S. González-Pintor

Time integration of the neutron diffusion equation on hexagonal geometries

To study the behavior of nuclear reactors like the Russian VVER reactors it is necessary to solve the time dependent neutron diffusion equation using a hexagonal mesh. Two methods are proposed to solve this equation. In both methods the spatial part of the equations is discretized using a high order spectral element method, based on assuming that the neutron flux can be expanded in terms of the modified Dubiners polynomials. For the time discretization of the equations two different strategies have been considered. For the first a finite differences one-step implicit method has been used and the other method is based on a modal expansion of the flux in terms of the dominant Lambda modes of the reactor core. The performance of both methods has been tested for an hypothetical transient in a 2-dimensional VVER 440 reactor.

Updating the Lambda modes of a nuclear power reactor

Starting from a steady state configuration of a nuclear power reactor some situations arise in which the reactor configuration is perturbed. The Lambda modes are eigenfunctions associated with a given configuration of the reactor, which have successfully been used to describe unstable events in BWRs. To compute several eigenvalues and its corresponding eigenfunctions for a nuclear reactor is quite expensive from the computational point of view. Krylov subspace methods are efficient methods to compute the dominant Lambda modes associated with a given configuration of the reactor, but if the Lambda modes have to be computed for different perturbed configurations of the reactor more efficient methods can be used. In this paper, different methods for the updating Lambda modes problem will be proposed and compared by computing the dominant Lambda modes of different configurations associated with a Boron injection transient in a typical BWR reactor.

Schwarz type preconditioners for the neutron diffusion equation

Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem.In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account.The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the preconditioner is less than the improvement in the convergence rate. To solve the neutron diffusion equation many linear systems has to be solved using preconditioned Krylov methods.Traditional preconditioners based on incomplete factorizations are expensive in terms of memory.Domain decomposition preconditioners are studied including substructuring preconditioners and additive Schwarz preconditioners.2D and 3D benchmarks have been studied obtaining better performance results than usual preconditioners.

Preconditioning the solution of the time-dependent neutron diffusion equation by recycling Krylov subspaces

Spectral preconditioners are based on the fact that the convergence rate of the Krylov subspace methods is improved if the eigenvalues of the smallest magnitude of the system matrix are ‘removed’. In this paper, two preconditioning strategies are studied to solve a set of linear systems associated with the numerical integration of the time-dependent neutron diffusion equation. Both strategies can be implemented using the matrix–vector product as the main operation and succeed at reducing the total number of iterations needed to solve the set of systems.

Pin-wise homogenization for SP N neutron transport approximation using the finite element method

The neutron transport equation describes the distribution of neutrons inside a nuclear reactor core. Homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem. The generalized equivalence theory for homogenization looks for discontinuous solutions through the introduction of discontinuity factors at the boundaries of the homogenized subdomains. In this work, the generalized equivalence theory is extended to the Simplified P N equations using the finite element method. This extension proposes pin discontinuity factors instead of the usual assembly discontinuity factors and the use of the simplified spherical harmonics approximation rather than diffusion theory. An interior penalty finite element method is used to discretize and solve the problem using discontinuity factors. One dimensional numerical results show that the proposed pin discontinuity factors produce more accurate results than the usual assembly discontinuity factors. The proposed pin discontinuity factors produce precise results for both pin and assembly averaged values without using advanced reconstruction methods. Also, the homogenization methodology is verified against the calculation performed with reference discontinuity factors.

Optimized Eigenvalue Solvers for the Neutron Transport Equation

A discrete ordinates method has been developed to approximate the neutron transport equation for the computation of the lambda modes of a given configuration of a nuclear reactor core. This method is based on discrete ordinates method for the angular discretization, resulting in a very large and sparse algebraic generalized eigenvalue problem. The computation of the dominant eigenvalue of this problem and its corresponding eigenfunction has been done with a matrix-free implementation using both, the power iteration method and the Krylov-Schur method. The performance of these methods has been compared solving different benchmark problems with different dominant ratios.

Using proper generalized decomposition to compute the dominant mode of a nuclear reactor

Abstract Proper generalized decomposition is a recently developed technique that solves some multidimensional problems using auxiliar models in a lower dimensional space. A method based on this technique together with the Newton method for the computation of eigenvalues is proposed to obtain the dominant eigenvalue of a nuclear power reactor core and its corresponding eigenvector. The method is tested using the bidimensional reactor Biblis.

Modified Block Newton iteration for the lambda modes problem in hexagonal geometry

To study the behavior of nuclear reactors like the Russian VVER reactors it is necessary to solve the time dependent neutron diffusion equation using a hexagonal mesh. This problem can be solved by means of a modal method, which uses a set of dominant modes to expand the neutron flux. To obtain this set of modes a differential eigenvalue problem for a steady state has to be solved. The spatial part of the equations is discretized using a high order finite element method, based on the fact that the neutron flux can be expanded in terms of the modified Dubiners polynomials. For the transient calculations using the modal method with a moderate number of modes, these modes must be updated each time step to maintain the accuracy of the solution. A Modified Block Newton iteration is studied to update the modes. The performance of the method has been tested for a hypothetical transient in a 2-dimensional VVER 440 reactor.