# Using Genetic Algorithms with Multi-Objective Optimization to Adjust Finite Element Models of Welded Joints

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## Abstract

**:**

## 1. Introduction

## 2. Determination of the Optimum Material Parameters for the Finite Element Method (FEM) by Genetic Algorithms (GA)

## 3. Genetic Algorithms with Multi-Objective Functions

_{1}, …, x

_{n}) and that we wish to find a solution (x

_{1}, …, x

_{n}) that maximizes or minimizes that function or experiments. In the current paper, this function may be the temperature field and the field of angular distortion of the welded joint FE models. First, a basic genetic algorithm creates randomly an initial generation of N individuals. Each solution is encoded in chromosomes and evaluated using, in this case, the FE model G0 (x

_{k1}, …, x

_{kn}) in such a way that the best individuals will be those whose result approaches the desired maximum or minimum result. A new generation is created and evaluated based on the best individuals (those that provide the best solution when the intention is to achieve a single objective, or to achieve several objectives simultaneously). This process is repeated until the best individual in each generation has a value that is close to the desired optimum. Each generation of individuals Gn when n ≠ 0 is usually created from the previous generation. Approximately 25% of new individuals have the best (i.e., the “fittest”, much like the theory of natural selection). A higher percentage of the individuals (e.g., 75%) result from cross-over of chromosomes from the previous generation (sexual reproduction). The remainder (10%) arises from random alteration of selected genes of the previous generation’s chromosomes (mutation). The adjustment process and the proposed methodology for development of the cross-over and mutations for the FE models that are proposed are described in Section 5.2 and Section 5.3.

## 4. Experimental Procedure

_{2}) and 80% argon (Ar). Figure 1a shows the welding parameters that were commonly used in GMAW. In addition, the distributed Heat Flux [KJ/mm] was calculated by the following equation:

#### 4.1. Measuring the Temperature Field

#### 4.2. Measurement of Angular Distortion

#### 4.3. Characterization of the Welded Joint and Measurement of Weld Bead Geometry

## 5. Finite Element Proposed for Modeling the Welded Joints

#### 5.1. Parameterization of the Welded Joints FE Models

#### 5.2. Adjusting the Welded Joints FE Models

_{Temp}and J

_{Dist}based on Equations (2) and (3) for each of the individuals, generated as follows:

_{Temp}and J

_{Dist}were defined in a way that errors in the modeling of the temperature and in the modeling of angular distortion could were comparable and of a similar order of magnitude. The objective function J

_{Tempj}(also known as Relative Absolute Error (RAE)) was defined as the average difference between the temperature obtained from each of the key nodes (n) that belong to the FE models (T

_{TFE(i)}) and the temperature obtained from the thermographic camera (T

_{EXP(i)}) at the homologous key points of the samples. In this case, the key nodes, P1, P5, P6, P10, P11, and P15, were not considered, as was mentioned in Section 4.1 due to a desire to avoid errors in measuring the temperature due to the transient starting and ending of the welding process. Instead, only the key nodes P2, P3, P4, P7, P8, P9, P12, P13, and P14 were considered for calculation of the temperature field. The J

_{Tempj}functions that were obtained for the cited nine key points were examined at 2-s intervals from the beginning of the welding process at t = 0 s. to t = 100 s (See Figure 2). The subscript j represents the numbering of the welding cord that was studied. In this case, they range from 1 to 3. T

_{EXP(i)}is the mean of experimental temperatures that were obtained at each of the nine key points that were studied. The objective function J

_{Distj}was defined as the difference between the absolute value of the average angular distortion obtained from a selected group of key nodes del FE model (α

_{FEM}) and the distortion angle that was obtained from the homologous key points of the tested samples (α

_{EXP}). Weights w

_{Temp}and w

_{Dist}were applied to each of the objective functions J

_{Tempj}and J

_{Distj}, respectively, in order to consider the different adjustment requirements for the parameters of the proposed FE models. In this case, w

_{Temp}was defined as being of a greater importance in the adjustment of the temperature field for the proposed FE model, whereas w

_{Dist}was defined as being of greater importance in the adjustment of the angular distortion of that FE model. For example, if a minor error in the adjustment of the temperature field for the proposed FE model is permitted, but a greater error is permitted in the adjustment of the angular distortion of that model, the w

_{Temp}and w

_{Dist}may be assigned the following weight (major importance level assigned to the temperature field).

_{Temp}and w

_{Dist}may be assigned the following weight (major importance level assigned to the angular distortion).

_{Temp}and w

_{Dist}weights have each been assigned a value of 1.0.

_{Total}, which is the function to minimize. It is defined as follows:

#### 5.3. Crossovers and Mutations

_{Totalj}for each of the welded joints that were studied was evaluated by use of a script that was written in “R” language [50]. The best individuals of this generation “0” were those that had the lowest value of the objective function J

_{Totalj}. The next generations (Generation 1 and subsequent generations) were created by crossing and mutation, as described below:

- 30% of the individuals with the lowest objective function J
_{Totalj}values of the previous generation became parents of the new generation. - 60% of individuals were created by crossing selected parents. The crossing was implemented by a script in “R” language [50]. It consists of a change in a random number of bits of chromosomes of two randomly selected best individuals. First, it was necessary to obtain two complete chromosomes from two selected parents. Then, each chromosome was coded to binary code to perform the process of crossing. Then, a random number of crossings (1, 2, 3, or 4) were selected. Also, for each crossing a position number that defined the position of the initial bit of each crossing was randomly selected too. In the same way, the longitude of the number of bits that form the crossing part of the chromosome was selected. With this information, some bits of the first parent were selected and the remaining bits were selected from the second parent to create a new chromosome for the next generation. Figure 7 shows the first and second positions 1 and 2, and the lengths (longitude 1) and (longitude 2) that were selected in this case for the crossing of two individuals from the generation “0”. In this case, position 1 was equal to “14” and position 2 was equal to “117”. Longitude 1 was equal to “12” and longitude was equal to “14” (all position and longitude values are in bits). Finally, the new chromosomes were decoded from their binary code to generate the new offspring that were formed by the parameters to be adjusted in order to model the thermo-mechanical behavior of the welded joints.
- The remaining 10% of individuals were obtained by mutation. A random number of bits (between one and the number that defines the longitude of the chromosome) are defined and random positions of the chromosome are selected and are commuted until reaching the number of the initial number of bits selected. Meanwhile, it was important to always check that the new randomly generated values were within the established ranges. The aim of generating individuals by mutation was to find new solutions in areas that had not been explored previously. This procedure was repeated separately for each of the welded joints that were studied for several generations until the objective function J
_{Totalj}no longer increased significantly.

#### 5.4. Correlation-Based Feature Selection

_{s}is the heuristic merit of a feature subset S containing k features, $\overline{{r}_{cf}}$ is the mean feature-class correlation, and $\overline{{r}_{ff}}$ is the mean feature-feature intercorrelation. The numerator indicates how a group of features predicts a numeric class and the denominator of the redundancy among the features. This technique was applied as a feature selector because of its low computational cost and simplicity of use.

## 6. Case Study

#### 6.1. Results That are Based on CFS

_{Total}objective function that was obtained from the three welded FE simulations. The CFS filter algorithm was used on a sample of 100 individuals or 100 FE simulations when considering the aforementioned 13 input parameters, which were randomly generated. Table 3 shows the range of values that were considered for each of the input parameters of the FE models studied to generate the 100 individuals. The range was the same for each of the three welded joints FE models studied and was based on experience that was gained in previous research work [11,19,20].

_{Totalj}” for each of the three FE models of welded joints proposed, and the 13 inputs parameters. The results show the most significant parameters of the FE models that were selected in each case.

_{Total}

_{1}. The second column of this table shows the most significant features that were selected for the second proposed FE model (current = 210 amps and voltage = 28.0 volts). It can be seen that it is the “rear_lenght” parameter in the input that most influences the objective function J

_{Total2}. Finally, the third column of this table shows the most significant features that were selected for the third proposed FE model (current = 260 amps and voltage = 35.0 volts). In this case, “rear_lenght”, “contac_p_End”, and “forward_lenght” are the parameters that most influence the objective function J

_{Total}

_{3}.

#### 6.2. Results of FE Model Adjustment

_{Total}were initially set at the mean value of the range studied and were not adjusted by GA. The second way of adjusting the parameters of the three welded joints that were studied considered the totality of the 13 parameters that were proposed. Figure 8 shows the development of the J

_{Totalj}of the best individual in each generation. As was mentioned in Section 5.2, the development assumes the same level of importance for the thermal field as for the angular distortion (w

_{Temp}= 1 and w

_{Dist}= 1). Figure 8a–c shows the development of the J

_{Total}for the best individual when considering the 13 parameters that define the thermo-mechanical behavior of the welded proposed joints FE models. Also, these figures show the development of the J

_{Totalj}for the best individual when only those parameters of most influence are considered when they are analyzed and selected by the CFS filter algorithm. An examination of all the figures revealed that the first welded joint FE model that was studied provided lower values of J

_{Total}(See Figure 8a) than the remaining welded joint FE models that were proposed. In addition, it is seen in these figures that the minimum values reached for J

_{Totalj}always correspond in cases when the 13 parameters that define the thermo-mechanical behavior of the proposed welded joints FE models are considered. Also, Figure 8a provided lower values of J

_{Total}than the remaining FE models that were proposed for the various requirements that were studied. This means that the first FE model’s settings of the 13 parameters are higher than those of the parameters in the other FE models. It was also noted that the stabilization of J

_{Total}was achieved within a few generations (five generations) in all of the cases studied, although the number of parameters of the FE model is very high.

_{Total}no longer increased significantly for each of the generations and for the two different ways that were proposed. For the 5th generation of the first specimen studied, it can be seen that the value of J

_{Total}with the 13 parameters reaches a value of 2.268, but reaches a value of 2.3416 with a selection of parameters by the CFS filter algorithm. This difference between J

_{Total}increases as the power supplied to the welded joint increases (See Table 1). So, for example, J

_{Total}reaches a value of 3.0838 for the second specimen studied with the 13 parameters, but reaches a value of 3.1654 with the parameters that were selected with CFS filter algorithm. For the third specimen, J

_{Total}reached values of 6.4968 and 6.4918 with the 13 parameters and the selection of parameters by the CFS filter algorithm, respectively. This difference in the values reached by J

_{Total}suggests that the parameters of the proposed welded joints FE models provide a better fit when the power supplied to the welded joint is lower. Analyzing the values that were achieved for the 13 parameters in their adjustment process (with the CFS filter algorithm), the unique parameter that is common to the three welded joints studied (rear_length) decreases as the power supplied to the welded joint increases (7.8, 7.4, and 5.2). However, analyzing the values achieved for the 13 parameters in their adjustment process (with the 13 parameters), it is seen that: higher melting point values (melting_point), thermal contacts between the weld bead and both plates for the central zone, and the end zone of the bead (contac_P_center and contac_P_end) and thermal contact between plates 1 and 2 with the ground (contact_P_G) generally increased as the power that was supplied to the welded increased. For example, the melting_point varied from 1427 to 1429, whereas the values for contact_P_end, contact_P_center, and contact_P_G varied from 19 to 247, from 10 to 662, and from 66 to 182, respectively. Furthermore, the values that achieve the objective function increased as the power supplied increased. This means that the models provide a better fit when the power that is supplied to the welded joint is lower. Also, analyzing the results of the adjustment of the two procedures proposed, rear_length is the unique parameter that is common to all of them, and, therefore, is one of the most influential parameters in the adjustment of the proposed welded joint FE models. As previously mentioned, rear_length is defined mainly to determine the weld flux rates per unit of volume in the weld pools, according to the double ellipsoidal theory, and is the weld pool length in the welding direction (back of the melting point). In this case, according to Figure 6c and Table 5, the rear_length is always much greater than the forward_length. This suggests a greater area of weld flux rates per unit of volume contributed in the weld pools behind the melting point. In turn, this suggests that the adjustment of the models of FE of the welded joints proposed could be mainly focused on the adjustment of the contribution of the weld flux rates behind in the back of the melting point.

## 7. Conclusions

_{Totalj}was defined for each of the proposed welded joints as the combination of an objective function based on the temperature field J

_{Tempj}and an objective function based on the angular distortion J

_{Distj}. Weights w

_{Temp}and w

_{Dist}were applied to each of the objective functions J

_{Tempj}and J

_{Distj}, respectively, in order to consider different adjustment requirements for the parameters of the proposed FE models. w

_{Temp}was defined as having greater importance in the adjustment of the temperature field, whereas w

_{Dist}was defined as having greater importance in the adjustment of the angular distortion. In this case, both the field of temperatures and the angular distortion were considered to be equally important in adjusting the parameters of the proposed FE models. Value of 1.0 was assigned to w

_{Temp}and w

_{Dist}respectively. Two different ways to adjust the parameters were proposed. They were (1) using a feature selection of the parameters that most influence the objective function J

_{Total}after applying to (CFS) filter algorithm; and, (2) using all of the 13 parameters for the three welded joint FE models. In analyzing the results of the adjustment of the two proposed procedures, it was seen that: (1) The rear_length is the parameter that is common to all, and, therefore, is one of the most influential parameters; (2) The differences in the values reached by J

_{Total}suggest that the parameters of the welded joints FE models provide a better fit when the power supplied to the welded joint is lower. In this case, the rear_length is always much greater than the forward_length. This suggests a greater area of weld flux rates per unit of volume contributed in the weld pools of the back of the melting point. In turn, this suggests that the adjustment of the models of FE of welded joints proposed could be focused mainly on the adjustment of the weld flux rates that contributed in the back of the melting point. Finally, we can say that the methodology that is proposed in this paper could be valid for adjusting the parameters that define the thermo-mechanical behavior of any type of welded joint FE models. In that case, the parameters that are needed to define the conduction and convection phenomena, as well as the weld flux of that welded joints FE models, would be defined differently than what this work proposes, and would depend on the type of welded joint that is studied.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Common welding parameters used in gas metal arc welding (GMAW); and, (

**b**) The welding process used to manufacture the butt joint single V-groove.

**Figure 2.**(

**a**) Points that were considered for the temperature field measurement of the butt joint with single V-groove welded joint; (

**b**) The temperature recorded by the thermographic camera at second 6; and, (

**c**) The temperature recorded by the thermographic camera at second 20.

**Figure 3.**(

**a**) The distortion angle measured in the butt joint single V-groove; (

**b**) Coordinate-measuring machine used to determine the angular distortion; and, (

**c**) Detail of the Key points selected.

**Figure 4.**(

**a**) Image obtained by the microscope showing the different phase changes in the welded joint; (

**b**) Ferrite and perlite in the weld cord; (

**c**) Ferrite and perlite in the base metal cord; and, (

**d**) Reduced area of the Heat-Affected Zone (HAZ).

**Figure 5.**(

**a**) Details of the mesh size considered for the proposed Finite Element (FE) models; and, (

**b**) Detail of the starting and ending points of the weld bead studied, which also was included in the FE models.

**Figure 6.**(

**a**) Thermal contacts proposed in the FE model to solve the conduction phenomena; (

**b**) Thermal convection coefficients proposed in the FE model to solve the convection phenomena; and, (

**c**) Double Ellipsoidal Weld Flux showing the forward_length, rear_length, width and depth.

**Figure 7.**Script development in R language to implement the optimization based on genetic algorithms (GA) and details of the crossovers and mutations of the individuals.

**Figure 8.**Development of objective function J

_{Total}for: (

**a**) First welded joint FE model; (

**b**) Second welded joint FE model; and, (

**c**) Third welded joint FE models proposed considering the same level of importance (w

_{Temp}= 1 and w

_{Dist}= 1).

**Figure 9.**Temperature: (

**a**) Of key node P14 without having adjusted the parameters of the FE model; (

**b**) Of key node P9 without having optimized the parameters of the FE model; (

**c**) Of key node P14 with the parameters of the FE model optimized; (

**d**) Of key node P9 with the parameters of the FE model optimized; (

**e**) Field after three seconds for the first welded joint FE model studied; and, (

**f**) Field after 30 s.

Inputs | Specimen 1 | Specimen 2 | Specimen 3 |
---|---|---|---|

Current (amps) | 140.0 | 210.0 | 260.0 |

Voltage (volts) | 26.0 | 28.0 | 35.0 |

Speed (mm/s) | 6.0 | 6.0 | 6.0 |

Heat Flux (KJ/mm) | 0.424 | 0.686 | 1.061 |

Inputs | Specimen 1 | Specimen 2 | Specimen 3 |
---|---|---|---|

Height (mm) | 1.3 | 1.5 | 2.5 |

Width (mm) | 9.5 | 8.7 | 12.0 |

Angular Distortion (°) | 4.64 | 4.723 | 4.934 |

Range of | Melt Point | Contact P_init | Contact P_center | Contact P_end | Contact P1_P2 | Contact P_G | Face_Film |

Parameters | (°C) | (N/s·°K) | (N/s·°K) | (N/s·°K) | (N/s·°K) | (N/s·°K) | (N/s·°K·mm) |

Min. | 1420 | 1 | 1 | 1 | 1 | 1 | 0.00001 |

Max. | 1440 | 300 | 700 | 300 | 200 | 200 | 0.00100 |

Step | 1 | 1 | 1 | 1 | 1 | 1 | 0.00001 |

Range of | Face_film2 | Face_film3 | Forward Length | Rear Length | Width | Depth | - |

Parameters | (N/s·°K·mm) | (N/s·°K·mm) | (mm) | (mm) | (mm) | (mm) | - |

Min. | 0.00001 | 0.00001 | 1.0 | 5.0 | 23.0 | 4.0 | - |

Max. | 0.00100 | 0.00100 | 2.0 | 10.0 | 26.0 | 6.0 | - |

Step | 0.00001 | 0.00001 | 0.1 | 0.1 | 0.1 | 0.1 | - |

**Table 4.**Results of using the correlation based feature selection (CFS) technique to analyze the relationship between the output feature or objective function J

_{Totalj}and the input parameters.

Inputs | First FE Model | Second FE Model | Third FE Model |
---|---|---|---|

melt_point | - | - | - |

contac_p_in. | - | - | - |

contac_p_midle | X | - | - |

contac_p_End | - | - | X |

contact_p1_p2 | - | - | - |

contact_p2_Ground | - | - | - |

face_film | - | - | - |

face_film2 | - | - | - |

face_film3 | - | - | - |

forward_lenght | - | - | X |

rear_lenght | X | X | X |

width | - | - | - |

depth | - | - | - |

**Table 5.**Results of the adjustment process for the three welded joints FE models: considering the 13 input parameters and considering the selected parameters by the CFS filter algorithm.

Ways to Adjust | FE | Gen. | Melt | Contact | Contact | Contact | Contact | Contact | Face | Face | Face | Forward | Rear | Width | Depth | J_{Total} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Weld | - | Point | P_init | P_center | P_end | P1_P2 | P_G | Film | Film_2 | Film_3 | Length | Length | - | - | - | |

- | - | (°C) | (N/(s·K)) | (N/(s·K)) | (N/(s·K)) | (N/(s·K)) | (N/(s·K)) | (N/(s·°K·mm)) | (N/(s·°K·mm)) | (N/(s·°K·mm)) | (mm) | (mm) | (mm) | (mm) | - | |

CFS algorithm | 1 | 01 | 1430 | 150.5 | 2 | 150.5 | 100.5 | 100.5 | 0.000505 | 0.000505 | 0.000505 | 1.5 | 9.7 | 24.5 | 5.0 | 3.6652 |

05 | 1430 | 150.5 | 13 | 150.5 | 100.5 | 100.5 | 0.000505 | 0.000505 | 0.000505 | 1.5 | 5.3 | 24.5 | 5.0 | 2.3416 | ||

2 | 01 | 1430 | 150.5 | 350.5 | 150.5 | 100.5 | 100.5 | 0.000505 | 0.000505 | 0.000505 | 1.5 | 6.7 | 24.5 | 5.0 | 3.1879 | |

05 | 1430 | 150.5 | 350.5 | 150.5 | 100.5 | 100.5 | 0.000505 | 0.000505 | 0.000505 | 1.5 | 6.5 | 24.5 | 5.0 | 3.1654 | ||

3 | 01 | 1430 | 150.5 | 350.5 | 163 | 100.5 | 100.5 | 0.000505 | 0.000505 | 0.000505 | 1.3 | 6 | 24.5 | 5.0 | 6.7602 | |

05 | 1430 | 150.5 | 350.5 | 174 | 100.5 | 100.5 | 0.000505 | 0.000505 | 0.000505 | 1.7 | 5.2 | 24.5 | 5.0 | 6.4968 | ||

13 parameters | 1 | 01 | 1431 | 102 | 2 | 219 | 193.0 | 62.0 | 0.00051 | 0.00047 | 0.00051 | 1.4 | 7.8 | 24.8 | 4.3 | 4.3917 |

03 | 1427 | 17 | 10 | 19 | 193.0 | 66.0 | 0.00052 | 0.00039 | 0.00041 | 1.0 | 7.8 | 24.9 | 4.3 | 2.2680 | ||

2 | 01 | 1423 | 164 | 107 | 130 | 115.0 | 118.0 | 0.00055 | 0.00004 | 0.00011 | 1.4 | 7.5 | 26 | 5.1 | 3.1938 | |

03 | 1425 | 257 | 65 | 189 | 195.0 | 176.0 | 0.00071 | 0.00039 | 0.00033 | 1.8 | 7.4 | 23.8 | 4.3 | 3.0838 | ||

3 | 01 | 1429 | 162 | 582 | 198 | 127.0 | 83.0 | 0.00095 | 0.0008 | 0.0007 | 1.6 | 5.4 | 25.3 | 4.6 | 6.5161 | |

03 | 1429 | 203 | 662 | 247 | 159.0 | 182.0 | 0.00093 | 0.0008 | 0.00068 | 1.6 | 5.2 | 23.1 | 5.4 | 6.4918 |

**Table 6.**Values of the angular distortion obtained with the proposed FE models and experimentally by use of a coordinate-measuring machine.

Specimen | FEM (°) | Experimental (°) | Error (%) |
---|---|---|---|

1 | 4.73 | 4.64 | 4.11 |

2 | 4.92 | 4.723 | 4.17 |

3 | 5.21 | 4.934 | 5.59 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lostado Lorza, R.; Escribano García, R.; Fernandez Martinez, R.; Martínez Calvo, M.Á.
Using Genetic Algorithms with Multi-Objective Optimization to Adjust Finite Element Models of Welded Joints. *Metals* **2018**, *8*, 230.
https://doi.org/10.3390/met8040230

**AMA Style**

Lostado Lorza R, Escribano García R, Fernandez Martinez R, Martínez Calvo MÁ.
Using Genetic Algorithms with Multi-Objective Optimization to Adjust Finite Element Models of Welded Joints. *Metals*. 2018; 8(4):230.
https://doi.org/10.3390/met8040230

**Chicago/Turabian Style**

Lostado Lorza, Rubén, Rubén Escribano García, Roberto Fernandez Martinez, and María Ángeles Martínez Calvo.
2018. "Using Genetic Algorithms with Multi-Objective Optimization to Adjust Finite Element Models of Welded Joints" *Metals* 8, no. 4: 230.
https://doi.org/10.3390/met8040230