2 State of the art
2.4 Multi-physical coupling approaches between codes
2.4.3 Temporal coupling
For both FA and pin-wise based coupling, the managing and well adaptation of the time steps between the TH and N codes is important since both codes have their own time step selection algorithm. The coupling of two different codes has to be implemented for both stationary and transient conditions. The most widely used option is the loose (“low”) coupled methods, which differently from tight (“high”) [Gaston01], provide certain independence between the coupled codes. Here, four kinds of solutions are encountered (illustrated in Fig. 5), semi-implicit [Jeong01], explicit (also referred as “marching solution method” or “asynchronous models” [Grandi01]), staggered Operator Splitting (OS) coupling and implicit schemes (also known as “synchronous models”). The nonlinear behavior of this coupled scheme between the N and TH models raise a challenge to find an accurate and efficient solution for LWR transients. For instance, the temporal and spatial discretizations affect stability pa- rameters such as decay ratios and key parameters of reactivity insertion accidents (powers peak, Dop- pler temperatures and coolant enthalpies), which are very sensitive to the degree of implicitness in the coupling between the N and the TH models. Therefore, several approaches for the temporal coupling between codes must be precisely understood. The following figures show these approaches [Ragu- sa01], [Watson01], where special emphasis is done for N and TH.
Fig. 5. Implicit (a), semi-implicit (b), explicit (c) and staggered OS (d) coupling schemes.
As can be seen in Fig. 5 (d), the N solution (NS) and the TH solution (THS) between two consecutive time steps are given by equations (2.22) and (2.23) respectively;
,
1,
1
t
N
t
NS
nTHS
nNS
, (2.22)THS
tTHt,NS
n,THS
n1
. (2.23) Multiple time steps marching schemes allow the solution of a selected code to proceed with several steps, while the other coupled code marches only with one large step. This temporal adaptive algo- rithm was developed to perform the synchronization and optimization of the performance of 3D N/TH FA and sub-channel analysis coupled code systems. However, it was noted [Grandi01] that applying automatic time steps selection to some models, did not increase the qualitative precision of the solu- tion compared to fix time steps implicit schemes, but it reduces significantly the computational cost. Coming back to the staggered OS [Zerkak01], the convergence of the solution is ruled by the time step where in general a master code controls the time advancement. Small time steps are required specifi- cally for the N (e.g. very small neutron generation time in the order of t = 10.0-4 s). Examples of this
type of coupling which involve OS schemes are TRACE/PARCS, RELAP5/PARCS and CRONOS2/FLICA4 [Royer02].
In the implicit approach (Fig. 5 (a)), convergence of the individual codes and of the feedback is re- quired. This method has the advantage of being the most accurate and stable out of the others. Several different approaches to solve the implicit coupling involve: Implicit-Runge-Kutta methods [Loza- no02], nested loops approximations and JFNK methods [Mousseau01], where improvements in the areas of convergence, dynamic time steps and non-linearities are envisaged. Lastly, semi-implicit methods, Fig. 5 (b), have the advantage of employing feedback parameters from old and new time steps (as done in TRAC-PF1/NEM). This might be of specific interest when dealing with power maps (PMs) as done with SCF. The main disadvantage of this method is its instability caused by the non- converged feedback parameters.
Implicit methods are required for many coupled problems due to the presence of non-smooth or rapid- ly varying transients, in the case of the neutronic solution. Especially in reactor analysis, these fast transients take into account the presence of inverse neutron velocity terms, which usually have con- vergence instabilities. In principle, it is desired to implement the temporal coupling with an implicit approach to ensure that the scheme will be stable, leading to the usage of longer time steps during the transient. Nevertheless, this type of approach is extensive in the way that each code requires complex modifications; therefore, other methods are encountered, such as the explicit method which presents limitations in the time step control to ensure stability.
2.4.3.1 Time step synchronization
For transient analysis the easiest way to couple these kinds of systems is by fixing the time step based on one of the two involved codes. This approach may lead to inaccurate predictions of the time de- pendent power behavior in continuous fast transients where the neutron flux (refer to Chapter 2.2.1) distribution and power shape change rapidly. In the counterpart, when the N code is the master, the computation time may increase due to the rapid repartition of time steps that could be easily managed with longer time step periods. Coupled time dependent algorithms can be found in the literature [Zerkak01]. For some time step control and calculation models, each code performs independently from each other. In this particular case, the vector containing the state variables is used to check the “global” time step and predict a new time step. A restart of the coupled code system from the previous time point is required if the criterion is not met.
In discussion of time step control schemes, it is instructive to consider the semi-implicit method. In order for the semi-implicit solution to be numerically stable, it must respect the material Courant limit, which dictates the maximum time step size based on the stability of computational parameters passing through a calculation node. This issue will be discussed in more detail in Chapter 7.1.3.
For some other coupled schemes (e.g. TRAC-BF1/NEM), variable time step algorithms have been developed in such a way that the kinetics algorithm has an automatic time step control routine that monitors temporal changes (global total power and local neutron flux) during a given time step and adjusts the time step size automatically. Other coupling approach involves the use of message passing interface (MPI) [Perez01] and the executive parallel virtual machine (PVM) [Geisto01] program, which is able to handle two of the major challenges for coupled calculations, exchange of boundary condition information between codes with different data structures and coordination of the time de- pendent advancement. The PVM philosophy is based on the definition of variables to be exchanged and the coordination of the advancement through time (calling for exchange variables between similar tasks, controlling of output generation and shutting down virtual memory after termination). The ex- ecutive program also allows for “synchronous” and “asynchronous” exchange of data between tasks. For “synchronous” exchange, the process is forced to have similar time steps, with interaction every time step [Jimenez01]. Some examples are found in RELAP5/PARCS, ATHLET/QUABOX, TRACE/PARCS and CRONOS/FLICA4. For “asynchronous” exchange, time steps are matched and data are exchanged at predefined time intervals only. Moreover, the executive program allows for other types of schemes, including “semi-implicit” coupling.