Fender: A finite element code for the solution of the diffusion equation in shield design applications

Abstract

Diffusion theory remains an important method of calculation for shield design. Using adjusted coefficients the method provides an inexpensive solution of adequate accuracy for survey and optimization studies in two- and three-dimensional geometries. Solved in the adjoint mode, the method provides an estimate of the importance function which may be used for the acceleration of generalized-geometry Monte Carlo calculations. A number of computer codes exist to solve the diffusion equation by a finite difference approximation in one-, two- and three-dimensions. The mesh systems used in such codes usually impose restrictions on the accuracy of representation of shields with complicated geometries. The computer code FENDER solves the diffusion equation for neutron or gamma transport using the finite element technique. At present the code is written for a two-dimensional problem in which the geometry is specified as an array of triangular or rectangular elements. This permits a good representation to be made of shields containing curved surfaces. The variation of the calculated particle fluxes within an element is assumed to be quadratic. FENDER may take details of the element structure from an external mesh generating package but also contains a semi-automatic mesh generating routine for use as a stand-alone code. Multigroup diffusion parameters may be either input directly or generated from material compositions. The code is capable of handling problems with at least 1000 elements which is roughly equivalent in size and attenuation to 10,000 finite difference meshes. A variety of boundary conditions may be specified. The paper includes an example of application to demonstrate the potential usefulness of the method and the code. The case chosen is the calculation of neutron fluxes in a stylized fast reactor.