---
category: neutron_sn
intro: |
  # Azmy’s problem
  
  As in
   
    * [Y. Y. Azmy, The Weighted Diamond-Difference Form of Nodal Transport Methods. NUCLEAR SCIENCE AND ENGINEERING: 98, 29-40 (1988). DOI: 10.13182/NSE88-6](https://doi.org/10.13182/NSE88-6)    
    
  ![Original problem formulation from 1988 paper](azmy-problem.png)
    
  ## Second-order complete structured rectangular grid
  
  This example solves the problem using a structured second-order grid.
  It computes the mean flux in each quadrant by integrating $\phi_1$ over each physical group using the instruction [`INTEGRATE`](https://www.seamplex.com/feenox/doc/feenox-manual.html#integrate).
  
  ![Second-order complete structured rectangular grid](azmy-structured-mesh.png)
  
terminal: | 
  $ gmsh -2 azmy-structured.geo
  $ feenox azmy-structured.fee 2
  LLQ = 1.653e+00 (ref 1.676e+0)
  LRQ = 4.427e-02 (ref 4.159e-2)
  URQ = 2.712e-03 (ref 1.992e-3)
  16900 unknowns for S2, memory needed = 0.2 Gb
  $ feenox azmy-structured.fee 4
  LLQ = 1.676e+00 (ref 1.676e+0)
  LRQ = 4.164e-02 (ref 4.159e-2)
  URQ = 1.978e-03 (ref 1.992e-3)
  50700 unknowns for S4, memory needed = 0.7 Gb
  $ feenox azmy-structured.fee 6
  LLQ = 1.680e+00 (ref 1.676e+0)
  LRQ = 4.120e-02 (ref 4.159e-2)
  URQ = 1.874e-03 (ref 1.992e-3)
  101400 unknowns for S6, memory needed = 2.7 Gb
  $
...
DEFAULT_ARGUMENT_VALUE 1 4
PROBLEM neutron_sn DIM 2 GROUPS 1 SN $1 MESH $0.msh

MATERIAL src S1=1 Sigma_t1=1 Sigma_s1.1=0.5
MATERIAL abs S1=0 Sigma_t1=2 Sigma_s1.1=0.1

PHYSICAL_GROUP llq MATERIAL src
PHYSICAL_GROUP lrq MATERIAL abs
PHYSICAL_GROUP urq MATERIAL abs
PHYSICAL_GROUP ulq MATERIAL abs

BC mirror   mirror
BC vacuum   vacuum

SOLVE_PROBLEM

# compute mean values in each quadrant
INTEGRATE phi1 OVER llq RESULT lower_left_quadrant
INTEGRATE phi1 OVER lrq RESULT lower_right_quadrant
INTEGRATE phi1 OVER urq RESULT upper_right_quadrant

PRINTF "LLQ = %.3e (ref 1.676e+0)" lower_left_quadrant/(5*5)
PRINTF "LRQ = %.3e (ref 4.159e-2)" lower_right_quadrant/(5*5)
PRINTF "URQ = %.3e (ref 1.992e-3)" upper_right_quadrant/(5*5)

WRITE_RESULTS
PRINTF "%g unknowns for S${1}, memory needed = %.1f Gb" total_dofs memory()