In the paper at hand, co-simulation approaches are analyzed for coupling two solvers. The solvers are assumed to be coupled by algebraic constraint equations. We discuss 2 different coupling methods. Both methods are semi-implicit, i.e. they are based on a predictor/corrector approach. Method 1 makes use of the well-known Baumgarte-stabilization technique. Method 2 is based on a weighted multiplier approach. For both methods, we investigate formulations on index-3, index-2 and index-1 level and analyze the convergence, the numerical stability and the numerical error. The presented approaches require Jacobian matrices. Since only partial derivatives with respect to the coupling variables are needed, calculation of the Jacobian matrices may very easily be calculated numerically and in parallel with the predictor step. For that reason, the presented methods can in a straightforward manner be applied to couple commercial simulation tools without full solver access. The only requirement on the subsystem solvers is that the macro-time step can be repeated once in order to accomplish the corrector step. Within the paper, we introduce methods for coupling mechanical systems. The presented approaches can, however, also be applied to couple arbitrary non-mechanical dynamical systems.

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The numerical stability and the convergence behavior of cosimulation methods are analyzed in this manuscript. We investigate explicit and implicit coupling schemes with different approximation orders and discuss three decomposition techniques, namely, force/force-, force/displacement-, and displacement/displacement-decomposition. Here, we only consider cosimulation methods where the coupling is realized by applied forces/torques, i.e., the case that the coupling between the subsystems is described by constitutive laws. Solver coupling with algebraic constraint equations is not investigated. For the stability analysis, a test model has to be defined. Following the stability definition for numerical time integration schemes (Dahlquist's stability theory), a linear test model is used. The cosimulation test model applied here is a two-mass oscillator, which may be interpreted as two Dahlquist equations coupled by a linear spring/damper system. Discretizing the test model with a cosimulation method, recurrence equations can be derived, which describe the time discrete cosimulation solution. The stability of the recurrence equations system represents the numerical stability of the cosimulation approach and can easily be determined by an eigenvalue analysis.

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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.

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