Post on 18-Jan-2018
description
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