Implicit co-simulation methods: Stability and convergence analysis for solver coupling approaches with algebraic constraints
The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is...
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| Published in | Zeitschrift für angewandte Mathematik und Mechanik Vol. 96; no. 8; pp. 986 - 1012 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Weinheim
Blackwell Publishing Ltd
01.08.2016
Wiley Subscription Services, Inc |
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| Abstract | The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single‐mass oscillator, a linear two‐mass oscillator is used here for analyzing the stability of co‐simulation methods. The two‐mass co‐simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co‐simulation test model with a linear co‐simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis.
The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. |
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| AbstractList | The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single‐mass oscillator, a linear two‐mass oscillator is used here for analyzing the stability of co‐simulation methods. The two‐mass co‐simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co‐simulation test model with a linear co‐simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis. The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single‐mass oscillator, a linear two‐mass oscillator is used here for analyzing the stability of co‐simulation methods. The two‐mass co‐simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co‐simulation test model with a linear co‐simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis. The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. The analysis of the numerical stability of co-simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well-known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force-, force/displacement- and displacement/displacement-decomposition. The stability analysis of co-simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single-mass oscillator, a linear two-mass oscillator is used here for analyzing the stability of co-simulation methods. The two-mass co-simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co-simulation test model with a linear co-simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis. The analysis of the numerical stability of co-simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well-known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force-, force/displacement- and displacement/displacement-decomposition. The stability analysis of co-simulation methods with algebraic constraints is inherently related to the definition of a test model. |
| Author | Schweizer, Bernhard Lu, Daixing Li, Pu |
| Author_xml | – sequence: 1 givenname: Bernhard surname: Schweizer fullname: Schweizer, Bernhard email: schweizer@sds.tu-darmstadt.de organization: Department of Mechanical Engineering, Institute of Applied Dynamics, Technical University Darmstadt, Germany – sequence: 2 givenname: Pu surname: Li fullname: Li, Pu organization: Department of Mechanical Engineering, Institute of Applied Dynamics, Technical University Darmstadt, Germany – sequence: 3 givenname: Daixing surname: Lu fullname: Lu, Daixing organization: Department of Mechanical Engineering, Institute of Applied Dynamics, Technical University Darmstadt, Germany |
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| Cites_doi | 10.4271/2011-01-0516 10.1016/0045-7825(72)90018-7 10.1002/zamm.201300191 10.1504/IJVSMT.2012.045308 10.1016/S1270-9638(03)00016-6 10.1007/s00419-011-0586-0 10.1109/92.609870 10.1007/BF01934907 10.1016/S0045-7825(97)00140-0 10.1016/j.cam.2007.09.005 10.1115/PVP2006-ICPVT-11-93184 10.1016/S0045-7825(02)00518-2 10.1002/(SICI)1097-0207(19970815)40:15<2841::AID-NME193>3.0.CO;2-S 10.1023/A:1009824327480 10.1115/1.4028503 10.1115/1.1648307 10.1007/s11044-010-9234-7 10.4271/2013-01-1191 10.1007/s11044-014-9422-y 10.1076/1387-3954(200006)6:2;1-M;FT093 10.1063/2.1301305 10.2478/v10180-011-0016-4 10.1115/1.4030508 |
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| References | J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comput. Methods Appl. Mech. Eng. 1, 1-16 (1972). J. Cuadrado, J. Cardenal, P. Morer, and E. Bayo, Intelligent Simulation of Multibody Dynamics: Space-State and Descriptor Methods in Sequential and Parallel Computing Environments, Multibody Syst. Dyn. 4, 55-73 (2000). V. Carstens, R. Kemme, and S. Schmitt, Coupled simulation of flow-structure interaction in turbomachinery, Aerosp. Sci. Technol. 7, 298-306 (2003). J. Ambrosio, J. Pombo, M. Pereira, P. Antunes, and A. Mosca, A computational procedure for the dynamic analysis of the catenary-pantograph interaction in high-speed trains, J. Theor. Appl. Mech. 50/3, 681-699 (2012), Warsaw. F. Gonzalez, M. A. Naya, A. Luaces, and M. Gonzalez, On the effect of multirate co-simulation techniques in the efficiency and accuracy of multibody system dynamics, Multibody Syst. Dyn. 2(4), 461-483 (2011). B. Gu and H. H. Asada, Co-simulation of algebraically coupled dynamic subsystems without disclosure of proprietary subsystem models, J. Dyn. Syst. Meas. Control 126, 1-13 (2004), DOI: 10.1115/1.1648307. Y. G. Liao and H. I. Du, Co-simulation of multi-body-based vehicle dynamics and an electric power steering control system, Proc. Inst. Mech. Eng. K, J. Multibody Dyn. 215, 141-151 (2001). F. Spreng, P. Eberhard, and F. Fleissner, An approach for the coupled simulation of machining processes using multibody system and smoothed particle hydrodynamics algorithms, Theor. Appl. Mech. Lett. 3(1), 8-013005 (2013). C. W. Gear and D. R. Wells, Multirate linear multistep methods, BIT 24, 484-502 (1984). A. Verhoeven, B. Tasic, T. G. J. Beelen, E. J. W. ter Maten, and R. M. M. Mattheij, BDF Compound-Fast Multirate Transient Analysis with Adaptive Stepsize Control, J. Numer. Anal. Ind. Appl. Math. 3(3-4), 275-297 (2008). R. Kübler and W. Schiehlen, Two methods of simulator coupling, Math. Comput. Model. Dyn. 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Lu, Predictor/Corrector co-simulation approaches for solver coupling with algebraic constraints, ZAMM - J. Appl. Math. Mech. 95, 911-938 (2015). M. Naya, J. Cuadrado, D. Dopico, and U. Lugris, An Efficient Unified Method for the Combined Simulation of Multibody and Hydraulic Dynamics: Comparison with Simplified and Co-Integration Approaches, Arch. Mech. Eng. LVIII, 223-243 (2011). M. Datar, I. Stanciulescu, and D. Negrut, A co-simulation environment for high-fidelity virtual prototyping of vehicle systems, Int. J. Veh. Syst. Model. Test. 7, 54-72 (2012). E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 3rd ed. (Springer, 2009). N. Negrut, D. Melanz, H. Mazhar, D. Lamb, and P. Jayakumar, Investigating Through Simulation the Mobility of Light Tracked Vehicles Operating on Discrete Granular Terrain, SAE Int, JPasseng. Cars - Mech. Syst. 6, 369-381, DOI: 10.4271/2013-01-1191, 2013. W. J. T. 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| References_xml | – reference: V. Savcenco, Comparison of the asymptotic stability properties for two multirate strategies, J. Comput. Appl. Math. 220, 508-524 (2008). – reference: J. Ambrosio, J. Pombo, M. Pereira, P. Antunes, and A. Mosca, A computational procedure for the dynamic analysis of the catenary-pantograph interaction in high-speed trains, J. Theor. Appl. Mech. 50/3, 681-699 (2012), Warsaw. – reference: F. Gonzalez, M. A. Naya, A. Luaces, and M. Gonzalez, On the effect of multirate co-simulation techniques in the efficiency and accuracy of multibody system dynamics, Multibody Syst. Dyn. 2(4), 461-483 (2011). – reference: B. Gu and H. H. Asada, Co-simulation of algebraically coupled dynamic subsystems without disclosure of proprietary subsystem models, J. Dyn. Syst. Meas. Control 126, 1-13 (2004), DOI: 10.1115/1.1648307. – reference: W. J. T. Daniel, Analysis and implementation of a new constant acceleration subcycling algorithm, Int. J. Numer. Methods Eng. 40, 2841-2855 (1997). – reference: V. Carstens, R. Kemme, and S. Schmitt, Coupled simulation of flow-structure interaction in turbomachinery, Aerosp. Sci. Technol. 7, 298-306 (2003). – reference: E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed. (Springer, 2010). – reference: M. Datar, I. Stanciulescu, and D. Negrut, A co-simulation environment for high-fidelity virtual prototyping of vehicle systems, Int. J. Veh. Syst. Model. Test. 7, 54-72 (2012). – reference: F. Spreng, P. Eberhard, and F. Fleissner, An approach for the coupled simulation of machining processes using multibody system and smoothed particle hydrodynamics algorithms, Theor. Appl. Mech. Lett. 3(1), 8-013005 (2013). – reference: S. Wuensche, C. Clauß, P. Schwarz, and F. Winkler, Electro-thermal circuit simulation using simulator coupling, IEEE Trans. Very Large Scale Integ. (VLSI) Syst. 5, 277-282 (1997). – reference: R. Kübler and W. Schiehlen, Two methods of simulator coupling, Math. Comput. Model. Dyn. Syst. 6, 93-113 (2000). – reference: J. Cuadrado, J. Cardenal, P. Morer, and E. Bayo, Intelligent Simulation of Multibody Dynamics: Space-State and Descriptor Methods in Sequential and Parallel Computing Environments, Multibody Syst. Dyn. 4, 55-73 (2000). – reference: B. Schweizer and D. Lu, Predictor/Corrector co-simulation approaches for solver coupling with algebraic constraints, ZAMM - J. Appl. Math. Mech. 95, 911-938 (2015). – reference: W. J. T. Daniel, A study of the stability of subcycling algorithms in structural dynamics, Comput. Methods Appl. Mech. Engrg. 156, 1-13 (1998). – reference: W. J. T. Daniel, A partial velocity approach to subcycling structural dynamics, Comput. Methods Appl. Mech. Engrg. 192, 375-394 (2003). – reference: N. Negrut, D. Melanz, H. Mazhar, D. Lamb, and P. 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| SubjectTerms | 04A25 Algebra algebraic constraints Approximation Co-simulation Computer simulation Constraint modelling Constraints Convergence Coupling Decomposition Displacement Eigenvalues implicit Mathematical analysis Mathematical models Methods Model testing Numerical stability Simulation solver coupling Stability Stability analysis subcycling Subsystems Time integration |
| Title | Implicit co-simulation methods: Stability and convergence analysis for solver coupling approaches with algebraic constraints |
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