Numerical Integration of Equations of Motion

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1 GraSMech course Computer-aided analysis of rigid and flexible multibody systems Numerical Integration of Equations of Motion Prof. Olivier Verlinden (FPMs) Prof. Olivier Brüls GraSMech Multibody 1 Modelling steps Choose the configuration parameters (q) Set up the kinematics: express position, velocity and acceleration (rotational and translational) in terms of q and its first and second time derivatives Express the forces in terms of q, its time derivatives and time t Build the differential equations of motion Numerical treatment of the equations (lesson 5) GraSMech Multibody 2 Layout of the presentation General principle of an integration scheme (O. Verlinden) ODEs (without constraints) principle of integration principal methods: multistep (Newmark), Runge-Kutta practical realization of a time step (explicit and implicit) DAEs (with constraints) Properties of integration methods (O. Brüls) ODEs Accuracy, stability Stiff ODEs DAEs The index Index reduction methods GraSMech Multibody 3 1

2 Concerned equations The equations of motion consist of n cp differential equations (dynamic equilibrium) n_c constraint equations at position level or at velocity level or at acceleration level => n c +n cp differential-algebraic equations (DAE) GraSMech Multibody 4 Residuals The equations of motion are considered in residual form From the formulation, we can only estimate the residuals f ( 0) for given values of q and its time derivatives, λ and t => It is the job of the numerical integration to draw them to zero by finding the right values of q(t) and λ! GraSMech Multibody 5 Principle of numerical integration We want to integrate the equations of motion without constraints Step by step procedure 3xn cp unknowns One or several states at and before t h=time step GraSMech Multibody 6 2

3 Basic equations The 3xn cp corresponding equations are the n cp residuals and the 2xn cp integrals The integrals are replaced by integration formulas example: Newmark formulas (0<β<0.5, 0<γ<1) GraSMech Multibody 7 First-order form Integration methods generally deal with first-order differential equations with integration formulas of the form N second order differential equations can be transformed into 2N equivalent first-order differential equations GraSMech Multibody 8 Implicit - explicit The integration formula is implicit if it involves the term in and explicit otherwise Number of steps: number of known configurations at and before time t involved in the integration formula GraSMech Multibody 9 3

Adams-Moulton integration formulas m=0 (Euler implicit) m=1

Multibody 10 BDF integration formulas m=1 (=ADAMS-0) m=2 (two

multistep formulas General form of multistep integration

ADAMS-2: α 0 =12, α 1 =-12, β 0 =5, β 1 =8, β 2 =-1 BDF-m: p=m

4 Adams-Moulton integration formulas m=0 (Euler implicit) m=1 (one step,=trapezoidal rule) m=2 (two steps) GraSMech Multibody 10 BDF integration formulas m=1 (=ADAMS-0) m=2 (two steps) m=3 (three steps) GraSMech Multibody 11 General form of multistep formulas General form of multistep integration formulas or method explicit if β 0 =0 ADAMS-m: p=0 and k=m ADAMS-2: α 0 =12, α 1 =-12, β 0 =5, β 1 =8, β 2 =-1 BDF-m: p=m and k=0 BDF-3: α 0 =11, α 1 =-18, α 2 =9, α 3 =-2, β 0 =6 GraSMech Multibody 12 4

5 Generating second-order formulas By applying recursively implicit first-order formulas you can generate second-order formulas => more efficient than working with 2N equations GraSMech Multibody 13 Equivalent 2 nd order ADAMS-Moulton m=0 m=1 m=2 GraSMech Multibody 14 Equivalent 2 nd order BDF formulas m=1 m=2 GraSMech Multibody 15 5

6 Consistent initial conditions The initial conditions are and must be given by the user The initial accelerations must verify the equations of motion and are then given by GraSMech Multibody 16 Resolution with explicit formulas The accelerations are calculated as for the initial accelerations GraSMech Multibody 17 Resolution with implicit formulas Once positions and velocities have been replaced by the integration formulas, the only unkowns are the accelerations at t+h solved by iterative procedure of Newton-Raphson with J the iteration matrix (Jacobian matrix) GraSMech Multibody 18 6

7 Computation of the iteration matrix The accelerations at t+h intervenes at three levels The iteration matrix is given by KT, CT= tangent stiffness and damping matrices GraSMech Multibody 19 Computation of iteration matrix Formulas of Newmark an approximate iteration matrix is sufficient to get convergence J tends to M when h decreases (always possible to get convergence with M by decreasing h -> not efficient with stiff systems) GraSMech Multibody 20 Integration with constraints The system of equations of motion is or or GraSMech Multibody 21 7

8 Constraints and initial conditions The equations of motion with constraints at acceleration level must be used The constraint equations at acceleration level must be used to get consistent intial accelerations The same relationship can be used with explicit integration formulas but unavoidable drift of the constraints at velocity and position levels! GraSMech Multibody 22 Constraints and implicit formulas The constraints simply consist of some more nonlinear equations in the accelerations (and Lagrange multipliers) at t+h whatever the constraint level which can be solved by Newton-Raphson GraSMech Multibody 23 Runge-Kutta methods So-called «stage» methods: simultaneous resolution of the equations of motion at s instants t i = t n + c i h between t and t+h with formulas => sxn set of equations to solve simultaneously GraSMech Multibody 24 8

9 Runge-Kutta methods The state at time t n+1 is given by other formulas GraSMech Multibody 25 Example: RKI 5/12 (Radau) 2 stages implicit scheme (γ=5/12 for RKI 5/12) GraSMech Multibody 26 Resolution for RKI 5/12 The set of 2N nonlinear equations is solved by the usual iterative procedure of Newton-Raphson and state at t n+1 is given by GraSMech Multibody 27 9

10 Integration second part Properties of integration methods => Olivier Brüls GraSMech Multibody 28 Time integration I. General principle of an integration scheme (O. Verlinden) II. Properties and specific methods (O. Brüls) ODEs Multistep (Newmark), Runge-Kutta Accuracy, stability Stiff ODEs DAEs The index Index reduction methods GraSMech Multibody 29 ODEs: Multistep methods Differential equation: + Adams 2: Algebraic equation: General form of a linear multistep method (k steps): Implicit : ( in the RHS nonlinear probl) BDF: Adams: GraSMech Multibody 30 10

11 ODEs: Multistep methods Newmark: (*) Implementation: Newmark formulae into (*) Theoretical analysis: Two-step method with a one-step implementation GraSMech Multibody 31 ODEs: Runge-Kutta methods Euler explicit: Mid-point rule? Intermediate Euler step: s stages: 1-step formula: ERK: IRK: GraSMech Multibody 32 ODEs: Accuracy Global error: Order p if when Order Adams with k steps k + 1 BDF with k steps k Newmark 1 or 2 RADAU5 (3 stages) 5 for intermediate h :! constant of errors GraSMech Multibody 33 11

12 ODEs: Accuracy Undamped oscillator: 2π Exact period: T = ω Numerical result: T 2 q + q = ω 0 Periodicity error: T T T GraSMech Multibody 34 ODEs: Stability Undamped oscillator Euler explicit Linear scalar test equation x Stability region -1 1 GraSMech Multibody 35 x ODEs: Stability Undamped oscillator Euler implicit Linear test equation The stability region includes the left half plane x -1 1 x GraSMech Multibody 36 12

ODEs: Stability Unconditional stability (A-stability): «stable solution for any stable linear system» (whatever h) Stability regions for BDFs BDF1 = Euler implicit Dahlquist s barrier for linear

13 ODEs: Stability Unconditional stability (A-stability): «stable solution for any stable linear system» (whatever h) Stability regions for BDFs BDF1 = Euler implicit Dahlquist s barrier for linear multistep methods: Unconditional stability is only possible for p 2 GraSMech Multibody 37 ODEs: Stability Linear multistep method for Linear test equation: Eigenvalues of the difference equation Spectral radius «amplification factor» at each time step Stability region: GraSMech Multibody 38 ODEs: Stability Spectral radius: on the imaginary axis 2 q + q = ω 0 jωhx = measure of the numerical damping GraSMech Multibody 39 13

14 13.6 Comparaison de quelques méthodes Available methods Matlab ode45 = Dormand/Prince method (explicit Runge-Kutta) ode15s = BDF of order 1 to 5 (implicit multistep) On the web dopri5.f, dopri5.c (Dormand/Prince method) lsode.f, dassl.f, cvode.c (BDF) radau5.f (implicit Runge-Kutta) GraSMech Multibody 40 Example 2 dof system Properties of the system k 1 = N/m and k 2 = 1961 N/m m =1 kg Eigen frequencies: 1 and 10 Hz Simulation from initial conditions q 1 =0 and q 2 =1 GraSMech Multibody 41 2 dof system 20 steps per period GraSMech Multibody 42 14

2 dof system 5 steps per period GraSMech Multibody 43 Stiff ODEs Stiff differential equations: slow & fast modes Example: FE model with thousands of dofs Accuracy is mostly required for the slow

15 2 dof system 5 steps per period GraSMech Multibody 43 Stiff ODEs Stiff differential equations: slow & fast modes Example: FE model with thousands of dofs Accuracy is mostly required for the slow modes Stability is also required for the fast modes Explicit methods Restricted stability region very small time steps Implicit methods Large stability domain larger time steps Price to pay: nonlinear equations to solve GraSMech Multibody 44 Stiff ODEs Limit of explicit methods Higher order explicit methods result in similar stability problems GraSMech Multibody 45 15

16 Newmark formulae: To be solved with: Stiff ODEs q = q + hq + h (0.5 β ) q + h βq 2 2 n+ 1 n n n n+ 1 q = q + h(1 γ ) q + hγ q n+ 1 n n n+ 1 M( q ) q + h( q, q, t ) = 0 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 From qn, q n, q n, we seek qn+ 1, q n+ 1, q n+ 1 satisfying one non-linear equation two linear equations Choice of parameters Undamped Damped 2nd-order accuracy GraSMech Multibody 46 Stiff ODEs Newmark formulae: q = q + hq + h (0.5 β ) a + h βa 2 2 n+ 1 n n n n+ 1 q = q + h(1 γ ) a + hγ a n+ 1 n n n+ 1 Generalized-α method [Chung & Hulbert 1993] (1 α ) a + α a = (1 α ) q + α q m n+ 1 m n f n+ 1 f n To be solved with: M( q ) q + h( q, q, t ) = 0 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 Two types of acceleration variables: Second-order accuracy + numerical damping GraSMech Multibody 47 Stiff ODEs Spectral radius: Newmark vs. Generalized-α GraSMech Multibody 48 16

17 ODEs: summary Multistep Dahlquist s barrier: stability vs. accuracy Variable step-size: implementation is not trivial Runge-Kutta Higher orders of precision are possible IRK: all stages are coupled (n x s unknowns!) Generalized-α One-step method Unconditional stability (numerical damping) Second-order accuracy GraSMech Multibody 49 DAEs Consider the scalar singular perturbation problem: if Limit case = DAE:, the system becomes very stiff Underlying ODE? Differentiation index of the original DAE = 1 Integration procedure: GraSMech Multibody 50 DAEs Multibody system Constraints at velocity level: Constraints at acceleration level: One more differentiation to obtain Differentiation index = 3! Hidden constraints Index reduction constraints at velocity level: index-2 constraints at acceleration level: index-1 drift!! GraSMech Multibody 51 17

18 DAEs General forms of DAEs Perturbation index: (singular matrix) High-index problems = highly sensitive! Multibody simulation Numerical problems Precision (constraints) Index-1 ~ ODE Large drift Index-2 Index-3 Highly sensitive No error at position level GraSMech Multibody 52 DAEs: Direct methods Generalized-α method [Arnold & B. 07] M( q ) q + h( q, q, t ) + B λ = 0 (1) T n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 Φ( q ) = 0 n+ 1 (2) q = q + hq + h (0.5 β ) a + h βa 2 2 n+ 1 n n n n+ 1 q = q + h(1 γ ) a + hγ a n+ 1 n n n+ 1 (1 α ) a + α a = (1 α ) q + α q m n+ 1 m n f n+ 1 f n GraSMech Multibody 53 DAEs: Direct methods Generalized-α method [Arnold & B. 07] Good prediction required for Newton process if no convergence, h may be reduced Stopping criterion Linearized problem Newton iterations GraSMech Multibody 54 18

19 DAEs: Direct methods Generalized-α method [Arnold & B. 07] Tolerance: tol = ATOL + RTOL( Mq + h + B T λ + Φ ) Iteration matrix : T ( t ) K = M( q) q + h( q, q, ) + B ( qλ ) / q t C = h( q, q, t) / q t Bad numerical conditioning for small h scaling strategy required GraSMech Multibody 55 DAEs: Direct methods Generalized-α for a double pendulum [Géradin & Cardona 01] Undamped scheme Damped scheme Weak instability for index-3 DAEs! stability + accuracy GraSMech Multibody 56 DAEs: Direct methods DAEs direct solvers Generalized-α: index-3 (numerical damping) IRK / RADAU5 (Hairer): index 3 BDF / DASSL (Petzold): index 1 Index reduction? Projection Stabilization (Baumgarte, GGL, overdetermined DAEs) Coordinate partitioning / splitting GraSMech Multibody 57 19

20 DAEs: Index reduction Projection Index-1 solution: After each time-step: project on the constraints with the condition with the condition GraSMech Multibody 58 DAEs: Index reduction Baumgarte stabilization: Any drift converges dynamically to zero Choice of the parameters? GGL formulation: [Gear, Leimkuhler, Gupta 85] Overdetermined DAE: ODASSL [Führer, Leimkuhler 91] GraSMech Multibody 59 DAEs: Index reduction Generalized coordinate partitioning System with n coordinates and m constraints Locally, select n - m independent coord. Solve for : Eliminate (+ derivatives) from the eq. of motion Underlying ODE: Efficient numerical implementation [Wehage & Haug 82] GraSMech Multibody 60 20

DAEs: Index reduction Coordinate splitting: null space matrix P: ok by construction Index-1 DAE: P =

nt space GraSMech Multibody 62 DAEs: etc Partitioned methods (e.g.

21 DAEs: Index reduction Coordinate splitting: null space matrix P: ok by construction Index-1 DAE: P = basis of the tangent space [Yen 93] GraSMech Multibody 61 DAEs: Index reduction Null space matrix (n = 3, m = 1) = independent coord. Span the tangent space GraSMech Multibody 62 DAEs: etc Partitioned methods (e.g. half-explicit methods) Implicit method for the «algebraic part» Explicit method for the «differential part» Energy conserving schemes, variational integrators 1st-integrals (energy, momenta) & symplecticity Nonlinear stability analysis Longer integration intervals are possible GraSMech Multibody 63 21

22 Time integration: Conclusion Multibody system = index-3 DAE Need sophisticated implicit algorithms Combine DAE solvers (RADAU5, DASSL) and Projection (velocity level) Stabilization (GGL, ODAE) Coordinate partitioning / splitting Flexible multibody dynamics Large & stiff systems Generalized-α scheme (MECANO: HHT) Energy conserving schemes, variational integrators GraSMech Multibody 64 Time integration: References Technical books E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I - Nonstiff Problems. Springer-Verlag, 2nd edition, E. Hairer and G. Wanner. Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems. Springer-Verlag, 2nd edition, K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, 2nd edition, Numerical methods in multibody dynamics M. Géradin and A. Cardona. Flexible Multibody Dynamics: A Finite Element Approach. John Wiley & Sons, New York, E. Eich-Soellner, Claus Führer: Numerical Methods in Multibody Dynamics, ECMI-Series, Teubner, Stuttgart, GraSMech Multibody 65 Time integration: References A few papers C.W. Gear, B. Leimkuhler, and G.K. Gupta. Automatic Integration of Euler- Lagrange Equations with Constraints. J. Comput. Appl. Mech. 12 & 13: 77-90, R. A. Wehage, and E. J. Haug. Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems. ASME J. Mech. Des., 104: , J. Yen. Constrained Equations of Motion in Multibody Dynamics as ODEs on Manifolds. SIAM J. Numer. Anal., 30: , J. Chung and G.M. Hulbert. A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. J. of Applied Mechanics, 60: , B. Owren and H.H. Simonsen. Alternative integration methods for problems in structural dynamics. Computer Methods in Applied Mechanics and Engineering, 122:110, M. Arnold and O. Brüls. Convergence of the generalized-alpha scheme for constrained mechanical systems. Multibody System Dynamics, 18(2): , M. Arnold. Numerical methods for simulation in applied mechanics. In: Simulation techniques for applied mechanics, Eds: M. Arnold and W. Schiehlen, Springer, GraSMech Multibody 66 22

Technische Universität Berlin

Technische Universität Berlin Institut für Mathematik M7 - A Skateboard(v1.) Andreas Steinbrecher Preprint 216/8 Preprint-Reihe des Instituts für Mathematik Technische Universität Berlin http://www.math.tu-berlin.de/preprints