**1. NAME AND TITLE**

FEM-2D: Two-Dimensional Diffusion Theory Code System Based on the Method of Finite Elements.

**AUXILIARY ROUTINE**

RSYM: Simulates parts of RSYST Code System.

**2. CONTRIBUTOR**

Institute for Nuclear Energy (IKE), University of Stuttgart, Stuttgart, Germany.

**3. CODING LANGUAGE AND COMPUTER**

FORTRAN IV and COMPASS; CDC 6600.

**4. NATURE OF PROBLEM SOLVED**

FEM-2D solves the two-dimensional diffusion equation in x-y geometry. This is done by the finite element method.

**5. METHOD OF SOLUTION**

FEM-2D uses triangular elements with first- and second-order Lagrange approximations. The systems equations are formulated in multigroup form and solved by a Cholesky procedure which operates only on nonzero elements. Various acceleration techniques are available for the outer iteration. Fluxes along various lines and rates in arbitrary zones may be output.

**6. RESTRICTIONS OR LIMITATIONS**

The code uses variable dimensioning. Thus the problem size is restricted by the largest array which usually is the systems matrix. Fluxes of all groups are kept in memory. This might become another restrictive data set for a large number of groups.

The validity of the results is restricted by the approximations used.

FEM-2D requires a finite element net which allows the approximation of fluxes at most parabolas. The node distribution should be more dense in areas of heavy flux changes (near absorbers or the reflector).

**7. TYPICAL RUNNING TIME**

The running time is determined by the number of elements, the number of nodes and the approximation used. Due to the source and rate calculations the number of nodes is the least important factor. Typical running times on a CDC 6600 for a two group problem are:

Nodes Elements Approximation Time/Iteration

in seconds

177 301 first order 0, 26

654 1204 first order 0, 70

501 228 second order 0, 48

654 301 second order 0, 61

746 345 second order 0, 70

**8. COMPUTER HARDWARE REQUIREMENTS**

FEM-2D was designed for and runs on the CDC 6600 computer using disk for multigroup problems and extended core storage for block data.

**9. COMPUTER SOFTWARE REQUIREMENTS**

A FORTRAN IV compiler is required. FEM-2D is written in overlay mode. It uses one utility and two mathematical routines written in COMPASS.

**10. REFERENCES**

H. P. Franke and F. A. R. Schmidt, "FEM-2DProgram for Solving the Two-Dimensional Diffusion Equation by the Method of Finite Elements," IKE-4-25 (ORNL-tr-2971) (July 1974).

F. A. R. Schmidt, "FEM-2DInput Description and Performance," IKE-4-43 (March 1975).

**11. CONTENTS OF CODE PACKAGE**

Included are the referenced documents and one (1.2MB) DOS diskette which contains the source code and sample problem input.

**12. DATE OF ABSTRACT**

September 1975.

**KEYWORDS: ** FINITE ELEMENT METHOD; NEUTRON; TWO-DIMENSIONS;
DIFFUSION THEORY