1. Name of Program:
   VARIANT: A new nodal transport and diffusion module for the DIF3D (Ref. 1, 2) neutronics code.
2. Computer for Which Program is Designed and Other Machine Version Packages Available:
   CRAY X-MP, SUN SPARCstations, IBM RS6000 series.
3. Description of Problem Solved:
   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.
4. Method of Solution:
   The multigroup neutron transport equations are solved using a variational nodal method (Refs. 3-7) 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 (Ref. 6).
5. Restrictions on the Complexity of the Problems:
   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 computational resources required for evaluation and storage of response matrices for problems involving large numbers of unique node types impose a practical limit on problem complexity. For highly heterogeneous problems involving thousands of different node types, calculation and storage of response matrices represents the primary computational cost.
6. Typical Running Time:
   The times provided apply to a three-dimensional isotropic problem for a small liquid-metal reactor (LMR) with 30 planar symmetry, 9 energy groups, 14 axial mesh planes and 16 rings of hexagons. The problem consisted of 1694 nodes with 24 compositions and 216 unique node types. Each outer iteration required 70 inner iterations (5 groups required 10 inner iterations and 4 groups required 4 inner iterations). The diffusion calculation required 18 outer iterations and the transport calculation required 19 outer iterations. The diffusion calculation iterations used 41 CPU seconds on a CRAY X-MP14, 47 seconds on an IBM RS6000, and 107 seconds on a SPARC 20/50. The transport calculation for this problem (with a P3 angular expansion) required 231 seconds on the CRAY X-MP/14, 1046 seconds on an IBM RS6000, and 2183 seconds on a SPARC 20/50.
7. Unusual Features of the Program:
   Variational nodal methods incorporate a number of attractive features. These include a standard hierarchy of space-angle approximation, well behaved small mesh limits, and the absence of both ray effects and artificial diagonal streaming depressions. Dimensionless parts of the response matrices involving integrals in space and angle are pre-computed once and for all using MATHEMATICA for each geometry option. The results are stored in FORTRAN data statements and used to generate response matrix sets for unique nodes (defined by cross section and dimension data) prior to fission source iteration. Anisotropic scattering (up to order P5) is also available. VARIANT achieves near Monte Carlo accuracy at a fraction of the cost.
8. Related and Auxiliary Programs:
   VARIANT reads and writes the standard interface files specified by the Committee on Computer Code Coordination (CCCC). It is implemented as a computational module in the DIF3D code system at Argonne National Laboratory.
9. Status:
   VARIANT is currently in use on the Reactor Analysis Division network which consists of SUN SPARCstations and IBM RS6000 series workstations. Capabilities for perturbation calculations, neutron kinetics applications, treatment of inhomogeneous nodes, and execution on parallel computing platforms are under development.
10. References:
    1. K. L. Derstine, "DIF3D: A Code to Solve One-, Two-, and Three-Dimensional Finite Difference Diffusion Theory Problems," Argonne-82-64, Argonne National Laboratory (1982).
    2. R. D. Lawrence, "The DIF3D Nodal Neutronics Option for Two- and Three-Dimensional Diffusion Theory Calculations in Hexagonal Geometry," Argonne-83-1, Argonne National Laboratory (1983).
    3. G. Palmiotti, C. B. Carrico, and E. E. Lewis, "Variational Nodal Methods with Anisotropic Scattering," Nuclear Science and Engineering, 115, 223-243, (Nov. 1993).
    4. J. Y. Doriath, F. Malvagi, G. Palmiotti, J. M. Ruggieri, C. B. Carrico, E. E. Lewis, and G. Gastaldo, "Variational Nodal Method (VNM) to Solve 3D Transport Equation: Applications to EFR Design," Proceedings of Mathematical Methods and Supercomputing in Nuclear Applications, I-571, Karlsruhe, Germany, (April 1993).
    5. C. B. Carrico, E. E. Lewis, and G. Palmiotti, "Three-Dimensional Variational Nodal Transport Methods for Caretsian, Triangular and Hexagonal Criticality Calculations," Nuclear Science and Engineering, 111, pp. 168­179, (June 1992).
    6. C. B. Carrico and E. E. Lewis, "Variational Nodal Solution Algorithms for Multigroup Criticality Problems," Proc. Int. Topl. Mtg. Advances in Mathematics, Computations and Reactor Physics, April 28­May 1 1991, Pittsburgh, Pennsylvania.
    7. E. E. Lewis, C. B. Carrico, and G. Palmiotti, "Variational Nodal Formulation for the Spherical Harmoncis Equations," Nuclear Science and Engineering, 122, 194-203 (1996).
11. Machine Requirements:
    At least 8 Mb of memory are recommended for program and file buffer storage on short word machines like SPARCstations and RS6000 workstations. 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.
12. Programming Languages Used:
    FORTRAN 77 is used. The program can be executed entirely in FORTRAN. Dynamic memory allocation routines on the workstations are written in C or supplied from host machine libraries.
13. Operating System:
    No special requirements or restrictions.
14. Other Programming or Operating Information or Restrictions:
    The stand-alone source code contains approximately 120,000 lines of FORTRAN statements (including DIF3D driver and edit routines).
15. Name and Establishment of Author or Contributor:
    • G. Palmiotti
      Nuclear Engineering Division
      Argonne National Laboratory
      9700 South Cass Avenue
      Argonne, Illinois 60439
    • E. E. Lewis
      Department of Mechanical Engineering
      Northwestern University
      2145 Sheridan Road
      Evanston, Illinois 60208
16. Materials Available:
    Distribution may be restricted.
    
    • User's Manual
    • Magnetic Tape Cartridge containing:
        - Source Code
        - Sample Problem Data Card-Images
        - Sample Problem Output
        - Code Dependent BCD and Binary Card Image File Descriptions
17. Sponsor:
    U.S. Department of Energy, Office of Nuclear Energy, Science, and Technology.

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