Investigation of efficient and reliable numerical algorithms for coupled reactor calculations - XTREAM project - Tasks 3a, 3b and 4a

Rapport, 2017

In this report, some of the work performed in the framework of the X-TREAM project (with X-TREAM standing for neXt generation numerical Techniques for deterministic REActor Modelling) is reported. The work was supported by the Nordic Thermal Hydraulic Network (NORTHNET). The focus of the report is on the tasks 3a and 3b (elaboration of 1-D test cases and comparisons of non-linear inconsistent/consistent methods) and on the task 4a (recommendations for non-linear consistent reactor safety simulations).

A simplified one-dimensional Boiling Water Reactor (BWR) model in steady-state conditions was developed to study the efficiency of the Jacobian-Free Newton Krylov (JFNK) method to solve strongly non-linear multi-physic problems, such as the one corresponding to BWR behaviour, where the interdependence between neutron transport, fluid dynamics and heat transfer needs to be resolved. The modelling assumptions were chosen so that the physics of BWRs could be properly accounted for with an as simple as possible model.

It was found that there is little advantage of solving each of the mono-physics problems using a JFNK approach and iterating between the various solvers until convergence is reached. As noticed with classical operator splitting approaches, such a solver is prone to oscillatory behaviour due to the strong physical coupling between coolant density/fuel temperature and the distributions of neutrons in space/energy. Such oscillations can only be overcome by using some relaxation factor, which results in slow convergence of the overall multi-physics problem.

On the other hand, it was also demonstrated that solving the entire multi-physics problem simultaneously using JFNK is a very powerful technique. The robustness of this technique nevertheless relies on two important aspects: the necessity to create a good enough initial guess to be used by the JFNK algorithm and the need to properly precondition the problem. For the former, using a non-linear Gauss-Seidel technique was proven to provide such an acceptable guess. For the latter, an efficient preconditioner was found to have the following characteristics: separate preconditioning of the neutron transport problem, no preconditioning of the heat transfer problem, preconditioning of the fluid dynamics problem where the cross-dependencies between the void fraction and the phasic velocities are resolved and where the pressure field is treated independently. Based on these characteristics, an analytical preconditioner using first-order (i.e. linear) approximations of the balance equations and relying on a simpler formulation of the problem solved was derived and demonstrated to be efficient. The development of such an analytical preconditioner, which is the key condition to be fulfilled for a multi-physics problem to be solved using a JFNK technique, is nevertheless a major undertaking. The possibility to use a linearly-approximated preconditioner, knowing the structure of the desired preconditioner, was also proposed. It was found to be an interesting alternative when the development of an analytical Jacobian is either too difficult (due to complex models being used in the problem being solved) or when the models are not accessible (“black-box” approach).