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Publié par	Enfea
Nombre de lectures	14
Langue	English

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Basics on numerical algorithms for
Non Smooth Dynamical Systems
VincentAcary,FrédéricDubois
TuturialLecture
ThirdSICONOSGeneralMeeting. September15-17,Bristol
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.1/32Ü
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Outline
1–Introdution
1.1–Scope
1.2–LinearComplementaritySystems(LCS)
1.3–LinearLagrangiansystemswithContactandFriction
• 2–Event–Driven
• 3–Time–stepping
• 4–Comparison
• 5–Illustrations
• 6–Conclusion
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.2/32h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Scope
Only Initial Value Problems (IVP).
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.3/32h
h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Scope
Only Initial Value Problems (IVP).
Two typical examples of Non Smooth Dynamical Systems (NSDS) :
• Linear Complementarity Systems
• Lagrangian Dynamical Systems with contact and friction
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.3/32h
h
h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Scope
Only Initial Value Problems (IVP).
Two typical examples of Non Smooth Dynamical Systems (NSDS) :
• Linear Complementarity Systems
• Lagrangian Dynamical Systems with contact and friction
Two major kinds of time integration scheme :
• Event–driven scheme. (the time–steps depend on the events)
• Time–stepping scheme (the time–step does not depend on the events)
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.3/32h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Linear Complementarity systems
The Linear Complementarity System (LCS) may be deﬁned by
8
>
x˙ =Ax+Bλ
<
(1)
y =Cx+Dλ
>
:
0≤y⊥λ≥ 0
n×n n×m m×n m×m
withA∈ IR ,B∈ IR ,C∈ IR ,D∈ IR , form constraints.
In the sequel, we consider the scalar case (m = 1)
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.4/32h
h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Linear Complementarity systems
The Linear Complementarity System (LCS) may be deﬁned by
8
>
x˙ =Ax+Bλ
<
(1)
y =Cx+Dλ
>
:
0≤y⊥λ≥ 0
n×n n×m m×n m×m
withA∈ IR ,B∈ IR ,C∈ IR ,D∈ IR , form constraints.
In the sequel, we consider the scalar case (m = 1)
Notion of Relative degreer
yλ
Deﬁning the Markov Parameters as
2
(D,CB,CAB,CA B,...)
the relative degree is the rank of the ﬁrst non zero Markov Parameter.
“The number of differentiation ofy to obtain explicitlyy in function ofλ.”
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.4/32h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Linear Complementarity systems (Continued...)
Relative degreer = 0, D > 0, Trivial case
yλ
−1
• The multiplierλ = max(0,−D Cx) is a Lipschitz continuous function ofx
• The numerical integration may be performed with any standard ODE solvers.
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.5/32h
h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Linear Complementarity systems (Continued...)
Relative degreer = 0, D > 0, Trivial case
yλ
−1
• The multiplierλ = max(0,−D Cx) is a Lipschitz continuous function ofx
• The numerical integration may be performed with any standard ODE solvers.
Relative degreer = 1, D=0, CB > 0
yλ
• The multiplierλ is a function of timet, not necessarily continuous, for instance,
of bounded variations (BV).
• The numerical integration have to be performed with speciﬁc solvers
(Event–Driven or Moreau’s Time–stepping)
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.5/32h
h
h
Introduction Event-Driven Time-stepping Comparison Illustrations Conclusion
Linear Complementarity systems (Continued...)
Relative degreer = 0, D > 0, Trivial case
yλ
−1
• The multiplierλ = max(0,−D Cx) is a Lipschitz continuous function ofx
• The numerical integration may be performed with any standard ODE solvers.
Relative degreer = 1, D=0, CB > 0
yλ
• The multiplierλ is a function of timet, not necessarily continuous, for instance,
of bounded variations (BV).
• The numerical integration have to be performed with speciﬁc solvers
(Event–Driven or Moreau’s Time–stepping)
Relative degreer = 1, D=0, CB = 0, CAB>0
yλ
• The system is not self-consistent : Need a re-initialization mapping
• The multiplierλ is a real measure.
• Speciﬁc solvers (Event–Driven or Moreau’s Time–stepping) as for Lagrangian
dynamical system with constraints
INRIARhône–Alpes
Basics on Numerical algorithms – Third SICONOS General Meeting. – p.5/32

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