Speeding up Composite Differential Evolution for structural optimization using neural networks

Figures & data

Figure 1. Illustration of CoDE and C²oDE. (a) CoDE, (b) C²oDE.

Figure 2. Typical architecture of Feed-Forward Neural Networks.

Figure 3. Workflow of surrogate assisted Composite Differential Evolution (SA-CoDE).

Figure 4. 10-bar truss.

Figure 5. 25-bar truss.

Table 1. Allowable stresses for 25-bar truss.

Group	Member	Allowable stress for compression (ksi)	Allowable stress for tension (ksi)
1	1	35.092	40.0
2	2, 3, 4, 5	11.590	40.0
3	6, 7, 8, 9	17.305	40.0
4	10, 11	35.092	40.0
5	12, 13	35.092	40.0
6	14, 15, 16, 17	6.759	40.0
7	18, 19, 20, 21	6.959	40.0
8	22, 23, 24, 25	11.082	40.0

Table 2. Load cases for 25-bar truss.

Node	Case 1			Case 2
	PX	PY	PZ	PX	PY	PZ
1	–	20.0	−5.0	1.0	10.0	−5.0
2	–	−20.0	−5.0	–	10.0	−5.0
3	–	–	–	0.5	–	–
6	–	–	–	0.5	–	–

Figure 6. 72-bar truss.

Table 3. Member group for the 72-bar truss.

Group	Member	Group	Member	Group	Member	Group	Member
1	1–4	5	19–22	9	37–40	13	55–58
2	5–12	6	23–30	10	41–48	14	59–66
3	13–16	7	31–34	11	49–52	15	67–70
4	17, 18	8	35, 36	12	53, 54	16	71, 72

Table 4. Load cases for the 72-bar truss.

Node	Load Case 1 (kips)			Load Case 2 (kips)
	PX	PY	PZ	PX	PY	PZ
17	5.0	5.0	−5.0	–	–	−5.0
18	–	–	–	–	–	−5.0
19	–	–	–	–	–	−5.0
20	–	–	–	–	–	−5.0

Figure 7. Convergence histories of the CoDE and the SA-CoDE for the 10-bar truss.

Figure 8. Convergence histories of the CoDE and the SA-CoDE for the 25-bar truss.

Figure 9. Convergence historyies of the CoDE and the SA-CoDE for the 72-bar truss.

Table 5. Results for the 10-bar truss.

Member	TLBO (Camp & Farshchin, 2014)	ODE-NNC (Pham, 2016)	DE(20)	DE(30)	CoDE	SA-CoDE
1 (in^2)	30.668	30.534	30.585	30.522	30.522	30.522
2 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
3 (in^2)	23.158	23.211	23.107	23.202	23.200	23.197
4 (in^2)	15.223	15.228	15.164	15.225	15.223	15.226
5 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
6 (in^2)	0.542	0.552	0.560	0.551	0.551	0.551
7 (in^2)	7.465	7.457	7.470	7.457	7.457	7.457
8 (in^2)	21.026	21.036	21.062	21.036	21.036	21.037
9 (in^2)	21.466	21.507	21.548	21.526	21.528	21.528
10 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
Best (lb)	5060.973	5060.857	5060.890	5060.854	5060.854	5060.854
Worst (lb)	–	–	5079.637	5076.672	5076.669	5076.669
Mean (lb)	5064.808	5060.892	5065.144	5063.414	5061.385	5061.964
SD	6.371	0.035	5.907	5.590	2.838	4.002
Constraint violation	None	None	None	None	None	None

Table 6. Results for the 25-bar truss.

Group	TLBO (Camp & Farshchin, 2014)	ODE-NNC (Pham, 2016)	DE(20)	DE(30)	CoDE	SA-CoDE
1 (in^2)	0.0100	0.010	0.010	0.010	0.010	0.010
2 (in^2)	1.9878	1.987	1.987	1.987	1.987	1.987
3 (in^2)	2.9914	2.994	2.993	2.993	2.993	2.993
4 (in^2)	0.0102	0.010	0.010	0.010	0.010	0.010
5 (in^2)	0.0100	0.010	0.010	0.010	0.010	0.010
6 (in^2)	0.6828	0.684	0.684	0.684	0.684	0.684
7 (in^2)	1.6775	1.677	1.677	1.677	1.677	1.677
8 (in^2)	2.6640	2.663	2.662	2.662	2.662	2.662
Best (lb)	545.175	545.163	545.163	545.163	545.163	545.163
Worst (lb)	–	545.165	546.807	545.271	545.163	545.163
Mean (lb)	545.483	–	545.274	545.167	545.163	545.163
SD	0.306	0.003	0.349	0.022	0.000	0.000
Constraint violation	None	None	None	None	None	None

Table 7. Results for the 72-bar truss.

Group	TLBO (Camp & Farshchin, 2014)	ODE-NNC (Pham, 2016)	DE(20)	DE(30)	CoDE	SA-CoDE
1 (in^2)	1.881	1.885	1.892	1.893	1.887	1.877
2 (in^2)	0.514	0.514	0.511	0.513	0.511	0.518
3 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
4 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
5 (in^2)	1.271	1.267	1.281	1.262	1.271	1.270
6 (in^2)	0.515	0.514	0.516	0.512	0.513	0.514
7 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
8 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
9 (in^2)	0.532	0.525	0.527	0.525	0.523	0.516
10 (in^2)	0.513	0.515	0.509	0.515	0.517	0.516
11 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
12 (in^2)	0.100	0.100	0.100	0.100	0.100	0.100
13 (in^2)	0.157	0.157	0.156	0.156	0.156	0.157
14 (in^2)	0.543	0.545	0.547	0.545	0.545	0.545
15 (in^2)	0.408	0.410	0.408	0.411	0.412	0.407
16 (in^2)	0.573	0.569	0.573	0.570	0.568	0.569
Best (lb)	379.632	379.618	379.639	379.617	379.617	379.623
Worst (lb)	–	–	380.430	379.907	379.662	382.306
Mean (lb)	379.759	379.642	379.888	379.667	379.631	379.860
SD	0.149	0.024	0.216	0.057	0.013	0.545
Constraint violation	None	None	None	None	None	None

Table 8. Required number of exact fitness evaluations.

Approach	10-bar truss	25-bar truss	72-bar truss
TLBO	13,767	12,199	21,542
ODE-NNC	7000	5000	10,000
DE(20)	10,000	10,000	10,000
DE(30)	15,000	15,000	15,000
CoDE	29,960	29,960	29,960
SA-CoDE	11,960	11,960	11,960

Table 9. Influence of the training data size.

Number of generations of the first stage iter1	10	25	50	100
Number of training samples	560	1460	2960	5960
Best (lb)	5060.855	5060.854	5060.854	5060.854
Mean (lb)	5063.275	5061.921	5061.964	5062.327
Worst (lb)	5076.670	5076.669	5076.669	5076.669
SD (lb)	4.999	3.942	4.002	4.419
Number of exact fitness evaluations	10,360	10,960	11,960	13,960

Camp, C. V., & Farshchin, M. (2014). Design of space trusses using modified teaching–learning based optimization. Engineering Structures, 62-63, 87–97. https://doi.org/https://doi.org/10.1016/j.engstruct.2014.01.020

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Pham, H. A. (2016). Truss sizing optimization using enhanced differential evolution with opposition-based mutation and nearest neighbor comparison. Journal of Science and Technology in Civil Engineering (STCE)-NUCE, 10(5), 3–10.

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