Smooth constraints for Spline Variational Modeling Julien Lenoir (1), Laurent Grisoni (1), Philippe...
-
Upload
brent-elliott -
Category
Documents
-
view
221 -
download
0
description
Transcript of Smooth constraints for Spline Variational Modeling Julien Lenoir (1), Laurent Grisoni (1), Philippe...
Smooth constraints for Spline Variational ModelingJulien Lenoir(1), Laurent Grisoni(1),
Philippe Meseure(1,2), Yannick Rémion(3), Christophe Chaillou(1)
(1) Alcove Project - INRIA Futurs - LIFL - University of Lille (France)(2) SIC, University of Poitiers (France)
(3) LERI, University of Reims champagne-Ardenne (France)
Outlines Introduction, Previous work & Objectives
A continuous model
Smooth constraints
Results
Introduction Context of variational modelling
Geometrical constraints Energy minimization
Relation with physically based modelling Lagrange formalism (energy minimization) Static vs dynamic
What we call « smooth constraint » Example: sliding point constraint
Previous work [Welch and Witkin 92] Variational surface
modeling Lagrange formalism – static simulation Lagrange multipliers Ponctual constraints & global constraints
[Witkin et al 87] Multiple object definition: parametric, implicit Energy function to minimize
Only parametric: Fixed point, surface attachment… Parametric and Implicit: Floating attachment
Objectives Propose
Dynamic solution for variational modeling Class of smooth constraint for parametric object
[Terzopoulos and Qin 94] D-NURBS for sculpting[Remion et al. 99] Dynamic spline Lagrange dynamics formalism Lagrange multipliers Baumgarte stabilization scheme
Outlines Introduction, Previous work & Objectives
A continuous model
Smooth constraints
Results
A continuous model (1D) [Lenoir et al. 2002]
Geometry defined as a spline:
Apply Physical Properties Homogeneous mass : m Kinetic energy :
External forces : Deformation energy : E Gravity :
n
kkk sbtts
0
)()(),,( qqP
s
n
kkk dssbtmtK
2
0
)()(21),( qq
)()(/ sbFsQ ii PF
s
n
kk
yg dssbtqmgtE
k0
)()(),( q
qk=(qkx,qky,qkz) position of the control pointsbk are the spline base functionst is the time, s the parametric abscissa
Resolution Simulation using the Lagrange formalism
We obtain the following system:
where :
M is band, symmetric and constant over time
i
i
iqEQ
qK
dtd
)(
BA Μ
MM
M
000000
Μ dssbsbm jsiij )()(Μ
izyx q ),,( AAAA
Including constraints Let g be a constraint:
Baumgarte technique [Baumgarte 72]
Overall equation includes the Lagrange multipliers
Each scalar equation requires a Lagrange multiplier
0),,( qtsg
0122
g
tgt
g
EB
λA
0LLM T
EA Lwritten as
Outlines Introduction, Previous work & Objectives
A continuous model
Smooth constraints
Results
We transform the constraint equation:0qg ),,( ts
Smooth constraints
Authorizing s to depend on time0qg ),),(( tts
Baumgarte scheme:
=> s dynamics is needed=> s is considered as a new unknown:
A Free Variable
0122
g
tgt
g
A new equation is needed to control the value of s Principle of Virtual Power:
A constraint must not work, so we get [Remion03]:
Smooth constraints
0gλ s
.
Force which ensuresthe constraint
ss
Pg
P(s(t),t)=P0
λ
Example on a sliding point constraint
Smooth constraints The dynamic system becomes:
Resolution with decomposition of the accelerations[Remion03] : Tendancy (without constraints) Usual constraints correction Smooth constraints correction
Time consuming method
E
B
λ
A0
0000
sLL
LLM
s
Ts
T
Smooth constraints v2
Force which ensuresthe constraint
ss
Pg
P(s(t),t)=P0
λ
0gλ s
.Normalcase :
ss
Pg
0gλ s
.
P(s(t),t)=P0
λ
Generalcase :
Example of sliding point constraint:
We simplify the equation by allowing the constraint to “work”. It enforces s to reach the solution: 0gλ
s
s ..
The overall system becomes:
Resolution by decomposition :
Same complexity but less stage (50%) of computation
E
B
λ
A0
00
0s
LLLLM
s
Ts
T
ct AAA
tsc
Ts
Tc
t
LsLLLsLM
M
AEAλλABA
tTss
T
Ts
Tc
t
LLLLLMLsLM
M
AEλλλABA
111
1
1
1
Smooth constraint v2
Smooth constraint v2 Examples of smooth constraint:
sliding point sliding tangent sliding curvature
Possibility to define multiple constraints relative to one free variable Example: Sliding point constraint with tangent control
Sliding point constrained to a point links to an object
Results Correct re-parametrization of the spline:
Results A shoelace:
Results A hang rope:
Results Sliding point constraint on a 2D spline:
Conclusion & Perspectives Proposition of smooth constraint class
sliding point constraint sliding tangent constraint sliding curvature constraint
Dynamic simulation => control of the end user
Correct re-parametrization of the curve
Use to introduce local friction