Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Mounia Haddouni ∗ , † , Vincent Acary ♦ , † , St´ ephane Garreau ∗ , Jean-Daniel Beley ∗ and Bernard Brogliato † 1st. Pan-American congress on Computational Mechanics (PANACM 2015) Buenos Aires, 27–29 April, 2014. ∗ ANSYS, Villeurbanne, France ♦ INRIA Chile. Las Condes, Santiago de Chile, Chile † INRIA Rhˆ one-Alpes, Centre de recherche Grenoble, St Ismier, France – 1/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Objectives & Motivations Motivations High-fidelity dynamical simulation of mechanisms Nonsmooth multi-body systems with ◮ Signorini unilateral contact, ◮ Coulomb friction, ◮ Newton (or Poisson) impact law, ◮ clearances in joints. Industrial context ◮ Real CAD geometries with edge discontinuities ◮ Robustness w.r.t large number of events: contact activation and deactivation finite accumulation of impacts stick/slip transitions. Objectives & Motivations – 2/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Objectives & Motivations Motivations Simulation of Circuit breakers (INRIA/Schneider Electric) Objectives & Motivations – 3/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Objectives & Motivations Motivations Simulation of watch chronograph mechanism (INRIA/ANSYS) Objectives & Motivations – 3/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Objectives & Motivations Objectives ◮ Time–integration methods in an event–driven framework ◮ Review of D.A.E. integrators with various indices (from 1 to 3). ◮ Standard comparisons on academical examples ◮ Performance profiles on industrial benchmarks Objectives & Motivations – 4/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Modeling framework Modeling framework Signorini unilateral contact and impact law Unilateral contact law : Body B 0 � g ( q ) ⊥ λ � 0 . (1) Newton Impact law: C B if g ( q ) � 0 , then U + = − eU − (2) n γ g γ U : normal relative velocity ( U = ˙ g ) e : kinetic coefficient of restitution C A T 2 T 1 Body A Figure: Signed distance between two bodies A and B at contact γ Modeling framework – 5/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Modeling framework Modeling framework (cont.) Equations of motion q = v ˙ v = F ( q , v , t ) + G T ( q ) λ M ( q ) ˙ g α ( q ) = 0 , α ∈ B (3) 0 � g β ( q ) ⊥ λ β � 0 , β ∈ U , if g β ( q ) � 0 , then U β, + = − eU β, − ◮ g ( q ) ∈ R m : vector of constraints ◮ B ⊂ N index set of bilateral constraints ◮ U ⊂ N index set of unilateral constraints ◮ G ( q ) = ∇ T g ( q ) ∈ R m × n Jacobian matrix of the constraints ◮ λ ∈ R m is the Lagrange multipliers vector associated to the constraints. Modeling framework – 6/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Modeling framework Modeling framework (cont.) Index sets of active constraints The set of all constraints is denoted by I 0 = B ∪ U . Closed contacts index set: I 1 = { γ ∈ I 0 , g γ ( q ) = 0 } Closed contacts index set for a non trivial period of time: I 2 = { γ ∈ I 0 , g γ ( q ) = 0 , ˙ g γ ( q ) = 0 } Position based constraints : index-3 differential algebraic equation. On the period over which I 2 is constant, we solve q = v ˙ v = F ( q , v , t ) + G T ( q ) λ (4) M ( q ) ˙ g γ ( q ) = 0 , γ ∈ I 2 . Modeling framework – 7/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Modeling framework Modeling framework (cont.) Lower index differential algebraic equation Velocity based constraints : index-2 differential algebraic equation. If the constraint g ( · ) is differentiated once with respect to time, one obtains the following index-2 DAE q = v ˙ v = F ( q , v , t ) + G T ( q ) λ (5) M ( q ) ˙ G γ ( q ) v = 0 , γ ∈ I 2 . Acceleration based constraints : index-1 differential algebraic equation. If g ( · ) is differentiated twice, one gets the index-1 DAE q = v ˙ v = F ( q , v , t ) + G T ( q ) λ M ( q ) ˙ (6) v + dG γ ( q ) G γ ( q ) ˙ v = 0 , γ ∈ I 2 . dt Modeling framework – 8/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Numerical time integration methods Time–stepping vs. Event-driven scheme Time–stepping schemes Principle of nonsmooth event capturing methods 1. A unique formulation of the dynamics is considered. For instance, a dynamics in terms of measures. mdu = dr q = u + ˙ (7) u + � 0 if q � 0 0 � dr ⊥ ˙ 2. The time-integration is based on a consistent approximation of the equations in terms of measures. For instance, � � du = ( u + ( t k +1 ) − u + ( t k )) ≈ ( u k +1 − u k ) du = (8) ] t k , t k +1 ] ] t k , t k +1 ] 3. Consistent approximation of measure inclusion. � p k +1 ≈ dr − dr ∈ N K ( t ) ( u + ( t )) (9) ] t k , t k +1 ] (10) ➜ p k +1 ∈ N K ( t ) ( u k +1 ) Numerical time integration methods – 9/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Numerical time integration methods Time–stepping vs. Event-driven scheme Event-driven schemes Principle of nonsmooth event tracking methods Time-decomposition of the dynamics in ◮ modes , time-intervals in which the dynamics is smooth ( I 1 and I 2 invariant), ◮ discrete events , times where the dynamics is nonsmooth (changes in I 1 and/or I 2 ). Comments On the numerical point of view, we need ◮ detect events with for instance root-finding procedure. ◮ Dichotomy and interval arithmetic ◮ Newton procedure for C 2 function and polynomials ◮ solve the non smooth dynamics at events with a reinitialization rule of the state, ◮ integrate the smooth dynamics between two events with any DAE solvers associated with a given index formulation. Numerical time integration methods – 10/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Numerical time integration methods Time–stepping vs. Event-driven scheme Comparison Numerical time–integration methods for Nonsmooth Multibody systems (NSMBS): Nonsmooth event capturing methods (Time–stepping methods) � robust, stable and proof of convergence � low kinematic level for the constraints � able to deal with finite accumulation � very low order of accuracy even in free flight motions Nonsmooth event tracking methods (Event–driven methods) � higher order accuracy integration of free flight motions � no proof of convergence � sensitivity to numerical thresholds � reformulation of constraints at higher kinematic levels. � unable to deal with finite accumulation Numerical time integration methods – 11/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Numerical time integration methods Mechanical D.A.E. integrators Index-1 DAE integrators Using the acceleration based constraints, we have to solve v I 0 0 ˙ q − G T ( q ) = F ( q , v , t ) 0 M ( q ) v ˙ (11) − dG ( q ) 0 G ( q ) 0 λ v dt The Lagrange multipliers λ ( v , q , t ) can be obtained for a given q and v by solving � G ( q ) M − 1 F ( q , v , t ) + dG ( q ) � � G ( q ) M − 1 ( q ) G T ( q ) � λ ( v , q , t ) = − v (12) dt The following index-1 DAE � � ˙ � I � � � 0 q v = (13) F ( q , v , t ) + G T ( q ) λ ( v , q , t ) 0 M ( q ) ˙ v can be numerically solved by a any solver for ODE. We use in the work embedded 4 / 5 order Runge–Kutta-Fehlberg (RKF45) method. Numerical time integration methods – 12/33
Numerical time integration schemes for nonsmooth multibody systems in the event-driven framework Numerical time integration methods Mechanical D.A.E. integrators Index-2 DAE integrators Half-explicit method of order 5 (HEM5) [Brasey and Hairer, 1993] 8 stages T i = t n + c i h M ( Q i ) ˙ V i = F ( Q i , V i , T i ) + G T ( Q i )Λ i ˙ (14) Q i = V i G ( Q i ) V i = 0 , a ij ˙ a ij ˙ At each stage, we solve Q i = q n + h � Q j , V i = v n + h � V j . j < i j < i � F ( Q i , V i , T i ) − G T ( Q i ) � � � ˙ � � M ( Q i ) V i = , (15) G ( Q i +1 ) 0 Λ i r i i − 1 where r i = − G ( Q i +1 ) � a i +1 , j ˙ ( v n + h V j ). ha i +1 , i j =1 Comments ◮ Exact enforcement velocity constraints G ( Q i ) V i = 0 , ∀ i = 1 . . . 8. ◮ Λ i is NOT an approximation of λ ( T i ) ◮ non symmetric matrix solver. Numerical time integration methods – 13/33
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