The development of a monte carlo code for the solution of generalized-geometry problems by tracking in a finite element mesh

Abstract

Practical shield design is concerned with problems of deep penetration in complicated geometries. Solutions may be obtained using approximate methods, such as diffusion theory with corrections based on kernel techniques, or transport theory. The most widely used transport methods employ the discrete-ordinate S n approximation which is restricted in practice to two-dimensional geometries. For the more complicated geometric models Monte Carlo remains the only practicable approach. The present generation of Monte Carlo codes permit description of the shield regions in fine detail but the method is still restricted in practice by the limitations of the available techniques for reducing the computing time. The most powerful acceleration techniques utilize importance data derived from adjoint calculations. A rigorous solution of the adjoint equation is as difficult to obtain as the forward solution and the success of this approach stems from the major improvements in the efficiency which can be gained using approximate importance distributions. These may be obtained using diffusion theory and in the ADC (adjusted-diffusion-coefficient) method developed at Winfrith, for example, the shortcomings of the diffusion approximation at deep penetration are compensated by fitting the diffusion coefficient to a series of benchmark calculations in slab geometry. Solutions of the adjoint equations are carried out in either RZ or XYZ orthogonal geometries or in finite element mesh but the approximate importance functions can be utilized most efficiently when the physical and importance regions are identical. If a generalized-geometry capability is to be retained, this can be most readily achieved by ray-tracing using a finite element (FE) representation. An accurate description of a complicated material region for Monte Carlo can be made using the well-known Combinatorial Geometry (CG) method which requires only a few simple geometric bodies. The equivalent FE description, however, requires a large number of elements. Thus the analysis of a trajectory within the region defined using a finite element mesh involves a much larger number of boundary crossings than would be required using the corresponding CG description. Less computation is involved at each crossing, however, since the dements are defined by plane rather than conic surfaces, and the adjacent element is known from the FE formulation without carrying out an elaborate search procedure. In order to evaluate finite element Monte Carlo tracking, a multidimensional MonteCarlo code, FEDRAN, has been written, utilizing a triangular mesh, which is coupled to a finite element diffusion code to provide automatic acceleration. The code is described and some samples of its application to radiation shielding problems in reactor design are given.