On an analytical evaluation of the flux and dominant eigenvalue problem for the steady state multi-group multi-layer neutron diffusion equation

Kerntechnik ◽  
2014 ◽  
Vol 79 (5) ◽  
pp. 430-435 ◽  
Author(s):  
C. Ceolin ◽  
M. Schramm ◽  
B. E. J. Bodmann ◽  
M. T. Vilhena ◽  
S. de Q. B.Leite
Keyword(s):  
Steady State ◽  
Eigenvalue Problem ◽  
Diffusion Equation ◽  
Dominant Eigenvalue ◽  
Neutron Diffusion ◽  
Analytical Evaluation ◽  
Neutron Diffusion Equation

scholarly journals Resolution of the Generalized Eigenvalue Problem in the Neutron Diffusion Equation Discretized by the Finite Volume Method

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Álvaro Bernal ◽  
Rafael Miró ◽  
Damián Ginestar ◽  
Gumersindo Verdú
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Eigenvalue Problem ◽  
Diffusion Equation ◽  
Finite Volume Method ◽  
Difference Method ◽  
Neutron Diffusion ◽  
Generalized Eigenvalue ◽  
Volume Method ◽  
Neutron Diffusion Equation

Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Element Method (FEM) and the Finite Volume Method (FVM) are easily applied to unstructured meshes. On the one hand, the FEM can be accurate for smoothly varying functions. On the other hand, the FVM is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm implemented in the ARB Partial Differential Equations solver has been used to discretize the neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library.

scholarly journals Block Preconditioning Matrices for the Newton Method to Compute the Dominant λ-Modes Associated with the Neutron Diffusion Equation

2019 ◽  
Vol 24 (1) ◽  
pp. 9
Author(s):  
Amanda Carreño ◽  
Luca Bergamaschi ◽  
Angeles Martinez ◽  
Antoni Vidal-Ferrándiz ◽  
Damian Ginestar ◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Diffusion Equation ◽  
Newton Method ◽  
Computational Time ◽  
Neutron Diffusion ◽  
The Matrix ◽  
Generalized Newton ◽  
Block Preconditioners ◽  
Matrix Free ◽  
Neutron Diffusion Equation

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.

Improved nodal expansion method for solving neutron diffusion equation in cylindrical geometry

2010 ◽  
Vol 240 (8) ◽  
pp. 1997-2004 ◽  
Author(s):  
Dengying Wang ◽  
Fu Li ◽  
Jiong Guo ◽  
Jinfeng Wei ◽  
Jingyu Zhang ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Expansion Method ◽  
Cylindrical Geometry ◽  
Neutron Diffusion ◽  
Nodal Expansion Method ◽  
Neutron Diffusion Equation
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