ADAPTIVE SOLUTION OF THE NEUTRON DIFFUSION EQUATION WITH HETEROGENEOUS COEFFICIENTS USING THE MIXED FINITE ELEMENT METHOD ON STRUCTURED MESHES

The neutron transport equation can be used to model the physics of the nuclear reactor core. Its solution depends on several variables and requires a lot of high precision computations. One can simplify this model to obtain the SPN equation for a generalized eigenvalue problem. In order to solve this eigenvalue problem, we usually use the inverse power iteration by solving a source problem at each iteration. Classically, this problem can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. In this article, we focus on the steady-state diffusion equation with heterogeneous coefficients discretized on Cartesian meshes. In this situation, it is expected that the solution has low regularity. Therefore, it is necessary to refine at the singular regions to get better accuracy. The Adaptive Mesh Refinement (AMR) is one of the most effective ways to do that and to improve the computational time. The main ingredient for the refinement techniques is the use of a posteriori error estimates, which gives a rigorous upper bound of the error between the exact and numerical solution. This indicator allows to refine the mesh in the regions where the error is large. In this work, some mesh refinement strategies are proposed on the Cartesian mesh for the source problem. Moreover, we numerically investigate an algorithm which combines the AMR process with the inverse power iteration to handle the generalized eigenvalue problem.

Efficient and Flexible Finite Element Procedure for Free and Forced Vibration Problems of a Plate Coupled With Fluid

Identifying the coupled system natural frequencies and dynamic behavior of systems in the presence of fluid-structure interaction is one of the most important issues in the engineering design of buildings, road vehicles and aircraft. This paper presents an efficient and flexible finite element procedure using fully vectorized codes for the free and forced vibration analysis of a rectangular plate in contact with fluid. The 4-node MITC plate finite element (MITC4) based on the Mindlin plate theory is used to simulate the plate, while the 8-node acoustic pressure element is used to simulate the fluid. The derived system of equations using structural displacements and fluid pressures yields a non-symmetric system of equations. Solving the generalized eigenvalue problem for the non-symmetric system is more computationally intensive compared to solving the generalized eigenvalue problem for symmetric systems. The modal expansion technique is used to reduce the model size. Then the reduced non-symmetric system is symmetrized by right eigenvectors. The Newmark method is used to solve the forced vibration problem of the coupled systems. The effect of the height of the fluid on the natural frequencies is discussed. The natural frequencies and transient responses are in good agreement with those obtained from the commercial finite element software. Moreover, the technique is proved to be effective to solve the coupled system.