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Frontiers of Structural and Civil Engineering

ISSN 2095-2430

ISSN 2095-2449(Online)

CN 10-1023/X

Postal Subscription Code 80-968

2018 Impact Factor: 1.272

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (5) : 1150-1170    https://doi.org/10.1007/s11709-019-0543-5
RESEARCH ARTICLE
Optimal dome design considering member-related design constraints
Tugrul TALASLIOGLU()
Department of Civil Engineering, Osmaniye Korkut Ata University, Osmaniye 8000, Turkey
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Abstract

This study proposes to optimize the design of geometrically nonlinear dome structures. A new Multi-objective Optimization Algorithm named Pareto Archived Genetic Algorithm (PAGA), which has an ability of integrating the nonlinear structural analysis with the provisions of American Petroleum Institute specification is employed to optimize the design of ellipse and sphere-shaped dome configurations. Thus, it is possible to investigate how the qualities of optimal designations vary considering the shape, size, and topology-related design variables. Furthermore, the computing efficiency of PAGA is evaluated considering six multi-objective optimization algorithms and eight quality measuring indicators. It is shown that PAGA has a capability of both exploring an increased number of pareto solutions and predicting a pareto front with a higher convergence degree. Moreover, the inclusion of shape-related design variables leads to a decrease in both the weights of dome structures and their load-carrying capacities. However, the designer easily determines the most requested optimal design through the archiving feature of PAGA. Thus, it is also demonstrated that the proposed optimal design procedure increases the correctness degree in the evaluation of optimal dome designs through the tradeoff analysis. Consequently, PAGA is recommended as an optimization tool for the design optimization of geometrically nonlinear dome structures.

Keywords dome structure      geometric nonlinearity      multi-objective optimization      API RP2A-LRFD     
Corresponding Author(s): Tugrul TALASLIOGLU   
Just Accepted Date: 10 May 2019   Online First Date: 19 June 2019    Issue Date: 11 September 2019
 Cite this article:   
Tugrul TALASLIOGLU. Optimal dome design considering member-related design constraints[J]. Front. Struct. Civ. Eng., 2019, 13(5): 1150-1170.
 URL:  
https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0543-5
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I5/1150
Fig.1  (a) Top and (b) side view of dome structure used for a conceptual model
member cross-sectional properties geometrical configuration 1 geometrical configuration 2 geometrical configuration 3
cross-sectional properties of longitudinally arched members same different different
cross-sectional properties of horizontally arched members different different different
cross-sectional properties of diagonal members not included different not included
structural analysis-related parameter names structural analysis related parameter values
convergence tolerance (see CNVTOL*) 0.00001
load step (see NSUBST*) 500
Arc-length mult. (see ARCLEN*) 50
Tab.1  Three geometrical configurations used to arrangement of arched and diagonal members for sphere and ellipse shaped dome structures
design variable names item design variable values
varying design variables fixed design variables
size-related design variables ParDV ParDVL = 1<<ParDVU = 37
shaper-related design variables ParSDVx ParSDVxL = 19 m<<ParSDVxU = 21 m ParSDVxL = ParSDVxU = 20 m
ParSDVy ParSDVyL = 19 m<<ParSDVyU = 21 m ParSDVyL = ParSDVyU = 20 m
ParSDVz ParSDVzL = 19 m<<ParSDVzU = 21 m ParSDVzL = ParSDVzU = 20 m
topology-related design variables ParLDN ParLDNL = 2<<ParLDNU = 5
ParHDN ParHDNL = 2<<ParHDNU = 5
Tab.2  Design variables and values governed the proposed optimal design approach
structural analysis-related parameter names structural analysis related parameter values
convergence tolerance (see CNVTOL*)
load step (see NSUBST*)
Arc-length mult. (see ARCLEN*)
0.00001
500
50
Tab.3  Structural analysis-related variables and values governed the proposed optimal design approach
Fig.2  Visualization of sub-populations utilized by PAGA
Fig.3  Flow chart for PAGA
Fig.4  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark function ZDT1)
Fig.5  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark function DTLZ2)
Fig.6  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark function DTLZ7)
Fig.7  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark truss example with 756 bars)
Fig.8  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark truss example with 200 bars)
Fig.9  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark truss example with 4 bars)
Fig.10  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark truss example with 2 bars (see Ref. [35]))
Fig.11  (a) True pareto front and (b) pareto fronts obtained by employed MEAs (benchmark truss example with 2 bars (see Ref. [36]))
names of MEAS DM GD HV IGD NHV PD spacing spread runtime
benchmark function: ZDT1 NSGAII 0.73 0.83 0.00 2.24 0.00 1605.30 0.20 0.90 0.00
SMSEMOA 0.71 0.85 0.00 2.29 0.00 1609.45 0.20 0.91 0.00
MOEAD 0.68 0.92 0.00 2.25 0.00 1608.48 0.22 0.88 0.00
PESAII 0.70 0.86 0.00 2.24 0.00 1664.76 0.20 0.90 0.00
SPEA2 0.71 0.86 0.00 2.23 0.00 1600.40 0.18 0.87 0.01
eMOEA 0.78 0.94 0.00 2.31 0.00 1304.88 0.19 0.92 0.00
PAGA 0.65 0.00 0.83 0.03 0.68 1828.92 0.01 0.61 0.81
benchmark function: DTLZ2 NSGAII 0.53 0.16 0.01 0.40 0.01 1901.66 0.13 0.71 0.00
SMSEMOA 0.49 0.17 0.01 0.43 0.01 1969.63 0.15 0.76 0.00
MOEAD 0.53 0.16 0.01 0.42 0.01 2130.83 0.13 0.73 0.00
PESAII 0.48 0.17 0.01 0.42 0.01 1917.34 0.14 0.73 0.00
SPEA2 0.49 0.16 0.01 0.42 0.01 1980.29 0.17 0.74 0.01
eMOEA 0.57 0.18 0.02 0.41 0.02 1743.31 0.17 0.73 0.00
PAGA 0.47 0.00 0.38 0.03 0.31 2346.35 0.02 1.14 0.32
benchmark function: DTLZ7 NSGAII 0.74 2.36 0.00 2.53 0.00 1914.18 0.96 0.98 0.00
SMSEMOA 0.76 2.57 0.00 2.74 0.00 1866.81 1.14 1.12 0.00
MOEAD 0.68 2.32 0.00 2.81 0.00 1738.48 0.83 0.98 0.00
PESAII 0.72 2.41 0.00 2.47 0.00 1807.44 1.04 1.03 0.00
SPEA2 0.75 2.35 0.00 2.71 0.00 2189.70 0.77 0.97 0.01
eMOEA 0.81 2.35 0.00 2.95 0.00 1327.14 1.26 1.49 0.00
PAGA 0.56 0.01 0.97 0.04 0.23 1796.35 0.06 1.36 0.81
Tab.4  A summary of quality measuring indicator values considering benchmark mathematical functions
names of MEAS DM GD HV IGD NHV PD spacing spread runtime
benchmark design example: truss with 4 bars NSGAII 0.74 0.78 48.02 66.58 0.33 21550.81 69.62 0.93 0.01
SMSEMOA 0.74 0.99 47.14 72.47 0.32 20910.55 70.83 0.91 0.00
MOEAD 0.76 1.78 47.33 74.23 0.32 19854.58 81.96 0.92 0.00
PESAII 0.71 0.95 47.30 77.58 0.32 20862.89 65.70 0.93 0.00
SPEA2 0.75 0.88 47.30 72.73 0.32 19494.03 67.65 0.91 0.02
eMOEA 1.00 1.47 17.50 468.49 0.12 16544.26 91.26 0.98 0.00
PAGA 0.71 0.35 58.81 9.42 0.40 47652.82 10.63 0.84 0.43
benchmark design example: truss with 2 bars NSGAII 0.75 46.83 4098.50 2255.24 0.68 591981.98 5322.47 1.17 0.01
SMSEMOA 0.75 43.27 4114.76 2031.32 0.68 589446.42 5520.16 1.17 0.00
MOEAD 0.76 37.12 4164.50 2320.06 0.69 512188.97 4624.75 1.12 0.00
PESAII 0.76 35.07 4159.03 2465.07 0.69 537298.51 4827.55 1.14 0.00
SPEA2 0.74 46.96 4160.18 2221.58 0.69 543605.76 5311.99 1.18 0.02
seMOEA 0.78 10.59 385.10 15991.40 0.06 926.68 6127.34 1.37 0.00
PAGA 0.64 8105462.87 4478.66 410.37 0.74 3493847065.14 12843949.95 1.87 0.25
benchmark design example: truss with 2 bars NSGAII 0.76 0.08 11.04 3.37 0.52 7396.84 4.24 0.74 0.01
SMSEMOA 0.77 0.21 10.98 3.47 0.51 7196.58 4.83 0.78 0.00
MOEAD 0.77 0.14 10.95 3.90 0.51 7390.63 4.90 0.79 0.00
PESAII 0.80 0.14 11.04 3.36 0.52 7458.80 4.85 0.75 0.00
SPEA2 0.75 0.01 10.94 3.72 0.51 7570.65 3.97 0.79 0.02
eMOEA 0.84 0.02 5.07 56.10 0.24 395.68 5.736 0.84 0.00
PAGA 0.57 160.47 11.62 1.47 0.54 623889.60 11.98 1.26 0.84
benchmark design example: truss with 200 bars NSGAII 0.78 856.589 2457891.89 10235.56 0.62 52789.56 22589.59 1.59 0.89
SMSEMOA 0.73 758.578 1248952.91 9875.68 0.58 51236.08 18569.23 1.42 0.56
MOEAD 0.65 756.268 785631.46 8956.45 0.54 47238.58 17862.22 1.39 0.45
PESAII 0.81 725.213 924562.02 11245.96 0.48 36214.23 14502.36 1.68 0.48
SPEA2 0.83 805.126 712568.25 23568.56 0.38 32589.21 25589.69 1.89 2.58
eMOEA 0.56 506.89 688922.56 118956.59 0.29 12256.03 68792.38 2.12 0.29
PAGA 0.98 404.56 6795981.80 3638.91 0.84 7690385.99 10456.54 0.98 10.29
benchmark design example: truss with 756 bars NSGAII 0.75 828.234 2549026.89 78945.21 0.68 183692.28 89756.59 1.58 3.56
SMSEMOA 0.56 1125.565 1256891.03 56872.09 0.59 112586.89 64562.23 1.45 4.12
MOEAD 0.72 945.568 897869.23 28697.72 0.57 89546.25 78236.12 1.38 4.01
PESAII 0.63 2456.890 2568956.69 33698.47 0.58 78956.33 48964.59 1.86 4.89
SPEA2 0.46 1124.561 789236.26 32589.65 0.45 68956.31 61235.2 1.68 7.89
eMOEA 0.36 987.563 564891.21 456982.48 0.39 9875.56 92568.94 3.15 3.58
PAGA 0.87 703.29 7944542.53 10156.298 0.85 20860564.30 19105.54 0.79 45.69
Tab.5  A summary of quality measuring indicator values considering benchmark structural design problem
Fig.12  True pareto front and random solutions. Ellipse-shaped dome structure with (a1–b1) geometrical configurations 1; (a2–b2) geometrical configurations 2; (a3–b3) geometrical configurations 3
Fig.13  True pareto front and random solutions. Sphere-shaped dome structure with (a1–b1) geometrical configurations 1; (a2–b2) geometrical configurations 2; (a3–b3) geometrical configurations 3
Fig.14  True pareto front and random solutions. Sphere-shaped dome structure with (a1–b1) geometrical configurations 1; (a2–b2) geometrical configurations 2; (a3–b3) geometrical configurations 3 (see Ref. [3])
names of geometrical configurations topology-related design variables
ParLDN ParHDN
GC1* correspon. to Min. weight 2 2
GC1 correspon. to Max. N. Disp. 3 3
GC1 correspon. to Max. El. force 2 2
GC2 correspon. to Min. weight 2 3
GC2 correspon. to Max. N. Disp. 2 2
GC2 correspon. to Max. El. force 2 2
GC3 correspon. to Min. weight 2 2
GC3 correspon. to Max. N. Disp. 2 2
GC3 correspon. to Max. El. force 2 2
Tab.6  Values of topology-related design variables obtained by use of ellipse-shaped dome structure
names of geometrical configurations shape-related design variables
ParSDVx ParSDVy ParSDVz
GC1 correspon. to Min. weight 19.85557 20.82589 19.43820
GC1 correspon. to Max. N. Disp. 20.36540 20.71619 20.98582
GC1 correspon. to Max. El. Force 20.09272 19.46097 20.01900
GC2 correspon. to Min. weight 19.00387 20.20425 19.90336
GC2 correspon. to Max. N. Disp. 19.91150 20.98851 20.42992
GC2 correspon. to Max. El. force 19.60113 19.96712 19.69775
GC3 correspon. to Min. weight 19.31799 20.31915 19.75669
GC3 correspon. to Max. N. Disp. 20.00000 20.00000 20.00000
GC3 correspon. to Max. El. force 20.30754 19.87931 20.39669
Tab.7  Values of shape obtained by use of ellipse-shaped dome structure
names of geometrical configurations size-related design variables
1 2 3 4 5 6
GC1 correspon. to Min. weight PIPST19 PIPST19 - - - -
GC1 correspon. to Max. N. Disp. PIPEST25 PIPEST25 PIPDEST64 - - -
GC1 correspon. to Max. El. force PIPEST13 PIPDEST152 - - - -
GC2 correspon. to Min. weight PIPEST13 PIPST19 PIPST51 PIPST19 - -
GC2 correspon. to Max. N. Disp. PIPST32 PIPEST76 PIPEST13 PIPDEST76 - -
GC2 correspon. to Max. El. force PIPST64 PIPDEST127 PIPEST76 PIPEST64 - -
GC3 correspon. to Min. weight PIPST25 PIPEST19 PIPEST13 - - -
GC3 correspon. to Max. N. Disp. PIPST254 PIPEST76 PIPEST13 - - -
GC3 correspon. to Max. El. force PIPST25 PIPDEST152 PIPDEST102 - - -
Tab.8  Values of size-related design variables obtained by use of ellipse-shaped dome structure
names of geometrical configurations entire weight (kN) Elem. force (kN) nodal Def. (mm) maximum load step
GC1 correspon. to Min. weight 3.35260 969.91775 1.30914 3
GC1 correspon. to Max. N. Disp. 44.03088 23055.86543 9.35454 6
GC1 correspon. to Max. El. force 95.92513 161111.77648 3.48130 6
GC2 correspon. to Min. weight 9.86727 1001.94235 1.85393 4
GC2 correspon. to Max. N. Disp. 44.44543 3518.11966 25.29756 5
GC2 correspon. to Max. El. force 62.32958 15268.67893 13.71436 7
GC3 correspon. to Min. weight 4.20822 3980.52871 6.28181 4
GC3 correspon. to Max. N. Disp. 57.43795 17505.42250 24.46849 5
GC3 correspon. to Max. El. force 75.16784 89511.33387 5.86092 6
Tab.9  Values of objective functions obtained by use of ellipse-shaped dome structure
names of geometrical configurations topology-related design variables
ParLDN ParHDN
GC1 correspon. to Min. weight 2 2
GC1 correspon. to Max. N. Disp. 5 3
GC1 correspon. to Max. El. force 2 2
GC2 correspon. to Min. weight 2 2
GC2 correspon. to Max. N. Disp. 5 4
GC2 correspon. to Max. El. force 2 2
GC3 correspon. to Min. weight 2 3
GC3 correspon. to Max. N. Disp. 2 2
GC3 correspon. to Max. El. force 2 2
Tab.10  Values of topology-related design variables obtained by use of sphere-shaped dome structure
names of geometrical configurations shape-related design variables
ParSDVx ParSDVy ParSDVz
GC1 correspon. to Min. weight 19.14188 19.14188 19.14188
GC1 correspon. to Max. N. Disp. 20.35520 20.35520 20.35520
GC1 correspon. to Max. El. force 20.58711 20.58711 20.58711
GC2 correspon. to Min. weight 20.83281 20.83281 20.83281
GC2 correspon. to Max. N. Disp. 20.91648 20.91648 20.91648
GC2 correspon. to Max. El. force 20.94760 20.94760 20.94760
GC3 correspon. to Min. weight 19.77586 19.77586 19.77586
GC3 correspon. to Max. N. Disp. 20.91008 20.91008 20.91008
GC3 correspon. to Max. El. force 20.09626 20.09626 20.09626
Tab.11  Values of shape-related design variables obtained by use of sphere-shaped dome structure
names of geometrical configurations size-related design variables
1 2 3 4 5 6
GC1 correspon. to Min. weight PIPST25 PIPEST13 - - - -
GC1 correspon. to Max. N. Disp. PIPST32 PIPST89 PIPDEST102 PIPEST76 PIPDEST64 -
GC1 correspon. to Max. El. force PIPDEST51 PIPDEST127 -
GC2 correspon. to Min. weight PIPEST38 PIPST38 PIPEST19 PIPST25 - -
GC2 correspon. to Max. N. Disp. PIPST305 PIPEST127 PIPEST19 PIPEST203 PIPDEST152 PIPDEST152 (6)
PIPDEST64 (7)
PIPEST64 (8)
PIPEST19 (9)
PIPEST89 (10)
PIPDEST51 (11)
PIPEST102 (12)
PIPST38 (13)
GC2 correspon. to Max. El. force PIPEST38 PIPEST13 PIPEST38 PIPEST254 - -
GC3 correspon. to Min. weight PIPST25 PIPEST25 PIPEST19 - - -
GC3 correspon. to Max. N. Disp. PIPST64 PIPDEST64 PIPEST13 - - -
GC3 correspon. to Max. El. force PIPST32 PIPDEST102 PIPDEST76 - - -
Tab.12  Values of size-related design variables obtained by use of sphere-shaped dome structure
names of geometrical configurations entire weight (kN) Elem. force (kN) nodal Def. (mm) maximum load step
GC1 correspon. to Min. weight 3.73945
(6.3865)*
3940.12712
(1304.2828)
6.60662
(1.2521)
4
GC1 correspon. to Max. N. Disp. 89.72802 7611.97809 9.07981 6
GC1 correspon. to Max. El. force 81.96421 160002.81483 4.91042 6
GC2 correspon. to Min. weight 11.21544
(357.1588)
1235.21542
(38846.2703)
4.34045
(27.9652)
4
GC2 correspon. to Max. N. Disp. 311.83751 2514.27344 26.61241 6
GC2 correspon. to Max. El. force 103.67229 15310.77152 8.91583 5
GC3 correspon. to Min. weight 6.88694
(350.1250)
2895.17655
(1562597.4696)
3.00763
(19.7811)
4
GC3 correspon. to Max. N. Disp. 20.89693 18536.57314 25.66832 5
GC3 correspon. to Max. El. force 44.07446 79771.87314 9.17463 6
Tab.13  Values of objective functions obtained by use of sphere-shaped dome structure
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