Development of a finite element method for neutron transport equation approximations

Abstract

Supervisors: Damián Ginestar Peiró Gumersindo Verdú Martín Ací em pariren i ací estic. I com que he investigat certes coses, ací les conte, ací les dic. AbstractThe neutron transport equation describes the neutron population and the nuclear reactions inside a nuclear reactor core. First, this equation is introduced and its assumptions are stated. Then, the stationary neutron diffusion equation which is the most useful approximation of this equation, is studied. This approximation leads to a differential eigenvalue problem. To solve the neutron diffusion equation, a h ≠ p finite element method is investigated. To improve the efficiency of the method a Restricted Additive Schwarz preconditioner is implemented.Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To test the behaviour of the method, rod ejection accidents are numerically simulated. However, a non-physical behaviour appears when a cell is partially rodded: this is, the rod cusping effect, which is solved by using a moving mesh scheme. In other words, the mesh follows the movement of the control rod. Numerical results show that the rod cusping effect is corrected with this scheme.After that, the simplified spherical harmonics approximation, SP N , is developed to solve the steady state problem. This approximation extends the spherical harmonics approximation, P N , in one dimensional geometries to multidimensional geometries with strong assumptions. It improves the diffusion theory results but does not converge as N tends to infinity. The advantages and limitations of this approximation are tested on several one-, two-and three-dimensional reactors.Finally, the spatial homogenization in the context of the finite element method is studied. Homogenization consists in replacing heterogeneous subdomains by homogeneous ones, in such a way that the homogenized problem provides fast and accurate average results. Discontinuous solutions were proposed in the Generalized Equivalence Theory. Here, a discontinuous Galerkin finite element method where the jump condition for the neutron flux is imposed in a weak sense using interior penalty terms is introduced. Also, the use of discontinuity factors for the correction of the homogenization error when using the SP N equations is investigated.v ResumenLa ecuación del transporte neutrónico describe la población de neutrones y las reacciones nucleares dentro de un reactor nuclear. Primero, introducimos esta ecuación y las aproximaciones de la misma. Entonces, estudiamos la ecuación de la difusión neutrónica, la aproximación al transporte más utilizada. Para el caso estacionario, esta aproximación da lugar a un problema diferencial de valores propios. Para resolver la ecuación de la difusión se ha desarrollado un método de elementos finitos h≠p. Para mejorar la eficiencia del método se ha implementado un precondicionador del tipo Restricted Additive Schwarz.Una vez hemos obtenido la distribución neutrónica en ...

Lambda modes comparison for different approximations of the neutron transport equation: Diffusion, SN and SP3

“…Different neutron transport codes are used as a reference to compare the results of the developed SP 3 code. Next sections make use of some standard codes as reference values, such as the Discrete Ordinates based method code DANTSYS, as well as FEMFUSSION, a finite element SP 3 method developed by A. Vidal-Ferrndiz [12]. Another code based on the discrete ordinates method mentioned before is used as a reference, which was validated in [11].…”

“…3. It was compared with a neutron transport code using S 16 and with another SP 3 code called FEMFFUSION which uses the finite element method to discretize the spatial variables [12]. The flux curves calculated with FDM SP c 3 and FDM SP E 3 are overlapping in fig.…”

Contributions to solve the Multi-group Neutron Transport equation with different Angular Approaches

“…Different neutron transport codes are used to validate the results of the developed SP 3 code. Next sections make use of some standard codes as reference values, such as the Discrete Ordinates based method code DANTSYS, as well as FEMFUSSION, a finite element SP 3 method developed by Vidal Ferràndiz [2018]. Furthermore, analytical solutions from Capilla et al [2008] are shown to compare with the results calculated by the algorithms implemented in this work.…”

“…5.12. The results were compared with the results obtained with the FDM neutron transport code using S 96 , n-DOTEC and with another SP 3 code called FEMFFUSION which uses the finite element method for the spatial discretization Vidal Ferràndiz [2018]. The curves calculated with the FDM cell-centered scheme SP C 3 and the edge-centered scheme FDM SP E 3 coincide.…”