs. Lagrangian
  • Eulerian

Simulation

  • Discretization
  • f space
  • Simulation mesh required
  • Better guarantees / operator consistency
  • Conservation of mass problematic
  • Arbitrary boundary conditions hard
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SLIDE 68

Eulerian

  • vs. Lagrangian
  • Lagrangian

Simulation

  • Discretization
  • f the

material

  • Meshless

simulation

  • No guarantees on

consistency

  • Mass preserved

automatically (particles)

  • Arbitrary boundary

conditions easy (per particle)

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SLIDE 69

Navier‐Stokes Equations

  • Momentum equation:
  • Continuity equation:
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SLIDE 70

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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SLIDE 71

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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SLIDE 72

Momentum Equation

  • Instance of Newton’s Law
  • Right‐hand side consists of
  • Pressure forces
  • Viscosity forces
  • External forces
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SLIDE 73

Density Estimate

  • SPH has concept of density built in
  • Particles carry mass
  • Density computed from particle density

ρi =

X

j

wijmj

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SLIDE 74

Pressure

  • Pressure acts to equalize density differences
  • CFD: γ

= 7, computer graphics: γ = 1

  • large K and γ

require small time steps

p = K(

Ã

ρ ρ0

!γ

− 1)

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SLIDE 75

Pressure Forces

  • Discretize
  • Use symmetric SPH gradient approximation
  • Preserves linear and angular momentum

ap = −∇p

ρ

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SLIDE 76

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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SLIDE 77

Viscosity

  • Discretize

using SPH Laplace approximation

  • Momentum‐preserving
  • Very unstable
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SLIDE 78

XSPH (artificial viscosity)

  • Viscosity an artifact, not simulation goal
  • Viscosity needed for stability
  • Smoothes velocity field
  • Artificial viscosity: stable smoothing
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SLIDE 79

Integration

  • Update velocities
  • Artificial viscosity
  • Update positions
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SLIDE 80
  • Apply to individual particles
  • Reflect off boundaries
  • 2‐way coupling
  • Apply inverse impulse to object

Boundary Conditions

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SLIDE 81

Surface Effects

  • Density estimate breaks down at boundaries
  • Leads to higher particle density
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SLIDE 82

Surface Extraction

  • Extract iso‐surface of density field
  • Marching cubes
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SLIDE 83

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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SLIDE 85

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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SLIDE 86

Application 2:

Elastic Solid Simulation

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SLIDE 87

Goal

Simulate elastically deformable objects

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SLIDE 88

Goal

Simulate elastically deformable objects efficient and stable algorithms ~ different materials

elastic, plastic, fracturing

~ highly detailed surfaces

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SLIDE 89

Elasticity Model

What are the strains and stresses for a deformed elastic material?

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SLIDE 90

Elasticity Model

Displacement field

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SLIDE 91

Elasticity Model

Gradient of displacement field

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SLIDE 92

Elasticity Model

Green‐Saint‐Venant non‐linear strain tensor

symmetric 3x3 matrix

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SLIDE 93

Elasticity Model

Stress from Hooke’s law

symmetric 3x3 matrix

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SLIDE 94

Elasticity Model

For isotropic materials

Young’s modulus E Poisson’s ratio v

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SLIDE 95

Elasticity Model

Strain energy density Elastic force

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SLIDE 96

Elasticity Model

Volume conservation force

prevents undesirable shape inversions

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SLIDE 97

Elasticity Model

Final PDE

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SLIDE 98

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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SLIDE 101

Surface Animation – Alternative 1

Simply use MLS approximation

  • f deformation field

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

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SLIDE 102

Surface Animation – Alternative 1

Vertex 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 1

Easy GPU Implementation

scalars remain constant  only have to send particle deformations to the GPU

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Surface Animation – Alternative 2

Use weighted first‐order Taylor approximation for 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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SLIDE 106

Plasticity

Include plasticity effects

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SLIDE 107

Plasticity

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

strain state variable

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SLIDE 108

Plasticity

Strain state variables updated by absorbing some of the elastic strain

Absorb some of the elastic strain: Limit amount of plastic strain:

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SLIDE 109

Plasticity

Update the reference shape and store the plastic strain state variables

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SLIDE 110

Ductile Fracture

Initial statistics:

2.2k nodes 134k surfels

Final statistics:

3.3k nodes

144k surfels

Simulation time:

23 sec/frame

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Modeling Discontinuities

Only visible nodes should interact

crack

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Modeling Discontinuities

Only visible nodes should interact

  • collect nearest neighbors

crack

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Modeling Discontinuities

Only visible nodes should interact

  • collect nearest neighbors
  • perform visibility test

crack

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Modeling Discontinuities

Only visible nodes should interact

  • collect nearest neighbors
  • perform visibility test

crack

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Modeling Discontinuities

Only visible nodes should interact Discontinuity along the crack surfaces

crack

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Modeling Discontinuities

Only visible nodes should 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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SLIDE 118

Modeling Discontinuities

Solution: transparency method1

  • nodes in vicinity of crack

partially interact

  • by modifying the weight

function  crack becomes transparent near 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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SLIDE 119

Modeling Discontinuities

Weight function Shape function

Visibility Criterion Transparency Method

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SLIDE 120

Demo

(demo‐shapefunctions)

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SLIDE 121

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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SLIDE 122

Re‐sampling: Example

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SLIDE 123

Brittle Fracture

Initial statistics:

4.3k nodes 249k surfels

Final statistics:

6.5k nodes

310k surfels

Simulation time:

22 sec/frame

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SLIDE 124

Summary

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SLIDE 125

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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SLIDE 126

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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SLIDE 127

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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SLIDE 128

Application 3:

Shape & Motion Modeling

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SLIDE 129

Shape Deformations

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SLIDE 130

Shape Deformations: Objective

Find a realistic shape deformation given the user’s input constraints.

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SLIDE 131

Shape Deformations

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SLIDE 132

Shape Deformations

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SLIDE 133

Shape Deformations

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SLIDE 134

Deformation Field Representation

Use meshless shape functions to define a continuous deformation field.

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SLIDE 135

Deformation Field Representation

Complete linear basis in 3D

Precompute for every node and neighbor

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SLIDE 136

Deformation Field Optimization

We are optimizing the displacement field

nodal deformations unknowns to solve for

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SLIDE 137

Deformation Field Optimization

The displacement field should satisfy the input constraints. Position constraint

 quadratic in the unknowns

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SLIDE 138

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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SLIDE 139

Deformation Field Optimization

The total energy to minimize Solve using LBFGS

  • unknowns: nodal displacements
  • need to compute derivatives

with respect to unknowns

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SLIDE 140

Nodal Sampling & Coupling

Keep number of unknowns as low as possible.

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SLIDE 141

Nodal Sampling & Coupling

Ensure proper coupling by using material distance in weight functions.

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SLIDE 142

Nodal Sampling & Coupling

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

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SLIDE 143

Nodal Sampling & Coupling

Grid‐based fast marching to compute material distances.

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SLIDE 144

Nodal Sampling & Coupling

Resulting sampling is roughly uniform

  • ver the material.

Resulting coupling respects the topology

  • f the shape.
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SLIDE 145

Surface Deformation

Use Alternative 1 of the surface animation algorithms discussed before Vertex positions and normals updated on the GPU

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SLIDE 146

Shape Deformations

100k vertices, 60 nodes  55 fps

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SLIDE 147

Shape Deformations

500k vertices, 60 nodes  10 fps

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SLIDE 148

Demo

(demo‐dragon)

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SLIDE 149

Deformable Motions

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SLIDE 150

Deformable Motions: Objective

Find a smooth path for a deformable object from given key frame poses.

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SLIDE 151

Deformation Field Representation

shape functions in space shape functions in time

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SLIDE 152

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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SLIDE 153

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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SLIDE 154

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 constraints volume preservation constraints acceleration constraints

  • bstacle avoidance constraints
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SLIDE 155

Deformation Field Optimization

We want a smooth motion. Acceleration constraint

for all nodes in all frames

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SLIDE 156

Deformation Field Optimization

We want a collision free motion. Obstacle avoidance constraint

for all nodes in all frames

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SLIDE 157

Deformable Motions

solve time: 10 seconds, 25 frames 59 nodes 500k vertices 2 keyframes

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SLIDE 158

Adaptive Temporal Sampling

Number of unknowns to solve for: 3NT

 keep as low as possible!

Constraints only imposed at frames

 what 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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SLIDE 159

Adaptive Temporal Sampling

Solve only at the key frames.

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SLIDE 160

Adaptive Temporal Sampling

Evaluate over whole time interval.

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SLIDE 161

Adaptive Temporal Sampling

Introduce new frame where energy highest and solve.

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SLIDE 162

Adaptive Temporal Sampling

Evaluate over whole time interval.

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SLIDE 163

Adaptive Temporal Sampling

Iterate until motion is satisfactory.

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SLIDE 164

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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SLIDE 165

Demo

(demo‐towers)

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SLIDE 166

Demo

(demo‐animation‐physics)

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SLIDE 167

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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SLIDE 168

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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SLIDE 169

Conclusions

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SLIDE 170

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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SLIDE 171

Website

All material available at

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

Contact information

bart.adams@cs.kuleuven.be wicke@stanford.edu

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SLIDE 172

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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SLIDE 173

Thank You!