# Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MOEA Algorithm and Improvement

#### 2.1. Basic Concept of MOEA

_{1}< 1.0 and 0.6 < f

_{2}< 1.0; the blue circle region shows the selected optimal solutions.)

#### 2.2. Improvement to MOEA

#### 2.2.1. Mathematical Model

#### 2.2.2. Pareto Dominate Relations

_{a}and x

_{b}be two different solutions. If x

_{a}dominates x

_{b}, the two conditions of Equations (6) and (7) must be satisfied.

_{a}is no worse than x

_{b}:

_{a}⊱ x

_{b}“⊱”represents dominate relations.

## 3. Numerical Method Verification

#### 3.1. Function Test

#### 3.1.1. Test Function MOP1

_{1}and ƒ

_{2}, and this optimal problem has a convex Pareto frontier:

#### 3.1.2. Test Function MOP2

#### 3.2. Test Results

## 4. Analysis of Typical Discrete Structure Optimization

#### 4.1. Space Truss Optimization

#### 4.2. Space Tower Optimization

## 5. Conclusions

- (1)
- In this study, an improved MOEA optimization method was adopted to realize optimization of spatial discrete structures using moving bar node coordinates.
- (2)
- In contrast to the traditional optimization, which uses weight as the only object, the improved MOEA multi-objective optimization method is a combination of the structural weight and the maximum displacement of the spatially discrete structure. This method uses the strength rank idea and obtains a group of optimal solutions, not an individual value. Then, the optimal case can be confirmed according to other constraint conditions, such as the boundary value and the user’s preferences.
- (3)
- This algorithm was tested using standard mathematical functions, and the results indicate that the rate of convergence is fast and that no inferior solution evenly distributes in the solution space, which maintains good diversity without converging to a local part. Thus, we used the combination of this method with ANSYS to analyze a three-dimensional truss and tower structure, and the results confirmed that the modified MOEA algorithm appears to be feasible and superior with respect to the optimization of the discrete structure design.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Pareto curve of the function test: (

**a**) MOP1 Pareto curve and scatterplots and (

**b**) MOP2 Pareto curve and scatterplots (MOP: Multi-objective Optimization).

**Figure 8.**Truss chart of the evolution generation scatterplot: (

**a**) Step-5 evolution Pareto frontier curve; (

**b**) Step-16 evolution Pareto frontier curve; (

**c**) Step-25 evolution Pareto frontier curve; (

**d**) Step-31 evolution Pareto frontier curve; (

**e**) Step-42 evolution Pareto frontier curve; and (

**f**) Step-50 evolution Pareto frontier curve.

**Figure 11.**Shape change diagram of truss: (

**a**) Step-5 shape diagram of the truss; (

**b**) Step-16 shape diagram of the truss; (

**c**) Step-25 shape diagram of the truss; (

**d**) Step-31 shape diagram of the truss; (

**e**) Step-42 shape diagram of the truss; and (

**f**) Step-50 shape diagram of the truss.

**Figure 15.**Tower evolution diagrams: (

**a**) Step-5 evolution diagram; (

**b**) Step-16 evolution diagram; (

**c**) Step-25 evolution diagram; (

**d**) Step-31 evolution diagram; (

**e**) Step-42 evolution diagram; (

**f**) Step-50 evolution diagram.

**Figure 16.**Tower chart of evolution generation scatterplot: (

**a**) Step-5 evolution Pareto frontier curve; (

**b**) Step-16 evolution Pareto frontier curve; (

**c**) Step-25 evolution Pareto frontier curve; (

**d**) Step-31 evolution Pareto frontier curve; (

**e**) Step-42 evolution Pareto frontier curve; and (

**f**) Step-50 evolution Pareto frontier curve.

Population Size | Terminate Generation | Crossover Probability | Mutation Probability |
---|---|---|---|

40 | 30 | 0.4 | 0.02 |

Population Size | Terminate Generation | Crossover Probability | Mutation Probability |
---|---|---|---|

50 | 50 | 0.4 | 0.02 |

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**MDPI and ACS Style**

Nan, B.; Bai, Y.; Wu, Y.
Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement. *Appl. Sci.* **2020**, *10*, 1964.
https://doi.org/10.3390/app10061964

**AMA Style**

Nan B, Bai Y, Wu Y.
Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement. *Applied Sciences*. 2020; 10(6):1964.
https://doi.org/10.3390/app10061964

**Chicago/Turabian Style**

Nan, Bo, Yikui Bai, and Yue Wu.
2020. "Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement" *Applied Sciences* 10, no. 6: 1964.
https://doi.org/10.3390/app10061964