**1. NAME AND TITLE**

DIF3D 8.0/VARIANT8.0: Code System Using Variational Nodal Methods and Finite Difference Methods to Solve Neutron Diffusion and Transport Theory Problems.

**2. CONTRIBUTOR**

Argonne National Laboratory, Argonne, Illinois.

**3. CODING LANGUAGE AND COMPUTER**

Fortran 77 and C; Unix Workstations (Sun SparcStation, IBM RS/6000), (C00649/MFMWS/01).

**4. NATURE OF PROBLEM SOLVED**

The DIF3D8.0/VARIANT8.0 release differs from the previous DIF3D7.0 release in that it includes a significantly expanded set of solution techniques using variational nodal methods. DIF3D's nodal option solves the multigroup steadystate neutron diffusion equation in two- and three-dimensional hexagonal and cartesian geometries and solves the transport equation in two-and three-dimensional cartesian geometries. Eigenvalue, adjoint, fixed source and criticality (concentration) search problems are permitted as are anisotropic diffusion coefficients. Flux and power density maps by mesh cell and region-wise balance integrals are provided. Although primarily designed for fast reactor problems, upscattering and for finite difference option only internal black boundary conditions are also treated.

VARIANT solves the multigroup steady-state neutron diffusion and transport equations in two- and three-dimensional Cartesian and hexagonal geometries using variational nodal methods. The transport approximations involve complete spherical harmonic expansions up to order P5. Eigenvalue, adjoint, fixed source, gamma heating, and criticality (concentration) search problems are permitted. Anisotropic scattering is treated, and although primarily designed for fast reactor problems, upscattering options are also included.

Related and Auxiliary Programs: DIF3D reads and writes the standard interface files specified by the Committee on Computer Code Coordination (CCCC). DIF3D is included in the REBUS-3 code package and can thus be used to provide the neutronics solutions required in REBUS-3 depletion calculations.

**5. METHOD OF SOLUTION**

The neutron diffusion and transport equations are solved using a variational nodal method
with one mesh cell (node) per hexagonal assembly (Cartesian geometry node sizes are specified by
the user). The nodal equations are derived from a functional incorporating nodal balance, and
reflective and vacuum boundary conditions through Lagrange multipliers. Expansion of the
functional in orthogonal spatial and angular (spherical harmonics) polynomials leads to a set of
response matrix equations relating partial current moments to flux and source moments. The
equations are solved by fission source iteration in conjunction with a coarse mesh rebalance
acceleration scheme. The inner iterations are accelerated by a partitioned matrix scheme equivalent
to a synthetic diffusion acceleration method*. *

**6. RESTRICTIONS OR LIMITATIONS**

Problem dimensions are all variable. Enough memory must be allocated to contain all the
information for at least one energy group. Flux and source expansions of up to sixth order are
allowed. Partial current expansions up to second order are allowed. Angular and scattering
expansions of up to P5 are allowed. The typical limiting factor for a problem lies in the storage of
response matrices for problems involving large numbers of unique node types. For highly
heterogeneous problems involving thousands of different node types, calculation and storage of
response matrices represent the primary computational cost*.*

**7. TYPICAL RUNNING TIME**

A three-dimensional nodal calculation for a small LMR with 60 degree planar symmetry, 9
energy groups, 14 axial mesh planes and 16 rings of hexagons required 22 CPU seconds on a Sun
SPARCstation 20 (61 seconds on a SPARCStation 5, 18 seconds on an IBM RS/6000), to perform
14 outer iterations with 28 inners/outer and a convergence criteria of 10^{-6}. All of the test cases
completed in less than 5 minutes on a IBM RS/6000 Model 270 and on a UltraSparc 60.

**8. COMPUTER HARDWARE REQUIREMENTS**

The modular version of the code is in production use at Argonne National Laboratory on Unix Workstations Sun SPARCStation. External data storage must be available for approximately 40 scratch and interface files. Fourteen of these files are random access scratch files (grouped into 6 file groups), and the remainder are sequential access files with formatted or unformatted record types.

**9. COMPUTER SOFTWARE REQUIREMENTS**

No special requirements are made on the operating system (SOLARIS 2.6 for SPARCStations and AIX 4.3.3 on the IBM). The included installation procedure requires Fortran 77 and C compilers. With modifications the program can be executed entirely in FORTRAN. Optional dynamic memory allocation and timing routines supplied from host machine libraries or code in "C" may be used on Unix workstations. Although developed on the Cray and IBM 30xx, the current version is tailored to Sun SparcStations and IBM AIX RS/6000.

**10. REFERENCES**

**a. Included on CD in DOC/C649.PDF:**

K. L. Derstine, "DIF3D: A Code to Solve One-, Two-, and Three-Dimensional Finite-Difference Diffusion Theory Problems," ANL-82-64 (1984).

R. D. Lawrence, "The DIF3D Nodal Neutronics Option for Two- and Three-Dimensional Diffusion Theory Calculations in Hexagonal Geometry," ANL-83-1 (1983).

G. Palmiotti, E. E. Lewis, and C. B. Carrico, "VARIANT: VARIational Anisotropic Nodal Transport for Multidimensional Cartesian and Hexagonal Geometry Calculation," ANL-95/40 (October 1995).

C. H. Adams, et.al., "The Utility Subroutine Package Used by Applied Physics Division Export Codes," ANL-83-3 (May 1992).

**b: Background information:**

** **R. D. Lawrence, "Progress in Nodal Methods for the Solution of the Neutron Diffusion
and Transport Equations," *Prog. Nucl. Energy, 17, 3, 271* (1986).

** **P. J. Finck and K. L. Derstine, "The Application of Nodal Equivalence Theory to
Hexagonal Geometry Lattices," Proceedings of the International Topical Meeting Advances in
Mathematics, Computations and Reactor Physics, Pittsburgh, Pa., Vol. 4, pp 16.1 4-1 (1991).

D. O'Dell, "Standard Interface Files and Procedures for Reactor Physics Codes, Version IV," UC-32, Los Alamos Scientific Laboratory (September 1977).

B.J. Toppel, "A Users Guide for the REBUS-3 Fuel Cycle Analysis Capability," ANL-83-2 (1983 revised October 1990).

** **

**11. CONTENTS OF CODE PACKAGE**

Included are referenced documents (10.a in pdf format) on one CD which also contains a UNIX tar file including source code, sample problem input and output, code dependent BCD and binary card image file descriptions, scripts, and a readme file.

**12. DATE OF ABSTRACT**

March 1997, revised March 2001.

**KEYWORDS:** DIFFUSION THEORY; MULTIGROUP; COMPLEX GEOMETRY;
CRITICALITY CALCULATIONS; CCCC INTERFACE FORMAT;
REACTOR PHYSICS