1. POLI di MI tecnicolano OPTIMAL PRECONDITIONERS FOR THE SOLUTION OF CONSTRAINED MECHANICAL SYSTEMS IN INDEX THREE FORM Carlo L. Bottasso, Lorenzo Trainelli Politecnico di Milano, Italy Daniel Dopico University of La Coruña, Spain ACMD 2006 The University of Tokyo Komaba, Tokyo, August 1-4, 2006

2. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Outline Introduction and background, with specific reference to Newmarks method; Conditioning and sensitivity to perturbations of index 3 DAEs; Optimal preconditioning by scaling; Conclusions: Index 3 DAEs can be made as easy to integrate as well behaved ODEs!

3. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Constrained mechanical (multibody) systems in index 3 form: where Displacements: Velocities: Lagrange multipliers: Constraint Jacobian: Index 3 Multibody DAEs M v 0 = f ( u ; v ; t ) + G ( u ; t ) ¸ ; u 0 = v ; 0 = © ( u ; t ) ; uv ¸ G : = © ; T u

4. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Newmarks method Newmarks method in three-field form (Newmark 1959): and similarly for HHT, generalized-α, etc. Remarkother integration methods Remark: the analysis is easily extended to other integration methods, for example BDF-type integrators (Bottasso et al. 2006). Newmarks Method 1 ° h M ( v n + 1 ¡ v n ) = f n + 1 + G n + 1 ¸ n + 1 ¡ µ 1 ¡ 1 ° ¶ M a n ; u n + 1 ¡ u n = h µ ¯ ° v n + 1 + µ 1 ¡ ¯ ° v n ¶¶ ¡ h 2 2 µ 1 ¡ 2 ¯ ° ¶ a n ; 0 = © n + 1 ;

5. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Newtons method Solving with Newtons method at each time step: where the iteration matrix is and the right hand side is Linearization of Newmarks Method DisplacementsVelocities Multipliers { { { Kinematic eqs. Equilibrium eqs. Constraint eqs. { { { J q = ¡ b ; J = 2 6 6 6 4 X 1 ° h U ¡ G I ¡ ¯h ° I 0 G T 00 3 7 7 7 5 X : = ¡ ( f ; u ) n + 1 ¡ ³ ( G ¸ ) ; u ´ n + 1 ; Y : = ¡ ( f ; v ) n + 1 ; U : = M + ° h Y : b = 0 B B @ 1 ° h M ( v n + 1 ¡ v n ) ¡ ( f n + 1 + G n + 1 ¸ n + 1 ) + ³ 1 ¡ 1 ° ´ M a n u n + 1 ¡ u n ¡ h ³ ¯ ° v n + 1 + ³ 1 ¡ ¯ ° v n ´´ + h 2 2 ³ 1 ¡ 2 ¯ ° ´ a n © n + 1 1 C C A :

6. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO finite precision arithmetics Model effects of finite precision arithmetics by means of the small parameter : Perturbation Analysis " ¯ ¯ ¯ ¯ l i m h ! 0 q i ( h ; " ) ¯ ¯ ¯ ¯ · ° ° ° ° l i m h ! 0 J ¡ 1 ( h ; 0 ) ° ° ° ° 1 ° ° ° ° l i m h ! 0 b ( h ; " ) ° ° ° ° 1 : J q = ¡ b l i m h ! 0 q ( h ; 0 ) = 0 l i m h ! 0 b ( h ; 0 ) = 0 Insert into and neglect higher order terms to get

7. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Interpretation of the main result accuracy will be very poor for small time stepsspoil the order of convergence If and/or depend on negative powers of, accuracy will be very poor for small time steps (this will spoil the order of convergence of the integration scheme). Perturbation Analysis ¯ ¯ ¯ ¯ l i m h ! 0 q i ( h ; " ) ¯ ¯ ¯ ¯ · ° ° ° ° l i m h ! 0 J ¡ 1 ( h ; 0 ) ° ° ° ° 1 ° ° ° ° l i m h ! 0 b ( h ; " ) ° ° ° ° 1 : J ¡ 1 b h

8. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Apply perturbation analysis to three-field form: Accuracy Accuracy of the solution: Remark Remark: severe loss of accuracy (ill conditioning) for small ! Perturbation Analysis of Newmark h ¯ ¯ ¯ ¯ l i m h ! 0 ¢ u n + 1 ¯ ¯ ¯ ¯ · O ( h 0 ) ; ¯ ¯ ¯ ¯ l i m h ! 0 ¢ v n + 1 ¯ ¯ ¯ ¯ · O ( h ¡ 1 ) ; ¯ ¯ ¯ ¯ l i m h ! 0 ¢ ¸ n + 1 ¯ ¯ ¯ ¯ · O ( h ¡ 2 ) : l i m h ! 0 b = 8 < : O ( h ¡ 1 ) O ( h 0 ) O ( h 0 ) 9 = ; :

9. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Perturbation Analysis of Newmark Perturbation analysis for other possible forms of Newmarks method:

10. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Index 3 Multibody DAEs Solutions proposed in the literature: Index two form Index two form (velocity level constraints), but drift and need for stabilization; Position and velocity level constraints Position and velocity level constraints (GGL of Gear et al. 1985, Embedded Projection of Borri et al. 2004), but increased cost and complexity; Scaling Scaling, simple and straightforward to implement (related work: Petzold Lötstedt 1986, Cardona Geradin 1994, Negrut 2005).

11. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Scale equationsunknowns Scale equations and unknowns of Newtons problem: with where = left preconditioner, scales the equations; = right preconditioner, scales the unknowns. Remark Remark: scale back unknowns once at convergence of Newtons iterations for the current time step. Left and Right Preconditioning ¹ J ¹ q = ¡ ¹ b ; ¹ J : = D L JD R ; ¹ q : = D ¡ 1 R q ; ¹ b : = D L b ; D L D R

12. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Left scaling: Right scaling: Accuracy of the solution: Remark Remark: accuracy (conditioning) independent of time step size, as for well behaved ODEs! Same results for all other forms of the scheme. Optimal Preconditioning Scale equilibrium eqs. by D L = 2 4 ¯h 2 I 00 0 I 0 00 I 3 5 h 2 D R = 2 6 6 6 4 I 00 0 ° ¯h I 0 00 1 ¯h 2 I 3 7 7 7 5 Scale velocities by Scale Lagrange multipliers by h 2 h ¯ ¯ ¯ ¯ l i m h ! 0 ¢ u n + 1 ¯ ¯ ¯ ¯ · O ( h 0 ) ; ¯ ¯ ¯ ¯ l i m h ! 0 ¢ v n + 1 ¯ ¯ ¯ ¯ · O ( h 0 ) ; ¯ ¯ ¯ ¯ l i m h ! 0 ¢ ¸ n + 1 ¯ ¯ ¯ ¯ · O ( h 0 ) :

13. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Numerical Example Andrews squeezing mechanism: At each Newton iteration, solve first non-decreasing Stop at first non-decreasing Newton correction, i.e.: tightest achievable convergence A measure of the tightest achievable convergence of Netwon method. J j q j = ¡ b j k q j + 1 k ¸ k q j k

14. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Numerical Example

15. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Numerical Example

16. Optimal Scaling of Index Three DAEs POLITECNICO di MILANO Conclusions DAEs can be made as easy to solve numerically as well behaved ODEs DAEs can be made as easy to solve numerically as well behaved ODEs; Recipe for Newmarks scheme: - Use any form of the algorithm (two field forms are more computationally convenient); left andright scaling - Use left and right scaling of primary unknowns; Recover - Recover scaled quantities at convergence. Simple to implement in existing codes.