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Laser welding (LW) becomes one of the most economical high quality joining processes. LW offers the advantage of very controlled heat input resulting in low distortion and the ability to weld heat sensitive components. To exploit efficiently the benefits presented by LW, it is necessary to develop an integrated approach to identify and control the welding process variables in order to produce the desired weld characteristics without being forced to use the traditional and fastidious trial and error procedures. The paper presents a study of weld bead geometry characteristics prediction for laser overlap welding of low carbon galvanized steel using 3D numerical modelling and experimental validation. The temperature dependent material properties, metallurgical transformations and enthalpy method constitute the foundation of the proposed modelling approach. An adaptive 3D heat source is adopted to simulate both keyhole and conduction mode of the LW process. The simulations are performed using 3D finite element model on commercial software. The model is used to estimate the weld bead geometry characteristics for various LW parameters, such as laser power, welding speed and laser beam diameter. The calibration and validation of the 3D numerical model are based on experimental data achieved using a 3 kW Nd:Yag laser system, a structured experimental design and confirmed statistical analysis tools. The results reveal that the modelling approach can provide not only a consistent and accurate prediction of the weld characteristics under variable welding parameters and conditions but also a comprehensive and quantitative analysis of process parameters effects on the weld quality. The results show great concordance between predicted and measured values for weld bead geometry characteristics, such as depth of penetration, bead width at the top surface and bead width at the interface between sheets, with an average accuracy greater than 95%.

The reduction of energy consumption and atmospheric pollution as well as the improvement of safety have led the automotive industry reconsider their design methods in order to decrease the weight of vehicles by using high performance materials. The tailored welded banks are now the major components constituting the new vehicle body. This design approach aims to optimize the weight of the vehicle structure by joining sheets from various nuances of high resistance material and different thicknesses. Therefore, the quantity of welds in the car body was more than doubled in the last years. In these conditions, the aptitude of the welds to resist to diverse solicitations remains the main concern of engineers. This aptitude is conditioned largely by the weld cross section geometry characteristics [

The modeling of laser welding has evolved through several stages, starting with fundamental heat equations based simple analytical models to the most complex numerical models taking into account practically all physical phenomena involved in the welding process [

More recently, the combination of two types of heat sources, such as Gaussian conical combined with cylindrical heat source is proposed [

Another study reported that convection heat transfer plays a very important role in the welding process and considered the most important factor influencing the shape and the geometry of the weld cross-section [

Oussaid et al. conducted an experimental investigation to evaluate the effects of various laser welding parameters on the welding quality [

Indeed, when a fast and efficient prediction model is needed, ANN offers many advantages especially in the case of computationally intensive predictions and real-time applications where numerical models are very slow and not adapted. ANN models have been used with success to model many welding process. However, ANN model can accurately predict the weld geometry as well as the change in weld pool profiles like between conduction mode and keyhole mode only if trained with good and appropriate data. Moreover, producing an accurate and flexible ANN model requires very large data to ensure efficient ANN learning and validation processes. On the other hand, the generation of the needed data using experiment is relatively long and expensive. Therefore, experimentally validated 3D finite element method based models can be used to generate acceptable and cost effective data.

The present paper describes the development of a 3D numerical model to be used for generating the additional data needed to build the most accurate, flexible and efficient ANN based model for predicting the weld bead geometry characteristics in laser overlap welding of low carbon galvanized steel. An integrated approach combining 3D numerical modeling and experimental validation is proposed to produce the most consistent predictions. The welding process simulation is carried out by 3D finite element model using Comsol Multiphysics software. The numerical modelling is built on simple heat transfer model based on a moving heat source in a finite medium volume to evaluate the temperature field, and thus to estimate the melting zone dimensions and the weld geometry characteristics for various welding parameters and conditions. The model is based on an improved adaptive 3D heat source used to simulate both key-hole and conduction laser welding process mode. The volumetric heat source is calibrated using specific factors defined as function of the laser parameters to adapt respectively the weld bead width and the absorption depth of the laser beam. The latent heat of fusion is taken into account by adapting the material temperature dependent specific heat in the temperature range situated between solidus and liquidus. The calibration and validation of the model are achieved using experimental data produced on a commercial 3 kW Nd:Yag laser system performed according to a structured experimental design and confirmed statistical analysis tools. The results reveal that the 3D numerical model is able to provide not only a consistent and accurate prediction of the weld bead geometry characteristics under variable welding parameters and conditions but also a comprehensive analysis of the average effects of the welding parameters on the weld quality. The results show great concordance between predicted and measured values of the weld characteristics.

The paper is organized in four parts as follows. The first presents an introduction, a brief review and the objectives of the study. The second outlines succinctly the numerical method, describes the model, the governing equations and the simulation parameters and conditions. The third discusses the numerical and experimental results. The fourth presents the conclusion of the work and outline the potential direction of future research.

The 3D simulation model of the laser welding process in overlap joint configuration consists to irradiate two overlapped low carbon galvanized steel sheets with respective thickness of 1 and 2 mm, as presented in

The proposed model is however built under various simplifying assumptions such as: 1) the influence of zinc vapor diffusion on the thermal properties of

weld zone is ignored. 2) The experimental investigations revealed a relatively limited effect of the gap size on different weld geometric characteristics. The maximum observed contribution of the gap in these characteristics variation represents approximately 5%. Therefore, the gap size is not considered as variable in the present model. 3) The material is considered homogeneous and isotropic. 4) The welded parts are considered free from any geometrical imperfection, and finally 5) Fluid flows, Marangoni effect and the latent heat of vaporization are not considered in order to simplify the model and to avoid long simulations.

The proposed model is based on conduction heat transfer including melting phase change. Equation (1) called Fourier’s second law, describes the transient heat conduction in the solid. Solving this well-known equation permit to obtain the temperature field distribution in welded sheets at specific time and spatial coordinates.

ρ C P d T d t = k ∇ 2 T + Q ( x , y , z , t ) (1)

where ρ, k, C_{p} and Q Denote respectively the material density, the thermal conductivity, heat capacity and the heating power per unit volume.

The melting phase change is included into the model both in term of latent heat of fusion and the temperature dependant properties. The enthalpy method is used to model the phase transition by modifying the temperature dependent heat capacity as denoted in Equation (2) and illustrated in

C P = C P * ( T ) + 1 Δ T √ π e − T − T f Δ T 2 L f (2)

With T f the melting point, L f tent heat of fusion and ∆T a temperature range of solid/liquid phase change set to 72 K.

At the initial time, sheets temperature is uniform and set at the room temperature T_0 = 295 K. Heat exchange between the welded parts and their external environment is integrated into the numerical model as a heat loss, which is expressed by convection and radiation. The convection occurs in the top surface and frontal surfaces. The convection is expressed as:

Q c o n v = h ( T − T 0 ) (3)

With h the natural convection coefficient and T 0 the room temperature. The radiation is occurred at the top surface and expressed by the Stephan Boltzmann law:

Q r a d = σ ε ( T 4 − T 0 4 ) (4)

With σ Stephan-Boltzmann constant and ε the material emissivity.

The bottom surface is supposed thermally insulated according to the following equation:

− n ( k ∇ T ) = 0 (5)

Since the geometry and laser beam are symmetric across the y-axis, the symmetry boundary condition suggests that only half of the geometry is modeled. This symmetry simplification ignores the laser beam misalignment during welding and the geometrical imperfections of the parts. The symmetry simplification has drastically reduced the computational time.

The identification of adequate heat source is the most important step in welding process model building. According to the literature, a conical heat source with Gaussian distribution is the simplest and the most appropriate for predicting the shape of the laser weld bead cross-section [

In General, to differentiate between the two welding modes during modeling, simply assign a heat source to each mode. A 2D surface heat source (Equation (6)) to the conduction mode and a 3D volumetric heat source (Equation (7)) to the keyhole mode. The transition between the two modes must take place at the vaporisation temperature, but this increases the model complexity, because it can happen, for the same laser conditions, to start the simulation with 2D heat source and finish with 3D heat source.

Q ( x , y , t ) = Q 0 A ( 1 − R ) 2 π r 2 ⋅ exp ( − ( x − x 0 ) 2 2 r 2 − ( y − v t ) 2 2 r 2 ) (6)

Q ( x , y , z , t ) = Q 0 A ( 1 − R ) 2 π r 2 ⋅ exp ( − ( x − x 0 ) 2 2 r 2 − ( y − v t ) 2 2 r 2 − A c | z | ) (7)

Since the difficulty to set the transition threshold according to laser parameters, an adaptive 3D heat source, moving in the y axis direction, is used to simulate both keyhole and conduction mode (Equation (8)). This volumetric heat source is calibrated by introducing two coefficients (m and n) to adapt respectively the weld bead width and the absorption depth of the laser beam. These coefficients are related to the laser parameters with the most significant contribution to the variation of weld characteristics [

Q ( x , y , z , t ) = Q 0 A ( 1 − R ) 2 π r 2 exp ( − ( x − x 0 ) 2 ( m r ) 2 − ( y − v t ) 2 2 r 2 − n A c | z | ) (8)

where Q 0 , v , A , R and r denote respectively laser power, welding speed, material absorptivity, material reflectivity and laser beam radius.

The coefficients m and n are estimated as function of the laser parameters using the following empirical relationships.

− n ( k ∇ T ) = 0 m = 1.194 − 17310 − 6 Q 0 [ W ] + 18.33 v [ m s ] + 649 d [ m ] (9)

n = 0.565 − 8310 − 6 Q 0 [ W ] + 9.44 [ m s ] + 123 d [ m ] (10)

The geometry is discretized using an extremely refined tetrahedral mesh at the neighborhood of the weld line and large mesh elements in the rest of the geometry. A mesh convergence study is conducted to determine the optimal mesh size, by simulating the welding process using the same laser parameters setting (2500 [W] laser power, 55 [mm/s] welding speed and 395 [μm] laser beam diameter) and different mesh sizes around the weld line.

As illustrated in

In order to validate the proposed model, typical ASTM A635 CS galvanized steel sheets with A40 coating type commonly used in the automotive industry are used to perform the overlap joint welds. The temperature dependant properties of this low carbon steel, which are density, thermal conductivity and specific heat, integrated in model are summarized in

A series of simulations and tests of validation were carried out simultaneously, according to comprehensive and structured experimental design. As indicated in

Temperature [K] | Density [kg/m^{3}] | Specific heat [J/(kg∙K)] | Thermal conductivity [W/(m∙K)] |
---|---|---|---|

298 | 7863 | 446 | 74.7 |

962 | 7639 | 903 | 35.7 |

996 | 7648 | 886 | 34.1 |

1039 | 7647 | 882 | 32.2 |

1126 | 7661 | 600 | 28.3 |

1273 | 7587 | 623 | 29.6 |

1723 | 7317 | 729 | 33.6 |

1785 | 7287 | 738 | 34.2 |

1804 | 7132 | 785 | 34.8 |

1806 | 7030 | 817 | 35 |

2503 | 6449 | 795 | 35 |

Property | Symbol | Value | Unit |
---|---|---|---|

Temperature liquidus | T_{L} | 1806 | K |

Temperature solidus | T_{S} | 1734 | K |

Latent heat of fusion | L_{f} | 244 | kJ/kg |

Material emissivity | ε | 0.6 | |

Material Absorptivity | A | 800 | 1/m |

Material reflectivity | R | 0.3 | - |

Laser Parameter | Symbol | Unit | Level 1 | Level 2 | Level 3 |
---|---|---|---|---|---|

Power | Q0 | W | 2000 | 2500 | 3000 |

Speed | v | mm/s | 40 | 55 | 70 |

Diameter | d | μm | 300 | 395 | 490 |

power, welding speed and laser beam diameter). These parameters and levels are fixed following a series of preliminary tests [

The total computation time depends on the welding speed: time_end = L/v, while the time step depends on the laser beam diameter, such the position of the laser beam at time t_i is located at a distance s = 2/3 *d from its previous position, ie at time t_(i-1).

Where L is the model geometry length, d the laser beam diameter and v the welding speed.

The contours of the weld cross sections are determined from the temperature fields based on the location of points reaching the liquidus temperature. The weld penetration depth and the weld seam width, respectively at the surface and at the interface are measured on the cross sections of the weld using an adapted Matlab indexing algorithm.

A number of methods are available to evaluate the models performance by estimating the prediction errors. It is generally recommended to use several criteria to

obtain a satisfactory prediction error estimate. The following four traditional estimates from forecasting techniques are often used: 1) the coefficient of determination (R2), 2) the Mean Absolute Error (MAE), 3) the Mean Absolute Percent Error (MAPE), 4) the Mean Root Squared Error (MRSE), and 5) the Relative Mean Root Squared Error (RMRSE). They can be defined mathematically as:

R 2 = 1 − ∑ i = 1 n ( y i − y ^ i ) 2 ∑ i = 1 n ( y i − y ¯ ) 2 (11)

M A E = 1 n ∑ i = 1 n | y i − y ^ i | (12)

A P E = ( 1 n ∑ i = 1 n | y i − y ^ i y i | ) 100 ( % ) (13)

M R S E = 1 n ∑ i = 1 n ( y i − y ^ i ) 2 (14)

R M R S E = M R S E y ¯ 100 ( % ) (15)

The combination of these statistical tools provides a very clear picture of the model performance by providing a satisfactory prediction error estimates. The performance of the model, based on the selected five criteria, is presented at ^{2} less than 90% the three weld characteristics. R^{2} reaches 96%. The maximum observed MAPE remains less than 7.5% representing about 7%, 2.5% and 7.5 for DOP, WS and WI respectively. The maximum observed RMRSE remains less than 3.5% representing 3.04%, 1.09% and 3.32 for DOP, WS and WI respectively. The best model performance is obtained for WS with prediction errors 3 times lower than for DOP and WI. The highest errors are obtained for WI. This is explained by slight variation of the gap during the experiments. Globally, with such good results, the model is considered adequate to be used without difficulty in predictive strategies.

Another dimension of the model evaluation consists to compare the laser parameters effects on the weld bead geometry characteristics extracted from ANOVA performed on the L_{9} experimental data with those extracted from ANOVA performed on simulation data. The simulation data are generated from a structured L_{27} factorial design using the same parameter as in the experimentation. The choice of L_{27} is motivated by the possibility to evaluate the effect of the laser parameters interaction on the variation weld bead geometry characteristics. The interaction cannot be evaluated by a fractional design.

R^{2} | MAE | MAPE | MRSE | RMRSE | |
---|---|---|---|---|---|

DOP | 90.94% | 146.33 | 6.94% | 60.22 | 3.04% |

WS | 96.09% | 30.56 | 2.44% | 13.84 | 1.09% |

WI | 94.12% | 102.11 | 7.28% | 40.75 | 3.32% |

Source | DF | DOP | WS | WI | |||
---|---|---|---|---|---|---|---|

% C | F-Value | % C | F-value | % C | F-Value | ||

P | 2 | 28.53 | 960.01 | 47.03 | 796.22 | 28.24 | 82.13 |

D | 2 | 12.61 | 424.09 | 10 | 169.25 | 22.23 | 71.45 |

V | 2 | 56.14 | 1888.73 | 40.91 | 692.63 | 45.18 | 145.23 |

P*D | 4 | 0.13 | 2.23 | 0.28 | 2.41 | 1.50 | 2.42 |

P*V | 4 | 2.33 | 39.24 | 1.34 | 11.32 | 0.48 | 0.78 |

V*D | 4 | 0.14 | 2.3 | 0.21 | 1.76 | 3.83 | 0.28 |

Error | 18 | 0.12 | 0.24 | 1.24 | |||

Total | 26 | 100 | 100 | 100 |

simulation data and an L27 simulation data. These graphs reveal small differences between the three designs. The largest difference (less than 10%) is observed for the power contribution in WI variation. This is probably due to the fact of not considering the gap in the simulations.

As established following the experimentation phase, the welding speed and the laser power are the most influential factors on the weld bead geometric characteristics, while the laser beam diameter effect remains relatively limited. The evaluation of the laser parameters interaction shows also non-significant effects. The total contribution of all interactions does not exceed 6%. It represents 2.6%, 1.83% and 5.81 for DOP, WS and WI respectively. These results are validated by the graphs of effects of the laser parameters on the weld bead geometry characteristics presented in

The present paper presents an integrated approach for building a forecasting 3D finite element model able to predict weld bead geometry characteristics for laser welding of low carbon galvanized steel in overlap configurations. An improved adaptive 3D heat source is used for simulating both keyhole and conduction mode. The implementation of the model is supported by a heat source calibration technique using specific modelling factors defined as function of the laser parameters to adapt respectively the weld bead width and the absorption depth of the laser beam.

A commercial 3 kW Nd:Yag laser system, a structured experimental design and improved statistical analysis tools are used to evaluate the modelling approach and to confirm the prediction model accuracy. Extensively numerical simulations carried out through 3D finite element method reveal great concordance between modelling and experimental results. The comparison of predicted and measured weld geometric characteristics reveals an average accuracy greater than 95%. The prediction errors originate principally from experimental errors as well as from the considered assumptions. Globally, the results demonstrate that the numerical simulation can effectively lead to a consistent and accurate weld bead geometry characteristics prediction model under variable welding parameters and conditions and provide an appropriate framework for comprehensively qualitative and quantitative analysis of the effects of welding process parameters on the weld quality.

With the encouraging results obtained using this model, the laser overlap welding of low carbon galvanized steel will be the subject of additional and intensive investigations to generate more simulation and experimental data in order to develop an efficient ANN predictive modelling approach.

The authors declare no conflicts of interest regarding the publication of this paper.

Oussaid, K. and El Ouafi, A. (2019) A Three-Dimensional Numerical Model for Predicting the Weld Bead Geometry Characteristics in Laser Overlap Welding of Low Carbon Galvanized Steel. Journal of Applied Mathematics and Physics, 7, 2169-2186. https://doi.org/10.4236/jamp.2019.710149