DOI: 10.3303/CET23107104 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 03 April 2023; Revised: 26 July 2023; Accepted: 27 October 2023 
Please cite this article as: Grubits P., Dániel G., Habashneh M., Szép J., Movahedi Rad M., 2023, Advanced Numerical Simulation and 
Modeling of Welding Processes: Stochastic representation of parameters for Improved Fabrication, Chemical Engineering Transactions, 107, 
619-624  DOI:10.3303/CET23107104 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 107, 2023 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Bohong Wang, Petro Kapustenko 

Copyright © 2023, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-07-6; ISSN 2283-9216 

Advanced Numerical Simulation and Modeling of Welding 

Processes: Stochastic representation of parameters for 

Improved Fabrication 

Péter Grubits, Gosztola Dániel, Muayad Habashneh, János Szép, Majid Movahedi 

Rad* 

Faculty of Architecture, Civil engineering and Transport Sciences, Department of Structural and Geotechnical Engineering, 

Széchenyi István University, H-9026 Győr, Hungary 

majidmr@sze.hu 

Numerical simulations play a pivotal role in advancing fabrication processes and welding technologies, enabling 

the pursuit of sustainable practices. By employing the finite element method, crucial insights regarding welded 

specimens can be derived, encompassing deformed shapes, residual stresses, and even microstructural 

properties such as phase proportions and hardness. This study focuses on the modeling framework of welding 

processes, emphasizing the influence of various welding parameters on sustainable outcomes, including 

reduced environmental impact and enhanced resource efficiency. The investigation delves into the 

characterization of heat sources, accounting for temperature-dependent material properties and developing a 

comprehensive thermo-mechanical analysis. By incorporating sustainability considerations and utilizing our 

Finite Element (FE) model, we conducted further analysis to elucidate the stability behavior, aligning with 

sustainable objectives. By considering welding current, arc voltage, and welding speed as random variables 

with mean values and standard deviations, the study aims to identify a model that effectively accounts for the 

inherent randomness of the welding process. This research contributes to the growing body of knowledge on 

sustainable welding practices by merging numerical simulations, advanced modeling techniques, and 

sustainability principles. The outcomes of this study have the potential to inform industry stakeholders and 

decision-makers about the most effective strategies for achieving sustainable welding processes and minimizing 

the ecological footprint of the welding operations. 

1. Introduction 

The growing need for responsible and efficient practices has fuelled significant advancements in numerical 

simulations and modeling techniques for welding processes. These advancements have allowed researchers 

to gain crucial insights into welded specimens, encompassing deformed shapes, residual stresses, and 

microstructural properties. 

The finite element analysis (FEA) plays a pivotal role in this work, serving as a powerful tool that empowers 

researchers to conduct comprehensive thermo-mechanical analyses of welding processes. In this context, Deng 

et al. (2007) proposed a FEA approach that incorporates the inherent strain theory to accurately forecast welding 

distortion. Numerical simulations by adopting FEA to enhance fabrication processes and welding technology 

were proposed by Kollár et al. (2017). Zhao et al. (2018) presented finite element analysis design of welding 

structure. To enhance the performance of welding, Zhou et al. (2023) developed technique involves the 

combination of arc welding and in-situ laser shock forging, which can also be modelled with FEM.   

Beyond the deterministic approach, where traditional methods may have limitations in fully capturing the 

complex behaviors of structural elements, researchers have increasingly recognized the significance of 

considering the inherent randomness in various structural engineering applications such as structural topology 

optimization (Habashneh and Rad, 2023) and reinforced concrete designs at elevated temperatures (Szép et 

al., 2023). Dániel et al. (2022) proposed a reliability-based analysis, considering uncertainties for numerical 

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modeling of glulam beams. Similarly, Yin et al. (2023) used deterministic and probabilistic analyses to investigate 

mono piles for offshore wind turbines based on the FEM method.  

In their study, Sulaiman et al. (2022) proposed that during the welding process, the quality of the weld is 

significantly influenced by the welding conditions, encompassing key parameters such as welding speed, 

voltage, and current. The study suggests that these input variables can be effectively modeled by a normal 

distribution, with both mean and standard deviation subject to randomization.  

The current study endeavors to merge probabilistic approach, numerical simulations, advanced modeling 

techniques, and sustainability principles to further enhance welding processes. By taking into account the factors 

of welding speed, arc voltage, and welding current as random variables, with mean values and standard 

deviations, the research aims to comprehensively capture the inherent randomness of welding processes, 

acknowledging that accurately accounting for these uncertainties is crucial for obtaining reliable and realistic 

results. 

2. Finite element modelling and welding simulation 

2.1 Geometry modelling 

An implementation of a 3D mechanical model was made necessary by the inherent three-dimensionality of the 

loads, boundary conditions, and typical welding process geometry. It is additionally practical to know the 

temperatures at any moment and at any location by using the 3D model. This makes it possible to conduct a 

thorough process analysis (Ling et al., 2023). In this study, a lap-joint specimen made of two steel plates that is 

welded on both sides serves as the test model. The FE model's plate dimensions for the ordinary plate and the 

flange plate are 300x160x20 mm and 360x160x20 mm. The weld length is 160 mm, which is the longitudinal 

length of the plates, and the size of the pool is 2 mm. As seen in Figure 1, the FE model comprises of five 

objects. The foundation plates, which are welded on both sides, are the beam's flange plate (white part) and the 

regular plate (grey part). Additionally, a section of the web-plate (green part) was developed for the modelling 

of boundary conditions. The model is comprised of 39,680 hexagonal components, each consisting of eight 

nodes, resulting in a total of 58,208 nodes.  

 

Figure 1: 3D FE model 

2.2 Thermal analysis 

The fundamental premise behind thermal analysis is the preservation of energy. The computation of the 

temperature evolution at each individual node was performed using the heat transfer theory to simulate the 

welding operation. The primary component of this concept is the flux of energy q Eq(1) (Goldak and Akhlaghi, 

2005), where the 𝜅 is the material property and ∇𝑇 represents the temperature gradient. Thermal conductivity 

has a direct relationship with flux of energy.  

𝑞 = −𝜅∇𝑇  (1) 

By utilizing the heat conduction formulation Eq(2) (Goldak and Akhlaghi, 2005) and having access to the 

information regarding q, it becomes feasible to ascertain the temperature in every node. In this equation, 𝜌 

represents the density of the material, 𝐶𝑝  denotes the specific heat capacity, T signifies an increment in 

temperature, q represents the heat flux vector, Q denotes the term for heat generation, 𝑡 represents time, and 

𝛻 represents the spatial gradient operator. 

𝜌𝑐𝑝𝑑𝑇

𝑑𝑡
+ ∇𝑞 + 𝑄 = 0   (2) 

The use of the double ellipsoid heat source model is employed in this work to replicate the heat produced by 

welding. The power density distribution q inside the front and the rear quadrant is given as (Goldak et al., 1984): 

𝑞(𝑥, 𝑦, 𝑧, 𝑡) =
6√3𝑓𝑓𝑄

𝑎𝑏𝑐𝜋√𝜋
𝑒−3𝑥

2/𝑎2𝑒−3𝑦
2/𝑏2𝑒−3𝑧

2/𝑐𝑓
2
   (3) 

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𝑞(𝑥, 𝑦, 𝑧, 𝑡) =
6√3𝑓𝑟𝑄

𝑎𝑏𝑐𝜋√𝜋
𝑒−3𝑥

2/𝑎2𝑒−3𝑦
2/𝑏2𝑒−3𝑧

2/𝑐𝑟
2
  (4) 

The semi-axes of the ellipsoid parallel to the coordinate axes x, y, and z are represented by the parameters a, 

b, and c in Eq(3) and Eq(4) (Goldak and Akhlaghi, 2005), which correspond to the pool's width, depth, and 

length. The ff and fr are the fraction factor of the heat, where ff+fr=2. Q is the energy input, which is expressed 

as: 

𝑄 = 𝜂𝑈𝐼   (5) 

where η is the heat source efficiency, U is voltage, and I is the welding current. The used parameters for the 

simulation are shown in Table 1. 

Table 1: Heat source parameters 

Parameter name Value Unit 

Front length, cf 2.50 mm 

Rear length, cr  5.00 mm 

Width, a  2.00 mm 

Depth, b  2.00 mm 

Welding efficiency, η  0.60 - 

Welding voltage, U 23.00 V 

Welding current, I  260.00 A 

 

Natural boundary conditions have been defined to solve the differential equation in Eq(1). It is illustrated in Eq(6) 

(Goldak et al., 1984), where 𝜅  is the thermal conductivity normal to the surface, q denotes the given flux, h 

signifies the heat transfer coefficient for convection, σ represents the Stefan-Boltzmann constant, ε represents 

the emissivity, and T0 is the ambient temperature for convection and radiation.  

𝜅𝑛
𝑑𝑇

𝑑𝑛
+ 𝑞 + ℎ(𝑇 − 𝑇0) + 𝜎𝜀(𝑇4 − 𝑇0

4) = 0    (6) 

For the simulation, the used parameters are shown in Table 2. While the emissivity of steel changes with 

temperature, it is assumed in this study that it is constant at the ambient temperature. Considering the steel 

material used in the experimental test, the value of the emission coefficient ε can be taken as 0.6. 

Table 2: Thermal properties coefficients 

Coefficient Value Unit 

Thermal conductivity normal to the surface, 𝜅 20 W/m2K 

Coefficient for convection, h 100 W/m2K 

Coefficient for emission, ε 0.6 - 

2.3 Mechanical analysis 

There is a variation in the distribution of heat between the immediate vicinity of the welding site and its surrounds, 

which causes expansion and contraction. This impact produces significant stresses due to the thermal strains 

of the material (Anca et al., 2011). In the simulation nonlinear deformation was calculated, for which the 

equilibrium equations, constitutive stress-strain relations and geometric compatibility are fundamental. During 

the analysis typical elasto-plastic material model with the von Mises yield criterion is used, where the 

displacement field is interpolated over an element given as (Lindgren, 2007): 

𝑢( 𝑥, 𝑡) = 𝑁𝑚( 𝑥) 𝑢(𝑡)𝑖
𝑛+1

𝑖
𝑛+1

𝑖
𝑛+1

𝑖
𝑛+1     (7) 

where the left superscript n is the time increment counter, the subscript i is an iteration counter, 𝑁𝑚( 𝑥, 𝑡)𝑖
𝑛+1  is 

a matrix with the interpolation, shape functions and 𝑢(𝑡)𝑖
𝑛+1  is the current estimate of the element displacement 

vector at the end of the increment and the corresponding coordinate is 𝑥𝑖
𝑛+1 .  

2.4 Welding simulation 

The welding process was simulated with ABAQUS software (Smith, 2009), for which coupled thermal-stress 

analysis was used. In order to address this issue, it is necessary to use elements that include degrees of freedom 

pertaining to both temperature and displacement. This approach enables the concurrent determination of both 

thermal and mechanical solutions. Under the assumption of a weakly coupled analysis, the welding simulation 

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is conducted by performing thermal analysis independently from mechanical analysis, where thermal analysis 

is performed first, followed by mechanical analysis.  

2.5 Simulation results 

The primary objective of the first section of this study was to validate the temperatures that obtained from the 

experimental tests. For this simplified cross-section of the pool was implemented, where the pool size was 2 

mm. To ensure the technology and quality of the weld, the plates were preheated to 100 °C. During the 

experimental test, the welding process have several phases, of which the first two are modelled, in which the 

two sides of the plate are welded together. The lap joint is welded on the right side in the first stage and on the 

left in the second step. The overall simulation time was 81.5 s, and there was a 16 s cooling period after each 

welding cycle. The resulting temperatures were assessed at the locations as shown in Figure 2, located directly 

above the welding-pool on the opposite face of the regular plate, including the beginning, middle, and end point 

of the pool's length. Due to the second phase welding processes, the heat from the welding reached these 

locations.  

 

Figure 2: The measuring points used for validation 

After the simulation was complete, the temperature was compared to the outcomes of the experiment. The 

temperatures from the model produced the same outcome as the experiment at the highlighted points with a 10 

% margin of error, as shown in Figure 3. The weld's highest temperature during the experiment was 160 °C, 

and 162 °C in the numerical simulation. Furthermore, it is obvious from Figure 3 that the temperature of the first 

welding-phase had no effect on the zone under research. 

 

Figure 3: Temperatures obtained at the designated measuring points 

3. Integrating probabilistic analysis 

Achieving a comprehensive understanding of the behavior and performance of welded structures requires 

acknowledging the inherent uncertainties associated with input parameters. Deterministic approaches often 

overlook these uncertainties, which can greatly impact the reliability and accuracy of the results. To address this 

limitation, a probabilistic analysis framework is employed in this study to comprehensively evaluate the effects 

of random variables on the welding process.  

In this analysis, the welding current, arc voltage, and welding speed are considered as key random variables. 

These variables exhibit inherent variations due to various factors such as material properties, environmental 

conditions, and operator skills. To capture the probabilistic nature of these variables, they are modeled as 

independent random variables following the normal distribution, which is commonly used for continuous 

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variables with symmetric distributions. The mean values and standard deviations for each variable are presented 

in Table 3 which were determined within the specified bounds. 

Table 3: The considered random variables 

Parameter Mean value Standard deviation 

Welding current 260 10% 

Arc voltage 22.95 5% 

Welding speed 8.50 30% 

 

Figure 4 displays the simulation results obtained from integrating probabilistic analysis. These results 

demonstrate how the incorporation of random variables affects the temperatures obtained in the second phase 

compared to those obtained through deterministic analysis. For instance, the highest welding temperature in 

the case of deterministic design was 160 °C, while it is reduced to 138 °C in the case of the fifth probabilistic 

simulation. Additionally, the lowest temperature obtained after the second cooling phase was 145 °C in the case 

of deterministic design, whereas in the fifth probabilistic simulation, it is reduced to 131 °C. It is worth noting that 

since the temperature of the first welding phase did not affect the zone under investigation, there were no 

changes during the first welding phase in the case of probabilistic analysis as well. 

 

Figure 4: Obtained temperatures from probabilistic analysis at the measuring point (8cm) 

In our probabilistic formulation, we incorporated standard deviations based on established standards and 

guidelines. These standard deviations represent the inherent variability and uncertainty in the parameters and 

data used in our analysis. When we computed the probabilistic solution, we considered the combined effects of 

these uncertainties. It's noteworthy that while some of our probabilistic results closely aligned with the 

deterministic solution, we also observed larger discrepancies between the probabilistic results and the 

experimental data, as well as the deterministic solution. It is crucial to note that the probabilistic approach aims 

to capture the full spectrum of possible outcomes, accounting for variations in input parameters and data. While 

it may seem counterintuitive that this approach sometimes resulted in larger discrepancies, it actually highlights 

the importance of understanding the contributions of individual probabilistic parameters. 

4. Conclusions 

This study utilized advanced numerical simulation and modeling techniques to investigate welding processes 

and their implications for sustainable practices. The welding process was meticulously simulated, including 

preheating to 100 °C, multiple welding phases, and cooling intervals. Temperature assessments were 

conducted at key points above the welding pool, revealing a remarkable agreement with experimental results. 

Our numerical model closely approximated experimental outcomes, achieving an agreement within a 10 % 

margin of error.  

Incorporating probabilistic analysis into our research framework allowed us to recognize the significance of 

inherent uncertainties in input parameters. Key random variables including welding current, arc voltage, and 

welding speed, are considered in this analysis. By modeling these variables as independent random variables 

conforming to a normal distribution, we effectively captured their probabilistic nature. Our results from 

probabilistic simulations illuminated the profound impact of incorporating random variables on temperatures 

observed during the second welding phase. This valuable insight highlights the complexity of capturing the 

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complete spectrum of potential outcomes and emphasizes the importance of understanding individual 

probabilistic parameters.  

Our research contributes to the ongoing discourse surrounding welding processes and lays the foundation for 

further investigations in this domain. Future research endeavors can build upon our work to refine and expand 

our understanding of the behavior and performance of welded structures. 

Nomenclature

𝜅 – material property, J/mm s ºC 

∇𝑇 – temperature gradient, - 

q – heat flux, W/mm2 

𝜌 – density of the material, g/mm3 

𝐶𝑝 – specific heat capacity, J/(g ºC) 

T – increment in temperature, ºC 

Q – source of heat generation, W/mm3 

t – time, s 

η – heat source efficiency, - 

U – voltage, V 

I – welding current, A 

ε – Coefficient for emission, - 

h – Coefficient for convection, W/m2K

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