Adaptive resolution of 1D mechanical B-spline

Julien Lenoir, Laurent Grisoni, Philippe Meseure, Christophe Chaillou

Problem

Real-time physical simulation of a knot

Fixed resolution simulation Goal: adaptive resolution simulation

Related work

1D model and knot tying [Wang et al 05] Mass-spring model, not adaptive [Brown et al 04] Non physics based model, ‘follow

the leader’ rules, not adaptive Generality on multiresolution physical model

Discrete model: [Luciana et al 95, Hutchinson et al 96, Ganovelli et al 99]

Continuous model: [Wu et al 04, Debunne et al 01, Grinspun et al 02,Capell et al 02]

Outline

Physical simulation of 1D B-spline Geometric subdivision of a B-spline Mechanical multiresolution Results Side effect Conclusion

Physical model: Lagrange formalismVariational formulation

+ Mechanical system defined via DOF

= Energy minimization relatively to DOFs

Physical simulation of 1D B-spline

Geometric model: B-spline

n

kkk sbtts

0

)()(),( qPqk=(qkx,qky,qkz) position of the kth control pointsbk are the spline base functions

t is the time, s the parametric abscissa

Physical simulation of 1D B-spline Physical Model:

Definition of the DOFs:iq

Physical simulation of 1D B-spline Physical Model:

Definition of the DOFs: Lagrange equations applied to B-spline:

i

i

iq

EQ

q

K

dt

d

)(

iq

Kinetic energyEnergies not derived from a potential(Collisions, Frictions…)

Potential energies(Deformations, Gravity…)

Physical simulation of 1D B-spline Physical Model:

Definition of the DOFs: Lagrange equations applied to B-spline:

i

i

iq

EQ

q

K

dt

d

)(

iq

i

zyx q ),,( AAAA

M

M

M

00

00

00

Μ dssbsbm j

s

iij )()(Μ

BA Μ

),,( zyx BBBB

Generalized mass matrix:

Gather the and termsi

Q iq

E

Physical simulation of 1D B-spline Continuous deformation energies

Stretching [Nocent01]: Green-Lagrange tensor allows large deformations Piola-Kirchhoff elasticity constitutive law

Bending: in progress… Twisting: not treated (need to extend the model to

a 4D model)

dss

sl

s

sl

s

tslrYtE

end

begin

stretch

)(

.1)(

/),(

8

.)( 0

22

0

Physical simulation of 1D B-spline Constraints by Lagrange multipliers (λi):

Direct integration into the mechanical system:

E

B

λ

A

0L

LM T

Physical simulation of 1D B-spline Constraints by Lagrange multipliers (λi):

Direct integration into the mechanical system:

λi links a scalar constraint g(s,t) to the DOFs

E

B

λ

A

0L

LM T

Physical simulation of 1D B-spline Constraints by Lagrange multipliers (λi):

Direct integration into the mechanical system:

λi links a scalar constraint g(s,t) to the DOFs L and E are determined via the Baumgarte

scheme:

E

B

λ

A

0L

LM T

012

2

g

tgt

g

Time step

=> Possible violation but no drift

Physical simulation of 1D B-spline

Resulting physical simulation: - 6 constraint equations - 33 DOF

Lack of DOF in some area

Geometric problem

Geometric subdivision of a B-spline Exact insertion in NUB-spline (Oslo algorithm):

NUBS of degree d

Knot vectors:

i

idjij 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

Mechanical multiresolution

Insertion and suppression: Reallocate the data structure: pre-allocation Shift the pre-computed data and re-compute the

missing part (example: , ) Continuous stretching deformation energies:

Pre-computed terms: 4D array Sparse Symmetric

ds

sbq

sbsbsbsbB

n

jjj

qpmiimpq

3

1

)()0(

)()()()(

Μ 1Μ

Avoid multiple computationStorage in an 1D array

Mechanical multiresolution

Criteria for insertion: Geometric problem => geometric criteria Problem appears in high curvature area Fast curvature evaluation based on control points

Criteria for suppression: Segment rectilinear

1.

2

1

11

11

iiii

iiiii qqqq

qqqqC

Results

Adaptive resolution

Low resolution

High resolution

Geometric property: Multiple insertion at the same location decrease

locally the continuity => ‘Degree’ insertions

= C-1 local continuity = Cutting

Mechanical property: Dynamic cutting

without anythingspecial to handle

Side effect

Side effect

Example of multiple cutting:

Conclusion

Real-time adaptive 1D mechanical model Continuous model (in time and space)

=> Stable over time=> Can handle sliding constraint [Lenoir04]

Dynamic cutting appears as a side effect Future works:

Enhance the deformation energies:Better bending + Twisting (4D model, cosserat [Pai02])

Handle length constraint