Error analysis for co‐simulation with force‐displacement coupling

Abstract

Co-simulation is a simulation technique for time dependent coupled problems in engineering that restricts the data exchange between subsystems to discrete communication points in time. In the present paper we follow the block-oriented framework in the recently established industrial interface standard FMI for Model Exchange and Co-Simulation v2.0 and study local and global error of co-simulation algorithms for systems with force-displacement coupling. A rather general convergence result for the co-simulation of coupled systems without algebraic loops shows zero-stability of co-simulation algorithms with forcedisplacement coupling and proves that order reduction of local errors does not affect the order of global errors. The theoretical investigations are illustrated by numerical tests in the novel FMI-compatible co-simulation environment SNiMoWrapper.

Energy conservation and power bonds in co-simulations: non-iterative adaptive step size control and error estimation

“…where P 0 kα and P kα are the exact solution and the cosimulation result, respectively, where P kα (t i ) = P 0 kα (t i ) is assumed. For input extrapolation of order m, the local error in the inputs is 22,23 ∆u k = O(∆t m+1 ) for sufficiently smooth problems and constant step sizes ∆t i = ∆t. Then, from considering Eq.…”

“…24 Outside the realm of co-simulations, energy errors are used as a measure of quality in hybrid earthquake simulations 25,26 or in molecular dynamics simulations, for example. Additionally, the method proposed here naturally solves the issue of the numerical values of the outputs lying on very different scales for force-displacement 9,10,23 coupling, or the force-velocity coupling we shall investigate in the next section: The outputs representing forces will typically have much larger numeric values than the ones representing displacements or velocities. As a result, local errors from some simulators may be given weightings which are much too large compared to others.…”

“…Because it can be considered two coupled Dahlquist equations, the use of a linear 2-DOF oscillator as a co-simulation test case has been proposed. [8][9][10][11]23,24,[30][31][32] This test system is also a simplification of numerous relevant real engineering systems. One such example is the quarter car model 33 depicted in Fig.…”

Semantic adaptation for FMI co-simulation with hierarchical simulators

“…Error control : Richard extrapolation 9,47,66 can be implemented by creating a semantic adaptation that runs a whole scenario at twice the rate of the original one; multi-order input extrapolation 59,67 amounts to implementing two approximation schemes (see item above) run in parallel; the embedded method 68 requires that a semantic adaptation is implemented to perform a discretized numerical integration of some of the signals in the internal scenario; energy-based techniques 69 can be implementing by coding semantic adaptations which monitor for energy dissipativity in some of the signals in the internal FMUs.…”

Collaborative simulation method with spatiotemporal synchronization process control

“…Liang [5] proposed a novel combinative algorithm for communication among domain models in multidisciplinary collaborative simulation by using a proper model encapsulation method and a matched RTI control strategy. Arnold [6][7][8][9] discussed multidisciplinary simulation problems and the current algorithms used in both mono and multi-disciplinary simulation of vehicle system dynamics, presented some numerical methods together with estimation of errors for coupling simulation, and developed the Functional Mock-Up Interface (FMI) for supporting multi-disciplinary modeling and simulation. Huang [10] studied the algebraic loop problem in multi-domain simulation to reveal the relationship between simulation stability and system topologies, and proposed two algebraic loop compensation algorithms using numerical iteration and approximate function to simulate the forging process.…”