1D deformable model for real time physical simulation Julien Lenoir november 23 rd 2004 GRAPHIX-LIFL...

1D deformable model forreal time physical simulation

Julien Lenoirnovember 23rd 2004

GRAPHIX-LIFLChristophe ChaillouPhilippe Meseure, Laurent Grisoni

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Objectives

IRCAD

Simulation group-CIMIT

3

Outlines

1) 1D deformable model

2) Constraints

3) Multi-resolution

4

Physically-based animation: generalities

Possible independent geometry-physics

Time discretization => time step

Collision detection and response

Physical resolution: static vs dynamic Static: calculates the new positions

Dynamic: computes the accelerations Numerical integration => velocities and positions

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General context

Mechanical model = Model defined by physical laws Discrete => physical equations are linked to some

specific location

Continuous => physical equations are formulated in a continuous domain with spatial discretization without spatial discretization

6

Outlines


2) Constraints

3) Multi-resolution

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Real/Interactive time for manipulation

Realistic behavior

stretching bending twisting

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1) 1D deformable modelPrevious works

Geometric model: piecewise linear, subdivision, spline…

Physical model: Discrete models:

Mass-spring [Desbrun et al.99] [Casiez01] dynamic

Continuous models: Finite difference [Terzo87]

dynamic, spatial discretization (regular grid) Lagrange [QinTerzo96][Remion99]

dynamic, non spatial discretization, spline Cosserat elasticity - static resolution [Pai02]

static

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1) 1D deformable modelChoices

Geometry: Discrete model (points) Subdivision curve Continuous model (spline…)

Physics: Discrete models: Mass-spring (Newton) Continuous models:

Spatial discretization: FDM, LEM… Any DOF: Modal analysis, Lagrange formalism

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1) 1D deformable modelOur Choices

Geometry: Discrete model (points) Subdivision curve Continuous model (spline…)

Physics: Discrete models: Mass-spring (Newton) Continuous models:

Spatial discretization: FDM… Any DOF: Modal analysis, Lagrange formalism

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1) 1D deformable modelOur Choices: Geometry

Spline model: interpolation/approximation of n control points weight given by basis functions abscissa parameter

iq

ib

0q

1q

2q

3q

n

i

i sbs1

)()( iqP

s

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Important properties of a spline:

Vector of particular parametric values

Continuity: C0, C1(Catmull-Rom), C2(cubic Bspline),…

For most models: locality of order m:Each control point influences at most m spline segments

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Bezierbasis functions

Uniform cubic Bsplinebasis functions

No locality Locality of order 4

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1) 1D deformable modelOur Choices: Physics

Lagrange formalism: Set of n virtual DOF

Defines and

Principle of virtual works

Lagrangian

Principle of minimum action

=> minimization of the Lagrangian action

)(tqi

),,( tqqE iic ),,( tqqE iip

0)(

ii q

L

q

L

dt

d

),,(),,(),,( tqqEtqqEtqqL iipiicii

ii

c

i

c Qq

E

q

E

dt

d

)(

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DOF = coordinates of the control points => equations of motion:

Defining kinetic energy:

1) 1D deformable modelDynamic simulation of a spline

)(tqi

dstststqqE iic 2),(),(2

1),,( P

i

ii

Qq

E

q

E

dt

d cc

)(

mts ),(

Parametric density of mass

n

i

i sbttst

ts1

)()(),(),( iqP

P Velocity of a point

dstsm 2),(2

1P

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Matrix form:

where is the generalized mass matrixand is the DOF acceleration vector


n

jjji

cc tqdssbsbmq

E

q

E

dt

d

i

ii 1

)()()()(

,z}y,{x,{1..n},

A

Tzyx qqqA

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Properties of : Axis independence => 3 sub-systems symmetric Band of width 2m-1 (m=locality order) Constant over time

=> LU pre-computation of(constant and band)


n

jjji

cc tqdssbsbmq

E

q

E

dt

d

i

ii 1

)()()()(

,z}y,{x,{1..n},

dssbsbmM jiij )()(M

M

M

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Right part of Lagrange equations: Gravity Ambient viscosity Interactions with other objects, users or itself Deformation energies

=> Gathered in a vector System resolution:

LU band => O(n) Numerical integration

=> getting new velocities and positions

i

Q

B

BA M

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1) 1D deformable modelDeformation energies

Type: Discrete

Continue

Deformations: Stretching

Bending

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Discrete deformation energies Parametric sampling of the curve

points = extremities of the spline segments

Point sampling on spline segments

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Discrete deformation energies Stretching springs

Bending springs


Continuous distribution of masses (Lagrange)=> not a mass-spring model

Angular springs

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Continuous deformation energies Stretching [Nocent01]

elasticity given by the Young modulus Bending:

Estimated by the second derivative of the position

n

i

i sbttss 1

2

2

)()(),( iqP

Tangent

CurvatureCT

PF bats

sk

),(2

2

Force:

not a pure bending energy

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1) 1D deformable modelTest framework

Visualization using generalized cylinder, implicit surface,...

Animation software framework: SPORE Approximation using spheres for collision detection

=> decomposition of the spline into spheres: Curvilinear distribution of spheres

=> dynamic algorithm

Extremity of a spline segment

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1) 1D deformable modelResults (PIV 2.4Ghz, 512 Mb)

Computation time for one simulation time step (1ms), including numerical integration (RK4): Cubic NUBS, 50 points, discrete energies: 7.69ms Cubic UBS, 50 points, discrete energies: 5.73ms

spline having a uniform knot vector get pre-computed parts

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Computation time for one simulation time step (1ms), including numerical integration (RK4): 4 cubic NUBS, 15 points, discrete energies: 6.9ms 4 cubic NUBS, 15 points, continuous energies: 8.9ms

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Computation time for one simulation time step (1ms), including numerical integration (RK4): 11 cubic NUBS, 15 points, discrete energies: 7.1ms 4 cubic NUBS, 10 points, continuous energies: 5ms

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1) 1D deformable modelApplication: Intestine simulation

Rendering: Implicit surface - punctual skeleton [Triquet01]

Simulation: 165 points, 14ms for one time step of 1msSpline with uniformknot vector =>pre-computation

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1) 1D deformable modelContribution summary

Real-time simulation of 1D dynamic spline Proposition of a continuous bending energy Extension to 2D and 3D objects

29

Outlines


2) Constraints

3) Multi-resolution

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2) ConstraintsWhy using constraints ?

Some interactions cannot be handled by collisions: Linked objects (articulated bodies) Strong interactions (suture...)

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2) ConstraintsPrevious Works

Penalty Methods [Platt Bar88]

Kinematic reduction

Projection [Gascuel92]

Lagrange multipliers

[Baraff96,Cotin99,Remion99]

Post-stabilization methods

[Ascher94,Faure98,Cline02]

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2) ConstraintsPrevious Works

Penalty Methods [Platt Bar88]

Kinematic reduction

Projection [Gascuel92]

Lagrange multipliers

[Baraff96,Cotin99,Remion99]

Post-stabilization methods

[Ascher94,Faure98,Cline02]

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2) ConstraintsSimple constraints

What do we call “a simple constraint” ? Interaction without any other object (or considered as

immobile) Absolute constraint in space Examples:

pendulum,rope moving along a stick, ...

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Dynamic equations extended by:

One Lagrange multiplier λ per constraint equation Extended matrix system

Faster resolution using acceleration decomposition:

0),,(

.)(1

tqqC

Lq

EQ

q

E

q

E

dt

d

j

c

j

jijp

i

c

i

c

i

i

EA

BλA

L

LM T

ct AAA

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Constraint must be rewritten as a function of the DOF acceleration: Baumgarte scheme:

=> possible violation but no drift

Examples of constraints: Fixed point, point on a plane, point on an axis Fixed tangent Fixed curvature vector ...

0),( tqC

0),(),(),( tqcCtqCbtqCa

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Results: 30 points, 6 constraints = 2 ms

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2) ConstraintsSliding constraints

Sliding movement through a hole

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0),,,( tsqqg

General constraint equation:


A new class of constraints Example on a fixed point constraint:

0),(),,( 0 PtsPtqqg A

Let be mobile, determined by the object dynamic !!As

0)),((),,,( 0 PttsPtsqqg

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becomes a new unknown, neither a DOF, nor a lagrange multiplier, a free variable

=> requires a new equationLagrange multipliers formalism gives:

s

0.

λs

g T

λ Force ensuring the constraint g T

sg

P(s,t) s(t)

Example on a sliding point constraint:

0)),((),,( PttsPtsqg

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Acceleration of the resolution:allowing the constraint to work:

=> direct relation to compute=> accelerates the global resolution=> controls the dynamic of (« friction-like »)

0.

λs

gs

T

s

s

theoretical framework => no friction00 accelerates the dynamic => inverse friction0 brakes the dynamic => normal friction

ss

P(s,t) s(t)

λT

sg

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2) ConstraintsResults

Shoelace: 28 points, discrete energy, 10ms

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Suture

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Slipknot... Sliding point constrained on another point of the curve

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2) Constraints

Tensor product: extension to 2D and 3D Two types of constraints:

simple constraints sliding constraints

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2) ConstraintsConstraints among different objects

Link objects of different nature Rigid bodies,

deformable objects... Example: swing

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Global system equations:


How to link several objects involving different physical formalisms ?

Hypothesis:Each object animation can be defined in a matrix form

DOF unknowns:(positions, displacements, velocities, accelerations…)

BA M

1

1

n

1

B

B

A

A

nM

M

0

01

FU K

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Constraint method: Lagrange multipliers

Proposition of a software framework: Interface: ArticulableObject and Constraint Articulation: composition of ArticulableObject and set

of Constraint Optimization of the computation:

Articulation solves the system of equations Updating matrices if required (done by each object)

EA

BλA

L

LM T

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3x10 points, 42 constraints= 3.14 ms

4x8 points + mass spring (25 nodes) + rigid body + 54 constraints

= 10.3 ms

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2) ConstraintsContribution summary

Practical set of constraints based on Lagrange

multipliers

Extension of the theoretical Lagrange multipliers

to handle sliding constraints

Software framework for the simulation of

articulations

All results easily extensible to 2D and 3D objects

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Outlines


2) Constraints

3) Multi-resolution

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3) Multi-resolution

Problem: Bending properties of the model depend on the

distribution of the control points Increases the number of control points

=> increases the computation and the memory cost

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3) Multi-resolution

Requirements: Overcoming the curvature limitation on the spline

control (example: knot tightening) concentration of control points in a small parametric area

Keeping acceptable computation times and reasonable memory costs

Proposal: Dynamic control of the spline resolution (geometric

and mechanical)

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3) Multi-resolutionPrevious Works

Mass-spring [Hutchinson et al.96]

FEM multi-resolution [Debunne et al.00]

Lagrange reduction parameters [Nocent et al.01]

Subdivision [Grinspun et al.02]: function

refinement

FEM with hierarchical function and non regular

grid [Capelle et al.02]

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Well known results about geometric subdivision of Non Uniform B-splines exact insertion algorithm (Oslo)

3) Multi-resolutionLocal refinement of BSplines

NUBS of degree d

Knot vectors:

i

id

jij bb~

,

sinon

~si

0

1 10,

jjjji

ttt

rji

iri

rjrirji

iri

irjrji tt

tt

tt

tt,1

11

1,

1,

~

~

~

01,

0,

1 )1( idiii

diii qqq

)(sbit it 1it

insertion

)(~

sbit~ it~

1~it 2

~it

The simplification of BSplines is often an approximation

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3) Multi-resolutionProposition

Extension of the spline subdivision scheme to the mechanical model=> dynamic number of DOF=> local modification (due to locality property)

Local adaptation of the resolution

Adaptation criteria based on curvature

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Deformation energies:

discrete energy related to one resolution=> needs a re-distribution=> inadequate

continuous energy not linked to the resolution but to the shape=> adequate

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Continuous stretching energy: data structure 4D array:

Sparse Symmetric

Proposition: Efficient encoding: compression into a 1D array No redundance, no null values Access with 4 indirection tables => O(1)

ds

sbq

sbsbsbsbB

n

jjj

qpmiimpq

3

1

)()0(

)()()()(

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3) Multi-resolutionResults

NUBS, 11 points, continuous stretching

Without multi-resolution2.16ms

With automatic insertion12ms

stability with 20 points

With automatic insertionand suppression

11msstability with 18 points

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Interesting side effect: cuttingeach insertion on the same Bspline knot decreases the continuity => C-1 after enough number of insertions => cutting

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3) Multi-resolutionContribution summary

Automatic adaptation of the resolution

Knot tightening

Cutting

No extension to 2D or 3D:

tensor product not adequate !!

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Conclusion & Future Works

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Conclusion & Future WorksContribution summary

Spline-based real-time deformable model: Set of optimization strategies for real-time Continuous bending energy

Constraints based on Lagrange multipliers: Set of practical constraints Sliding constraints Software framework for articulation

Multi-resolution: Dynamic and automatic adaptation of the resolution Dynamic cutting

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Conclusion & Future Works

Improving the continuous bending energy Developing a continuous twisting energy Exploring other possibilities of the sliding

constraints Accelerating the multi-resolution procedure:

Sherman-Morrisson scheme for inverse matrix Studying other type of geometry in 2D and 3D

like subdivision surfaces

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Thank you !

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1) 1D deformable modelWhat about visualization ?

2 slides: Modèle 1D => pas de volume (2 videos pour transition) Epaisseur donnée par la collision (correlation modele

collision – modele visu)

Broken Lines Continuity C0

No depth perception

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1) 1D deformable modelRendering 4/4

Implicit surfaces Ponctual model with marching cubes Convolution model

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Right part of Lagrange equation: Gravity: Ambient viscosity:

Gathered in a vector (except the ambient viscosity)

System resolution:

LU band => O(n) Numerical integration

i

Q

dssbmgQ ii )(

n

jjij qdssbsbCQ

i1

)()(

B

MVm

CQ

jqV j

VBA Mm

CM

with

BVA )(m

CM

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Goal (video en haut d’où on en tire les buts pt par pt) on le vire

Interactive 1D deformable model evolving in an environement (=> collision with other objects)

Links between different objects ( collision) Enough flexibility to tight a knot difficile à faire !!

Citer les source (ou le noter) Citer que c ’est du réel !!

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2) ConstraintsSmooth constraints

Coulomb friction on a sliding point constraint: adherence vs sliding

fixed point vs sliding point Two states: adherence-sliding

If adherence try simulation with a fixed point constraint

if force needed to ensure the constraint to high transform the fixed point into a sliding point

and re-itere the resolution with introducing a friction forcec

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!! « Adequate » à la place de « adapte » !! « Involving » remplacer par « based on »

ou « using »

1D deformable model for real time physical simulation Julien Lenoir november 23 rd 2004 GRAPHIX-LIFL...

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