Numerical Integration of Equations of...

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GraSMech – Multibody 1

Numerical Integration of Equations of Motion

Prof. Olivier Verlinden (FPMs)

[email protected]

Prof. Olivier Brü[email protected]

GraSMech course 2009-2010

Computer-aided analysis of rigid

and flexible multibody systems


Modelling steps

� Choose the configuration parameters (q)

� Set up the kinematics: express position, velocity and acceleration (rotational and translational) in terms of qand 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)


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

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Concerned equations

The equations of motion consist of

� ncp differential equations (dynamic equilibrium)

� n_c constraint equations

�at position level

�or at velocity level

�or at acceleration level

=> nc+ncp differential-algebraic equations (DAE)


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, λλλλ and t

=> It is the job of the numerical integration to draw them to zero by finding the right values of q(t) and λλλλ !


Principle of numerical integration

� We want to integrate the equations of motion without constraints

� Step by step procedure

One or several states at and before t

h=time step

3xncp unknowns

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Basic equations

� The 3xncp corresponding equations are

� the ncp residuals

� and the 2xncp integrals

� The integrals are replaced by integration formulas

example: Newmark formulas (0<β<0.5, 0<γ<1)


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


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

4


Adams-Moulton integration formulas

� m=0 (Euler implicit)

� m=1 (one step,=trapezoidal rule)

� m=2 (two steps)


BDF integration formulas

� m=1 (=ADAMS-0)

� m=2 (two steps)

� m=3 (three steps)


General form of multistep formulas

� General form of multistep integration formulas

or

method explicit if β0=0

� ADAMS-m: p=0 and k=mADAMS-2: α0=12, α1=-12, β0=5 , β1=8 , β2=-1

� BDF-m: p=m and k=0BDF-3: α0=11, α1=-18, α2=9, α3=-2 , β0=6

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Generating second-order formulas

� By applying recursively implicit first-order formulas

you can generate second-order formulas

=> more efficient than working with 2N equations


Equivalent 2nd order ADAMS-Moulton

� m=0

� m=1

� m=2


Equivalent 2nd order BDF formulas

� m=1

� m=2

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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


Resolution with explicit formulas

The accelerations are calculated as for the initial accelerations


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)

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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


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)


Integration with constraints

� The system of equations of motion is

� or

� or

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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 !


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


Runge-Kutta methods

� So-called « stage » methods: simultaneous resolution of the equations of motion at s instants ti = tn + ci hbetween t and t+h

with formulas

� => sxN set of equations to solve simultaneously

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Runge-Kutta methods

� The state at time tn+1 is given by other formulas


Example: RKI 5/12 (Radau)

� 2 stages implicit scheme (γ=5/12 for RKI 5/12)


Resolution for RKI 5/12

� The set of 2N nonlinear equations is solved by the usual iterative procedure of Newton-Raphson

� and state at tn+1 is given by

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Integration – second part

� Properties of integration methods

=> Olivier Brüls


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


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:

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ODEs: Multistep methods

� Newmark: (*)

� Implementation: Newmark formulae into (*)

� Theoretical analysis:

� Two-step method with a one-step implementation


ODEs: Runge-Kutta methods

� Euler explicit:

ERK:

IRK:

Intermediate Euler step:

1-step formula:

� Mid-point rule?

�s stages:


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!

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ODEs: Accuracy

Periodicity error:

2 0q qω+ =��

2T

π

ω=

T

T T

T

−

Exact period:

Numerical result:

Undamped oscillator:


ODEs: Stability

� Undamped oscillator Euler explicit

-1 1

x

x

� Linear scalar test equation

Stability region


ODEs: Stability

� Undamped oscillator Euler implicit

-1 1

x

x

� Linear test equation

The stability region

includes the left half plane

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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


ODEs: Stability

� Linear multistep method for

� Linear test equation:

� Eigenvalues of the difference equation

� Spectral radius

« amplification factor » at each time step

� Stability region:


ODEs: Stability

Spectral radius:on the imaginary axis

= measure of the numerical damping

xjωωωωh

2 0q qω+ =��

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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)

13.6 Comparaison de quelques méthodesAvailable methods


Example – 2 dof system

� Properties of the system

�k1 = 79.15 N/m and k2 = 1961 N/m

�m =1 kg

�Eigen frequencies: 1 and 10 Hz

� Simulation from initial conditions q1=0 and q2=1


2 dof system – 20 steps per period

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2 dof system – 5 steps per period


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


Stiff ODEs

Higher order explicit methods result in similar stability problems

Limit of explicit methods

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Stiff ODEs

Newmark formulae:

To be solved with:

� From , we seek satisfying

� one non-linear equation

� two linear equations

2 2

1 1

1 1

(0.5 )

(1 )

n n n n n

n n n n

h h h

h h

β β

γ γ+ +

+ +

= + + − +

= + − +

q q q q q

q q q q

� ��

� � ��

1 1 1 1 1( ) ( , , )n n n n nt+ + + + ++ =M q q h q q 0��

, ,n n nq q q� �� 1 1 1, ,n n n+ + +q q q� ��

� Choice of parameters

�Undamped

�Damped 2nd-order accuracy


Stiff ODEs

Newmark formulae:

To be solved with:

2 2

1 1

1 1

(0.5 )

(1 )

n n n n n

n n n n

h h h

h h

β β

γ γ+ +

+ +

= + + − +

= + − +

q q q a a

q q a a

�

� �

1 1 1 1 1( ) ( , , )n n n n nt+ + + + ++ =M q q h q q 0��

1 1(1 ) (1 )m n m n f n f n

α α α α+ +− + = − +a a q q��

Generalized-αααα method [Chung & Hulbert 1993]

� Two types of acceleration variables:

� Second-order accuracy + numerical damping


Stiff ODEs

� Spectral radius: Newmark vs. Generalized-α

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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


DAEs

� Consider the scalar singular perturbation problem:

if , the system becomes very stiff

� Limit case = DAE:

� Underlying ODE?

Differentiation index of the original DAE = 1

� Integration procedure:


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 !!

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DAEs

� General forms of DAEs(singular matrix)

Numerical problems Precision (constraints)

Index-1 ~ ODE Large drift

Index-2

Index-3 Highly sensitive No error at position level

Multibody

simulation

� Perturbation index:

High-index problems = highly sensitive !


DAEs: Direct methods

� Generalized-α method [Arnold & B. ’07]

2 2

1 1

1 1

(0.5 )

(1 )

n n n n n

n n n n

h h h

h h

β β

γ γ+ +

+ +

= + + − +

= + − +

q q q a a

q q a a

�

� �

1 1(1 ) (1 )m n m n f n f n

α α α α+ +− + = − +a a q q��

1 1 1 1 1 1 1

1

( ) ( , , )

( )

T

n n n n n n n

n

t+ + + + + + +

+

+ + =

=

M q q h q q B λ 0

Φ q 0

�� (1)

(2)




Good prediction required for Newton process

⇒ if no convergence, h may be reduced

Newton iterations

Stopping criterion

Linearized problem

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Bad numerical conditioning for small h

⇒ scaling strategy required

�Iteration matrix :


( )( ) ( , , ) ( ) /

( , , ) /

T

t

t

t

t

= ∂ + + ∂

= ∂ ∂

K M q q h q q B q λ q

C h q q q

��

� �

�Tolerance: ( )Ttol ATOL RTOL= + + + +Mq h B λ Φ��



� Generalized-α for a double pendulum [Géradin & Cardona ‘01]

Undamped scheme Damped scheme

stability

+

accuracy

Weak instability for

index-3 DAEs!



� 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

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DAEs: Index reduction

� Projection

� Index-1 solution:

�After each time-step: project on the constraints

with the condition

with the condition



� Baumgarte stabilization:

�Any drift converges dynamically to zero

�Choice of the parameters?

[Gear, Leimkuhler, Gupta ’85]

ODASSL

[Führer, Leimkuhler ’91]

� GGL formulation:

� Overdetermined DAE:



� 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]

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� Coordinate splitting:

ok by construction

[Yen ’93]

� null space matrix P:

P = basis of the tangent space…

� Index-1 DAE:



Span the tangent space

Null space matrix (n = 3, m = 1)

= independent coord.


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

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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


Time integration: References

Technical books� E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary

Differential Equations I - Nonstiff Problems. Springer-Verlag, 2nd edition, 1993.

� E. Hairer and G. Wanner. Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems. Springer-Verlag, 2nd edition, 1996.

� K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, 2nd edition, 1996.

Numerical methods in multibody dynamics� M. Géradin and A. Cardona. Flexible Multibody Dynamics: A Finite

Element Approach. John Wiley & Sons, New York, 2001.� E. Eich-Soellner, Claus Führer: Numerical Methods in Multibody

Dynamics, ECMI-Series, Teubner, Stuttgart, 1998.


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, 1985.

� R. A. Wehage, and E. J. Haug. Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems. ASME J. Mech. Des., 104: 247-255, 1982.

� J. Yen. Constrained Equations of Motion in Multibody Dynamics as ODEs on Manifolds. SIAM J. Numer. Anal., 30:553-568, 1993.

� 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, 60:371-375, 1993.

� B. Owren and H.H. Simonsen. Alternative integration methods for problems in structural dynamics. Computer Methods in Applied Mechanics and Engineering, 122:110, 1995.

� M. Arnold and O. Brüls. Convergence of the generalized-alpha scheme for constrained mechanical systems. Multibody System Dynamics, 18(2):185-202, 2007.

� M. Arnold. Numerical methods for simulation in applied mechanics. In: Simulation techniques for applied mechanics, Eds: M. Arnold and W. Schiehlen, Springer, 2008.

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