<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Determination of the reactor kinetic characteristics is very important for the design and development of a new reactor system. In this sense, the computation of lambda modes associated to a nuclear power reactor has interest since these modes can be used to analyze the reactor criticality and to develop modal methods to analyze transient situations in the reactor. In this paper, the lambda problem has been discretized using a high order finite element method to obtain a generalized algebraic eigenvalue problem. A multilevel method is proposed to solve this generalized eigenvalue problem combining a hierarchy of meshes with a Modified Block Newton method. The Krylov-Schur method is used to compare the efficiency of the multilevel method solving several benchmark problems.</p></abstract>

Determination of the reactor kinetic characteristics is very important for the design and development of a new reactor system. In this sense, the computation of lambda modes associated to a nuclear power reactor has interest since these modes can be used to analyze the reactor criticality and to develop modal methods to analyze transient situations in the reactor. In this paper, the lambda problem has been discretized using a high order finite element method to obtain a generalized algebraic eigenvalue problem. A multilevel method is proposed to solve this generalized eigenvalue problem combining a hierarchy of meshes with a Modified Block Newton method. The Krylov-Schur method is used to compare the efficiency of the multilevel method solving several benchmark problems.

Multilevel method to compute the lambda modes of the neutron diffusion equation

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Multilevel method to compute the lambda modes of the neutron diffusion equation"}

Determination of the reactor kinetic characteristics is very important for the design and development of a new reactor system. In this sense, the computation...

Input: Initial approximation U = [u₁,...,u_d] for the eigenvectors.
Output: Diagonal matrix of eigenvalues Λ and matrix V with the eigenvectors as its columns.
1:	Orthonormalize U	▹ MODIFIED GRAM-SCHMIDT
2:	Compute the Ritz approximations U, W	▹ RAYLEIGH-RITZ GEN.
3:	conv=0 (conv indicates the number of converged eigenvalues)
4:	while conv < q do
5:	Compute $Δ U = [Δ u_{1}^{λ}, \dots, Δ u_{end}^{λ}]$ $\begin{array}{} \displaystyle \Delta U =[\Delta u^{\lambda}_{1},\dots,\Delta u^{\lambda}_{\text{end}}] \end{array}$	▹ CORRECTION NEWTON EQ. (16)
6:	U = U − ΔU
7:	Orthonormalize(U)	▹ MODIFIED GRAM-SCHMIDT
8:	Compute the Ritz approximatisons U, W	▹ RAYLEIGH-RITZ GEN.
9:	fori =conv → q do
10:	if $\frac{‖ M u_{i} - w_{i} L u_{i} ‖}{‖ u_{i} ‖} < t o l_{i}$ ${{M{u_i} - {w_i}L{u_i}} \over {{u_i}}} < to{l_i}$then	▹ CONVERGENCE CHECKING
11:	Λ = [Λ,w_i]	▹ LOCKING CONVERGED EIGENPAIRS
12:	V = [V,u_i]
13:	else
14:	break
15:	end if
16:	end for
17:	conv=i
18:	W = [w_conv,...,w_end]
19:	U = [u_conv,...,u_end]
20:	end while

	DoFs		CPU Time
FED	Fine	Coarse	Krylov-Schur	Multilevel (CM)	its. MBNM	Improv.
2	334510	125730	237s	180s (35s)	5	24%
3	1106180	411992	1864s	1596s (463s)	4	14%

	DoFs		CPU Time
FED	Fine	Coarse	Krylov-Schur	Multilevel (CM)	Improv.
2	137842	18522	106s	87s (3.8s)	17%
3	453962	59582	794s	555s (41s)	30%

Multilevel method to compute the lambda modes of the neutron diffusion equation

Data publikacji: 24 cze 2017

Zakres stron: 225 - 236

Otrzymano: 16 kwi 2017

Przyjęty: 24 cze 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00019

Słowa kluczowe
Multilevel method, Finite element method, Modified block Newton method, Lambda modes, Generalized eigenvalue problem

© 2017 A. Carreño, A. Vidal-Ferrándiz, D. Ginestar, G. Verdú, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

MBNM

Computational times of Ringhals reactor

Computational times of cuboid reactor

Multilevel method to compute the lambda modes of the neutron diffusion equation

Data publikacji: 24 cze 2017

Zakres stron: 225 - 236

Otrzymano: 16 kwi 2017

Przyjęty: 24 cze 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00019

Słowa kluczoweMultilevel method, Finite element method, Modified block Newton method, Lambda modes, Generalized eigenvalue problem

© 2017 A. Carreño, A. Vidal-Ferrándiz, D. Ginestar, G. Verdú, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

MBNM

Computational times of Ringhals reactor

Computational times of cuboid reactor

Słowa kluczowe
Multilevel method, Finite element method, Modified block Newton method, Lambda modes, Generalized eigenvalue problem