(The) p-version of the finite element method for the solution of neutron diffusion equations

Abstract

The finite element method, one of the widely used numerical methods for solving certain types of differential equations, is based on approximating the solution by piecewise smooth functions, specifically polynomials. In the method, the problem domain is divided into small elements, and a polynomial basis function is specified in each element. Thus polynomial basis functions are local in character, and well suited for handling complex geometries. In general, there are two kinds of finite element method: the h--version and the A-version. In the h-version, polynomial basis functions are fixed over each element and accuracy is achieved by refining the mesh size h. In the p-version, the mesh is fixed and accuracy is achieved by increasing the degree p of polynomial basis functions. Based on the hierarchical structure of the basis functions, two types of the A-version of the finite element code are developed for solving the two-dimensional neutron diffusion equations. One is the conventional p--type FEM(pFEM1), and the reactor domain is discretized into finite elements, and each element matrices are assembled and solved. In the other, with the domain decomposition approach, the reactor domain is decomposed into subdomains and each subdomain is solved independently. They are coupled by use of incoming and outgoing partial currents along the interfaces, which are obtained analytically from fluxes. The codes use power method in outer iterations and two acceleration schemes are implemented. For pFEM1, Chebyshev one parameter acceleration scheme is used to reduce computing times. For pFEM2, coarse group rebalance method (CGR) is used, since in pFEM2 each subdomain is solved independently in the whole reactor domain and balance equation in the subdomain can be constructed. Since domain decomposition is used in pFEM2 code, the computations are implemented on parallel computers. Based on the multi -p V cycle method, multilevel acceleration is implemented into both of the met...