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Research Papers

Stabilized Implicit Cosimulation Method: Solver Coupling With Algebraic Constraints for Multibody Systems

[+] Author and Article Information
Bernhard Schweizer

Department of Mechanical Engineering,
Institute of Applied Dynamics,
Technical University Darmstadt,
Otto-Berndt-Strasse 2,
Darmstadt 64287, Germany
e-mail: schweizer@sds.tu-darmstadt.de

Pu Li, Daixing Lu, Tobias Meyer

Department of Mechanical Engineering,
Institute of Applied Dynamics,
Technical University Darmstadt,
Otto-Berndt-Strasse 2,
Darmstadt 64287, Germany

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 27, 2014; final manuscript received April 29, 2015; published online August 26, 2015. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 11(2), 021002 (Aug 26, 2015) (18 pages) Paper No: CND-14-1224; doi: 10.1115/1.4030508 History: Received September 27, 2014

In this manuscript, an implicit cosimulation method is analyzed, where the solvers are coupled by algebraic constraint equations. We discuss cosimulation approaches on index-2 and on index-1 level and investigate constant, linear and quadratic approximation functions for the coupling variables. The key idea of the method presented here is to discretize the Lagrange multipliers between the macrotime points (extended multiplier approach) so that the coupling equations and their time derivatives can simultaneously be fulfilled at the macrotime points. Stability and convergence of the method are investigated in detail. Following the stability analysis for time integration schemes based on Dahlquist's test equation, an appropriate cosimulation test model is used to examine the numerical stability of the presented cosimulation method. Discretizing the cosimulation test model by means of a linear cosimulation approach yields a system of linear recurrence equations. The spectral radius of the recurrence equation system characterizes the numerical stability of the underlying cosimulation method. As for time integration methods, 2D stability plots are used to graphically illustrate the stability behavior of the coupling approach.

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Figures

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Fig. 1

Homogenous linear two-mass oscillator: test model for analyzing the stability of cosimulation methods

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Fig. 2

Cosimulation test model: force/force–decomposition approach

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Fig. 3

Extrapolation and interpolation functions for constant, linear, and quadratic approximation (index-2 cosimulation approach)

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Fig. 4

Extrapolation and interpolation functions for constant and linear approximation (index-1 cosimulation approach)

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Fig. 5

2D stability plots for the implicit index-2 cosimulation approach and the explicit cosimulation approach of Ref. [1] (approximation order k=0,1,2) for the symmetrical test model (αm21=αΛr21=αΛi21=1)

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Fig. 6

2D stability plots for the implicit index-2 cosimulation approach and the explicit cosimulation approach of Ref. [1] (approximation order k=0,1,2) for the unsymmetrical test model (αm21=αΛr21=αΛi21=10)

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Fig. 7

2D stability plots for the implicit index-1 cosimulation approach (approximation order k=0,1) for the symmetrical (αm21=αΛr21=αΛi21=1) and for the unsymmetrical (αm21=αΛr21=αΛi21=10) test model

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Fig. 8

Convergence plots for the stabilized implicit index-2 cosimulation approach (approximation order k=0,1,2)

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Fig. 9

Convergence plots for the stabilized implicit index-1 cosimulation approach (approximation order k=0,1)

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Fig. 10

Cosimulation results for the nonlinear two-mass oscillator: displacement x1(t) and velocity v1(t), Lagrange multiplier λc(t), constraint equations gc(t), g·c(t), and g··c(t)

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Fig. 11

Convergence plots for the nonlinear two-mass oscillator: global error of the position and velocity variables over the macrostep size H

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Fig. 12

Planar four-bar mechanism: interpretation as two double pendulums coupled by a fixed joint

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Fig. 13

Cosimulation results for the four-bar mechanism: displacements x2(t), y2(t), and angle ϕ2(t), Lagrange multiplier λcax(t), constraint equations gcax(t), g·cax(t), and g··cax(t)

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