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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)


## 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).


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()