Due to the fact that vastly different variables and constraints are simultaneously considered, truss layout optimization is a typical difficult constrained mixed-integer nonlinear program. Moreover, the computational cost of truss analysis is often quite expensive. In this paper, a novel fitness estimation based particle swarm optimization algorithm with an adaptive penalty function approach (FEPSO-AP) is proposed to handle this problem. FEPSO-AP adopts a special fitness estimate strategy to evaluate the similar particles in the current population, with the purpose to reduce the computational cost. Further more, a laconic adaptive penalty function is employed by FEPSO-AP, which can handle multiple constraints effectively by making good use of historical iteration information. Four benchmark examples with fixed topologies and up to 44 design dimensions were studied to verify the generality and efficiency of the proposed algorithm. Numerical results of the present work compared with results of other state-of-the-art hybrid algorithms shown in the literature demonstrate that the convergence rate and the solution quality of FEPSO-AP are essentially competitive.
As a typical real world project, truss structural analysis is considered to be computationally expensive [
Vanderplaats and Moses [
Due to the strong coupling between the variables, the search efficiency of multilevel methods is often limited [
More recently, researchers turn to hybridizing different techniques to further enhance the searching efficiency of metaheuristic algorithms. Lingyun et al. [
However, the common weakness of metaheuristic algorithms based structural optimization is that a huge number of structural analyses are required, which is quite time-consuming. In this paper, we propose a new hybridized algorithm, termed FEPSO-AP for truss layout optimization that aims to enhance the optimal efficiency by using the fitness estimations to partly substitute the computationally expensive fitness calculations. The finite element method (FEM) is adopted to evaluate the structural performance. Empirical results demonstrate that the proposed method is highly promising for truss layout optimization.
The main aim of truss layout optimization can be formulated as follows:
Different types of constraints might be considered simultaneously depending on the problem to be solved. Four typical design constraints involved in this work can be stated by
This section describes the fitness estimation based PSO algorithm with an adaptive penalty function approach (FEPSO-AP) developed in this research. As FEPSO-AP integrates PSO, FE, and AP, this section recalls the basic concept of PSO algorithm, fitness estimation strategy, and adaptive constraint handling approach. Finally, a framework of FEPSO-AP algorithm is presented.
As one of the most popular metaheuristic algorithms, PSO has found a wide application in real world projects for its structural concision and searching efficiency [
If the particle flies from its current position to the next position, its velocity and position are updated by
Sun et al. [
According to (
Use a virtual position
Illustration of the virtual position.
As mentioned in Section
In this work, we use penalty function methods (PEM) to transfer a constrained problem into an unconstrained problem by adding the influence of violated constraints to the initial fitness function. A pseudoobjective function is stated as follows:
As summarized in [
The penalty function
According to (
To transfer the formulated truss layout optimization problem into an unconstrained optimization problem, the adaptive penalty function, the fitness estimation strategy, and the PSO algorithm are assembled as follows.
Select particle Choose a particle If Repeat
The following four benchmark examples have been used to demonstrate the generality and efficiency of the FEPSO-AP algorithm: a planar 15-bar truss subjected to a single load condition and stress constraints, a spatial 25-bar truss subjected to a single load condition under stress and displacement constraints, a planar 37-bar truss subjected to multiple frequency constraints, a planar 47-bar truss subjected to three load conditions under stress and local buckling constraints.
Programs of FEPSO-AP algorithm and structural finite element method (FEM) algorithm are developed by using MATLAB R2013a. A personal computer with a Pentium E5700 processor and 2 GB memory under the Microsoft Windows 7 operating system has been used to run the optimization software.
For all benchmarks examined in this study, the FEPSO-AP algorithm parameters are set as the usual constants of standard PSO which are obtained by [
Twenty-five independent runs are performed with the best one being selected for each problem.
The original geometry of planar 15-bar truss is shown in Figure
Loading condition acting on the planar 15-bar truss.
Case | Node |
|
|
---|---|---|---|
1 | 8 | 0 | −10.0 |
Layout optimization of the planar 15-bar truss.
Geometry and element definitions of the planar 15-bar truss
Best solution of the planar 15-bar truss
All design variables are classified into 23 groups: sizing variables: shape variables:
Material parameters and design constraints are listed in Table
Material parameters, design constraints, and search range of the planar 15-bar truss optimization problem.
Category | Values |
|
|
Material Parameters | |
Density | 0.1 lb/in3 |
Modulus of elasticity |
|
Constraints | |
Stress | The allowable elements stress interval: |
Search range | |
Shape variables |
|
Sizing variables |
|
Figure
Comparison of optimized designs found for the planar 15-bar truss.
No. | Variable | FA [ |
FM-GA [ |
PSO [ |
CPSO [ |
SCPSO [ |
FEPSO-AP |
---|---|---|---|---|---|---|---|
1 |
|
0.954 | 1.081 | 0.954 | 1.174 | 0.954 | 1.081 |
2 |
|
0.539 | 0.539 | 1.081 | 0.539 | 0.539 | 0.539 |
3 |
|
0.220 | 0.287 | 0.270 | 0.347 | 0.270 | 0.270 |
4 |
|
0.954 | 0.954 | 1.081 | 0.954 | 0.954 | 0.954 |
5 |
|
0.539 | 0.539 | 0.539 | 0.954 | 0.539 | 0.539 |
6 |
|
0.220 | 0.141 | 0.287 | 0.141 | 0.174 | 0.111 |
7 |
|
0.111 | 0.111 | 0.141 | 0.141 | 0.111 | 0.111 |
8 |
|
0.111 | 0.111 | 0.111 | 0.111 | 0.111 | 0.111 |
9 |
|
0.287 | 0.539 | 0.347 | 1.174 | 0.287 | 0.347 |
10 |
|
0.440 | 0.440 | 0.440 | 0.141 | 0.347 | 0.347 |
11 |
|
0.440 | 0.539 | 0.270 | 0.440 | 0.347 | 0.440 |
12 |
|
0.220 | 0.270 | 0.111 | 0.440 | 0.220 | 0.287 |
13 |
|
0.220 | 0.220 | 0.347 | 0.141 | 0.220 | 0.287 |
14 |
|
0.270 | 0.141 | 0.440 | 0.141 | 0.174 | 0.111 |
15 |
|
0.220 | 0.287 | 0.220 | 0.347 | 0.270 | 0.270 |
16 |
|
114.9670 | 101.5775 | 106.052 | 102.287 | 137.222 | 100.009 |
17 |
|
247.0400 | 227.9112 | 239.025 | 240.505 | 259.909 | 248.078 |
18 |
|
125.9190 | 134.7986 | 130.356 | 112.584 | 123.501 | 131.524 |
19 |
|
111.0670 | 128.2206 | 114.273 | 108.043 | 110.002 | 123.211 |
20 |
|
58.2980 | 54.8630 | 51.987 | 57.795 | 59.936 | 54.077 |
21 |
|
−17.5640 | −16.4484 | 1.814 | −6.430 | −5.180 | −9.039 |
22 |
|
−5.8210 | −13.3007 | 9.183 | −1.801 | 4.219 | −14.905 |
23 |
|
31.4650 | 54.8572 | 46.909 | 57.799 | 57.883 | 54.084 |
Best weight (lb) | 75.5473 | 76.6854 | 82.2344 | 77.6153 | 72.6153 | 74.1673 | |
Maximum displacement (in.) | 24.9993 | 24.9992 | 24.9999 | 24.9909 | 24.9912 | 24.9999 | |
Number of structural analyses | 8,000 | 8,000 | 4,500 | 4,500 | 4,500 | 4,000 |
The original geometry of spatial 25-bar truss is shown in Figure
Loading condition acting on the spatial 25-bar truss.
Case | Node |
|
|
|
---|---|---|---|---|
1 | 1 | 1.0 | −10.0 | −10.0 |
2 | 0.0 | −10.0 | −10.0 | |
3 | 0.5 | 0.0 | 0.0 | |
6 | 0.6 | 0.0 | 0.0 |
Layout optimization of the spatial 25-bar truss.
Geometry and element definitions of the spatial 25-bar truss
Best solution of the spatial 25-bar truss
To ensure the structural symmetries, all design variables are classified into 13 groups: sizing variables: shape variables:
Material parameters and design constraints are listed in Table
Material parameters, design constraints, and search range of the spatial 25-bar truss optimization problem.
Category | Values |
---|---|
Material parameters | |
Density | 0.1 lb/in3 |
Modulus of elasticity |
|
Constraints | |
Stress | The allowable elements stress interval: |
Displacement | The allowable nodal displacement interval: |
Search range | |
Shape variables |
|
Sizing variables |
|
Figure
Comparison of optimized designs found for the spatial 25-bar truss.
No. | Variable | FA [ |
FM-GA [ |
PSO [ |
CPSO [ |
SCPSO [ |
FEPSO-AP |
---|---|---|---|---|---|---|---|
1 |
|
0.1 | 0.1 | 0.1 | 0.3 | 0.1 | 0.1 |
2 |
|
0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
3 |
|
0.9 | 1.1 | 1.1 | 1.0 | 1.0 | 1.0 |
4 |
|
0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
5 |
|
0.1 | 0.1 | 0.4 | 0.1 | 0.1 | 0.1 |
6 |
|
0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
7 |
|
0.1 | 0.2 | 0.4 | 0.2 | 0.1 | 0.1 |
8 |
|
1.0 | 0.8 | 0.7 | 0.9 | 0.9 | 0.9 |
9 |
|
37.3200 | 33.0487 | 27.6169 | 33.4976 | 36.9520 | 36.8958 |
10 |
|
55.7400 | 53.5663 | 51.6196 | 62.3735 | 54.5786 | 54.1337 |
11 |
|
126.6200 | 129.9092 | 129.9071 | 114.5945 | 129.9758 | 130.0000 |
12 |
|
50.1400 | 43.7826 | 42.5526 | 40.0531 | 51.7317 | 51.9924 |
13 |
|
136.4000 | 136.8381 | 132.7241 | 133.6695 | 139.5316 | 140.0000 |
Best weight (lb) | 118.83 | 120.1149 | 129.2076 | 123.5403 | 117.2271 | 117.3022 | |
Maximum displacement (in.) | 0.3500 | 0.3500 | 0.3503 | 0.3505 | 0.3518 | 0.3500 | |
Maximum stress (ksi) | 18.8302 | 17.1574 | 16.4391 | 15.5913 | 19.9702 | 20.0182 | |
Minimum stress (ksi) | −9.4017 | −6.4822 | −10.9931 | −6.4102 | −9.4049 | −9.4024 | |
Number of structural analyses | 6,000 | 10,000 | 4,500 | 4,500 | 4,500 | 4,500 |
The original geometry of planar 37-bar truss is shown in Figure
Layout optimization of the planar 37-bar truss.
Geometry and element definitions of the planar 37-bar truss
Best solution of the planar 37-bar truss
To ensure the structural symmetric about the sizing variables: shape variables:
Material parameters and design constraints are listed in Table
Material parameters, design constraints, and search range of the planar 37-bar truss optimization problem.
Category | Values |
---|---|
Material parameters | |
Density | 7800 kg/m3 |
Modulus of elasticity |
|
Constraints | |
Natural frequencies |
|
Search range | |
Shape variables |
|
Sizing variables |
|
Figure
Comparison of optimized designs found for the planar 37-bar truss.
No. | Variable | OC [ |
GA [ |
PSO [ |
RO [ |
FEPSO-AP |
---|---|---|---|---|---|---|
1 |
|
3.2508 | 2.8932 | 2.6797 | 3.0124 | 3.4197 |
2 |
|
1.2364 | 1.1201 | 1.1568 | 1.0623 | 0.9766 |
3 |
|
1.0000 | 1.0000 | 2.3476 | 1.0005 | 0.8313 |
4 |
|
2.5386 | 1.8655 | 1.7182 | 2.2647 | 2.8073 |
5 |
|
1.3714 | 1.5962 | 1.2751 | 1.6339 | 1.2997 |
6 |
|
1.3681 | 1.2642 | 1.4819 | 1.6717 | 1.6483 |
7 |
|
2.4290 | 1.8254 | 4.6850 | 2.0591 | 2.4972 |
8 |
|
1.6522 | 2.0009 | 1.1246 | 1.6607 | 1.5379 |
9 |
|
1.8257 | 1.9526 | 2.1214 | 1.4941 | 1.7590 |
10 |
|
2.3022 | 1.9705 | 3.8600 | 2.4737 | 2.7069 |
11 |
|
1.3103 | 1.8294 | 2.9817 | 1.5260 | 1.3046 |
12 |
|
1.4067 | 1.2358 | 1.2021 | 1.4823 | 1.4004 |
13 |
|
2.1896 | 1.4049 | 1.2563 | 2.4148 | 3.0476 |
14 |
|
1.0000 | 1.0000 | 3.3276 | 1.0034 | 0.5947 |
15 |
|
1.2086 | 1.1998 | 0.9637 | 1.0010 | 0.8756 |
16 |
|
1.5788 | 1.6553 | 1.3978 | 1.3909 | 1.2546 |
17 |
|
1.6719 | 1.9652 | 1.5929 | 1.5893 | 1.4446 |
18 |
|
1.7703 | 2.0737 | 1.8812 | 1.7507 | 1.5889 |
19 |
|
1.8502 | 2.3050 | 2.0856 | 1.8336 | 1.6480 |
Natural frequencies (Hz) |
|
20.0850 | 20.0013 | 20.0001 | 20.056 | 20.020 |
|
42.0743 | 40.0305 | 40.0003 | 40.035 | 40.022 | |
|
62.9383 | 60.0000 | 60.0001 | 60.030 | 60.233 | |
|
74.4539 | 73.0444 | 73.0440 | 74.387 | 72.137 | |
|
90.0576 | 89.8244 | 89.8240 | 85.929 | 84.065 | |
Best weight (kg) | 366.50 | 368.84 | 377.20 | 364.04 | 362.8812 | |
Number of structural analyses | — | — | 20,000 | 32,000 | 8,000 |
The original geometry of planar 47-bar truss is shown in Figure
Loading conditions acting on the planar 47-bar truss.
Case | Node |
|
|
---|---|---|---|
1 | 17 and 22 | 6.0 | −14.0 |
2 | 17 | 6.0 | −14.0 |
3 | 22 | 6.0 | −14.0 |
Layout optimization of the planar 47-bar truss.
Geometry and element definitions of the planar 47-bar truss
Best solution of the planar 47-bar truss
To ensure the structural symmetric about the sizing variables: shape variables:
Material parameters and design constraints are listed in Table
Material parameters, design constraints, and search range of the planar 47-bar truss optimization problem.
Category | Values |
---|---|
Material parameters | |
Density | 0.3 lb/in3 |
Modulus of elasticity |
|
Constraints | |
Stress | The allowable elements stress interval: |
Local buckling |
|
Search range | |
Shape variables |
|
Sizing variables |
|
Figure
Comparison of optimized designs found for the planar 47-bar truss.
No. | Variable | GA [ |
FSD-ES [ |
PSO [ |
CPSO [ |
SCPSO [ |
FEPSO-AP |
---|---|---|---|---|---|---|---|
1 |
|
2.50 | 2.70 | 2.80 | 2.60 | 2.50 | 3.20 |
2 |
|
2.20 | 2.50 | 2.70 | 2.50 | 2.50 | 2.80 |
3 |
|
0.70 | 0.70 | 0.80 | 0.70 | 0.80 | 0.70 |
4 |
|
0.10 | 0.10 | 1.10 | 0.30 | 0.10 | 0.10 |
5 |
|
1.30 | 0.90 | 0.80 | 1.20 | 0.70 | 0.60 |
6 |
|
1.30 | 1.10 | 1.30 | 1.10 | 1.40 | 1.60 |
7 |
|
1.80 | 1.80 | 1.80 | 1.60 | 1.70 | 1.90 |
8 |
|
0.50 | 0.70 | 0.90 | 0.80 | 0.80 | 0.90 |
9 |
|
0.80 | 0.90 | 1.20 | 1.10 | 0.90 | 1.00 |
10 |
|
1.20 | 1.30 | 1.40 | 1.30 | 1.30 | 2.00 |
11 |
|
0.40 | 0.30 | 0.30 | 0.30 | 0.30 | 0.10 |
12 |
|
1.20 | 1.10 | 1.40 | 0.80 | 0.90 | 0.40 |
13 |
|
0.90 | 1.00 | 1.10 | 1.00 | 1.00 | 1.40 |
14 |
|
1.00 | 0.90 | 1.20 | 1.00 | 1.10 | 1.50 |
15 |
|
3.60 | 0.80 | 1.60 | 0.90 | 5.00 | 1.20 |
16 |
|
0.10 | 0.10 | 1.00 | 0.10 | 0.10 | 0.70 |
17 |
|
2.40 | 2.70 | 2.80 | 2.70 | 2.50 | 5.00 |
18 |
|
1.10 | 0.80 | 0.80 | 0.90 | 1.00 | 1.00 |
19 |
|
0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 |
20 |
|
2.70 | 3.00 | 3.00 | 3.00 | 2.80 | 3.00 |
21 |
|
0.80 | 0.90 | 0.90 | 1.00 | 0.90 | 0.50 |
22 |
|
0.10 | 0.00 | 0.10 | 0.20 | 0.10 | 0.10 |
23 |
|
2.80 | 3.20 | 3.30 | 3.30 | 3.00 | 3.30 |
24 |
|
1.30 | 1.00 | 0.90 | 0.90 | 1.00 | 0.30 |
25 |
|
0.20 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 |
26 |
|
3.00 | 3.30 | 3.30 | 3.30 | 3.20 | 3.50 |
27 |
|
1.20 | 1.10 | 1.20 | 1.10 | 1.20 | 0.40 |
28 |
|
114.0000 | 100.9724 | 98.8628 | 99.3630 | 101.3393 | 87.7275 |
29 |
|
97.0000 | 80.4772 | 78.6595 | 83.4439 | 85.9111 | 72.4352 |
30 |
|
125.0000 | 136.8699 | 146.7331 | 126.3845 | 135.9645 | 162.6451 |
31 |
|
76.0000 | 64.3908 | 66.5231 | 69.5148 | 74.7969 | 67.2113 |
32 |
|
261.0000 | 247.0491 | 239.0901 | 218.2013 | 237.7447 | 218.2041 |
33 |
|
69.0000 | 55.2589 | 55.6936 | 58.0004 | 64.3115 | 50.6507 |
34 |
|
316.0000 | 338.4534 | 327.7882 | 322.2272 | 321.3416 | 375.4549 |
35 |
|
56.0000 | 48.7333 | 48.8641 | 51.4015 | 53.3345 | 36.6525 |
36 |
|
414.0000 | 409.7380 | 398.6775 | 401.5626 | 414.3025 | 408.7230 |
37 |
|
50.0000 | 43.4742 | 43.1400 | 46.8605 | 46.0277 | 36.9960 |
38 |
|
463.0000 | 472.1479 | 464.7831 | 458.3021 | 489.9216 | 483.4295 |
39 |
|
54.0000 | 44.8349 | 37.8993 | 46.8885 | 41.8353 | 37.9558 |
40 |
|
524.0000 | 512.1901 | 511.0450 | 527.8575 | 522.4161 | 535.7644 |
41 |
|
1.0000 | 3.8414 | 18.2341 | 16.2354 | 1.0005 | 4.6875 |
42 |
|
587.0000 | 591.1449 | 594.0710 | 610.8496 | 598.3905 | 599.7416 |
43 |
|
99.0000 | 84.5040 | 90.9369 | 98.3239 | 97.8696 | 101.4535 |
44 |
|
631.0000 | 630.3472 | 621.3943 | 624.9580 | 624.0552 | 605.4302 |
Best weight (lb) | 1925.7897 | 1842.6609 | 1975.8393 | 1908.8301 | 1864.0985 | 1799.7037 | |
Maximum stress (ksi) | 19.9528 | 20.0000 | 19.0636 | 19.3351 | 19.4735 | 19.9808 | |
Minimum stress (ksi) | −14.9973 | −15.0000 | −14.9999 | −14.9986 | −15.0000 | −14.9986 | |
Number of buckling elements | 0 | 0 | 0 | 0 | 0 | 0 | |
Number of structural analyses | 100,000 | 55,802 | 25,000 | 25,000 | 25,000 | 20,000 |
In this work, a new hybrid PSO algorithm is proposed to solve a quite challenging task in truss optimization area: truss layout optimization with multiple constraints.
Two computational techniques are adopted to further enhance the performance of PSO algorithm. In the first fitness estimation strategy, the evaluation of particles is partly substituted by the estimation of similar particles, with the purpose to reduce the computational cost of real world optimization problem. In the second adaptive penalty function approach, the iteration information is merged into the penalty function to find a good balance between the exploration and exploitation of the constrained design domain. The resulted algorithm is termed as FEPSO-AP.
Four benchmark truss layout optimization problems, subject to nodal displacement constraints, element stress constraints, natural frequency constraints, and local buckling constraints, are used to verify the performance of FEPSO-AP. Numerical results demonstrate that three out of four benchmarks, to which the FEPSO-AP based optimization is applied, delivered the best feasible designs to the author’s knowledge. Moreover, the convergence rate of the FEPSO-AP algorithm is quite competitive comparing to other state-of-the-art hybrid algorithms published in the former literatures.
The authors declare that there is no conflict of interests regarding the publication of this paper.