49 7 ( ) Vol.49 No.7 2018 7 Journal of Central South University (Science and Technology) July 2018

DOI: 10.11817/j.issn.1672-7207.2018.07.011

SVR

( 610065)

(SVR)SVR

NSGA-45 t 5

15.9%9.2%

TH213 A 1672�7207(2018)07�1650�07

Application of SVR response surface coupling evolutionary multi-objective optimization algorithm to structural optimization

TIAN Kun, HU Xiaobing, ZHAO Qingxiang, XU Yingli

(School of Manufacturing Science and Engineering, Sichuan University, Chengdu 610065, China)

Abstract: Aiming at the low efficiency and low precision in complex structure optimization design, an optimization system with support vector regression(SVR) response surface coupled with evolutionary multi-objective optimization algorithm was proposed. Based on structural risk minimization, the principle of SVR response surface was deduced. Orthogonal rotation combination design with contrasting predictive value was used to select sampling points to obtain the optimal experimental area. Based on the principle of NSGA- paradigm, the evolutionary optimization algorithm of interval preference was established and the optimization system framework was constructed. Taking the main girder of 45 t gantry crane as the research object, five geometrical parameters were adopted as design variables, and the maximum displacement and stress were constrained. The results show that by using the optimized system, the total mass of main girder and the first-order natural frequency are decreased by 15.9% and 9.2%, respectively. The high efficiency and feasibility of the optimization system are proved by the comparison of the results of different response surface models, the sensitivity analysis and the optimization scheme validation. Key words: support vector regression response surface; evolutionary multi-objective optimization algorithm; orthogonal rotation combination design; optimization system

( ) ( ) 1)

2017�07�16 2017�09�12 (Foundation item) (2015GZ0014 2016GZ0169 2017GZ0146) (Projects(2015GZ0014, 2016GZ0169, 2017GZ0146) supported by the Science and Technology Program of Sichuan Province)

CAD/CAPP/CAM E-mail: huxb@scu.edu.cn

7 SVR 1651

2) 3)

Pareto

ZHONG[1] Kriging

GOUDARZI [2]

QIAN [3]

SUN [4]

SVR

SVR

1 SVR 1.1 SVR

[5] (SVR)V-apnik

[6]

)(x,yP ) ,( RR �� yx n

,),,( 11 �yx )(),( RR �� nnn yx SVR

bxwxf i ��� )()( �

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)( ii xfy

y )(xf

1(a)

1(a) 1(b) �� � )(xfy

(a) (b) (x, y)

1 (x, y)

Fig. 1 belt and (x, y) loss values

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[9�10]

(quadratic regression orthogonal rotational combination design)[7]

mn 0rc mmmn ��� mc

m mr

r m0 0(mc mm 2r � r mc )

1 m[11]

1

Table 1 Parameters table of quadratic regression orthogonal

rotational combination design

m mc mr r n m0

2 4 4 1.414 16 8

3 8 6 1.682 23 9

4(1/2 ) 8 8 1.682 23 7

4 16 8 2.000 36 12

5(1/2 ) 16 10 2.000 36 10

5 32 10 2.378 59 17

6(1/4 ) 16 12 2.000 36 8

2 2.1 NSGA-

ParetoPareto

Pareto

[12]

NSGA- [13]

NSGA-

N NSGA-

Pareto

NSGA-

2.2

[14] NSGA-

1)

2) K-[15]

3) 4)

2

1) 2) 3) K- 4)

5) NSGA-

6)

2)3)

7 SVR 1653

2

Fig. 2 Flow chart of evolutionary optimization algorithm 2.3 SVR

SVR

3

3

Fig. 3 Flow chart of optimization system

3 3.1

4 45 t CAD2( X1

X2 X3 X4

X5 )

(a) (b)

4

Fig. 4 Gantry crane model and cross section of main girder

2

Table 2 Main parameters of gantry cranes

/m

/m

/t

/

t

/

GPa

/

(kg·m�3)

38 22 35 45 2.06 1.89 0.30

),,,,(min 54321 XXXXXG ; 1min� (6)

G 1�

1s.t. [ ]; [ ]f f� � (7)

][� 176.69 MPa ][ f31.43 mm

T54321 ],,,,[ XXXXX�X

3 3.2 SVR

( ) 49 1654

3

Table 3 Range of design variables for gantry crane

X1/mm 1 476 1 300~1 650

X2/mm 8 6~9

X3/mm 8 6~9

X4/mm 12.0 10.5~14.0

X5/mm 12.0 10.5~14.0

5 (5 ) 1 559

4

SVR 5

SVR(CCD)

Kriging ( 5 )

4

Table 4 Test of quadratic regression orthogonal rotation combination design

( )/mm ( ) ( ) X1 X2 X3 X4 X5

/MPa

/ mm

/

Hz /

kg

1 1 475.000 7.500 7.500 12.250 12.250 94.319 27.674 7.136 12 172.504

2 1 475.000 7.500 6.000 12.250 12.250 96.939 28.340 7.205 11 512.516

3 1 475.000 7.500 6.750 12.250 12.250 95.604 28.003 7.170 11 842.510

4 1 475.000 7.500 9.000 12.250 12.250 91.891 27.035 7.070 12 832.493

5 1 475.000 7.500 8.250 12.250 12.250 93.085 27.351 7.102 12 502.499

6 1 475.000 6.000 7.500 12.250 12.250 98.141 28.387 7.191 11 512.516

7 1 475.000 6.750 7.500 12.250 12.250 96.179 28.024 7.164 11 842.510

8 1 475.000 9.000 7.500 12.250 12.250 90.868 27.003 7.078 12 832.493

9 1 475.000 8.250 7.500 12.250 12.250 92.551 27.334 7.107 12 502.499

10 1 475.000 7.500 7.500 12.250 10.500 102.562 29.333 7.028 11 780.986

11 1 475.000 7.500 7.500 12.250 11.375 98.183 28.468 7.084 11 976.745

12 1 475.000 7.500 7.500 12.250 14.000 87.500 26.262 7.226 12 564.023

13 1 475.000 7.500 7.500 12.250 13.125 90.773 26.941 7.183 12 368.264

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46 1 562.500 6.750 8.250 13.125 11.375 91.704 24.599 7.430 12 570.548

47 1 518.750 7.125 7.875 12.688 11.813 93.008 26.064 7.288 12 371.526

48 1 562.500 8.250 6.750 13.125 11.375 90.763 24.561 7.447 12 570.548

49 1 518.750 7.875 7.125 12.688 11.813 92.567 26.045 7.294 12 371.526

50 1 387.500 8.250 8.250 13.125 11.375 103.313 31.340 6.675 12 408.348

51 1 431.250 7.875 7.875 12.688 11.813 98.495 29.405 6.905 12 300.214

52 1 562.500 6.750 6.750 13.125 13.125 86.958 23.854 7.596 12 262.927

53 1 518.750 7.125 7.125 12.688 12.688 90.577 25.662 7.372 12 227.503

54 1 387.500 6.750 8.250 13.125 13.125 98.515 30.347 6.824 12 179.030

55 1 431.250 7.125 7.875 12.688 12.688 96.218 28.951 6.982 12 175.767

56 1 387.500 8.250 6.750 13.125 13.125 97.687 30.308 6.832 12 179.030

57 1 431.250 7.875 7.125 12.688 12.688 95.808 28.932 6.986 12 175.767

58 1 562.500 8.250 8.250 13.125 13.125 81.472 22.731 7.483 13 661.208

59 1 518.750 7.875 7.875 12.688 12.688 87.628 25.048 7.311 12 907.068

7 SVR 1655

(a) (b) (c) (d)

5 Fig. 5 Responses of design variables to outputs

5

Table 5 Fitting accuracy and optimization results of different response surfaces

/s

/%

/%

/Hz

/kg

127 97.574 94.557 7.407 12 568.340

Kriging 158 98.965 96.533 7.156 11 326.765

SVR 122 99.998 97.660 6.997 10 514.376

(

) SVR

3.3

6

X1 X5 X4

( ) ( ) X3 X2 ( )( )

6

Fig. 6 Analysis results of outputs sensitivity 3.4

SVR

6 112 503.5 kg

7.705 Hz15.9%

9.2% 1

CAD ANSYS7

7 107.24 MPa30.406 mm 10.025

6

Table 6 Optimization solutions of main girder

( )/mm ( ) ( ) X1 X2 X3 X4 X5

/MPa

/ mm /

Hz /

kg 1 1 479.885 6.296 6.104 10.620 11.555 104.700 30.386 6.997 10 514.376

2 1 500.128 6.859 6.555 10.672 10.778 104.575 29.709 7.198 10 880.927

3 1 574.199 6.122 6.227 10.585 11.252 99.228 26.818 7.552 10 763.265

4 1 451.991 8.267 6.112 11.397 10.838 104.887 30.859 7.210 11 287.819

( ) 49 1656

(a) (b)

7 Fig. 7 Verification clouds of stress and displacement

4

1) SVR

2) NSGA-

SVR

3) SVR

15.9% 9.2%

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