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To realize the static and dynamic multiobjective topology optimization of joints in spatial structures, structural topology optimization is carried out to maximize the stiffness under static multiload conditions and maximize the first third-order dynamic natural frequencies. According to the single-objective optimization results, the objective function of the multiobjective topology optimization of joints is established by using the compromise programming method, and the weight coefficient of each static load condition is determined by using the analytic hierarchy process. Subsequently, under the constraint of the volume fraction, the multiobjective topology optimization of joints is realized by minimizing the multiobjective function. Finally, the optimized structure is smoothed to obtain a smoother joint, and its mechanical properties are compared with those of the hollow ball joint. The results indicate that the multiobjective topology optimization that considers the static stiffness and dynamic frequency can effectively improve the mechanical properties of the structure. Through the research on multiobjective topology optimization, a new type of spatial joint with reasonable stress, a novel form, and aesthetic shape can be obtained, which mitigates the shortcomings of single-objective topology optimization. In comparison to hollow spherical joints with the same weight, topology-optimized joints have a superior ability to resist deformation and improve low-order frequency, which verifies the feasibility of applying multiobjective topology optimization to the lightweight design of joints.

Joints are the key components that interconnect structural elements, such as large-span spatial trusses, grids, and reticulated shells [

Researchers have proposed different theoretical methods for the topology optimization of structural joints, which mainly focus on the single-objective optimization design of structures. Chen et al.[

To mitigate the aforementioned limitations, this paper proposes a multiobjective topology optimization method for spatial-structure joints based on the compromise programming method and combines the analytic hierarchy process (AHP) to determine the weights of the three load conditions axial load, shear load, and bending moment load. Next, the topology optimization of the spatial grid structure joint is carried out to maximize structural stiffness and low-order frequencies, and the initial optimization result is smoothed to obtain a new joint type that meets multiple goals and simultaneously has reasonable stress transmission.

Structural topology optimization is a mathematical method to rationally distribute materials in a design area. These methods include the homogenization method [_{min} = _{0}/1000 is usually assumed.

The topology optimization of static structures is a way to reasonably distribute materials in the design area to ensure maximum stiffness. In practice, joints are generally in complex stress states that include axial, shear, and bending moment stresses. Therefore, the static topology optimization problem aims to maximize the stiffness of the structure under multiple-load conditions, also known as the multistiffness topology optimization problem [

By combining topology optimization and the compromise programming method, the minimum flexibility is used instead of the maximum stiffness, and the volume fraction is taken as a constraint. The static multiload condition mathematical function is as follows:

The purpose of structural dynamic natural frequency topology optimization is to increase the low-order natural frequencies of the structure. The reciprocal of the weighted frequencies is often used to consider the topology optimization of the low-order natural frequencies of the structure, the weighted sum of the reciprocals of the frequencies of each order is as follows:

It can be seen from equation (_{0} = 0.

Multiobjective topology optimization methods can be divided into two types. The first transforms the multiobjective problem into one or a series of single-objective problems, and the optimized result is regarded as the solution of the multiobjective optimization problem. The second is to directly seek noninferior solutions and choose the better solution as the optimal solution.

Both stiffness and frequency goals need to be considered when carrying out multiobjective topology optimization on a structure to ensure that the low-order natural frequency and structural stiffness are sufficiently high. The main issue with multiobjective optimization lies in how to unify the dimensions of multiple objective functions. In this study, the compromise programming method is used to first solve the single-objective problem and obtain the optimal value, which is called the ideal point. Then, the multiobjective solution is undertaken according to the single-objective optimized result, and the weighted objective function is constructed to find the minimum distance from the ideal point. Taking the minimization of the weighted objective function as the goal, and the volume fraction as the constraint, according to equations (

The weight

The weight value given is only ascertained based on the designer’s experience and is not accurate.

The designer is required to have a wealth of engineering experience, which increases the difficulty of design.

The accurate weight of various load conditions cannot be given when there are several operating conditions.

To solve the above problems, the AHP [_{ij})_{m×m} (_{ij} is the degree of importance of load condition

Definition of importance scale of judgment matrix

Scaling (a_{ij}) | Importance |
---|---|

1 | |

3 | |

5 | |

7 | |

9 | |

1/3 | |

1/5 | |

1/7 | |

1/9 | |

2, 4, 6, 8 | The middle of each scale |

For matrix

Average random consistency index.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0 | 0 | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 | 1.49 |

Assuming that the vector

Based on the OptiStruct optimization program, the method of moving asymptotes is used as the optimization algorithm, and the lower-limit constraint and the perimeter constraint are used to control numerical instability. The compromise programming method is used to optimize the functions of the static stiffness and dynamic frequency. The optimized function is combined into a comprehensive objective function, and used in conjunction with the AHP to determine the weight of each load condition of the subobjective. Taking the volume fraction as the constraint and the minimization of the comprehensive objective function as the goal, the joint is optimized for multiobjective topology. The optimization process is shown in Figure

Multiobjective topology optimization process.

According to Figure

As shown in Figure ^{3}.

Finite element model and load arrangement of the joint. (a) Finite model. (b) Loads and boundary conditions.

The leftmost overhanging pipe is the fixed end, and the axial and shear load on each of the remaining seven tubes is 124.8 kN. Axial force (N) represents the axial tension in the middle four tubes and the axial pressure in the upper four tubes, the shear force (

In this paper, the joint action of axial, shear, and bending moment loads is considered according to the relative importance degree of each working condition. The axial load is obviously more important than the shear and bending moment loads, and the shear load is slightly more important than the bending moment load, according to Table

The maximum eigenvalue is found to be 3.06489, and the corresponding eigenvector ^{T}. By substituting the eigenvalue and

Equation (_{1} = 0.731, _{2} = 0.188, and _{3} = 0.091 can be obtained from equation (

For the topology optimization of spherical joints under static conditions, the objective function must minimize compliance (i.e., maximize stiffness). Further, the 40% volume fraction is a constraint, the penalty factor is set to 2, and the minimum member size is taken as 2.5 times the unit size, i.e., 25 mm. The stiffness models under different conditions are analyzed for topology optimization, and the iterative curves of compliance under different conditions are shown in Figure

Iteration step-compliance curve under various load conditions.

Figure

Maximum and minimum compliance under various load conditions.

Conditions | Max compliance/10^{4} N·mm | Min compliance/10^{4} N·mm |
---|---|---|

Axial | 28.68 | 3.29 |

Shear | 13.80 | 2.04 |

Bending moment | 32.68 | 4.93 |

The maximum and minimum compliance in Table

When performing topology optimization on the spherical joint shown in Figure

Variation curve of the first three natural frequencies in the optimization process.

It can be seen from Figure

Maximum and minimum frequencies and modes in dynamic frequency optimization.

Order | Max frequency/Hz | Min frequency/Hz | Formation |
---|---|---|---|

1 | 277.33 | 109.19 | Along the - |

2 | 284.40 | 110.16 | Along the - |

3 | 440.90 | 207.89 | Twisting along the |

By substituting the maximum and minimum frequencies shown in Table

In the multiobjective topology optimization study, the weight coefficient of each load condition is obtained according to equation (

Change curve of the objective function in the optimization process.

It can be seen from Figure

Compliance and frequency variation curve in the optimization process. (a) Compliance variation curve. (b) Frequency variation curve.

Figure

Table ^{4} N mm). This shows that multiobjective topology optimization significantly improves structural rigidity.

Initial compliance and optimized compliance

Load conditions | Initial compliance (10^{4} N·mm) | Optimized compliance (10^{4} N·mm) |
---|---|---|

Axial | 21.49 | 7.11 |

Shear | 10.35 | 6.00 |

Bending moment | 24.87 | 15.45 |

Table

Description of frequency before and after optimization and formation.

Order | Initial frequency (Hz) | Optimized frequency (Hz) | Formation |
---|---|---|---|

First | 109.19 | 166.98 | Along the - |

Second | 110.16 | 185.11 | Along the - |

Third | 207.89 | 289.89 | Twisting along the |

The density distribution of the joint obtained after optimization is shown in Figure

Density distribution of the optimized joint.

To determine the final optimized shape, the topology optimization results are further processed according to the density threshold. The density threshold controls the selection of the minimum density after the optimization of the structure, with a value between 0 and 1. The selection of the density threshold is crucial to determine the final structural shape.

According to the “Code for Design of Steel Structures” GB 50017-2017, for cast steel joints, the stress under complex loads should be less than 1.1fy, that is, less than 379.5 MPa. It can be seen from Figure

Mechanical property curves corresponding to different thresholds. (a) Maximum displacement. (b) von Mises stress. (c) Compliance. (d) Frequency.

Steel consumption and change rate at different thresholds.

T | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 |

Steel consumption/kg | 46.78 | 37.79 | 32.43 | 28.24 | 24.54 | 20.59 |

Change rate% | −19.22 | −14.18 | −12.92 | −13.10 | −16.10 |

Table

Figure

It can be seen in Figure

It can be seen from Figure

Figure

According to the above analysis, it is clear that when the density threshold is 0.5, the steel consumption of the optimized joint will be greatly reduced, and the structural performance is better. To obtain a topological form with better mechanical properties, the density threshold is taken as 0.5, and the corresponding structural, topological density distribution is shown in Figure

Topological density distribution of spherical joints with a threshold value of 0.6. (a) Top view. (b) Sectional half-structure view. (c) Three-dimensional view.

It can be seen that the optimized structural topology is bidirectionally symmetrical along the central axis, with a large hole in the middle and a small hole on each side. The structure does not have directionality. Figure

As shown in Figure

Optimized and Smoothed joint. (a) Initial optimization. (b) Smoothed optimization.

For spatial grid structures, hollow sphere joints are commonly used. To verify the performance of the optimized joints, a welded hollow sphere joint is created. The thickness of the hollow sphere is 36.5 mm, and the material properties of both joints are the same. The steel consumption and mechanical properties of the two joints are compared in Table

Comparison of the joints’ performance.

Parameters | Hollow joint | Optimized joint | Difference% |
---|---|---|---|

Weight/kg | 358.6 | 359.8 | 0.3 |

Axial condition stress/MPa | 445.0 | 276.6 | −37.8 |

Axial condition displacement/mm | 1.12 | 0.56 | −50.0 |

Shear condition stress/MPa | 259.0 | 144.4 | −44.2 |

Shear condition displacement/mm | 0.63 | 0.41 | −34.9 |

Bending moment condition stress/MPa | 224.3 | 146.4 | −34.7 |

Bending moment condition displacement/mm | 0.17 | 0.20 | 17.6 |

Combined condition stress/MPa | 369.6 | 229.4 | −37.9 |

Combined condition displacement/mm | 0.936 | 0.407 | −46.5 |

1st frequency/Hz | 159.13 | 213.91 | 34.4% |

Table

Through a comparison of the change rate of performance under the action of the three load conditions, it can be seen that the change rate of stress and displacement of the optimized joint under the action of axial, shear, and bending moment loads are −37.8% and −50.0%, −44.2% and −34.9%, and −34.7% and 17.6%, respectively, and the structural stress and displacement change rates under combined conditions are −37.9% and 46.5%, respectively. For spatial grid structures, the main mode of loading is axial loading. Through topology optimization, the weight of the axial load obtained using the AHP is 0.731. Through the process of topology optimization design, more materials are allocated to bear the axial load. The obtained joint can resist axial forces significantly better than it can resist shear forces and bending moments. The 1st frequency of the optimized joint is increased by 34.4%. In comparison to the traditional hollow spherical joint, the optimized joint has a more uniform stress distribution and greater rigidity.

Based on the above analysis, it can be concluded that in comparison to the hollow spherical joint, the joint designed through topology optimization has a better static and dynamic performance. Compared with the traditional single-objective topology optimization design, multiobjective optimization is more suitable for engineering practice since it considers both static and dynamic design objectives.

The most widely used casting method for the optimized joint is additive manufacturing [

Import the STereoLithography (STL) format file into the 3D printer, design the scanning path, and control the movement of the laser scanner and lifting platform through the generated data;

A laser beam irradiates the surface of the liquid photosensitive resin through the scanner controlled by a numerical control device according to the designed scanning path so that a layer of resin is solidified in the specific area of the surface. When processing of one layer is completed, the first section of the part is generated.

The lifting platform is lowered a certain distance and a new layer of liquid resin covers the cured layer. The second layer is scanned, and then the second cured layer is firmly bonded to the previously cured layer. In this way, a complete 3D workpiece prototype was formed through layer-by-layer superposition.

Finally, the prototypes are taken out of the resin and cured.

However, currently, the metal powder required for additive manufacturing is relatively expensive. To reduce this cost, researchers have proposed some methods to combine additive manufacturing with traditional castings, such as EPC-sand casting technology [

In this study, the multiobjective topology optimization method was applied to the topology optimization of the design of spatial-structure joints, and a design scheme that satisfied the minimum requisite compliance and maximized the low-order frequency was proposed. The main conclusions are as follows:

The compromise programming method is applied to the multiobjective topology optimization of spatial spherical joints, which comprehensively considers the multiobjective optimization of stiffness and frequency under static axial, shear, and bending moment loads, introducing the analytic hierarchy process to obtain the weight coefficient of load conditions. This mitigates the limitation of considering a single objective in the traditional topology optimization of joints and the empirical value of load conditions.

Minimizing the compliance and maximizing the first three-order frequencies to construct a multiobjective optimization function is comprehensively analyzed, taking the volume fraction as the constraint. By conducting multiobjective topology optimization for the joints, a new topology for the joint is obtained, which conforms to the load transmission path. The optimized joint has high static and dynamic performance and mitigates the oscillation problem caused by frequency alternation.

The optimized structure is smoothed to obtain an optimized joint with reasonable stress properties and improved aesthetics. The mechanical properties of the optimized joint and that of the welded hollow sphere joint are compared. In comparison to the hollow sphere joint, the optimized joint has better static and dynamic performance and uses the same amount of steel. In comparison to the traditional single-objective topology optimization design, multiobjective optimization is more suitable for engineering applications. The rationality of multiobjective topology optimization is verified, and an effective solution is provided for the lightweight design of joints.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work described in this paper has been funded in part by the National Natural Science Foundation of China (No. 51768024).