Stabilized overlapping modular time integration of coupled differential-algebraic equations

The modeling of technical systems often leads to differential equations that are coupled by constraints. This results in a set of coupled differential-algebraic equations (DAEs). For efficiency concerns the full system of DAEs is split in several subsystems. The solution proceeds in macro steps. In each macro step, the subsystems are solved separately by different time integration methods, while necessary data from other subsystems is approximated by extrapolation or interpolation. After each macro step, at discrete predefined or adaptively defined communication points, the data is updated. The approximation of data between these communication points (for the current macro step) leads to an additional error in the time integration and may furthermore cause instability. Different from modular time integration of ordinary differential equations (ODEs), this effect cannot be fixed by reducing the stepsize below a small stability bound. In this paper we discuss a stabilization strategy for the stable modular time integration of coupled DAEs. The application of this strategy is illustrated for a simple benchmark problem.

[1]  Martin Arnold,et al.  Coupled Simulation of Multibody Systems and Elastic Structures , 1999 .

[2]  P. Hartman.  Ordinary Differential Equations , 1965 .

[3]  Werner Schiehlen,et al.  Two Methods of Simulator Coupling , 2000 .

[4]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[5]  Michael Günther,et al.  Preconditioned Dynamic Iteration for Coupled Differential-Algebraic Systems , 2001 .

[6]  Martin Arnold,et al.  Numerical methods for simulation in applied dynamics , 2008 .

[7]  P. Rentrop,et al.  Multirate Partitioned Runge-Kutta Methods , 2001 .

[8]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[9]  Michael Valášek,et al.  Methodologies for Coupling Simulation Models and Codes in Mechatronic System Analysis and Design , 2021, The Dynamics of Vehicles on Roads and on Tracks.

[10]  P. Deuflhard,et al.  One-step and extrapolation methods for differential-algebraic systems , 1987 .

[11]  Kenneth R. Jackson,et al.  A survey of parallel numerical methods for initial value problems for ordinary differential equations , 1991 .

[12]  Martin Arnold,et al.  EFFICIENT SIMULATION OF BUSH AND ROLLER CHAIN DRIVES , 2005 .

[13]  H. Matthies,et al.  Partitioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction , 2002 .

[14]  Martin Arnold,et al.  Multi-Rate Time Integration for Large Scale Multibody System Models , 2007 .

[15]  Martin Arnold,et al.  Stability of Sequential Modular Time Integration Methods for Coupled Multibody System Models , 2010 .