A nodal collocation method for the calculation of the lambda modes of the PL equations

Capilla, M.T.; Talavera, C.F.; Ginestar, D.; Verdú, G.

doi:10.1016/j.anucene.2005.07.004

Cited by 22 publications

(50 citation statements)

References 7 publications

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“…Since P L equations (13) have a diffusive form, their spatial discretization can be done using a nodal collocation method, previously used for the neutron diffusion equation (Hébert, 1987;Verdú et al, 1994) and generalized for eigenvalue problems in multidimensional rectangular geometries (Capilla et al, 2005(Capilla et al, , 2008(Capilla et al, , 2012. We will only apply the method when the source term in Eq.…”

Section: The Nodal Collocation Methods For An Isotropic Sourcementioning

confidence: 99%

“…Furthermore, if X e (u 1 , u 2 , u 3 ) denotes the previously defined vector of l even moments that appears in Eq. (13) for node e, it is assumed that vector X e can be expanded in terms of (orthonormal) Legendre polynomials P k (u) (Capilla et al, 2005) up to a certain finite order M ,…”

Section: The Nodal Collocation Methods For An Isotropic Sourcementioning

confidence: 99%

“…In previous works (Capilla et al, 2005(Capilla et al, , 2008(Capilla et al, , 2012, a nodal collocation method was developed for the P L equations; the method was implemented into a computer code called SHNC (Spherical Harmonics Nodal Collocation) and then applied to criticality calculations. In this work, this method has been extended to study different neutron source problems in multidimensional geometries, comparing the obtained results with reference results provided in the literature.…”

Section: Introductionmentioning

confidence: 99%

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Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems

Capilla

Talavera

Ginestar

et al. 2016

Annals of Nuclear Energy

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A P L spherical harmonics-nodal collocation method is applied to the solution of the multidimensional neutron source transport equation. Vacuum boundary conditions are approximated by setting Marshak's conditions. The method is applied to several 1D, 2D and 3D problems with isotropic fixed source and with isotropic and anisotropic scattering. These problems are chosen to test this method in limit conditions, showing that in some cases a high order P L approximation is required to obtain accurate results and convergence. Results are also compared with the ones provided by several reference codes showing good agreement. It is also shown that Marshak's approximation to vacuum boundary conditions gives the same results that simulating vacuum with a purely absorbing medium and setting zero flux boundary conditions.

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Section: The Nodal Collocation Methods For An Isotropic Sourcementioning

confidence: 99%

Section: The Nodal Collocation Methods For An Isotropic Sourcementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems

Capilla

Talavera

Ginestar

et al. 2016

Annals of Nuclear Energy

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“…Inserting these expansions into equation (2.1) and with the aid of the orthogonality relations for the Legendre polynomials and the addition theorem for the associated Legendre functions we obtain the standard P L approximation in one-dimensional 24 2.1 One-dimensional geometries geometries [8,35] …”

Section: Spectral Element Methods (Sem)mentioning

confidence: 99%

“…We consider a homogeneous slab of length 2 cm [35], in the approximation of one group of energy and vacuum boundary conditions, that is, α − = 1, α + = 1, β − = 2 and β + = −2. The nuclear cross sections for this problem are: D = 1 3 , Σ a = 0.1, and νΣ f = 0.25.…”

Section: Homogeneous Eigenvalue Problemmentioning

confidence: 99%

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

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SummaryThe neutron diffusion equation describes the neutron population in a nuclear reactor core. This work deals with this model for nuclear reactors with hexagonal geometries. First, the stationary neutron diffusion equation is studied. This is a differential eigenvalue problem, called Lambda modes problem. To solve the Lambda modes problem, different methods have been compared in one-dimensional geometries, resulting as the best one the Spectral Element Method. The operators are then discretized using this scheme in two-and three-dimensional geometries, and the resulting algebraic eigenvalue problem is solved with the implicit restarted Arnoldi method.Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration neutron diffusion equation. Initially, a one step backard Euler method is used to discretize this equation in time. The transients to test the behaviour of the method are based on moving the control rods of the reactor, simulating an accident where a control rod is ejected and a scram is intialized to control the power evolution. An unphysical behaviour appears when a node is partially rodded, the rod cusping effect, which is corrected by weighting the cross sections with the flux of the previous time step. Good results are obtained for the power evolution and the rod cusping effect is corrected for different transients. To solve the algebraic systems arising in the backward method, a Krylov method is used, and different preconditioning strategies are tested. The first one consists of using the block structure obtained by the energy groups to solve the system by blocks, and different acceleration techniques for the block iterative scheme and a preconditioner using this block structure are proposed. Also, a spectral preconditioner, which makes use of the information in a Krylov subspace to preconditionate the next system is studied. Second and fourth order exponential methods are also proposed to integrate the time dependent neutron diffusion equation, where the exponential of the system matrix has to be multiplied by a vector. These schemes allow us to work without building explicitly the system matrix, and different methods are compared to calculate the product of the system matrix by a vector, such as a Krylov method, a Chebyshev approximation method, and a Leja Points method.Some situations arise in which a set of modes of a nuclear reactor core have to be updated, as in perturbative calculations or in the use of modal methods. Updating this set of modes can be very expensive when using the Arnoldi method, and for this reason several methods based on a Newton Iteration are proposed, such as the Modified Block Newton method, a One Sided Block Newton method and a Two Sided Block Newton method. As an alternative strategy, a reduced order model to update a set of modes based on the Proper Generalized Decomposition is also proposed. This method obtains an approximated solution as a sum of separable functions over the whole domain, reduc...

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A nodal collocation approximation for the multi-dimensional equations – 2D applications

Capilla

Talavera

Ginestar

et al. 2008

Annals of Nuclear Energy

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A nodal collocation method for the calculation of the lambda modes of the PL equations

Cited by 22 publications

References 7 publications

Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems

Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

A nodal collocation approximation for the multi-dimensional equations – 2D applications

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