Unstructured Grids and the Multigroup Neutron Diffusion Equation

Table 2

Results of the 2D IAEA PWR Benchmark problem obtained with the milonga code for the eighty proposed combinations of symmetry, meshing algorithm, basic shape, discretization scheme, and characteristic element/cell length. The reference solution is 𝜌=2874.9×10⁻⁵, which corresponds to 𝑘_eff=1.0296.

Case no.	Symmetry	Mesh algorithm	Basic shape	Solution method	ℓ𝑐; (cm)	Δ𝜌; (pcm)	max𝜙2; (—)	Total unknowns	Outer iter.	Linear iter.	Inner iter.	Residual norm	Relative error	Error estimate	Memory (Mb)	Page faults
1	1/4	Delaunay	▵	Volumes	4.0	−12.0	11.45	8264	3	32	1033	6𝑒−09	3𝑒−09	1𝑒−09	34	9470
2	1/4	Delaunay	▵	Volumes	3.0	8.5	11.35	15740	3	32	1967	2𝑒−08	1𝑒−08	4𝑒−09	48	13401
3	1/4	Delaunay	▵	Volumes	2.0	19.7	11.24	31188	3	32	3898	1𝑒−08	6𝑒−09	3𝑒−09	76	21932
4	1/4	Delaunay	▵	Volumes	1.0	15.3	11.23	127476	3	32	15934	7𝑒−09	3𝑒−09	3𝑒−09	273	81886
5	1/4	Delaunay	▵	Volumes	0.5	18.7	11.19	510676	3	32	63834	2𝑒−09	9𝑒−10	1𝑒−09	1148	352261
6	1/4	Delaunay	▵	Elements	4.0	17.3	11.00	4308	3	32	538	2𝑒−08	8𝑒−09	4𝑒−09	31	8794
7	1/4	Delaunay	▵	Elements	3.0	11.7	11.07	8108	3	32	1013	5𝑒−09	2𝑒−09	2𝑒−09	47	12946
8	1/4	Delaunay	▵	Elements	2.0	8.4	11.12	15936	3	32	1992	6𝑒−09	3𝑒−09	2𝑒−09	73	21347
9	1/4	Delaunay	▵	Elements	1.0	6.2	11.16	64478	3	32	8059	6𝑒−09	3𝑒−09	4𝑒−09	262	77105
10	1/4	Delaunay	▵	Elements	0.5	5.8	11.17	256700	3	32	32087	1𝑒−09	5𝑒−10	1𝑒−09	1091	331969
11	1/4	Delaunay	□	Volumes	4.0	5.3	11.49	5042	3	32	630	2𝑒−08	1𝑒−08	5𝑒−09	28	8007
12	1/4	Delaunay	□	Volumes	3.0	34.6	11.33	8560	3	32	1070	7𝑒−09	3𝑒−09	2𝑒−09	39	10895
13	1/4	Delaunay	□	Volumes	2.0	30.8	11.31	15576	3	32	1947	1𝑒−08	7𝑒−09	3𝑒−09	54	15645
14	1/4	Delaunay	□	Volumes	1.0	17.7	11.22	60774	3	32	7596	8𝑒−09	4𝑒−09	3𝑒−09	167	50115
15	1/4	Delaunay	□	Volumes	0.5	19.9	11.20	244936	3	32	30617	4𝑒−09	2𝑒−09	2𝑒−09	698	215678
16	1/4	Delaunay	□	Elements	4.0	19.6	11.00	5222	3	32	652	3𝑒−08	1𝑒−08	8𝑒−09	47	13098
17	1/4	Delaunay	□	Elements	3.0	12.3	11.07	8810	3	32	1101	2𝑒−08	9𝑒−09	7𝑒−09	69	18598
18	1/4	Delaunay	□	Elements	2.0	9.0	11.12	15922	3	32	1990	1𝑒−08	5𝑒−09	5𝑒−09	107	30591
19	1/4	Delaunay	□	Elements	1.0	6.3	11.16	61456	3	32	7682	5𝑒−09	2𝑒−09	4𝑒−09	389	113237
20	1/4	Delaunay	□	Elements	0.5	5.8	11.17	246332	3	32	30791	8𝑒−10	4𝑒−10	1𝑒−09	1676	498192
21	1/4	Delquad	▵	Volumes	4.0	−4.5	11.44	6228	3	32	778	4𝑒−08	2𝑒−08	7𝑒−09	30	8495
22	1/4	Delquad	▵	Volumes	3.0	57.0	11.53	11820	3	32	1477	3𝑒−08	1𝑒−08	4𝑒−09	40	10879
23	1/4	Delquad	▵	Volumes	2.0	45.9	11.03	24100	3	32	3012	2𝑒−08	1𝑒−08	5𝑒−09	62	17715
24	1/4	Delquad	▵	Volumes	1.0	51.2	11.04	96400	3	32	12050	2𝑒−08	1𝑒−08	9𝑒−09	207	61714
25	1/4	Delquad	▵	Volumes	0.5	52.8	11.06	385600	3	32	48200	2𝑒−09	1𝑒−09	2𝑒−09	838	255735
26	1/4	Delquad	▵	Elements	4.0	22.0	10.93	3288	3	32	411	3𝑒−08	2𝑒−08	6𝑒−09	30	8495
27	1/4	Delquad	▵	Elements	3.0	14.0	11.04	6148	3	32	768	4𝑒−08	2𝑒−08	8𝑒−09	39	11000
28	1/4	Delquad	▵	Elements	2.0	8.2	11.10	12392	3	32	1549	2𝑒−08	9𝑒−09	6𝑒−09	61	17067
29	1/4	Delquad	▵	Elements	1.0	6.3	11.15	48882	3	32	6110	2𝑒−09	1𝑒−09	1𝑒−09	197	58046
30	1/4	Delquad	▵	Elements	0.5	5.8	11.16	194162	3	32	24270	2𝑒−09	9𝑒−10	2𝑒−09	790	238769
31	1/4	Delquad	□	Volumes	4.0	−41.6	11.74	3132	3	32	391	5𝑒−08	3𝑒−08	8𝑒−09	26	7322
32	1/4	Delquad	□	Volumes	3.0	−24.8	11.51	5922	3	32	740	9𝑒−09	4𝑒−09	2𝑒−09	31	8779
33	1/4	Delquad	□	Volumes	2.0	−13.2	11.39	12050	3	32	1506	1𝑒−08	7𝑒−09	4𝑒−09	44	12239
34	1/4	Delquad	□	Volumes	1.0	−4.9	11.26	48200	3	32	6025	6𝑒−09	3𝑒−09	2𝑒−09	122	36094
35	1/4	Delquad	□	Volumes	0.5	−2.3	11.23	192800	3	32	24100	5𝑒−09	2𝑒−09	3𝑒−09	464	140228
36	1/4	Delquad	□	Elements	4.0	22.4	10.93	3294	3	32	411	3𝑒−08	1𝑒−08	7𝑒−09	35	9852
37	1/4	Delquad	□	Elements	3.0	14.1	11.04	6144	3	32	768	3𝑒−08	2𝑒−08	9𝑒−09	51	14018
38	1/4	Delquad	□	Elements	2.0	9.2	11.11	12392	3	32	1549	1𝑒−08	6𝑒−09	5𝑒−09	81	22526
39	1/4	Delquad	□	Elements	1.0	6.5	11.15	48882	3	32	6110	5𝑒−09	3𝑒−09	4𝑒−09	276	78431
40	1/4	Delquad	□	Elements	0.5	5.9	11.17	194162	3	32	24270	1𝑒−09	5𝑒−10	1𝑒−09	1104	318967
41	1/8	Delaunay	▵	Volumes	4.0	−9.9	11.39	4228	3	32	528	4𝑒−12	2𝑒−12	1𝑒−12	35	9630
42	1/8	Delaunay	▵	Volumes	3.0	5.0	11.32	7712	3	32	964	1𝑒−11	6𝑒−12	2𝑒−12	42	11578
43	1/8	Delaunay	▵	Volumes	2.0	17.7	11.20	15776	3	32	1972	2𝑒−12	1𝑒−12	7𝑒−13	55	15436
44	1/8	Delaunay	▵	Volumes	1.0	14.2	11.18	63902	3	32	7987	3𝑒−12	1𝑒−12	1𝑒−12	160	46691
45	1/8	Delaunay	▵	Volumes	0.5	17.6	11.15	254612	3	32	31826	2𝑒−12	8𝑒−13	1𝑒−12	555	168526
46	1/8	Delaunay	▵	Elements	4.0	18.3	10.96	2252	3	32	281	6𝑒−12	3𝑒−12	2𝑒−12	35	9589
47	1/8	Delaunay	▵	Elements	3.0	11.9	11.04	4040	2	24	505	2𝑒−08	1𝑒−08	7𝑒−09	41	11437
48	1/8	Delaunay	▵	Elements	2.0	8.5	11.08	8156	3	32	1019	2𝑒−12	8𝑒−13	6𝑒−13	55	15053
49	1/8	Delaunay	▵	Elements	1.0	6.3	11.12	32480	3	32	4060	1𝑒−12	6𝑒−13	8𝑒−13	151	43671
50	1/8	Delaunay	▵	Elements	0.5	5.8	11.13	128358	3	32	16044	7𝑒−13	3𝑒−13	7𝑒−13	527	157581
51	1/8	Delaunay	□	Volumes	4.0	3.7	11.44	2380	2	24	297	5𝑒−08	2𝑒−08	1𝑒−08	53	14173
52	1/8	Delaunay	□	Volumes	3.0	21.8	11.20	4194	3	32	524	7𝑒−12	4𝑒−12	2𝑒−12	36	10034
53	1/8	Delaunay	□	Volumes	2.0	34.5	11.10	7960	3	32	995	5𝑒−12	3𝑒−12	1𝑒−12	47	12879
54	1/8	Delaunay	□	Volumes	1.0	16.5	11.17	30652	3	32	3831	2𝑒−12	1𝑒−12	7𝑒−13	100	28851
55	1/8	Delaunay	□	Volumes	0.5	19.1	11.17	122066	2	24	15258	2𝑒−08	8𝑒−09	9𝑒−09	343	103825
56	1/8	Delaunay	□	Elements	4.0	17.5	10.97	2526	3	32	315	4𝑒−12	2𝑒−12	1𝑒−12	60	16159
57	1/8	Delaunay	□	Elements	3.0	11.4	11.04	4386	3	32	548	4𝑒−12	2𝑒−12	2𝑒−12	50	13768
58	1/8	Delaunay	□	Elements	2.0	9.0	11.08	8234	3	32	1029	2𝑒−12	1𝑒−12	1𝑒−12	73	20094
59	1/8	Delaunay	□	Elements	1.0	6.2	11.12	31186	2	24	3898	2𝑒−08	7𝑒−09	9𝑒−09	208	59589
60	1/8	Delaunay	□	Elements	0.5	5.8	11.13	123122	2	24	15390	9𝑒−09	5𝑒−09	1𝑒−08	807	236670
61	1/8	Delquad	▵	Volumes	4.0	−21.0	11.68	3224	3	32	403	1𝑒−11	5𝑒−12	2𝑒−12	43	11784
62	1/8	Delquad	▵	Volumes	3.0	36.9	11.50	6000	3	32	750	1𝑒−11	6𝑒−12	3𝑒−12	50	13502
63	1/8	Delquad	▵	Volumes	2.0	31.2	11.04	12316	3	32	1539	2𝑒−11	8𝑒−12	5𝑒−12	60	16395
64	1/8	Delquad	▵	Volumes	1.0	44.0	10.94	48714	3	32	6089	9𝑒−12	4𝑒−12	4𝑒−12	118	34441
65	1/8	Delquad	▵	Volumes	0.5	53.4	10.81	193980	3	32	24247	5𝑒−12	2𝑒−12	4𝑒−12	423	126768
66	1/8	Delquad	▵	Elements	4.0	24.0	10.90	1750	3	32	218	9𝑒−12	4𝑒−12	2𝑒−12	43	11786
67	1/8	Delquad	▵	Elements	3.0	15.0	11.00	3184	3	32	398	1𝑒−11	5𝑒−12	2𝑒−12	50	13509
68	1/8	Delquad	▵	Elements	2.0	8.9	11.06	6426	3	32	803	5𝑒−12	3𝑒−12	2𝑒−12	59	16047
69	1/8	Delquad	▵	Elements	1.0	6.4	11.11	24886	3	32	3110	2𝑒−12	1𝑒−12	1𝑒−12	113	32788
70	1/8	Delquad	▵	Elements	0.5	5.9	11.13	98052	2	24	12256	1𝑒−08	5𝑒−09	8𝑒−09	398	118153
71	1/8	Delquad	□	Volumes	4.0	−39.8	11.72	1650	3	32	206	1𝑒−11	5𝑒−12	2𝑒−12	36	9907
72	1/8	Delquad	□	Volumes	3.0	−24.9	11.46	3058	3	32	382	4𝑒−12	2𝑒−12	1𝑒−12	34	9349
73	1/8	Delquad	□	Volumes	2.0	−13.1	11.36	6228	2	24	778	3𝑒−08	1𝑒−08	5𝑒−09	40	10930
74	1/8	Delquad	□	Volumes	1.0	−5.1	11.23	24536	2	24	3067	3𝑒−08	2𝑒−08	1𝑒−08	78	22716
76	1/8	Delquad	□	Elements	4.0	23.7	10.90	1752	3	32	219	6𝑒−12	3𝑒−12	2𝑒−12	41	11213
77	1/8	Delquad	□	Elements	3.0	14.5	11.01	3184	3	32	398	6𝑒−12	3𝑒−12	2𝑒−12	43	11811
78	1/8	Delquad	□	Elements	2.0	9.4	11.07	6426	3	32	803	3𝑒−12	1𝑒−12	1𝑒−12	60	16220
79	1/8	Delquad	□	Elements	1.0	6.5	11.11	24874	3	32	3109	1𝑒−12	6𝑒−13	8𝑒−13	156	44000
80	1/8	Delquad	□	Elements	0.5	5.9	11.13	98042	2	24	12255	8𝑒−09	4𝑒−09	8𝑒−09	567	162168