The performance of TPPAA is examined by several structural optimization problems in this chapter. The comparison with previous approximation methods is also presented. The convergence criterion adopted in this chapter is that the relative difference of two consecutive objective function values is smaller than 0.1%. The specifications of computer used for the examples in this chapter is Intel i7-920 2.67GHz CPU with 4GB DDR3-1333 RAM.
4.1 2-bar truss
The 2-bar truss structure with a concentrated force acting on its top is shown in Fig.
4-1. The design data is listed in Table 4-1. The first design variable is the cross-section area of both bars and the second one is the supporting locations. The goal of the optimization is to minimize the total weight. This optimization problem considers sizing design and shape design simultaneously. The analytical expression can be derived as
1 22Fig. 4-1 2-bar truss structure
Table 4-1 Design data for 2-bar truss optimization
Constraints
Stress i 100 N2 for each bar
mm Size
2 2
1 2
0.2 cm 4.0 cm
0.1 m 1.6 m
x x
Move limit 0.75xi(k1)xi( )k 1.25xi(k1), i1, 2 Initial design x11.5 cm , 2 x2 0.5 m
Conservative approximation is adopted to construct the single-point approximate functions for the initial design. The SAO result of TPPAA is shown in Table 4-2 and Fig.
4-2, and the comparison with previous approximation methods is shown in Table 4-3.
The CPU time of optimization process is 16 seconds. The exact solution of design variables is ( ,x x1 2)(1.412, 0.377) and the objective function is 1.50865 (kgf).
Table 4-2 Result of TPPAA in 2-bar optimization x 1 x 2 Weight (kgf) Initial design 1.500 0.500 1.6771
Iteration 1 1.386 0.375 1.4707 Iteration 2 1.413 0.375 1.5088 Optimum design 1.410 0.381 1.5086
Fig. 4-2 Iteration history of weight of 2-bar truss
Table 4-3 Result comparison for 2-bar truss optimization Approximation method Iterations Weight (kgf)
TPPAA 3 1.5086
DQA 3 1.5087
TPEA 3 1.5086
Direct linear 3 1.5087
Linear-reciprocal 3 1.5086
4.2 3-bar truss optimization
The 3-bar truss structure with a concentrated force acting on it is shown in Fig. 4-3.
The design data are listed in Table 4-4. One of the design variables ( )x represents the 1 cross-section areas of bar 1 and bar 3, and the other one (x2) represents the cross-section area of bar 2. The total weight of the 3-bar structure is the objective
function to be minimized. In this simple case, the stress constraints can be expressed in analytical form, thus the exact solution can be obtained for comparison. The exact solution of design variables is ( ,x x1 2)(0.7887, 0.4082) and that of objective function is 26.3896 (lb).
Fig. 4-3 3-bar truss structure
Table 4-4 Design data for 3-bar truss optimization
Material property E30000 ksi, 0.1 lb in/ 3, 0.3
Conservative approximation is adopted to construct the single-point approximate functions for the initial design. The CPU time for the optimization process is 12 seconds.
The SAO result of TPPAA is shown in Table 4-5 and Fig. 4-4. The comparison with existing approximation methods is shown in Table 4-6.
Table 4-5 Result of TPPAA in 3-bar truss optimization
x 1 x 2 Weight (lb)
Initial design 2.000 1.000 66.569
Iteration 1 0.792 0.400 26.392
Optimum design 0.790 0.403 26.390
Fig. 4-4 Iteration history of weight of 3-bar truss
Table 4-6 Result comparison for 3-bar truss optimization Approximation method Iterations Weight (lb)
TPPAA 2 26.390
DQA 2 26.390
TPEA 2 26.390
Linear-reciprocal 2 26.390
CONLIN 2 26.390
4.3 4-bar truss optimization
The 4-bar truss structure with a concentrated force acting on its top is shown in Fig.
4-5, and the design data is listed in Table 4-7. The objective function is the total weight of the 4-bar structure with the cross-section area of each bar selected as design variable.
Fig. 4-5 4-bar truss structure
Table 4-7 Design data for 4-bar truss optimization
Material property E10000 ksi, 0.1 lb in/ 3, 0.29
Constraints
Stress 25 ksii 25 ksi for each bar Size 1.0 10 4 in2x ii, 1, 2,3, 4 Move
limit
( 1)
( ) ( 1)
6.5 , 1, 2,3, 4 6.5
k
k k
i
i i
x x x i
Initial design xi 2.0 in2, i1,2,3,4
Direct linear approximation is adopted to construct the single-point approximate functions for the initial design. The CPU time for the optimization process is 39 seconds.
Result of SAO with TPPAA is shown in Table 4-8 and Fig. 4-6 respectively. Moreover, the comparison with previous approximation methods is shown in Table 4-9.
Table 4-8 Result of 4-bar truss optimization
x 1 x 2 x 3 x 4 Weight (lb) Initial design 2.000 2.000 2.000 2.000 139.44
Iteration 1 2.451 2.190 2.243 0.308 118.35 Iteration 2 2.667 2.300 2.159 0.047 116.32 Iteration 3 2.662 2.296 2.161 0.007 115.39 Iteration 4 2.663 2.298 2.160 0.001 115.27 Optimum design 2.663 2.298 2.159 0.0002 115.25
Fig. 4-6 Iteration history of weight of 4-bar truss Table 4-9 Result comparison for 4-bar truss optimization Approximation method Iterations Weight (lb)
TPPAA 5 115.25
DQA 5 115.25
TPEA 4 115.27*
Direct linear 5 115.25
TPEA-GBMMA [27] 4 115.28*
ETPEA1 [28] 4 115.28*
(*: premature convergence)
4.4 6-bar truss optimization
The 6-bar truss structure with a concentrated force acting on its end is shown in Fig.
4-7. The design data is listed in Table 4-10. The objective function is the total weight of the 6-bar structure with the cross-section area of each bar selected as design variable.
Fig. 4-7 6-bar truss structure
Table 4-10 Design data for 6-bar truss optimization
Material property E10000 ksi, 0.1 lb in/ 3, 0.29 approximation is adopted to construct the single-point approximate functions for the initial design. The SAO result of TPPAA is shown in Table 4-11 and Fig. 4-8 respectively, and the comparison with previous approximation methods is shown in Table 4-12.
Table 4-11 Result of TPPAA in 6-bar truss optimization
x 1 x 2 x 3 x 4 x 5 x 6 Weight (lb) Initial design 2.000 2.000 2.000 2.000 2.000 2.000 106.23
Iteration 1 2.717 1.320 1.281 0.500 1.432 1.320 82.770 Optimum design 2.717 1.320 1.281 0.500 1.432 1.320 82.770
Fig. 4-8 Iteration history of weight of 6-bar truss Table 4-12 Result comparison for 6-bar truss optimization
Approximation method Iterations Weight (lb)
TPPAA 2 82.770
DQA 2 82.770
TPEA 2 82.770
Direct linear 2 82.770
4.5 10-bar truss optimization
The weight minimization problem of 10-bar truss structure with a concentrated force acting on its end is shown in Fig. 4-9 (a=360 in). The design data is listed in Table 4-13. The objective function is the total weight of the 10-bar structure with the cross-section area of each bar selected as design variable.
1 2
Fig. 4-9 10-bar truss structure
Table 4-13 Design data for 10-bar truss optimization
Material property E10000 ksi, 0.1 lb in/ 3, 0.29 approximation is adopted to construct the single-point approximate functions for the initial design. The SAO result of TPPAA is shown in Table 4-14 and Fig. 4-10. The comparison with previous approximation methods is shown in Table 4-15. TPAEA with large is also tested in this example. is selected to be large enough so that the convex substitute functions are always adopted for approximation. The result shows that with too large adopted, the SAO process is stable but inefficient.
Table 4-14 Result of TPPAA in 10-bar truss optimization
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Weight
(lb)
Initial design 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 2098.2
Iteration 1 8.404 1.667 8.393 2.882 3.139 1.667 6.458 7.557 2.590 2.294 1903.6
Iteration 2 7.556 0.556 8.578 3.558 1.046 0.556 6.420 6.275 2.898 0.899 1626.2
Iteration 3 7.687 0.234 8.335 3.766 0.349 0.234 6.187 5.117 3.431 0.300 1507.3
Iteration 4 7.930 0.100 8.076 3.900 0.116 0.100 5.780 5.592 3.664 0.147 1501.0
Iteration 5 7.899 0.100 8.101 3.900 0.100 0.100 5.800 5.513 3.677 0.141 1497.6
Optimum design 7.900 0.100 8.100 3.900 0.100 0.100 5.798 5.515 3.677 0.141 1497.6
Fig. 4-10 Iteration history of weight of 10-bar truss Table 4-15 Result comparison for 10-bar truss optimization
Approximation method Iterations Weight (lb)
TPPAA 6 1497.6
TPPAA (large ) 20 1497.6
DQA 6 1497.6
Direct linear 7 1497.6
TPEA-GBMMA [27] 6 1497.6
ETPEA1 [28] 6 1497.6
(*: different convergence criterion)
4.6 25-bar truss optimization
The 25-bar truss structure is shown in Fig. 4-11. There are 8 design variables which are the cross-section areas of the 25 bars, as shown in Table 4-16. The load
structure is selected as objective function for weight minimization.
Fig. 4-11 25-bar truss structure
Table 4-16 Selection of design variables of 25-bar optimization Design
Table 4-17 Design data for 25-bar truss optimization
Material property E10000ksi, 0.1 /lb in3, 0.29
The CPU time for the optimization process is 82 seconds. Conservative approximation is adopted to construct the single-point approximate functions for the initial design. The SAO result of TPPAA is shown in Table 4-18 and Fig. 4-12, and the comparison with previous approximation methods is shown in Table 4-19. In this case, the additional converge criterion is introduced that the relative difference of the design variable in two consecutive iterations is required to be less than 1%.
Table 4-18 Result of TPPAA in 25-bar optimization
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x Weight (lb) 8 Initial design 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 661.44
Iteration 1 0.500 2.013 2.829 0.500 0.500 0.760 1.740 2.821 576.42 Iteration 2 0.125 2.030 3.216 0.125 0.125 0.512 1.549 2.792 546.45 Iteration 3 0.031 2.053 3.051 0.031 0.031 0.694 1.592 2.649 545.77 Iteration 4 0.008 2.025 2.983 0.008 0.008 0.685 1.647 2.668 544.92 Iteration 5 0.002 2.036 3.010 0.002 0.002 0.684 1.625 2.669 544.76 Iteration 6 0.001 2.037 3.007 0.001 0.001 0.685 1.626 2.669 544.73 Optimum design 0.001 2.037 3.007 0.001 0.001 0.685 1.626 2.669 544.73
Fig. 4-12 Iteration history of weight of 25-bar truss
Table 4-19 Result comparison for 25-bar optimization Approximation method Iterations Weight (lb)
TPPAA 7 544.73
TPPAA (large ) 7 544.74
TPEA-GBMMA [27] 5 545.74*
ETPEA1 [28] 5 544.78*
(*: premature convergence)
4.7 Multi-section circular beam optimization
The structure of multi-section circular beam is shown in Fig. 4-13. It consists of 10 circular beam elements with the same length. The design data are listed in Table 4-20. A static loading is exerted at the end of this cantilever beam. Total weight is selected as objective function for the purpose of minimizing the weight of beam structure and the diameter D of each beam section is selected as design variables.
L=10in
P=150kips
D
Fig. 4-13 Multi-section circular beam structure
Table 4-20 Design data for multi-section circular beam optimization Material property E30000 ksi, 0.1 lb in/ 3, 0.29
The CPU time for the optimization process is 70 seconds. Conservative approximation is adopted to construct the single-point approximate functions for the initial design. The SAO result with TPPAA is shown in Table 4-21 and Fig. 4-14. The comparison with previous approximation methods is shown in Table 4-22.
Table 4-21 Result of TPPAA in multi-section circular beam optimization
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Weight
(lb) Initial design 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 4.9087
Iteration 1 2.481 2.391 2.288 2.167 2.025 1.854 1.646 1.387 1.054 1.000 2.8342 Iteration 2 2.481 2.396 2.304 2.203 2.093 1.970 1.831 1.673 1.512 1.161 3.1500 Iteration 3 2.481 2.396 2.304 2.203 2.093 1.969 1.828 1.661 1.457 1.152 3.1312
Optimum
design 2.481 2.396 2.304 2.203 2.093 1.969 1.828 1.661 1.451 1.152 3.1299
Fig. 4-14 Iteration history of weight of multi-section circular beam Table 4-22 Result comparison for multi-section circular beam optimization
Approximation method Iterations Weight (lb)
TPPAA 4 3.1299
TPEA 3 3.1299
CONLIN 4 3.1299
Direct linear 5 3.1300
TPEA-GBMMA [27] 3 3.1293
ETPEA1 [28] 3 3.1299
4.8 Multi-section tube beam optimization
The structure of multi-section tube beam is shown in Fig. 4-15. It consists of 10 tubular beam elements with the same length. The design data are listed in Table 4-23.
The ratio of outer diameter D to the thickness of the tube T is constant (D/T=12). A static loading is exerted at the end of this cantilever beam. Total weight is selected as objective function for the purpose of minimizing the weight of beam structure and the diameter D of each beam section is selected as design variable.
L=10in
P=150kips
D T
Fig. 4-15 Multi-section tube beam structure
Table 4-23 Design data for multi-section tube beam optimization Material property E30000 ksi, 0.1 lb in/ 3, 0.29 approximation is adopted to construct the single-point approximate functions for the initial design. The SAO result of TPPAA is shown in Table 4-24 and Fig. 4-16 respectively. The comparison with previous approximation methods is shown in Table 4-25.
Table 4-24 Result of TPPAA in multi-section tube beam optimization
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Weight
(lb) Initial design 3.600 3.600 3.600 3.600 3.600 3.600 3.600 3.600 3.600 3.600 3.1102
Iteration 1 3.016 2.875 2.716 2.536 2.330 2.092 2.000 2.000 2.000 2.000 1.3674 Iteration 2 3.090 2.984 2.869 2.746 2.611 2.465 2.285 2.069 1.777 1.111 1.4643 Iteration 3 3.090 2.984 2.869 2.744 2.606 2.453 2.277 2.069 1.807 1.454 1.4846
Optimum
design 3.090 2.984 2.869 2.744 2.606 2.453 2.277 2.069 1.807 1.435 1.4833
Fig. 4-16 Iteration history of weight of multi-section tube beam
Table 4-25 Result comparison for multi-section tube beam optimization Approximation method Iterations Weight (lb)
TPPAA 4 1.4833
TPEA 3 1.4832
Linear-reciprocal 4 1.4832
Direct linear 5 1.4832
TPEA-GBMMA [27] 3 1.4831
ETPEA1 [28] 3 1.4832
4.9 Multi-section rectangular beam optimization
The structure of multi-section rectangular beam is shown in Fig. 4-17. It consists of 10 rectangular beam elements with the same length, and the design data are listed in Table 4-26. The ratio of rectangular height H to the width W is constant (H/W=3). A static loading is exerted at the end of this cantilever beam. Total weight is selected as objective function for the purpose of minimizing the weight of beam structure and the height H of each beam section is selected as the design variable.
Fig. 4-17 Multi-section rectangular beam structure
Table 4-26 Design data for multi-section rectangular beam optimization Material property E30000 ksi, 0.1 lb in/ 3, 0.29 approximation is adopted for the approximation of the initial design. The SAO result of TPPAA is shown in Table 4-27 and Fig. 4-18, and the comparison with previous approximation methods is shown in Table 4-28.
Table 4-27 Result of TPPAA in multi-section rectangular beam optimization
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Weight
(lb) Initial design 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.0000
Iteration 1 1.765 2.893 2.769 2.625 2.455 2.250 2.000 1.688 1.286 1.200 1.7653 Iteration 2 1.953 2.896 2.785 2.664 2.530 2.382 2.213 2.021 1.822 1.405 1.9532 Iteration 3 1.942 2.896 2.785 2.664 2.530 2.381 2.210 2.008 1.760 1.393 1.9423
Optimum
design 1.942 2.896 2.785 2.664 2.530 2.381 2.210 2.008 1.754 1.392 1.9416
Fig. 4-18 Iteration history for multi-section rectangular beam
Table 4-28 Result of multi-section rectangular beam Approximation method Iterations Weight (lb)
TPPAA 4 1.9416
TPEA 3 1.9416
Linear-reciprocal 4 1.9416
Direct linear 5 1.9416
TPEA-GBMMA [27] 3 1.9418
ETPEA1 [28] 3 1.9418