VELOSO, L.J.P.; ALFARO VIGO, D.G.; ROSSETTO, S.

doi:10.5540/tema.2016.017.01.0093

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Articles • TEMA (São Carlos) 17 (1) • Jan-Apr 2016 • https://doi.org/10.5540/tema.2016.017.01.0093 copy

Melhorando o Desempenho Computacional de um Esquema de Diferenças Finitas para as Equações de Maxwell

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Maxwell's equations play a crucial role in electromagnetic theory and applications. However, it is not always possible to solve these equations analytically. Consequently, we have to use numerical methods in order to get approximate solutions of the Maxwell's equations. The FDTD (Finite-diference Time-Domain) method, proposed by K. Yee, is widely used to solve Maxwell's equations, due to its efficiency and simplicity. However, this method has a high computational cost. In this paper, we propose a parallel implementation of the FDTD method to run on GPUs by using CUDA platform. Our goal is to reduce the processing time required, allowing the use of the FDTD method in the simulation of electromagnetic wave propagation. We evaluate the proposed algorithm considering two different kind of boundary conditions: a Dirichlet type boundary conditions and absorbing boundary conditions. We get a performance gain ranging from 7 to 8 times when comparing the proposed parallel implementation with an optimized sequential version.

Keywords:
Maxwell's equations; Yee algorithm; Parallel Programming with GPU

j	Δtj	Δxj	ǁEǁ∞,j	q (empírico)
1	4 × 10−3	6,35 ×10−2	1,34 ×10−3	−
2	4 × 10−3	3,16 ×10−2	3,22 ×10−4	2.053
3	4 × 10−3	2,10 ×10−2	1,34 ×10−4	2.161
4	4 × 10−3	1,57 ×10−2	6,78 ×10−5	2.347
5	4 × 10−3	1,26 ×10−2	3,74 ×10−5	2.652
6	4 × 10−4	1,05 ×10−2	2,10 ×10−5	3.167
7	4 × 10−4	8,99 ×10−3	1,11 ×10−5	4.136
8	4 × 10−4	7,86 ×10−3	4,46 ×10−6	6.483

	C.C. tipo Dirichlet				PML
	O0	O1	O2	O3	O0	O1	O2	O3
300	2.53	0.74	0.65	0.60	6.17	2,92	2.73	2.27
1500	63.45	25.85	25.71	25.31	94.14	40.47	39.65	38.38
2100	124.62	51.23	50.89	50.43	178.71	74.77	72.51	72.00
2700	205.50	81.73	80.21	79.46	292.79	122.91	121.96	121.06
3300	307.25	122.49	120.46	120.20	429.85	174.33	172.33	170.18
3900	428.98	173.68	171.69	170.24	597.89	250.90	249.52	247.83
4500	570.81	232.64	229.93	228.73	791.64	334.84	327.62	325.17
5100	732.46	298.28	294.58	292.24	1014.02	431.95	427.09	421.52
5700	920.01	389.77	383.73	383.54	1263.45	525.73	515.99	511.94
6300	1117.02	445.81	438.78	438.28	1545.79	633.14	629.19	620.51
6900	1344.99	558.39	552.73	549.94	1874.89	901.37	876.68	882.07

	C.C. tipo Dirichlet				PML
	1×128	1×256	1×512	1×1024	1×128	1×256	1×512	1×1024
300	0,23	0,18	0,20	0,28	0.41	0.27	0.28	0.38
1500	4.24	3.70	3.67	3.99	4.88	4.78	4.94	5.13
2100	7.16	6.99	7.39	8.21	9.33	8.98	9.38	10.34
2700	12.17	11.83	11.64	12.18	16.70	15.32	15.06	15.74
3300	18.47	17.44	17.38	18.51	23.64	22.57	22.27	23.71
3900	24.70	23.82	23.81	24.68	31.54	30.43	30.68	31.93
4500	33.17	31.82	31.75	33.75	43.01	40.99	40.73	43.53
5100	42.52	41.45	40.74	41.96	55.20	53.70	53.61	56.53
5700	53.29	51.80	51.37	53.09	69.20	66.72	65.59	68.41
6300	65.76	63.85	63.39	66.34	83.45	80.80	80.56	85.25
6900	78.20	75.79	74.75	77.26	100.18	97.36	95.63	99.66

	Sequencial(O3)			Cuda(1×512)
	Dirichlet	PML	Dirichlet	PML
Speed[MCells/s]	89,04 (7.03)	61,08 (10.02)	663,13 (42.05)	486,55(42.39)
Speedup	-	-	7,08 (0,64)	7.98 (1,58)

Sociedade Brasileira de Matemática Aplicada e Computacional Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 , 13561-120 São Carlos - SP, Tel. / Fax: (55 16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

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[1] Autor correspondente: Leandro J. Pereira Veloso.