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60 3 MODELLING AND SIMULATION OF MIXED SYSTEMS

interval is specified for all participating simulators. Particularly for systems with
very different time constants this hinders an efficient processing of the simulation.
In recent years a whole range of simulator couplings have been developed in the
form of variations on the conservative method, e.g. Bechtold et al. [20], Buck et al.
[52], Patterson [316], Sung and Ha [392], Todesco and Meng [402], Zwoli´ nski
et al. [441], although frequent task changes between simulators can still gives rise
to performance problems in these approaches.
In the optimistic case, every simulator processes its internal events until no more
activity can be determined, which in the ideal case is by far the most efficient way.
Unfortunately, it may occur that another simulator generates an event for the first
in this period. Then all of the first simulator’s results from the moment in question
must be discarded. To achieve this the simulator in question must perform a leap
backwards (timewarp) and then start again at the time point in question. Depend-
ing upon the system under consideration this is associated with a high storage
requirement for the saving of old states. Furthermore, depending upon the nature
of the system under investigation, these timewarps can themselves become a per-
formance problem. Normally, however, electronics simulators [402] and mechanics
simulators do not provide the option of performing a timewarp, so that only the con-
servative approach and variations upon it remain. However, this is not necessarily
the case for the co-simulation of hardware and software, see Chapter 5.
In addition to the synchronisation between two simulator cores, the question
of the convergence of the solution also requires some consideration. This is par-
ticularly relevant for the coupling of two analogue cores, see Klein and Gerlach
[196]. The reason for this lies in the back-coupling between the two simulator
cores, which we will call A and B here. A maps its input x A to its output y A
using a function f A . B does the same with the function f B . In the simplest case,
the Gauss–Seidel iteration, the rule for the (k+1)th iteration step is:
k+1 k k
x = y = f A (x )
B A A (3.14)
x k+1 = y k+1 = f B (x k+1 )
A B B
In this case oscillations may occur. In the worst case the iteration does not converge
at all. Numerically more demanding methods, such as, for example, the Newton
procedure, tend to converge better, but are not universally applicable due to the
costly calculation of the required Jacobi matrices, see [196].

3.5.3 Examples of the simulator coupling
Introduction

In what follows the options and limitations of simulator coupling will be illustrated
in more detail on the basis of a few examples from mechatronics and micromecha-
tronics. This description will include the direct coupling between two simulators
as well as the systematic consideration of several simulators with a backplane.