Abstract

Efficiency and accuracy of a co-simulation may considerably be increased by using a variable communication-time grid. Therefore, an error estimator for controlling the macro-step size is required. Here, error estimators are derived and investigated for explicit and implicit co-simulation approaches. The paper focuses on mechanical co-simulation models. The basic results may, however, also be applied to arbitrary, non-mechanical co-simulation models.

References

References
1.
D'Silva
,
S.
,
Sundaram
,
P.
, and
Ambrosio
,
J.
,
2006
, “
Co-Simulation Platform for Diagnostic Development of a Controlled Chassis System
,”
SAE
Technical Paper No. 2006-01-1058.10.4271/2006-01-1058
2.
Fancello
,
M.
,
Morandini
,
M.
, and
Masarati
,
P.
,
2014
, “
Helicopter Rotor Sailing by Non-Smooth Dynamics Co-Simulation
,”
Archive Mech. Eng.
,
61
(
2
), pp.
253
268
.10.2478/meceng-2014-0015
3.
Gonzalez
,
F.
,
Gonzalez
,
M.
, and
Mikkola
,
A.
,
2010
, “
Efficient Coupling of Multibody Software With Numerical Computing Environments and Block Diagram Simulators
,”
Multibody Syst. Dyn.
,
24
(
3
), pp.
237
253
.10.1007/s11044-010-9199-6
4.
Quaranta
,
G.
,
Masarati
,
P.
, and
Mantegazza
,
P.
,
2002
, “
Multibody Analysis of Controlled Aeroelastic Systems on Parallel Computers
,”
Multibody Syst. Dyn.
,
8
(
1
), pp.
71
102
.10.1023/A:1015894729968
5.
Solcia
,
T.
, and
Masarati
,
P.
,
2011
, “
Efficient Multirate Simulation of Complex Multibody Systems Based on Free Software
,”
ASME
Paper No. DETC2011-47306.10.1115/DETC2011-47306
6.
Anderson
,
K.
, and
Duan
,
S.
,
1999
, “
A Hybrid Parallelizable Low-Order Algorithm for Dynamics of Multi-Rigid-Body Systems—Part I: Chain Systems
,”
Math. Comput. Modell.
,
30
(
9–10
), pp.
193
215
.10.1016/S0895-7177(99)00190-9
7.
Cuadrado
,
J.
,
Cardenal
,
J.
,
Morer
,
P.
, and
Bayo
,
E.
,
2000
, “
Intelligent Simulation of Multibody Dynamics: Space-State and Descriptor Methods in Sequential and Parallel Computing Environments
,”
Multibody Syst. Dyn.
,
4
(
1
), pp.
55
73
.10.1023/A:1009824327480
8.
Lacoursière
,
C.
,
2007
, “
A Parallel Block Iterative Method for Interactive Contacting Rigid Multibody Simulations on Multicore PCs
,”
Appl. Parallel Comput. State Art Sci. Comput.
, 1, pp.
956
965
.
9.
Lacoursiere
,
C.
,
Nordfeldth
,
F.
, and
Linde
,
M.
,
2014
, “
A Partitioning Method for Parallelization of Large Systems in Realtime
,”
Proceedings of the Third Joint International Conference on Multibody System Dynamics (IMSD 2014) and the Seventh Asian Conference on Multibody Dynamics (ACMD 2014)
, Busan, Korea, June 30–July 3, pp. 1–6.
10.
Malczyk
,
P.
, and
Fraczek
,
J.
,
2009
, “
Evaluation of Parallel Efficiency in Modeling of Mechanisms Using Commercial Multibody Solvers
,”
Archive Mech. Eng.
,
VI
(
3
), pp.
237
249
.
11.
Negrut
,
D.
,
Melanz
,
D.
,
Mazhar
,
H.
,
Lamb
,
D.
,
Jayakumar
,
P.
, and
Letherwood
,
M.
,
2013
, “
Investigating Through Simulation the Mobility of Light Tracked Vehicles Operating on Discrete Granular Terrain
,”
SAE Int. J. Passeng. Cars—Mech. Syst.
,
6
(
1
), pp.
369
381
.10.4271/2013-01-1191
12.
Negrut
,
D.
,
Tasora
,
A.
,
Mazhar
,
H.
,
Heyn
,
T.
, and
Hahn
,
P.
,
2012
, “
Leveraging Parallel Computing in Multibody Dynamics
,”
Multibody Syst. Dyn.
,
27
(
1
), pp.
95
117
.10.1007/s11044-011-9262-y
13.
Negrut
,
D.
,
Serban
,
R.
,
Mazhar
,
H.
, and
Heyn
,
T.
,
2014
, “
Parallel Computing in Multibody System Dynamics: Why, When How
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
4
), p.
041007
.10.1115/1.4027313
14.
Serban
,
R.
,
Melanz
,
D.
,
Li
,
A.
,
Stanciulescu
,
I.
,
Jayakumar
,
P.
, and
Negrut
,
D.
,
2015
, “
A GPU-Based Preconditioned Newton-Krylov Solver for Flexible Multibody Dynamics
,”
Int. J. Numer. Methods Eng.
,
102
(
9
), pp.
1585
1604
.10.1002/nme.4876
15.
Serban
,
R.
,
Olsen
,
N.
,
Negrut
,
D.
,
Recuero
,
A.
, and
Jayakumar
,
P.
,
2017
, “
A Co-Simulation Framework for High-Performance, High-Fidelity Simulation of Ground Vehicle-Terrain Interaction
,”
Proceedings of the NATO AVT-265 Specialists Meeting
, Vilinus, Lithuania.
16.
Valasek
,
M.
, and
Mraz
,
L.
,
2012
, “
Massive Parallelization of Multibody System Simulation
,”
Acta Polytech.
,
52
(
6
), pp. 94–98.
17.
Ambrosio
,
J.
,
Pombo
,
J.
,
Rauter
,
F.
, and
Pereira
,
M.
,
2009
, “
A Memory Based Communication in the Co-Simulation of Multibody and Finite Element Codes for Pantograph-Catenary Interaction Simulation
,”
Multibody Dynamics: Computational Methods and Applications
,
C. L.
Bottasso
, ed.,
Springer
, Dortrecht, The Netherlands, pp.
231
252
.
18.
Ambrosio
,
J.
,
Pombo
,
J.
,
Pereira
,
M.
,
Antunes
,
P.
, and
Mosca
,
A.
,
2012
, “
A Computational Procedure for the Dynamic Analysis of the Catenary-Pantograph Interaction in High-Speed Trains
,”
J. Theor. Appl. Mech.
,
50
(
3
), pp.
681
699
.
19.
Pombo
,
J.
, and
Ambrosio
,
J.
,
2012
, “
Multiple Pantograph Interaction With Catenaries in High-Speed Trains
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(
4
), p.
041008
.10.1115/1.4006734
20.
Alioli
,
M.
,
Morandini
,
M.
, and
Masarati
,
P.
,
2013
, “
Coupled Multibody-Fluid Dynamics Simulation of Flapping Wings
,”
ASME
Paper No. DETC2013-12198.10.1115/DETC2013-12198
21.
Naya
,
M.
,
Cuadrado
,
J.
,
Dopico
,
D.
, and
Lugris
,
U.
,
2011
, “
An Efficient Unified Method for the Combined Simulation of Multibody and Hydraulic Dynamics: Comparison With Simplified and Co-Integration Approaches
,”
Archive Mech. Eng.
,
58
(
2
), pp.
223
243
.10.2478/v10180-011-0016-4
22.
Eberhard
,
P.
,
Gaugele
,
T.
,
Heisel
,
U.
, and
Storchak
,
M.
,
2008
, “
A Discrete Element Material Model Used in a Co-Simulated Charpy Impact Test and for Heat Transfer
,”
Proceedings First International Conference on Process Machine Interactions
, Hannover, Germany, Sept. 3, pp. 3–4.
23.
Lehnart
,
A.
,
Fleissner
,
F.
, and
Eberhard
,
P.
,
2009
, “
Using SPH in a Co-Simulation Approach to Simulate Sloshing in Tank Vehicles
,”
Proceedings SPHERIC4
, Nantes, France, May 27–29, pp. 1–2.
24.
Spreng
,
F.
,
Eberhard
,
P.
, and
Fleissner
,
F.
,
2013
, “
An Approach for the Coupled Simulation of Machining Processes Using Multibody System and Smoothed Particle Hydrodynamics Algorithms
,”
Theor. Appl. Mech. Lett
,
3
(
1
), p.
013005
.10.1063/2.1301305
25.
Datar
,
M.
,
Stanciulescu
,
I.
, and
Negrut
,
D.
,
2012
, “
A Co-Simulation Environment for High-Fidelity Virtual Prototyping of Vehicle Systems
,”
Int. J. Veh. Syst. Modell. Test.
,
7
(
1
), pp.
54
72
.10.1504/IJVSMT.2012.045308
26.
Liao
,
Y. G.
, and
Du
,
H. I.
,
2001
, “
Co-Simulation of Multi-Body-Based Vehicle Dynamics and an Electric Power Steering Control System
,”
Proc. Inst. Mech. Eng. K, J. Multibody Dyn.
,
215
(
3
), pp.
141
151
.10.1243/1464419011544420
27.
Gomes
,
C.
,
Thule
,
C.
,
Broman
,
D.
,
Larsen
,
P. G.
, and
Vangheluwe
,
H.
,
2018
, “
Co-Simulation: A Survey
,”
ACM Comput. Surv.
,
51
(
3
), pp.
1
33
.10.1145/3179993
28.
Gomes
,
C.
,
2019
, “
Property Preservation in Co-Simulation
,” Doctoral dissertation, University of Antwerp, Antwerp, Belgium.
29.
Schweizer
,
B.
,
2019
, “
IUTAM Symposium on Solver-Coupling and Co-Simulation
,”
Proceedings of the IUTAM Symposium on Solver-Coupling and Co-Simulation 2017
, Darmstadt, Germany, Sept. 18–20, 2017, pp.
1
290
.
30.
Hairer
,
E.
,
Norsett
,
S. P.
, and
Wanner
,
G.
,
2009
,
Solving Ordinary Differential Equations I: Nonstiff Problems
, 3rd ed.,
Springer
, Berlin.
31.
Hairer
,
E.
, and
Wanner
,
G.
,
2010
,
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
, 2nd ed.,
Springer
, Berlin.
32.
Shampine
,
L. F.
,
1973
, “
Local Extrapolation in the Solution of Ordinary Differential Equations
,”
Math. Comput.
,
27
(
121
), pp.
91
97
.10.1090/S0025-5718-1973-0331803-1
33.
Stetter
,
H. J.
, and
Weinmüller
,
E.
,
1981
, “
On the Error Control in ODE Solvers With Local Extrapolation
,”
Comput.
,
27
(
2
), pp.
169
177
10.1007/BF02243551.
34.
Milne
,
W. E.
,
1926
, “
Numerical Integration of Ordinary Differential Equations
,”
Am. Math. Mon.
,
33
(
9
), pp.
455
460
.10.2307/2299609
35.
Arnold
,
M.
,
Clauss
,
C.
, and
Schierz
,
T.
,
2013
, “
Error Analysis and Error Estimates for Co-Simulation in FMI for Model Exchange and Co-Simulation in V2
,”
Arch. Mech. Eng.
,
60
(
1
), pp.
75
94
10.2478/meceng-2013-0005.
36.
Rahikainen
,
J.
,
Gonzalez
,
F.
, and
Naya
,
M. A.
,
2020
, “
An Automated Methodology to Select Functional Co-Simulation Configurations
,”
Multibody Syst. Dyn.
,
48
(
1
), pp.
79
103
.10.1007/s11044-019-09696-y
37.
Sadjina
,
S.
, and
Pedersen
,
E.
,
2016
, “
Energy Conservation and Coupling Error Reduction in Non-Iterative co-Simulations
,”
Eng. Comput.
, 35, pp.
1
9
.10.1007/s00366-019-00783-4
38.
Sadjina
,
S.
,
Kyllingstad
,
L. T.
,
Skjong
,
S.
, and
Pedersen
,
E.
,
2017
, “
Energy Conservation and Power Bonds in co-Simulations: Non-Iterative Adaptive Step Size Control and Error Estimation
,”
Eng. Comput.
,
33
(
3
), pp.
607
620
.10.1007/s00366-016-0492-8
39.
Zhao
,
J.
,
Wang
,
H.
, and
Zhang
,
H.
,
2019
, “
A Regression-Based Collaborative Filtering Recommendation Approach to Time-Stepping Multi-Solver Co-Simulation
,”
IEEE Access
,
7
, pp.
22790
22806
.10.1109/ACCESS.2019.2897486
40.
Busch
,
M.
,
2016
, “
Continuous Approximation Techniques for co-Simulation Methods: Analysis of Numerical Stability and Local Error
,”
Z. Angew. Math. Mech.
,
96
(
6
), pp.
1061
1081
.10.1002/zamm.201500196
41.
Gonzalez
,
F.
,
Gonzalez
,
M.
, and
Cuadrado
,
J.
,
2009
, “
Weak Coupling of Multibody Dynamics and Block Diagram Simulation Tools
,”
ASME
Paper No. DETC2009-86653.10.1115/DETC2009-86653
42.
Gonzalez
,
F.
,
Naya
,
M. A.
,
Luaces
,
A.
, and
Gonzalez
,
M.
,
2011
, “
On the Effect of Multirate co-Simulation Techniques in the Efficiency and Accuracy of Multibody System Dynamics
,”
Multibody Syst. Dyn.
,
25
(
4
), pp.
461
483
.10.1007/s11044-010-9234-7
43.
Schweizer
,
B.
, and
Lu
,
D.
,
2014
, “
Semi-Implicit Co-Simulation Approach for Solver Coupling
,”
Arch. Appl. Mech.
,
84
(
12
), pp.
1739
1769
.10.1007/s00419-014-0883-5
44.
Schweizer
,
B.
,
Li
,
P.
, and
Lu
,
D.
,
2015
, “
Explicit and Implicit Co-Simulation Methods: Stability and Convergence Analysis for Different Solver Coupling Approaches
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p.
051007
.10.1115/1.4028503
45.
Arnold
,
M.
,
2010
, “
Stability of Sequential Modular Time Integration Methods for Coupled Multibody System Models
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
3
), pp.
1
9
.10.1115/1.4001389
46.
Kübler
,
R.
, and
Schiehlen
,
W.
,
2000
, “
Two Methods of Simulator Coupling
,”
Math. Comput. Modell. Dyn. Syst.
,
6
(
2
), pp.
93
113
.10.1076/1387-3954(200006)6:2;1-M;FT093
47.
Meyer
,
T.
,
Li
,
P.
,
Lu
,
D.
, and
Schweizer
,
B.
,
2018
, “
Implicit co-Simulation Method for Constraint Coupling With Improved Stability Behavior
,”
Multibody Syst. Dyn.
,
44
(
2
), pp.
135
161
.10.1007/s11044-018-9632-9
48.
Schweizer
,
B.
, and
Lu
,
D.
,
2014
, “
Predictor/Corrector Co-Simulation Approaches for Solver Coupling With Algebraic Constraints
,”
ZAMM
,
95
(
9
), pp.
911
938
.10.1002/zamm.201300191
49.
Schweizer
,
B.
, and
Lu
,
D.
,
2015
, “
Stabilized Index-2 Co-Simulation Approach for Solver Coupling With Algebraic Constraints
,”
Multibody Syst. Dyn.
,
34
(
2
), pp.
129
161
.10.1007/s11044-014-9422-y
50.
Schweizer
,
B.
,
Li
,
P.
, and
Lu
,
D.
,
2015
, “
Implicit co-Simulation Methods: Stability and Convergence Analysis for Solver Coupling With Algebraic Constraints
,”
ZAMM
,
96
(
8
), pp.
986
1012
.10.1002/zamm.201400087
51.
Tomulik
,
P.
, and
Fra˛czek
,
J.
,
2011
, “
Simulation of Multibody Systems With the Use of Coupling Techniques: A Case Study
,”
Multibody Syst. Dyn.
,
25
(
2
), pp.
145
165
.10.1007/s11044-010-9206-y
52.
Tseng
,
F.
, and
Hulbert
,
G.
,
1999
, “
Network-Distributed Multibody Dynamics Simulation-Gluing Algorithm
,”
Advances in Computational Multibody Dynamics
,
J.
Ambrósio
and
W.
Schiehlen
, eds.,
IDMEC/IST Lisbon
,
Portugal
, pp.
521
540
.
53.
Wang
,
J.
,
Ma
,
Z. D.
, and
Hulbert
,
G.
,
2003
, “
A Gluing Algorithm for Distributed Simulation of Multibody Systems
,”
Nonlinear Dyn.
,
34
(
1/2
), pp.
159
188
.10.1023/B:NODY.0000014558.70434.b0
54.
Gu
,
B.
, and
Asada
,
H. H.
,
2004
, “
Co-Simulation of Algebraically Coupled Dynamic Subsystems Without Disclosure of Proprietary Subsystem Models
,”
ASME J. Dyn. Syst. Meas. Control
,
126
(
1
), pp.
1
13
10.1115/1.1648307.
55.
Schneider
,
F.
,
Burger
,
M.
,
Arnold
,
M.
, and
Simeon
,
B.
,
2017
, “
A New Approach for Force‐Displacement co‐Simulation Using Kinematic Coupling Constraints
,”
ZAMM
,
97
(
9
), pp.
1147
1166
.10.1002/zamm.201500129
56.
Li
,
P.
,
Lu
,
D.
,
Schmoll
,
R.
, and
Schweizer
,
B.
,
2019
, “
Explicit Co-Simulation Approach With Improved Numerical Stability
,”
IUTAM Symposium on Solver-Coupling and Co-Simulation
, Darmstadt, Germany, Sept. 17–20, pp.
153
201
.
57.
Li
,
P.
,
Yuan
,
Q.
,
Lu
,
D.
,
Meyer
,
T.
, and
Schweizer
,
B.
,
2020
, “
Improved Explicit co-Simulation Methods Incorporating Relaxation Techniques
,”
Archive Appl. Mech.
,
90
(
1
), pp.
17
30
.10.1007/s00419-019-01597-y
58.
Peiret
,
A.
,
Gonzalez
,
F.
,
Kövecses
,
J.
, and
Teichmann
,
M.
,
2018
, “
Multibody System Dynamics Interface Modelling for Stable Multirate co-Simulation of Multiphysics Systems
,”
Mechanism Mach. Theory
,
127
, pp.
52
72
.10.1016/j.mechmachtheory.2018.04.016
59.
Ben Khaled-El Feki
,
A.
,
Duval
,
L.
,
Faure
,
C.
,
Simon
,
D.
, and
Ben Gaid
,
M.
,
2017
, “
CHOPtrey: Contextual Online Polynomial Extrapolation for Enhanced Multi-Core co-Simulation of Complex Systems
,”
Simulation
,
93
(
3
), pp.
185
200
.10.1177/0037549716684v 026
60.
ADAMS Manual
,” MSC Software, accessed Dec. 3, 2020, https://www.mscsoftware.com
61.
Gonzalez
,
F.
,
Arbatani
,
S.
,
Mohtat
,
A.
, and
Kövecses
,
J.
,
2019
, “
Energy-Leak Monitoring and Correction to Enhance Stability in the co-Simulation of Mechanical Systems
,”
Mechanism Mach. Theory
,
131
, pp.
172
188
.10.1016/j.mechmachtheory.2018.09.007
62.
Holzinger
,
F.
, and
Benedikt
,
M.
,
2019
, “
Optimal Trigger Sequence for Non-Iterative Co-Simulation
,”
Proceedings of the Nineth International Conference on Simulation and Modeling Methodologies, Technologies and Applications
, Prague, Czech Republic, July 29–31, pp.
80
87
.10.5220/0007833800800087
63.
Holzinger
,
F. R.
, and
Benedikt
,
M.
,
2019
, “
Hierarchical Coupling Approach Utilizing Multi-Objective Optimization for Non-Iterative Co-Simulation
,”
Proceedings of the 13th International Modelica Conference
, Regensburg, Germany, Mar. 4–6, Vol.
157
, pp. 1–6.10.3384/ecp19157735
64.
De Jalon
,
J. G.
, and
Bayo
,
E.
,
2012
,
Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge
,
Springer Science & Business Media
, Berlin.
65.
Busch
,
M.
,
2012
, “
Zur Effizienten Kopplung Von Simulationsprogrammen—On the Efficient Coupling of Simulation Software
,” Ph.D. thesis,
University of Kassel
, Kassel, Germany.
66.
Rodriguez
,
B.
,
Gonzalez
,
F.
,
Naya
,
M. A.
, and
Cuadrado
,
J.
,
2019
, “
A Test Framework for the co-Simulation of Electric Powertrains and Vehicle Dynamics
,”
Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics
, Duisburg, Germany.
67.
Schweizer
,
B.
,
Lu
,
D.
, and
Li
,
P.
,
2015
, “
Co-Simulation Method for Solver Coupling With Algebraic Constraints Incorporating Relaxation Techniques
,”
Multibody Syst. Dyn.
,
34
(
2
), pp.
129
161
.10.1007/s11044-015-9464-9
68.
Gustafsson
,
K.
,
Lundh
,
M.
, and
Söderlind
,
G.
,
1988
, “
A PI Step Size Control for the Numerical Solution of Ordinary Differential Equations
,”
BIT Numer. Math.
,
28
(
2
), pp.
270
287
.10.1007/BF01934091
69.
Hindmarsh
,
A. C.
,
Brown
,
P. N.
,
Grant
,
K. E.
,
Lee
,
S. L.
,
Serban
,
R.
,
Shumaker
,
D. E.
, and
Woodward
,
C. S.
,
2005
, “
SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers
,”
ACM Trans. Math. Software (TOMS)
,
31
(
3
), pp.
363
396
.10.1145/1089014.1089020
70.
TU Darmstadt,
Lichtenberg High Performance Computer of TU Darmstadt
,” TU Darmstadt, accessed Dec. 3, 2020, https://www.hhlr.tu-darmstadt.de/hhlr/index.en.jsp
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