Numerical Methods for Neutron Transport Calculations of Nuclear Reactors

The objective of this thesis, which in clearly inspired by an industrial framework, is to try and narrow the gap between theoretical neutron modelling and application in the context of nuclear reactor design. This thesis is divided into three main chapters, preceded by a general overview. This structure reflects the three main topics which were chosen for this research project. The first topic develops the Spectral Element Method (SEM) approach and its use in conjunction with transport approximations. As it is documented in the specialized numerical analysis handbooks and in previous works by the author, the method has an excellent convergence rate which outperforms most classical schemes, but it has also some important drawbacks which sometimes seem to discourage its use for linear transport problems applied to nontrivial benchmarks. In order to elaborate the methodology of the specific problems encountered in reactor physics, three aspects are addressed looking for improvements. The first topic analyzed is related to the convergence order, whose value is less straightforward to define a priori by means of functional analysis than other numerical schemes. The adjective "spectral" refers in fact to the maximum order claimed, exponential with respect to the average size of the mesh. A comprehensive set of convergence tests is carried out applying SEM to a few transport models and with the aid of manufactured solutions, thus isolating the numerical effects from the deviations which are due only to modelling approximations. The hypothesis of grid conformity is also relaxed, replacing the classical Galerkin variational formulation with the Discontinuous Galerkin theory, characterized by a more flexible treatment of the mesh interfaces; this scheme allows local grid refinement and opens the way, in perspective, to mesh adaption. Finally, a simple and sufficiently flexible technique to deform the boundaries of each mesh is introduced and applied, in order to adapt the grid to curved geometries. In this way, the advantages of SEM can be applied to a vast class of common problems like lattice calculations. Moreover, thanks to a change of the basis functions used in SEM, it is possible to obtain elements with three sides (straight or deformed), that are a typical war horse of the Finite Element approach. The second topic is essentially devoted to the most "industrial" part of the thesis, developed entirely during the stay of the author in the AREVA NP headquarters in Paris. In AREVA, and in all other nuclear engineering enterprises, neutron diffusion is still the preferred neutronic model for full-core studies. Better approximations are reserved for library preparation, fuel studies and code validation, none of these being typically too much time or budget-constrained. Today needs start to require a certain level of improvement also in full-core analyses, trying to fitly model localized dis-homogeneities and reduce the penalizing engineering margins which are taken as provisions. On the other hand, a change in the model does not mean only an effort to write a new code, but has huge follow-ups due to the validation processes required by the authorities. Second-order transport may support the foreseen methodology update because it can be implemented re-using diffusion routines as the computational engine. The AN method, a second-order approximation of the transport equation, has been introduced in some studies, and its effect is discussed. Moreover, some effort has been reserved to the introduction of linear anisotropy in the model. The last topic deals with ray effects; they are a known issue of the discrete ordinate approach (SN methods) which is responsible for a reduction in the accuracy of the solution, especially in penetration problems with low scattering, like several shielding calculations performed for operator safety concerns. Ray effects are here characterized from a formal point of view in both static and time dependent situations. Then, quantitative indicators are defined to help with the interpretation of the SN results. Based on these studies, some mitigation measures are proposed and their efficacy is discussed