Finite element (FE) coupled thermal-mechanical analysis is widely used to predict the deformation and residual stress of wire arc additive manufacturing (WAAM) parts. In this study, an innovative single-layer multi-bead profile geometric modeling method through the isosceles trapezoid function is proposed to build the FE model of the WAAM process. Firstly, a straight-line model for overlapping beads based on the parabola function was established to calculate the optimal center distance. Then, the isosceles trapezoid-based profile was employed to replace the parabola profiles of the parabola-based overlapping model to establish an innovative isosceles trapezoid-based multi-bead overlapping geometric model. The rationality of the isosceles trapezoid-based overlapping model was confirmed by comparing the geometric deviation and the heat dissipation performance index of the two overlapping models. In addition, the FE-coupled thermal-mechanical analysis, as well as a comparative experiment of the single-layer eight-bead deposition process show that the simulation results of the above two models agree with the experimental results. At the same time, the proposed isosceles trapezoid-based overlapping models are all straight-line profiles, which can be divided into high-quality FE elements. It can improve the modeling efficiency and shorten the simulation calculation time. The innovative modeling method proposed in this study can provide an efficient and high-precision geometric modeling method for WAAM part FE coupled thermal-mechanical analysis.

Wire arc additive manufacturing (WAAM) is a metal additive manufacturing (AM) technology that uses the electric arc as the heat source to melt the metal wire and add material layer by layer along the preset path [

However, the high material deposition efficiency of WAAM technology is caused by the high heat input that occurred in this process. Moreover, the moving heat source results in repeated localized heating and uneven cooling, which will lead to complex thermal cycles, distinctive local thermal gradients, and high residual stress in the WAAMed part. The high residual stress reduces the mechanical properties and geometric accuracy of the formed parts and even provokes pronounced distortions and cracks. This is considered one of the significant challenges of the WAAM process [

The finite element (FE) coupled thermo-mechanical analysis can reflect the transient thermodynamic information during the WAAM process in real time [

The geometric modeling of the deposition layer is essential for FE analysis. In most current studies, the cross-section profile of a single weld bead is usually simplified to a rectangle function. Sun et al. [

In this study, to take into account both geometric accuracy and computational efficiency, an innovative single-layer multi-bead geometric modeling method based on the isosceles trapezoid function was proposed to establish the FE model of the WAAM process. Firstly, a straight-line overlapping model was selected based on a parabola function to calculate the optimal overlapping center distance. Then, a multi-bead isosceles trapezoid-based overlapping model was established based on the optimal center distance and the isosceles trapezoid model of a single bead. The comparisons of geometrical deviation and the heat dissipation performance index between two overlapping models were performed to verify the validity of the trapezoid-based overlapping model. In addition, the computational efficiency and error of the isosceles trapezoid-bead overlapping model were investigated through single-layer eight-bead deposition numerical simulation and corresponding experiment. The main innovations of this study are as follows: 1) A single-layer multi-bead overlapping model based on the isosceles trapezoid function was used to establish the finite element thermodynamic coupling simulation model of WAAM. 2) The proposed modeling method can provide a reference and technical idea for high efficiency and high precision modeling of WAAM finite element thermodynamic coupling simulation and other metal additive manufacturing technologies.

During the FE analysis of the WAAM process, the accuracy of a geometric model is directly related to simulation accuracy and validity of the calculation results. Ding et al. [

The principle of the straight-line overlapping model can be illustrated in _{1}(_{n}(_{2} is the leftmost point of the _{2}(_{1}_{1}(_{2}. Point _{2} is the peak point of the _{2}_{1} is the intersection of the _{1}_{2}_{1} and _{2}.

The first two bead functions are defined by

The optimal overlapping center distance of the parabola-based single-bead profile geometry model is determined by the equal area criterion. As shown in _{1}, _{1}, _{2}, _{2}, _{1,} and _{2} are defined by _{1}, _{2}, _{2,} and _{2} can be expressed as

According to the ideal overlapping, the overlapping area of the two parabolas equals the area of the critical valley. According to this, the area

The curved-side triangle area

The function

When _{1} = 3

Number | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

0.65 |
0.7 |
0.75 |
0.8 |
0.86 |
0.9 |

The cross-section profiles of the specimens are shown in

In finite element geometric modeling of WAAM deposition layer, the cross-section profile of a single weld bead is usually simplified to a rectangle function or fitted by curvilinear equation (such as parabola function, cosine function, and arc function). However, the rectangle function method is an oversimplified method which is difficult to guarantee the accuracy of modeling and calculation. In the meantime, the curvilinear function method has high precision. However it has low efficiency for finite element geometric modeling, and it is difficult to partition into high-quality finite element mesh. Then, the isosceles trapezoid-based profile model is proposed to replace the parabola-based profile model in this study (as shown in _{t}

_{t}

The area of the pentagon

To guarantee the equivalence, the cross-sectional area of the parabola model must be equal to the cross-sectional area of the isosceles trapezoid-based model. This means that the area of the pentagon

Consequently, _{t}

Then, _{t}

The parabola-based overlapping model is a fitting model with high accuracy. In contrast, the isosceles trapezoid-based overlapping model is a simplified model. Therefore, compared with the parabola-based model, the isosceles trapezoid-based model has a particular geometrical deviation. As shown in _{d} can be defined by the area of the non-overlapping area between the two models as:

_{d}

The average geometric deviation _{d(n)}

Previous researches [^{2}.

During the deposition process, the heat dissipation mechanism of the weld beads to the atmosphere includes convection and radiation. The heat dissipation performance index is directly related to the contact area between the beads and air [

According to the geometric relationship in _{R}_{(P)} and _{R}_{(T)} values can be expressed by

The error _{D}

The three-dimensional diagram and contour map of the error of the heat dissipation performance index at n = 2 are shown in

The three-dimensional diagram and contour map of the error of the heat dissipation performance index at n ≥ 3 are shown in

To sum up, the geometric deviation and the error of heat dissipation performance index of the overlapping model between the isosceles trapezoid-based model and the parabolic-based model are less than 7.5%, which means that the overlapping modeling method based on the isosceles trapezoid function has acceptable accuracy error and is suitable for establishing the FE coupled thermal analysis model.

To verify the validity of the proposed innovative FE geometric modeling method, a single-layer eight-bead deposition experiment and simulation are performed.

The single-layer eight-bead coupled thermo-mechanical models based on the isosceles trapezoidal and parabola functions are constructed in ABAQUS. The FE model is displayed in

As indicated in

where _{f}_{r}_{f}_{r}

Parameter | Value |
---|---|

Welding speed | 4 mm/s |

Welding time of each layer | 25 s |

Interval time | 120 s |

Cooling time | 10000 s |

Initial temperature | 20°C |

Parameter | Value |
---|---|

5 mm | |

4 mm | |

_{f} |
4 mm |

_{r} |
11 mm |

_{f} |
0.6 |

_{r} |
1.4 |

2300 W | |

Efficiency coefficient | 0.7 |

Parameter | Value |
---|---|

Convective heat transfer coefficient | 20 W/(m^{2}⋅°C^{4}) |

Radiated emissivity | 0.7 |

Ambient temperature | 20°C |

During the deposition experiment, a thermocouple sensor is set at point A (as shown in

In order to verify the quantitative comparison results of the residual stress on the substrate obtained by the experiment and simulation, the residual stress values of four points on the substrate (as shown in

The residual stress is extracted from paths on the surface of the substrate and bead to compare the residual stress simulation results of the two simulation models.

In addition, the calculation of the parabola-based FE model takes 580 min, while the trapezoid-based FE model takes 403 min. This means that the computation efficiency is improved by 30.5%, and the isosceles trapezoid-based model is computationally more effective than the parabolic-based model. Namely, the FE geometric modeling method based on the isosceles trapezoid proposed in this study has significant advantages in enhancing computational efficiency.

A geometric modeling method based on the isosceles trapezoidal curve for the multi-bead overlapping wire arc additive manufacturing deposition is proposed to replace the fitting curve based on the parabola modeling method. The effectiveness of the proposed modeling method based on the isosceles trapezoidal curve is verified by a variety of comparison methods.

1. The max average geometric deviation quotient of the isosceles trapezoid-based model is 7.5%. The geometric deviation and the error of the heat dissipation performance index of the overlapping model between the isosceles trapezoid-based model and the parabolic-based model are less than 7.5%, which means that the overlapping modeling method based on the isosceles trapezoid function has a high degree of geometric consistency and acceptable accuracy error than the parabolic-based model.

2. The thermal cycling curves, residual stress of different points of FE simulation and experiments, and residual stress distribution of FE simulation along the various paths indicate that the two models are highly consistent together and have high calculation precision. The isosceles trapezoid-based overlapping model can effectively replace the traditional parabolic-based model in the FE-coupled thermal analysis.

3. Compared with the fitting curve model, the computational efficiency of the finite element simulation of the isosceles trapezoid-based model is increased by 30.5% under the same calculation conditions. This means that the isosceles trapezoid-based model has significant advantages in computational modeling.

4. The proposed isosceles trapezoid-based modeling method can provide a reference and technical idea for high efficiency and high precision modeling of the finite element thermodynamics of the WAAM coupling simulation and other metal additive manufacturing technologies.

_{n}

Peak point of the _{n}(

_{f}

Length of the front ellipsoid of the heat source

_{r}

Length of the rear ellipsoid of the heat source

_{n}

Intersection point between the tangent line of _{n}(_{n−1}(

Width of the heat source

_{n}

Intersection point of _{n}_{n−1}

Depth of the heat source

_{n}

Bottom point of the pipeline of the _{n}

Overlapping center distance

_{n}

Leftmost point of the _{n}(

_{n}

Rightmost point of the _{n}(

_{f}

Fraction factor of the heat flux in the front parts

_{r}

Fraction factors of the heat flux in the rear parts

_{n}

Left endpoint of _{n}(

_{n}

Intersection point of _{n}_{n} and _{n−1}_{n}

Height of single bead

_{t}

Overlapping height of the isosceles trapezoid model

_{d}

Geometric deviation of the isosceles trapezoid-based overlapping model

_{n}

Intersection points of _{n}(_{n−1}(

_{R(P)}

Heat dissipation performance index of the parabola-based overlapping model

_{R(T)}

Heat dissipation performance index of the isosceles trapezoid-based overlapping model

_{n}

Intersection point of _{n}_{n} and _{n−1}_{n}

_{n}

Right endpoint of _{n}(

_{n}

Bottom point of the pipeline of the _{n}

_{n(x)}

Profile function of bead

Power input of FE model

Width of single bead

_{t}

Length of the upper side of the isosceles trapezoid model

The authors are sincerely grateful to the anonymous referees and the editor for their time and effort in providing constructive and valuable comments and suggestions that have led to a substantial improvement in the paper.

This research was funded by the National Natural Science Foundation of China (Grant No. 51705287) and the Scientific Research Foundation of Hubei Provincial Education Department (Grant No. D20211203).

The authors confirm contribution to the paper as follows: Study conception and design: Xiangman Zhou, Jingping Qin, Seyed Reza Elmi Hosseini; data collection: Jingping Qin, Zichuan Fu, Min Wang; analysis and interpretation of results: Xiangman Zhou, Jingping Qin, Youlu Yuan; draft manuscript preparation: Zichuan Fu, Junjian Fu, Haiou Zhang. All authors reviewed the results and approved the final version of the manuscript.

None.

The authors declare that they have no conflicts of interest to report regarding the present study.