Implicit co-simulation methods: Stability and convergence analysis for solver coupling approaches with algebraic constraints

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 96; no. 8; pp. 986 - 1012
• Main Authors: Schweizer, Bernhard, Li, Pu, Lu, Daixing
• Format: Journal Article
• Language: English
• Published: Weinheim Blackwell Publishing Ltd 01.08.2016; Wiley Subscription Services, Inc
• Subjects: 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
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201400087

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.