Abstract
In nuclear engineering, the λ -modes associated with the neutron diffusion equation are applied to study the criticality of reactors and to develop modal methods for the transient analysis. The differential eigenvalue problem that needs to be solved is discretized using a finite element method, obtaining a generalized algebraic eigenvalue problem whose associated matrices are large and sparse. Then, efficient methods are needed to solve this problem. In this work, we used a block generalized Newton method implemented with a matrix-free technique that does not store all matrices explicitly. This technique reduces mainly the computational memory and, in some cases, when the assembly of the matrices is an expensive task, the computational time. The main problem is that the block Newton method requires solving linear systems, which need to be preconditioned. The construction of preconditioners such as ILU or ICC based on a fully-assembled matrix is not efficient in terms of the memory with the matrix-free implementation. As an alternative, several block preconditioners are studied that only save a few block matrices in comparison with the full problem. To test the performance of these methodologies, different reactor problems are studied.
Highlights
The neutron transport equation is a balance equation that describes the behavior of the neutrons inside the reactor core
Some approximations are considered such as the multigroup neutron diffusion equation by relying on the assumption that the neutron current is proportional to the gradient of the neutron flux by means of a diffusion coefficient
Given a configuration of a nuclear reactor core, its criticality can be forced by dividing the production operator in the neutron diffusion equation by a positive number, λ, obtaining a neutron balance equation: the λ-modes problem
Summary
The neutron transport equation is a balance equation that describes the behavior of the neutrons inside the reactor core. Other Krylov methods have been used to compute these modes for other approximations of the neutron transport equation [7,13] Applying these kinds of methods requires either transforming the generalized problem (2) into an ordinary eigenvalue problem or applying a shift and invert technique. This convergence only depends on the separation of the group of target eigenvalues from the rest of the spectrum Another advantage is that these methods do not require solving as many linear systems as the previous methods. We use the Modified Generalized Block Newton Method (MGBNM) presented in [16], and we propose several ways to precondition the linear systems that need to be solved in this method in an efficient way.
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