Optimization of Welding Sequence for Frame Structures Based on Discrete Particle Swarm Optimization to Mitigate Welding Deformation

Abstract

Welding deformation in large thick-plate structures can severely compromise manufacturing quality, making its prediction and control a critical engineering challenge. This study focuses on an engineering vehicle frame, for which a finite element model was developed to investigate the effects of welding sequence and direction on residual deformation, using a local-global strain mapping approach. Thermo-elasto-plastic simulations were first performed on T-joints and fillet joints to extract local plastic strains, which were subsequently mapped onto the global elastic model to compute overall structural deformation. The simulation results showed good agreement with experimental measurements, with a deviation of approximately 5.6%, confirming the reliability of the proposed method for predicting welding-induced deformation and stress in complex assemblies. To further optimize the welding strategy, a surrogate model was constructed based on Design of Experiments (DOE), and a Discrete Particle Swarm Optimization (DPSO) algorithm was employed. The optimized welding sequence and direction reduced the maximum deformation by 43%, while significantly lowering computational cost without sacrificing accuracy. This integrated approach offers valuable guidance for welding process design in engineering vehicle frames and other large welded structures.

1. Introduction

Engineering vehicles are essential machinery in construction projects, significantly enhancing operational efficiency and ensuring the quality of construction outcomes. To ensure safe and stable operation during construction activities, it is crucial to maintain strict control over the structural integrity of key components. Among these, the vehicle frame is one of the most critical load-bearing structures, and its safety and stability play a vital role in the overall performance of the engineering vehicle.

The frame structure of engineering vehicles typically adopts a ladder-type design, composed of two longitudinal beams and several cross beams welded into an integrated unit. Both the longitudinal and cross beams have large dimensions and considerable thickness. As a result, the frame is a large-scale thick-plate welded structure, characterized by high manufacturing costs and long production cycles, making full-scale structural testing difficult to carry out. Meanwhile, welding parameters during the manufacturing process—such as current, voltage, speed, and sequence—directly affect the residual welding deformation [1]. Therefore, accurately predicting and controlling welding deformation in large thick-plate structures is critically important.

With the exponential growth of computing power and advancements in numerical algorithms, it has become feasible to simulate real-world welding processes through computer modeling. Finite Element Method (FEM) simulation serves as a valuable complement to experimental techniques, providing insights into the behavior and interactions of complex physical phenomena occurring during welding [2]. A variety of FEM approaches have been rapidly developed, among which the most commonly used are the thermo-elasto-plastic method, the inherent strain method, and the local-global method [3].

Yahiaoui, K [4] employed the thermo-elasto-plastic method to investigate the effect of welding sequence on the residual stress distribution in the welds of multi-pass welded pipe branch joints. High residual stress is formed in the vicinity of the weld region irrespective of the sequence of the welding. Deshpande, A.A. [5] conducted simulations of butt welding and post-weld heat treatment of two Inconel 718 plates using SYSWELD version 2009, a welding-specific simulation tool, and the general-purpose finite element software ABAQUS. The results from both software packages—such as thermal histories, residual stresses, nodal displacements, and stress relaxation—were compared and found to be in good agreement. Manurung, Y.H.P. et al. [6] conducted an analysis of the welding sequence effect on induced angular distortion based on the thermal-elastic-plastic approach with low manganese carbon steel S355J2G3 as specimen material. Heinze et al. [7] investigated the influences of mesh density and different CCT behaviors on the result quality of the calculation of welding-induced distortion.

The thermo-elasto-plastic method conducts comprehensive heat transfer and elasto-plastic analyses on the entire structural model, providing relatively high accuracy in predicting welding-induced thermal and mechanical responses. However, when applied to large-scale structures, this approach demands a fine mesh to capture local gradients and stress concentrations accurately. As a result, the computational cost becomes significantly high, often requiring substantial processing time and resources. This limitation poses challenges for practical applications, especially when multiple simulations are needed for design optimization or process parameter tuning.

The inherent strain method [8] considers inherent strain as the primary source of welding-induced residual stress and deformation. In this approach, the inherent strain is introduced into the structure as an initial strain, and an elastic analysis is subsequently performed to predict the resulting welding deformation and residual stress distribution.

Woo et al. [9] analyzed the welding deformation of ship side plate structures using a finite element method based on the inherent strain approach, incorporating interface elements and multi-point constraint functions. The numerical simulations conducted effectively validated the proposed systematic method, demonstrating its capability to determine the optimal welding sequence that minimizes welding-induced displacements. Deng et al. [10], based on inherent strain theory, developed an elastic finite element method to accurately predict welding distortion during the assembly process of ship structures. The effectiveness of the proposed elastic FEM was validated through comparison with experimental results. Zhu et al. [11] analyzed the welding distortions of a large beam structure using computational welding mechanics (CWM) techniques, including the inherent strain method and the shrinkage method combined with the lumping approach. The residual stress estimates obtained from the inherent strain method were more accurate than those derived from the shrinkage and lumping approaches. Furthermore, the study found that adopting an appropriate welding sequence can effectively minimize the welding distortion of the beam structure.

The inherent strain method circumvents the computational difficulties posed by high-temperature and nonlinear transient analyses typically encountered in welding simulations, thereby significantly reducing the required computation time. This efficiency makes it a practical choice for predicting welding deformation and residual stress in complex structures. However, the accuracy and applicability of the method heavily depend on a comprehensive and up-to-date database of inherent strains for various welding processes and parameters [12]. Since new welding techniques and materials continually emerge, the need to expand and refine this inherent strain database remains an ongoing challenge. Consequently, the method may face limitations in fully satisfying both theoretical analysis demands and practical engineering requirements.

Overall, the aforementioned methods have primarily been applied to typical joint structures of plates and pipes [13,14] such as T-joints, butt joints, and lap joints. They have been used to study welding processes on different materials, including gas tungsten arc welding (GTAW) of AA5251 aluminum plates [15] laser and arc welding of aerospace aluminum [16] and TIG welding of 316LN stainless steel [17]. Additionally, these methods have found applications in auto-motive and shipbuilding components with numerous parts, such as automotive bumpers [18] and ship hull structures [19]. However, due to the extensive computational time required, these methods are generally inapplicable for simulating welding distortion in large welded structures and have relatively limited use in complex, large-scale assemblies. Moreover, there is comparatively little research addressing challenges related to large-scale structures and welding sequences.

Over the past decades, numerous numerical methods have been developed to pre-dict welding-induced residual stresses and distortions. Souloumiac et al. [20] proposed a local–global approach in which thermo-elastic-plastic analysis is restricted to the weld region, while the remaining structure is treated elastically, significantly reducing computational cost without loss of accuracy. This method has been successfully ap-plied to various weld configurations, including T-joints and multipass welds.

The present local-global mapping approach is fundamentally different from the global-local submodeling technique reported by Perić et al. [21]. In the latter, global thermo-mechanical analysis is performed first, and the resulting temperature and displacement fields are transferred to a refined local model to improve accuracy near the weld. In contrast, the proposed method introduces residual plastic strains obtained from detailed local welding simulations as initial strains into the global structural model, thereby avoiding repeated global thermo-mechanical analyses. This feature makes the approach particularly efficient and well-suited for welding sequence optimization of large welded structures.

Tsirkas et al. [22] developed a local-global finite element approach for evaluating welding-induced distortions in laser-welded shipbuilding components. The local model, implemented using Sysweld (version 2001) software, incorporates the effects of metallurgical transformations based on temperature-dependent material properties and continuous cooling transformation (CCT) diagrams. The residual plastic strains and weld stiffness calculated from the local model are subsequently transferred to the global model to analyze the overall welding deformation of the structure. This approach has proven to be reliable when validated against experimental data and has been successfully adopted in industrial applications, offering advantages in both computational efficiency and storage requirements. Similarly, Duan et al. [23] proposed a local-global method capable of predicting welding-induced residual distortions in large-scale structures. Rong [24] validated a combined heat source model and employed a local-global mapping method to accurately predict welding deformation and optimize the welding sequence for a large marine propeller nozzle.

Furthermore, the optimization of welding processes to minimize welding residual stress and distortion in the final structure has constituted an active research area for several decades [25]. In experimental and numerical optimization studies, methods such as Design of Experiments (DOE), Genetic Algorithms (GA), Multi-Objective Particle Swarm Optimization (MOPSO), and Artificial Neural Networks (ANN) are among the most commonly employed techniques.

Islam et al. [26] and Choi et al. [27] integrated Genetic Algorithms (GA) with a three-dimensional thermo-elasto-plastic finite element model to optimize welding parameters and sequences for thin-walled tube structures, with the objective of minimizing welding-induced distortion. Yuan et al. [28] investigated the optimization of welding sequences based on the Artificial Immune Algorithm. Numerical simulation results demonstrated that, compared with the Immune Genetic Algorithm (IGA), Immune Clonal Algorithm (ICA), and traditional Genetic Algorithm (GA), the welding sequence optimized by the proposed Immune Cloning Algorithm yielded the least deformation, indicating its superior performance in minimizing welding-induced distortion. Wu et al. [29] developed a welding sequence optimization framework that integrates a coupled swarm intelligence algorithm with an Artificial Neural Network (ANN) to minimize welding-induced distortion in thin-walled aluminum alloy tube components. Wang et al. [30] proposed an improved Multi-Objective Particle Swarm Optimization (MOPSO) algorithm and applied it to optimize both the welding path length and the total welding deformation, demonstrating enhanced efficiency and effectiveness in welding process optimization.

As demonstrated by the findings of the literature review, numerical design optimization holds significant potential for improving various aspects of the welding manufacturing process. Among the available techniques, the intelligent Particle Swarm Optimization (PSO) algorithm is widely adopted due to its notable advantages, including a simple structure, minimal parameter tuning, and low dependency on the mathematical characteristics of the objective function [31]. Such optimization algorithms can be effectively applied in welding engineering to refine process parameters, design optimal welding paths, and integrate with conventional algorithms to enhance computational accuracy and efficiency.

Therefore, this study employs a Discrete Particle Swarm Optimization (DPSO) algorithm to optimize the welding sequence. By integrating the DPSO algorithm with a local–global finite element approach, the proposed method effectively evaluates the objective function, enabling accurate and efficient optimization of the welding sequence.

2. Welding Sequence Optimization Process

To achieve the optimization of the welding sequence in this study, two main components are involved: the calculation of welding deformation and the application of an intelligent optimization algorithm. The steps for optimizing the welding sequence of the large structure are schematically illustrated in Figure 1. For the welding deformation calculation, the thermo-elastic-plastic finite element method is employed to simulate the welding process of joints (local model) in the engineering vehicle. Subsequently, a local-global approach is applied to predict the welding distortion of the large welded structure based on the residual plastic strain obtained from the local model. Experimental tests are then conducted to validate the simulation results.

For the optimization algorithm, an improved discrete particle swarm optimization (DPSO) algorithm is used to determine the optimal welding sequence and direction for the vehicle frame, aiming to minimize residual deformation. The optimization results are presented and discussed.

3. Experimental Procedure and Finite Element Modeling

3.1. Experimental Procedure

To verify the effectiveness of the proposed methodology, the frame of the engineering vehicle was fabricated at Jiangyin Lvsong Machinery Co., Ltd., Wuxi, China, and its welding deformations were measured using a three-coordinate measurement system. The frame is constructed by welding multiple steel plates into a box-shaped structure and is primarily composed of an upper cover plate, longitudinal beams, and cross beams. It is made of structural steel Q345B, with plate thicknesses ranging from 25 mm to 42 mm. The overall dimensions of the frame are 3400 mm in length, 750 mm in width, and 680 mm in height at the center. The geometry of the structure is shown in Figure 2.

During fabrication, tack welding was first performed on a fixture at intervals of 150 mm, with each tack weld measuring 25 mm in length. Final welding was then carried out using a double-sided welding process. For the T-joint double-sided welds, one side was welded first, followed immediately by reverse welding on the opposite side [12].

In the frame model, there are three types of T-joints between the longitudinal beams and the cross beams, as well as one type of corner joint between the upper cover plate and the beams. The frame components are joined using Gas Metal Arc Welding (GMAW), and the corresponding welding parameters are listed in

Table 1. Welding condition.

Welded Joint	Size (mm × mm)	Welding (Leg Height) (mm)	Current (A)	Voltage (V)	Speed (mm/s)
Local Model A	25 × 25	8	200	30	5
Local Model B	40 × 25	8	230	30	5
Local Model C	25 × 6	6	180	28	7
Local Model D	42 × 25	10	250	30	5

. Welding was carried out using ER50-6 filler wire (Lvsong Machinery, Wuxi, China) with a diameter of 1.2 mm, a wire feeding speed of 9.5 m/min, and a shielding gas mixture of MIX 21 (20%CO₂/80%Argon) at a flow rate of 12 L/min. The time interval between successive welds was approximately 10 min. In addition, joint welding experiments were conducted. The welding process was performed by a robotic welding system, and the experimental setup is illustrated in Figure 3.

In the experiments, a three-coordinate measurement technique was employed to measure welding distortion. To ensure the reliability of the results and eliminate random errors during the welding process, welding deformation measurements were conducted on 10 frames. After the frames were fabricated, measurement points were marked on their surfaces, with the layout illustrated in Figure 2. Five measurement points were selected on each of the longitudinal beams. A single-arm 3D measuring instrument manufactured by Yancheng Changmin Intelligent Equipment Co., Ltd., Yancheng, China, was used to record the coordinates of the frame after tack welding and again after final welding. By comparing the 3D coordinate data collected before and after welding, the resulting welding distortion was calculated.

3.2. Finite Element Modeling

In applying the local-global method—where plastic strains are incorporated into an elastic finite element model as initial strains to predict welding distortion in large structures—it is critical to extract such plastic strains from diverse welded joints via detailed local three-dimensional models. In this section, the thermo-elastic-plastic method is employed to calculate the plastic strains in representative welded joints [32]. Local simulations of the welding process are performed using the welding-specific software Sysweld (version 2019), while the subsequent global analysis is conducted using PAM-ASSEMBLY (version 2019).

3.2.1. Local Model Thermal Analysis

During the welding process, the heat source exhibits characteristics of localized concentration, movement, and transient behavior, resulting in a highly non-uniform temperature field with steep spatial gradients. This uneven temperature distribution induces significant welding stresses and deformations. Therefore, the first step in local model analysis is to establish a physically reasonable and accurate heat source model. In arc welding, the volumetric heat source Q (in W/m³) is defined as the product of the arc power P (in W) and the process efficiency (ƞ).

Among various heat source models, the double-ellipsoidal heat flux distribution proposed by Goldak [6] has been widely adopted to predict welding temperature fields, residual stresses, and deformation. This model is particularly well-suited for Gas Metal Arc Welding (GMAW) processes. The schematic representation of the heat source model is shown in Figure 4, and its mathematical formulation is given in Equations (1) and (2).

$$Q_f(x,y,z)={\displaystyle \frac{6\sqrt{3}(f_{f}q)}{a_{f}bc\pi \sqrt{\pi }}}\exp (-{\displaystyle \frac{3x^2}{a_f^2}}-{\displaystyle \frac{3y^2}{b^2}}-{\displaystyle \frac{3z^2}{c^2}}), x\ge 0$$

(1)

$$Q_r(x,y,z)={\displaystyle \frac{6\sqrt{3}(f_{r}q)}{a_{r}bc\pi \sqrt{\pi }}}\exp (-{\displaystyle \frac{3x^2}{a_r^2}}-{\displaystyle \frac{3y^2}{b^2}}-{\displaystyle \frac{3z^2}{c^2}}), x<0$$

(2)

In this model, Q_f and Q_r represent the heat fluxes in the front and rear ellipses. f_f and f_r are the fractions of the heat deposited in the front and rear regions, satisfying the condition f_f + f_r = 2. Parameters a_f and a_r denote the lengths of the front and rear ellipsoids, respectively; b is the width of the molten pool, c is its depth, and q represents the total power of the welding process.

The heat source model was calibrated by iteratively adjusting Goldak’s heat source parameters until the simulated fusion zone geometry closely matched the experimental macrographs obtained from metallographic analysis, as illustrated for the T-joint and corner joint in Figure 5. The final parameters of the double-ellipsoidal heat source model are summarized in

Table 2. Double ellipsoid parameters.

Welded Joint	Qf W/m3	Qr W/m3	af mm	ar mm	b mm	c mm
Local Model A	22.96	19.14	3	4	5	6
Local Model B	22.9	19.69	3	4	4	6
Local Model C	32.55	27.12	3	4	3.5	4
Local Model D	16.3	13.66	3	4	5	7

.

Heat losses (q_c) due to convection are considered on all surfaces using Newton’s law of cooling:

$$q_e=-h_f(T_{sur}-T_0)$$

(3)

where h_f represents the convective heat transfer coefficient, T_sur and T₀ are the surface temperature and the ambient temperature. In this study, the film coefficient is assumed to be 25 W/(m²·°C), and the ambient temperature is set to 20 °C. Due to the close equivalence in chemical composition and thermo-mechanical properties between Q345B and S355J2G3 structural steels, the temperature-dependent thermal and mechanical properties of S355J2G3 [6] were adopted to represent Q345B in the welding simulations.

3.2.2. Local Model Mechanical Analysis

The mechanical analysis of the local model is performed using a three-dimensional solid mesh. Metallurgical transformations and nonlinear material properties are taken into account, and the weld region is simulated using the element birth and death technique to represent the progressive addition of weld metal to the workpiece [33]. The local model consists of plates with a length of 100 mm. To allow stress and strain to develop freely, minimal boundary constraints are applied, as shown by the arrows in Figure 6. Mechanical analysis was conducted using the aforementioned heat source model to obtain the stress and deformation states of the local model. The simulation time required to complete the full coupled analysis for each local model is approximately 0.5 h on a desktop computer equipped with a 2.30 GHz Intel^® Core™ i5-6500 CPU and 12 GB of RAM.

Mapping elements containing residual plastic strain and weld stiffness were extracted from the simulation results, as shown in Figure 7. These mapping elements represent the inherent strain field, which is primarily localized to the weld seam and its immediate vicinity. This localization reflects the fact that most plastic deformation occurs near the weld due to the thermal cycles and material phase changes during welding.

3.2.3. Global Model Calculation

After the entire structure is unclamped and fully cooled, the residual stress field can be expressed as:

$$\sigma =E(\epsilon -{\epsilon }^p)$$

(4)

where E denotes the elastic stiffness matrix, ε is the global strain, and ε^p represents the plastic strain obtained from the local model calculations.

Therefore, the plastic strains ε^p are projected as initial strains into the global structural model along the welding path using the Welding Macro Element (WME) method [14]. Each WME is connected to the global structure, which is modeled with shell elements. The residual distortions are then computed through a simple elastic analysis, making the local-global approach highly efficient—especially for optimizing the welding sequence in very large welded structures.

In the local thermo-mechanical analysis, a temperature-dependent elastic–plastic constitutive model with isotropic hardening is adopted. Plastic deformation follows the von Mises yield criterion with an associated flow rule, and temperature-dependent material properties are considered to capture welding-induced stress and deformation.

A finite element model of the frame was developed, consisting of both 3D and 2D elements, as illustrated in Figure 8 [34]. Frame structures often include irregular features such as fillets, bolt holes, and chamfers. However, in welding deformation analysis of large structures, these geometric details can be neglected due to their minimal influence on the residual welding deformation. The irregular bosses on the upper cover plate are simplified as uniform plates with equivalent thickness, ensuring that the reconstructed model maintains the same overall stiffness as the original structure [35]. The weld zones are meshed with high resolution, while the surrounding regions use coarser meshing for computational efficiency, as shown in Figure 9. Given that the length-to-thickness and width-to-thickness ratios of the frame plates are both greater than 20, the global finite element model was constructed using shell elements. The final model consists of 44,516 nodes and 43,080 elements.

The entire frame structure contains 24 welds, as illustrated in Figure 10. The welding process starts with the joint between a longitudinal beam and a cross beam on one side, followed by welding of the corresponding joint on the opposite side. This sequence results in a continuous longitudinal welding path along the X-axis, extending over the full length of the frame. The detailed welding sequence is provided in

Table 3. Welding sequences.

Step	Welding Sequence
1	WL1—WL9—WL2—WL10—WL5—WL6—WL11—WL12—WL7—WL3—WL08—WL4
2	WL01—WL09—WL02—WL010—WL06—WL011—WL012—WL07—WL8—WL03—WL06—WL04

. In addition, the boundary conditions applied in the simulation are consistent with those used in the experimental setup, and the constraint conditions are shown in Figure 9.

3.2.4. Validation of the FEM Model

To verify the accuracy of the local-global finite element (FE) model, a comparison between the simulation results and experimental data is essential. Figure 11 presents the simulated welding deformation of the frame. Welding deformation is strongly affected by the structural stiffness of the frame. Since there are more welds along the Z-direction on the longitudinal beams, bending deformation of the beams in the Y-axis direction is more pronounced, with a maximum deformation of 2.841 mm observed along the Y-axis and minimal deformation in other directions. Additionally, the substantial thickness of the upper cover plate contributes to its relatively small welding deformation.

Figure 12 presents the final welding displacements in the Y-axis direction, obtained from both experimental measurements and numerical simulations. With short standard error bars and an average deviation of approximately 5.6%, the finite element (FE) results show good agreement with the experimental data. To further clarify the reliability of the model, the potential sources of this discrepancy are analyzed as follows.

First, measurement uncertainty (e.g., sensor repeatability and positioning error) may introduce deviations in the recorded deformation values. Second, fixture constraints during welding—such as partial release or stiffness differences from the assumed ideal boundary conditions—can alter deformation paths but were simplified in the simulation. Third, material batch variability, including differences in yield strength, thermal expansion coefficient, and welding consumable performance, could contribute to the observed mismatch. Additionally, the FE model neglects certain local phenomena, such as the effects of weld start–stop regions, slight misalignment of components, and thermal loss variations in actual welding, which are difficult to reproduce precisely in the numerical model.

Considering these factors, the 5.6% deviation is within a reasonable engineering range, and the proposed local–global FE model remains sufficiently accurate for predicting welding distortion of large frame structures.

4. Discrete Particle Swarm Optimization-Based Welding Sequence Optimization

4.1. Principle of Discrete Particle Swarm Optimization (DPSO)

In practical production, many optimization problems involve discrete variables [36]. In this study, the optimization of the welding sequence and welding direction represents a typical discrete problem. To address this, each particle in the optimization algorithm represents a specific welding sequence and direction. An exchange strategy is employed to update the positions and velocities of the particles accordingly [30], it is given as follows:

$$v_i^{t+1}=\omega ·v_i^t+ c_1{r}_1·(p_i^t-x_i^t)+c_2{r}_2·(p_g^t-x_i^t)$$

(5)

$$x_i^{t+1}=x_i^t\oplus v_i^{t+1}$$

(6)

where $v_i^t$ and $x_i^t$ denote the velocity and position of particle i at iteration t, respectively. $p_i$ and $p_g$ represent the personal best position of particle i and the global best position found by the swarm, respectively. ω is the inertia weight coefficient, $c_1$ and $c_2$ are learning factors, and $r_1$, $r_2$ are random numbers uniformly distributed in the interval [0, 1].

4.2. Formulation of the Optimization Mode

Residual deformation and residual stress resulting from welding exert distinct influences on the performance and structural integrity of welded components. In practice, relatively simple and cost-effective pre-weld and post-weld treatment methods are commonly employed to effectively mitigate undesirable residual effects induced by the welding process. Islam et al. [26] employed a multi-variable optimization approach that incorporated not only the welding sequence and direction, but also welding speed, current, and voltage-key parameters directly influencing the heat input during the welding process. Among the various methods for controlling welding quality, the proper selection and arrangement of pre-weld parameters have a significant impact on the final results. However, the number of possible combinations of pre-weld control parameters-such as welding speed, current, and voltage-increases exponentially with the number of welds. As a result, these welding parameters are generally treated as fixed values during the early stages of product design. Therefore, this study prioritizes the optimization of welding sequence and welding direction, aiming to identify an optimal welding strategy under fixed parameter conditions.

If a welded structure contains n welds, the number of possible welding sequences can reach (n − 1)! due to the permutation of the weld order [27]. When the welding direction is also considered, the total number of combinations increases dramatically to (n − 1)! × 2ⁿ. For instance, in a structure with 24 welds, the total number of possible welding sequences and directions is approximately N = 24! × 2²⁴ ≈ 1 × 10³¹. This illustrates the extremely large solution space and the enormous computational effort required to identify the global optimum through exhaustive search. Therefore, it becomes critically important to construct an appropriate mathematical model that accurately captures the optimization process. Such a model enables efficient exploration of the design space with significantly reduced computational cost, facilitating rapid and effective optimization of the welding sequence and direction.

Due to the geometric symmetry of the frame model, only one side was selected for analysis. In accordance with typical welding sequence combinations commonly employed in engineering practice, the 12 welds on one side of the frame are divided into three groups. The pointer variables A and D₁ represent the welding sequence and welding direction, respectively, for the four welds on both sides (Welds 1–4). Variable B denotes the welding sequence of the four welds on the flange plate attachments (Welds 5–8), with the welding direction uniformly set from the inner joint (weld intersection) to the outer edge. Finally, variables C and D₃ correspond to the welding sequence and direction for the remaining four welds (Welds 9–12).

When variable A was set to “1234”, it indicates that Welds 1–4 are welded in the sequence 1-2-3-4. If these four welds are performed from top to bottom, the welding direction D₁ is set to “0”; otherwise, it is set to “1”. For example, if A is “3124” and D₁ is “1011”, this indicates that the welds are executed in the order 3-1-2-4, with corresponding welding directions of 1-0-1-1. For variable C, if the welds are carried out from right to left, the welding direction D₃ is set to “0”; otherwise, it is set to “1”. The actual values of A, D₁, B, C, and D₃ used in the study are summarized in

Table 4. Code value and true value correspond table.

Coded Value	Actual Value
	A	D1	B	C	D3
I	1234	0000	5678	9101112	0000
II	3241	0110	5867	1091112	0010
III	4132	1001	5768	9111012	1001
IV	4231	1011	5876	9121110	1110

according to the rules described above.

To rigorously evaluate the effectiveness of the optimized welding schemes, the maximum welding deformation of the frame is selected as the evaluation metric. An optimization mathematical model is developed to minimize the residual deformation and stress in the welded components by adjusting the welding sequence and direction, under a set of predefined constraints. The specific formulation of the optimization model is as follows:

$$\begin{array}{c}\min [{\delta }_t=F_1(x)]\\ x=[x_1,x_2,x_3,x_4,x_5]=[A_t,B_t,C_t,D_{1t},D_{3t}]\\ s.t.1\le A_t\le 4;1\le B_t\le 4;1\le C_t\le ;1\le D_{1t}\le 4;1\le D_{3t}\le 4\end{array}$$

(7)

In the equation, $A_t$, $B_t$ and $C_t$ represents the welding sequence pointers for weld groups A, B, and C, respectively, at the t-th iteration of the optimization process. Variables $D_{1t}$ and $D_{3t}$ denotes the welding direction pointers at the t-th iteration, while ${\delta }_t$ represent the maximum residual welding deformation at the t-th iteration.

4.3. Surrogate Model Construction

To enable efficient discrete particle swarm optimization (DPSO) of the welding model, a surrogate model for the optimization formulation (Equation (7)) is constructed based on statistical principles. An orthogonal experimental design is employed, focusing on two key factors: welding sequence and welding direction [37]. By arranging the corresponding pointers in different orders, an experimental design matrix is generated (

Table 5. Design of an experiment of the different welding sequence and directions.

No.	Actual Value					Total Welding Deformation $\mathit{\delta }$ (mm)
	A	D1	B	C	D3
1	I (0000)	I (5678)	I (9101112)	I (0000)	I (0000)	2.83
2	II (0110)	II (5867)	II (1091112)	II (0010)	II (0110)	2.82
3	III (1001)	III (5768)	III (9111012)	II (0010)	III (1001)	2.75
4	IV (1011)	IV (5876)	IV (9121110)	IV (1110)	IV (1011)	2.63
5	I (0000)	II (5867)	III (9111012)	IV (1110)	I (0000)	2.58
6	I (0000)	II (5867)	IV (9121110)	III (1001)	I (0000)	2.46
7	III (1001)	IV (5876)	II (1091112)	II (0010)	III (1001)	2.68
8	IV (1011)	III (5768)	II (1091112)	I (0000)	IV (1011)	2.91
9	I (0000)	III (5768)	IV (9121110)	II (1001)	I (0000)	2.36
10	II (0110)	IV (5876)	III (9111012)	I (0000)	II (0110)	2.87
11	III (1001)	I (5678)	II (1091112)	IV (1110)	III (1001)	2.76
12	IV (1011)	II (5867)	I (9101112)	III (1001)	IV (1011)	2.58
13	I (0000)	IV (5876)	II (1091112)	III (1001)	I (0000)	2.67
14	II (0110)	III (5768)	I (9101112)	IV (1110)	II (0110)	2.25
15	III (1001)	II (5867)	IV (9121110)	I (0000)	III (1001)	2.49
16	IV (1011)	I (5678)	III (9111012)	II (1001)	IV (1011)	2.17

). A total of 16 distinct combinations of welding sequence and direction are selected to identify the optimal configuration. Welding deformation analyses of the frame are conducted based on each combination listed in the table, and the resulting residual deformations are obtained. These discrete data points are then collected to form the training dataset for surrogate model construction.

Based on the 16 groups in the experimental design matrix, the surrogate model corresponding to the optimization formulation (Equation (7)) is constructed as follows:

$$\begin{array}{l}\delta =2.8995-0.0355A-0.0012D_1-0.0572B-0.001C+0.1679E\\ -0.0806A^2+0.1143D_1^2+0.1367B^2+0.1156C^2+0.1172D_3^2\\ -0.1159AD+0.1547AB+0.0273AC-0.0996AE\\ -0.1768D_{1}B+0.0764D_{1}C+0.0252D_1{D}_3\\ -0.1947BC-0.0721BE-0.17C{D}_3\end{array}$$

(8)

To evaluate the accuracy of the surrogate model developed in this study, the commonly used multiple correlation coefficient R and the adjusted coefficient of determination ${\overline{R}}^2$ are adopted to assess the quality of fit [30].

$$\mathrm{R}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(y_i-\overline{y})({\widehat{y}}_{\mathrm{i}}-\overline{y})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-\overline{y})}^2}}}}}$$

(9)

$${\overline{R}}^2=1-{\displaystyle \frac{n-1}{n-m-1}}(1-R^2)$$

(10)

In the equations, n represents the number of experimental samples, m is the number of design variables, $y_i$ denotes the observed value of response variable i (i.e., the total welding deformation listed in Table 5), ${\widehat{y}}_i$ is the predicted value of response variable i, and $\overline{y}$ is the mean of the observed values. The closer the multiple correlation coefficient R is to 1 within the range [0,1], the better the fitting performance of the surrogate model. After calculation, the multiple correlation coefficient R and the adjusted coefficient of determination ${\overline{R}}^2$ for the surrogate model of welding residual deformation are found to be 0.945 and 0.921, respectively. These results indicate that the surrogate model demonstrates excellent fitting performance.

4.4. Optimization of the Frame Welding Scheme

The discrete particle swarm optimization (DPSO) algorithm features simple parameter configuration, strong global search capability, and a straightforward optimization mechanism [38]. Moreover, it is easily combined with other optimization techniques. Due to its effectiveness in avoiding local optima in practical engineering problems [39], an improved version of the DPSO algorithm is adopted in this study to optimize the surrogate model for the welding process. The flowchart of the DPSO algorithm is illustrated in Figure 13, and the corresponding optimization strategy is presented as follows.

Step 1: The parameters of the particle swarm optimization algorithm are initialized as follows: the population size N is set to 50, the maximum number of iterations $T_{max}$ is 500, the initial inertia weight ω is 1, and the cognitive and social learning factors $c_1$ and $c_2$ are both set to 2. The maximum particle velocity $V_{max}$ is defined as 10.

Step 2: Based on the encoding scheme in Table 5, each particle is decoded into its corresponding welding sequence and welding direction.

Step 3: The fitness value $Fitness$ of each particle is calculated using the defined fitness function. The personal best position of each particle is stored, along with the best fitness value and corresponding global best position of the entire swarm.

Step 4: The velocity and position of each particle are updated according to the velocity and position update equations (Equations (5) and (6)).

Step 5: The fitness value of each updated particle is calculated. The current fitness value is compared with the fitness value at its historical best position. If the current value is better, the particle’s current position is recorded as its new personal best.

Step 6: For each particle, the fitness value at its personal best position is compared with the current global best fitness value. If the former is better, the global best position and fitness value are updated accordingly.

Step 7: It is determined whether the stopping criteria are met—either reaching the maximum number of iterations or achieving the desired level of accuracy. If the conditions are satisfied, the optimal solution is output; otherwise, the algorithm returns to Step 3 and continues the iterative process.

Step 8: Based on the optimized discrete variables, a suitable welding sequence and direction are selected to guide practical production.

By applying the Discrete Particle Swarm Optimization (DPSO) algorithm, an optimized result of the surrogate model was achieved. Following multiple optimization iterations, the optimal welding sequence and direction were determined as follows: IV(4231), III(1001), IV(5768), IV(9121110), and IV(1110). The corresponding prediction from the surrogate model indicated a total welding deformation of 1.536 mm. The optimal convergence trend of the specific optimization iterations is illustrated in Figure 14, where the mathematical optimization model converged after 80 iterations and reached the minimum value.

Based on these optimal parameters, a welding simulation of the frame was conducted, as illustrated in Figure 15. The simulation results indicate that the bending deformation of the crossbeam is relatively small, with a maximum value of 0.613 mm. The deformation of the upper cover plate remains below 0.475 mm. The bending deformation of the longitudinal beams exhibits a parabolic distribution along the X-axis, showing a consistent overall trend with minor variation. The maximum residual deformation of the frame is 1.619 mm, which is close to the predicted value from the surrogate model, thereby verifying its accuracy.

Compared with the original welding plan (Figure 11), the optimized scheme significantly reduces the deformation of the crossbeams, the upper cover plate, and longitudinal beams. The maximum welding deformation in the optimized case is reduced by 43% relative to the original design, demonstrating that the optimal solution effectively minimizes welding-induced deformation. This improvement is expected to enhance weldment quality and reduce overall production costs.

5. Conclusions

In this study, a computationally efficient framework combining a local–global finite element method with a discrete particle swarm optimization (DPSO) algorithm was developed to optimize the welding sequence of a large thick-plate engineering vehicle frame. The main conclusions can be summarized as follows:

(1)

A local–global welding simulation strategy was successfully established, in which detailed thermo-elastic–plastic analyses were performed only for representative local welded joints, while the overall frame deformation was predicted through an elastic global model by introducing residual plastic strains as initial strains. This approach significantly reduced computational cost while maintaining sufficient accuracy for large-scale welded structures.

(2)

The proposed finite element model demonstrated good agreement with experimental measurements. The predicted welding deformation of the frame showed an average deviation of only 5.6% compared with experimental results, confirming the reliability of the local–global mapping method for predicting welding-induced distortion in large engineering vehicle frames.

(3)

By integrating the local–global welding simulation with an improved DPSO algorithm, an efficient welding sequence optimization strategy was established. The optimized welding scheme effectively minimized residual deformation without modifying welding parameters such as current, voltage, or speed, making it highly practical for industrial applications.

(4)

Compared with the original welding sequence, the optimized scheme reduced the maximum welding deformation of the frame by approximately 43%. This substantial reduction demonstrates that welding sequence optimization alone can significantly improve structural accuracy and manufacturing quality for large welded frames.

(5)

The proposed framework avoids repeated global thermo-mechanical simulations during optimization and is therefore particularly well suited for problems involving extremely large solution spaces, such as welding sequence and direction optimization in complex welded structures.

Overall, the results indicate that the combination of a local–global finite element approach and intelligent optimization algorithms provides a powerful and practical tool for controlling welding deformation in large-scale welded structures. The proposed method can be readily extended to other complex welded assemblies in engineering vehicles, shipbuilding, and heavy equipment manufacturing, offering strong potential for improving manufacturing efficiency and reducing production costs.