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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (4) : 800-820     https://doi.org/10.1007/s11709-019-0517-7
RESEARCH ARTICLE
Novel decoupled framework for reliability-based design optimization of structures using a robust shifting technique
Mohammad Reza GHASEMI1(), Charles V. CAMP2, Babak DIZANGIAN3
1. Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan 9816745565, Iran
2. Department of Civil Engineering, The University of Memphis, Memphis, TN 38152, USA
3. Department of Civil Engineering, Velayat University, Iranshahr 9911131311, Iran
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Abstract

In a reliability-based design optimization (RBDO), computation of the failure probability (Pf) at all design points through the process may suitably be avoided at the early stages. Thus, to reduce extensive computations of RBDO, one could decouple the optimization and reliability analysis. The present work proposes a new methodology for such a decoupled approach that separates optimization and reliability analysis into two procedures which significantly improve the computational efficiency of the RBDO. This technique is based on the probabilistic sensitivity approach (PSA) on the shifted probability density function. Stochastic variables are separated into two groups of desired and non-desired variables. The three-phase procedure may be summarized as: Phase 1, apply deterministic design optimization based on mean values of random variables; Phase 2, move designs toward a reliable space using PSA and finding a primary reliable optimum point; Phase 3, applying an intelligent self-adaptive procedure based on cubic B-spline interpolation functions until the targeted failure probability is reached. An improved response surface method is used for computation of failure probability. The proposed RBDO approach could significantly reduce the number of analyses required to less than 10% of conventional methods. The computational efficacy of this approach is demonstrated by solving four benchmark truss design problems published in the structural optimization literature.

Keywords reliability-based design optimization      trusses      sensitivity analysis      shifting technique      cubic B-splines      response surface method     
Corresponding Authors: Mohammad Reza GHASEMI   
Online First Date: 04 March 2019    Issue Date: 10 July 2019
 Cite this article:   
Mohammad Reza GHASEMI,Charles V. CAMP,Babak DIZANGIAN. Novel decoupled framework for reliability-based design optimization of structures using a robust shifting technique[J]. Front. Struct. Civ. Eng., 2019, 13(4): 800-820.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0517-7
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I4/800
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Fig.1  Normal and lognormal PDF of Young’s modulus with different values of COV
Fig.2  Reaching procedure from d* to the first point in the safe space x* using PDF shifting technique
Fig.3  Flowchart for the three-phased RBDO approach
Fig.4  Flowchart of proposed PSA and ISAP based truss RBDO approach
Fig.5  10-bar planar truss, example 1
variable distribution unit mean value coefficient of variation (C.V.)
P1 normal kN 60 0.2
P2 normal kN 40 0.2
P3 normal kN 10 0.2
E normal GPa 200 0.1
L normal m 1 0.05
Tab.1  Statistical properties of 10-bar truss problem
optimal cross sectional area (×10−4 m2) cross sectional area (cm2)
Zhao and Qiu
(fmincon-RSM(b))
[29]
Dizangian and Ghasemi
(fmincon-MCS(c))
[32]
current work, PSA-ISAP
Phase 1-DDO Phases 2 and 3-RBDO
optimum d* Phase 2- reliable optimum x* Phase 3-ISAP
NSVi P at f (A)=55.0746 x**( α=0.825)
bar areas A1 10.705 10.000 6.2500 10.90850 0.19440 10.4822
A2 5.914 5.042 3.1250 4.39529 0.13880 4.4210
A3 14.424 14.028 8.6932 16.10410 0.16660 15.6850
A4 1.000 1.000 1.0000 1.00613 0.00000 1.0890
A5 1.000 1.000 1.0000 1.00000 0.00000 1.0000
A6 1.000 1.000 1.0000 1.00000 0.00000 1.0000
A7 5.531 9.000 6.2500 7.92476 0.05550 7.8510
A8 11.853 10.000 6.2500 10.28230 0.19444 10.0480
A9 1.000 1.000 1.0000 1.05930 0.02777 1.1210
A10 11.223 10.000 6.2500 9.93183 0.22220 9.6500
total area (cm2) 63.649 62.068 40.8182 63.6124 62.3472
No. of samples for sensitivity computation -- -- -- 1500
No. of Pf evaluation 1904 6 -- 4 8
No. of FE calls for reliability not given 6×106 258 524
βRSM
Pf (Φ(β))
0.090
(0.4641)
2.588
(0.00482)
2.501
(0.006192)
Pf G (MCS(a)) 0.002742 0.00615 0.00615
Tab.2  RBDO results comparison of 10-bar planar truss example
Fig.6  Convergence history for the 10-bar truss problem (Phase 2)
Fig.7  A close up view of the first optimum reliable design
Fig.8  Final fitted B-spline curves of the 10-bartruss (Phase 2 and 3)
Fig.9  Reliable optimum before and after ISAP on the fitted B-spline curves
item Iter. 1 Iter. 2 Iter. 3 Iter. 4
f(A) of start point of Phase 3 (×10−4) m2 55.0746 55.0746 55.0746 55.0746
extracted f(A) from cubic B-spline fitted curve corresponding to the target Pf (×10−4) 61.78 62.09 62.13 62.3472
modification factor α 0.785 0.8244 0.828 0.825
probability of failure* (RSM) 0.00681 0.00642 0.00640 0.00619
Tab.3  Computational summary of an ISAP, 10-bar planar truss example
Fig.10  Convergence history for the 10-bar truss problem (Phase 2 and 3)
Fig.11  A close-up convergence history for the 10-bar truss problem
Fig.12  13-bar bridge truss
group number variables
1 A1, A12
2 A2, A13
3 A3, A11
4 A4, A8
5 A5, A9
6 A6, A10
7 A7
Tab.4  Member grouping details of 13-bar truss
variable distribution mean value COV
load, P normal 66.726 kN 0.16
yield stress, Fy normal 248.221 MPa 0.12
Tab.5  Statistical properties of random variables, 13-bar truss
optimal cross sectional area (m2) ×10−3 cross sectional area (m2) ×10−3
Nakib and Frangopol
(IPF-ABT(b))
[40]
Ghorbani and Ghasemi
(ANFS-PSO-MCS(c))
[41]
Dizangian and Ghasemi
(fmincon-MCS(d))
[32]
current work, PSA-ISAP
Phase 1-DDO Phase 2 and 3-RBDO Phase 3-ISAP
Optimum
d *
Phase 2-reliable optimum x NSVi Pat
f (A)= 310.818
x**( α=0.711)
variable group A1 0.746 0.751 0.7299 0.4514 0.1000 0.16 0.7540
A2 1.219 1.191 1.2041 0.7307 1.1129 0.16 1.1678
A3 0.753 0.726 0.6723 0.4165 0.1965 0.24 0.7251
A4 0.745 0.741 0.7299 0.4236 0.3912 0.08 0.7125
A5 0.227 0.223 0.1300 0.0936 0.3675 0.16 0.1577
A6 0.840 0.810 1.0023 0.5142 1.0574 0.08 0.8329
A7 0.522 0.516 0.4774 0.2916 0.1000 0.12 0.4505
truss Mass (kg) 367.11 359.70 353.0202 210.225 356.782 347.536
No. of samples for sensitivity computation -- -- 1,400
No. of Pf evaluation 636 230 5 -- 5 9
No. of FE calls for reliability Not given Not given 5×107 394 712
βRSM
Pf (Φ (β))
1.97
(0.0244)
4.34
(7.124×10−6)
4.2653
(9.9817×10−6)
PfG (MCS(a)) 6.4×10−6 9.7×10−6 9.9×10−6 8.5×10−6
Tab.6  A comparison of RBDO results for the 13-bar bridge truss
Fig.13  Logarithmic convergence history for the 13-bar truss problem (Phase 2)
Fig.14  A close up view of the first optimum reliable design
Fig.15  Logarithmic fitted B-spline curves of 13-bar truss (Phase 2 and 3)
Fig.16  Reliable optimum before and after ISAP on the fitted B-spline curves
Fig.17  Convergence history for the 13-bar truss problem (Phase 2 and 3)
Fig.18  A close-up Convergence history for the 13-bar truss problem
Iter. 1 Iter. 2 Iter. 3 Iter. 4
f(A) of start point of Phase 3 (×10−3 kg) 310.818 310.818 310.818 310.818
extracted f(A) from cubic B-spline fitted curve corresponding to the target Pf (×10−3) 350.170 348.500 347.750 347.536
modification factor α 0.820 0.755 0.721 0.711
probability of failure* (RSM) 8.67×10−6 9.58×10−6 9.66×10−6 9.9817×10−6
Tab.7  Computational properties of an ISAP, 13-bar truss
Fig.19  The 72-bar space truss structure
variable distribution mean value coefficient of variation (C.V.)
Ai normal to be determined 0.05
E lognormal 104 ksi (68947.57 MPa ) 0.05
Px lognormal 5 kips (22.2411 kN) 0.1
Py lognormal 5 kips (22.2411 kN) 0.1
Pz lognormal −5 kips (−22.2411 kN) 0.1
Tab.8  Statistical properties of random variables, 72-bar space truss example
optimal cross sectional area (in2) cross sectional area (in2)
Shayanfar et al.
(GA-FORM(d)) [42]
current work, PSA-ISAP
Phase 1-DDO Phase 2 and 3-RBDO
optimum
d*
Phase 2- reliable optimum x* Phase 3-ISAP
NSVi Pat f (A)=532.63 x**( α=0.8)
variable group 1( A1 4) 0.1 (a) 0.1000 0.1000 0.00 0.1
2( A5 12) 0.9 0.7221 1.1129 0.08 1.1
3( A13 16) 0.3 0.1265 0.1965 0.21 0.3
4( A17,18) 0.5 0.3883 0.3912 0.00 0.5
5( A19 22) 1.1 0.3312 0.3675 0.18 0.3
6( A23 30) 0.9 0.6294 1.0574 0.17 1.1
7( A31 34) 0.1 0.1000 0.1000 0.00 0.1
8( A35,36) 0.1 0.1000 0.1218 0.00 0.1
9( A37 40) 1.5 0.7612 1.3747 0.10 1.3
10 ( A4148) 0.5 0.3930 0.7300 0.26 0.7
11 ( A4952) 0.3 0.1000 0.1000 0.00 0.1
12 ( A53,54) 0.3 0.1000 0.1000 0.00 0.1
13 ( A5558) 2.3 1.2754 2.5403 0.00 2.3
14 ( A5966) 0.9 0.5354 0.7248 0.00 0.7
15 ( A6770) 0.1 0.1000 0.1000 0.00 0.1
16 ( A71,72) 0.1 0.1000 0.1000 0.00 0.1
truss weight (b) (lb) 535.79 347.79 542.33 538.3454
No. of samples for sensitivity computation -- -- 4800
No. of Pf evaluations* 200 -- 14 20
No. of FE calls for reliability* 1008 1440
β G1RSM
Pf (Φ( β G1RSM))
3.18
(0.00081)
3.011
(0.00130)
β G2RSM
Pf (Φ( β G2RSM))
3.18
(0.00081)
3.011
(0.00130)
Pf G1 (MCS(c)) 0.00163 0.00101
Pf G2(MCS(c)) 0.00163 0.0010
Tab.9  RBDO results comparison of 72-bar space truss example
Fig.20  Logarithmic convergence history for the 72-bar truss problem (Phase 2)
Fig.21  A close up view of the first optimum reliable design
Fig.22  Logarithmic fitted B-spline curves of 72-bar truss (Phase 2 and 3)
Fig.23  Reliable optimum before and after ISAP on the fitted B-spline curves
Fig.24  Convergence history for 72-bar truss problem (Phase 2 and 3)
Fig.25  A close-up Convergence history for the 72-bar truss problem
item Iter. 1 Iter. 2 Iter. 3
f (A) of start point of Phase 3-lb(a) 532.63 532.63 532.63
extracted f (A) from cubic B-spline fitted curve corresponding to the target Pf 539.1 537.21 538.3454
modification factor α 0.79 0.778 0.8
probability of failure* (RSM) 0.00125 0.00128 0.00130
Tab.10  Computational summary of an intelligent self-adaptive procedure (Phase 3), 72-bar truss example
Fig.26  A 25-Bar space truss
node x y z
1 1.0 −10.0 −10.0
2 0 −10.0 −10.0
3 0.5 0 0
6 0.6 0 0
Tab.11  Loading condition (kips) for 25-bar space truss
optimal cross sectional area (in2) cross sectional area (in2)
Ho-Huu et al.
(SORA-ICDE)
[43]
Dizangian and Ghasemi
(DAF-RSM)
[33]
current work, PSA-ISAP
Phase 1-DDO Phase 2 and 3-RBDO
optimum
d*
Phase 2-reliable optimum x* Phase 3-ISAP
NSVi P at f (A)=600.94 x **( α=0.8)
variablegroup 1( A1) 0.1(a) 0.10 0.1 0.1 0.00 0.1
2( A2 5) 2.0 0.10 0.6 1.3 0.00 1.3
3( A6 9) 3.4 4.71 3.2 3.4 0.22 3.4
4( A10,11) 0.1 0.10 0.1 0.1 0.00 0.1
5( A12,13) 0.1 0.10 0.1 0.1 0.00 0.1
6( A14 17) 1.2 0.86 0.9 1.3 0.25 1.1
7( A18 21) 1.9 2.71 1.7 2.8 0.30 2.5
8( A22 25) 3.4 3.57 3.0 3.4 0.23 3.4
truss Mass (lb)(b) 659.52 660.804 520.32 695.44 659.22
No. of samples for sensitivity computation -- -- 800
No. of Pf evaluation not given 7 -- 4 7
No. of FE calls for reliability 25841 7×80 328 583
βRSM
Pf (Φ(β))
0.31
(0.495)
3.207(0.00067) 3.002
(0.00134)
Pf G (MCS(c)) 0.0023 0.00132 0.00129
Tab.12  RBDO results comparison of 25-bar space truss example
Fig.27  Logarithmic convergence history for the 25-bar truss problem (Phase 2)
Fig.28  Logarithmic fitted B-spline curves of 25-bar truss (Phase 2 and 3)
Fig.29  Reliable optimum before and after ISAP on the fitted B-spline curves
Fig.30  Convergence history for 25-bar truss problem (Phase 2 and 3)
Fig.31  A close-up convergence history for the 25-bar truss problem
Iter. 1 Iter. 2 Iter. 3
f(A) of start point of Phase 3-lb(a) 600.94 600.94 600.94
extracted f(A) from cubic B-spline fitted curve corresponding to the target Pf 680.95 673.71 659.22
modification factor α 0.95 0.9 0.8
probability of failure* (RSM) 0.001 0.00122 0.00134
Tab.13  Computational properties of an ISAP, 25-bar space truss
Phase 1 (DDO) Phase 2
(shifting technique)
Phase 3
(ISAP)
No. of D.Vs targeted failure probability FE calls for optimization FE calls for sensitivity FE calls for reliability FE calls for reliability total No. of FE calls
10-bar planar truss 10 6.21×10−3 1350 1500 258 266 3374
13-bar planar
truss
7 1×10−5 1050 1400 394 318 3162
72-bar space
truss
16 1.35×10−3 3200 4800 1008 432 9440
25-bar space
truss
8 1.35×10−3 950 800 328 255 2333
Tab.14  Summary of computational cost of four RBDO truss problems
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