| Nuclear energy is a kind of energy source which can provide a large amount of clean energy, with high development prospect. The simulation of nuclear power system,plays an important role in the design, development and security verification of nuclear power system. The solution of the neutron flux density distribution in the nuclear reactor core is an important part of the simulation of nuclear power system. As the priority step in the calculation of heat source, the solution of the neutron flux density distribution is also necessary for the thermal design and safety analysis of nuclear reactor.In this thesis, the RBF(Radial Basis Function) meshfree method is introduced to solve the critical problem of neutrion diffusion, in order to verify the feasibility and effectiveness of this method. And the RBF meshfree method is compared with the FVM(Finite Volume Method) and FDM(Finite Difference Method), to explore and display the advantages of RBF meshfree method. The neutron diffusion process in both the homogeneous and inhomogeneous medium are taken into consideration.To discuss the critical problem of neutron diffusion in homogeneous medium, the mathematical and physical model is established, the nodes of calculation area are generated and the solution formula is derived. And the single energy group, super critical double energy group, sub critical three energy group problems are solved. The results show that the RBF meshfree method is feasible and can get higher precision for different energy group conditions and different critical conditions. Heighest accuracy can be obtained when GA(Gaussian) kernel function is selected, followed by MQ(Multiquadrics) and IMQ(Inverse Multiquadrics). But on the numerical stability, MQ and IMQ are better than GA. For the uniform, part random and total random nodes generation mode, nodes are generated and programs are calculated for 1000 times. The results show that part random mode can get the heighest precision. Under all of the three MQ, IMQ and GA kernel function conditions, the RMS error of part random mode is 61.08%, 45.65% and 59.34% of the RMS error of uniform mode. It means that calculation precision is obviously improved when uniform mode is replaced by part random mode.To discuss the critical problem of neutron diffusion in inhomogeneous medium,two different types of cases are considered. One is a single group two media problems and the other is 2D-IAEA benchmark problems which is a simplified model of a typical pressurized water reactor. To solve these two cases, the nodes of calculation area are generated and the solution formula is derived. The calculation results show that RBF meshfree method is applicable to solve the inhomogeneous medium neutron diffusioncritical problem, and can get higher accuracy, especially for calculating global parameters such as the effective multiplication factor. The related error of the effective multiplication factor is 0.027%.The RBF meshfree method, FVM and FDM are used to solve the two-dimensional double group fission source problem and the two-dimensional single group additional source problem. Both the calculation process and results are compared and analysed.The results show that the accuracy of RBF meshfree method, is significantly higher than the FVM with FDM, but the iteration steps of RBF meshfree method which are needed for convergence is higher than FVM and FDM, in the calculation of the two-dimensional double group fission source problem. In the calculation of two-dimensional single group additional source problem, FDM gets the heighest ralated error. And when the number of nodes is small, the maximum relative error and average relative error of RBF meshfree method is less than FVM.The research of this thesis verified the application of RBF meshfree method in neutron diffusion problem solving. And the compariation with FVM and FDM explores and displays the advantages of RBF meshfree method. The RBF meshfree method could achieve better results in some specific problems. |
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