Block strategies to compute the lambda modes associated with the neutron diffusion equation

Given a configuration of a nuclear reactor core, the neutronic distribution of the power can beapproximated by means of the multigroup neutron diffusion equation. This is an approximationof the neutron transport equation that assumes that the neutron current is proportional to thegradient of the scalar neutron ux with a diffusion coeffcient [1]. This approximation is known asthe Fick's first law. To define the steady-state problem, the criticality of the system must be forced.In this work, the -modes problem is used. That yields a generalized eigenvalue problem whoseeigenvector associated with the dominant eigenvalue represents the distribution of the neutron uxin steady-state.The spatial discretization of the equation is made by a continuous Galerkin high order finite elementmethod is applied [2] to obtain an algebraic eigenvalue problem. Usually, the matrices obtainedfrom the discretization are huge and sparse. Moreover, they have a block structure given by the different number of energy groups. In this work, block strategies are developed to optimize thecomputation of the associated eigenvalue problems.First, different block eigenvalue solvers are studied. On the other hand, the convergence of theseiterative methods mainly depends on the initial guess and the preconditioner used. In this sense,different multilevel techniques to accelerate the rate of convergence are proposed. Finally, the sizeof the problems can be suffciently large to be unfeasible to be solved in personal computers. Thus,a matrix-free methodology that avoids the allocation of the matrices in memory is applied [3].Three-dimensional benchmarks are used to show the effciency of the methodology proposed.REFERENCES[1] Stacey, W. M. Nuclear reactor physics (Vol. 2). Weinheim: wiley-vch, 2018[2] Vidal-Ferrandiz, A., Fayez, R., Ginestar, D., and Verdú, G. Solution of the Lambda modesproblem of a nuclear power reactor using an h-p finite element method. Annals of NuclearEnergy, 72, pp. 338{349, 2018[3] Carreño Sánchez, A. M. Integration methods for the time dependent neutron diffusion equationand other approximations of the neutron transport equation. Doctoral dissertation, 2020.