Multi-objective process parameter optimization for induction heating curing of carbon fiber reinforced polymer wound circular tubes

Abstract

This study proposes an efficient electromagnetic induction heating approach to overcome the limitations of conventional curing processes for carbon fiber reinforced polymer (CFRP) wound circular tubes, such as complex procedures, high energy consumption, and high cost. Two external heating configurations, namely the copying coil and the cover-type coil, are investigated through an electromagnetic–thermal coupled finite element model validated by experiments. The model reveals the eddy current distribution, heating behavior, and temperature field evolution under different coil structures. The relative standard deviation (RSD) is introduced to evaluate temperature uniformity, while rotating speed, coil turns, and output current are selected as key process parameters. A multi-objective response surface model is developed with RSD, maximum temperature ( $T_{\max }$ ), and maximum temperature difference ( $\mathrm{\Delta }{T}_{\max }$ ) as response variables. By integrating weighted decomposition and non-dominated sorting, the Egret Swarm Optimization (ESO) algorithm is extended for multi-objective optimization to determine the Pareto-optimal parameter set. Results show that the optimized parameters significantly improve temperature uniformity while maintaining sufficient heating performance. This work provides theoretical and methodological guidance for efficient and energy-saving induction heating curing of CFRP wound circular tubes.

Introduction

Carbon Fiber Reinforced Polymer (CFRP), renowned for its high specific strength, high specific modulus, and excellent corrosion resistance, has been extensively applied in aerospace, energy, automotive, and pressure vessel industries.^1–4 However, its high manufacturing cost and low production efficiency have become major obstacles to large-scale application and sustainable development. The widely used two-step process of “filament winding–furnace curing” struggles to balance efficiency and energy consumption, thereby constituting a key bottleneck in industrialization.^5–7 Electromagnetic induction heating, as a non-contact, efficient, and controllable heating method, can rapidly and uniformly heat conductive materials and has gradually become a promising alternative technology for CFRP curing. ⁸

Extensive research has been conducted worldwide on the mechanisms, structural design, and process optimization of CFRP induction heating. Fu et al. established a coupled magnetic–thermal multiphysics model for CFRP and verified its accuracy through thermal imaging experiments on plain-weave composite plates, revealing that “coil–fiber texture coupling induces heat generation and heat conduction along fiber bundles”; they further highlighted that coil diameter and spacing significantly affect temperature field uniformity. ⁹ Gu et al. developed a “magnetic field–temperature field” mapping model and discovered the prevalent “edge-hot, center-cold” phenomenon during planar coil heating, proposing improvements by reducing coil spacing, introducing magnetic cores, and optimizing turn numbers. ¹⁰ Fu et al. further proposed a multi-coil variable-frequency and magnetic field superposition strategy for internal-coil heating of CFRP tubes, which significantly improved wall-thickness and circumferential temperature uniformity. ¹¹ Fink et al. experimentally validated an induction heating model for continuous carbon fiber composites, coupling intralayer potential distribution with through-thickness thermal conduction, and confirmed that polymer dielectric loss can dominate heat generation under specific conditions. ¹² Barazanchy and van Tooren investigated the induction welding mechanism of thermoplastic composites via finite element modeling, concluding that fiber Joule heating and matrix Joule heating are the dominant mechanisms, whereas inter-fiber contact resistance heating plays a minor role. ¹³ Li et al. numerically and experimentally analyzed the induction heating mechanism of CFRTP laminates and found that dielectric loss and contact resistance are the major heat sources, with dielectric loss being predominant. ¹⁴ Fink et al. subsequently introduced a resistive susceptor in composite induction welding to alleviate edge overheating and center cooling, thereby improving temperature uniformity and weld quality. ¹⁵ Kim et al. employed the lumped-element method to model heating behavior, revealing that stacking angle and interfacial contact conditions strongly affect in-plane heat transfer patterns. ¹⁶ Collectively, these studies have established a solid foundation for the mechanism and structural design of CFRP induction heating.

In terms of process parameter optimization, the Response Surface Methodology (RSM) has emerged as a powerful analytical tool. Li et al. investigated CFRP induction welding using a transient 3D finite element model, developed an RSM-based mapping between current, coil spacing, and temperature response, and determined optimal parameters validated by mechanical testing. ¹⁷ Zhang et al. combined laser and induction heating processes and applied the Box–Behnken Design (BBD) and WOA–BPNN hybrid model to optimize hardened layer depth and hardness, demonstrating its high-accuracy predictive performance. ¹⁸ Chen et al. designed a continuous carbon-fiber-reinforced polycarbonate prepreg manufacturing system based on resin melt impregnation theory and optimized its process parameters using orthogonal experiments and the TOPSIS–entropy weighting method. ¹⁹ Liu et al. developed a thermal field model for robot-assisted thermoplastic composite placement and analyzed heating distribution on curved surfaces. ²⁰ Samanis et al. characterized the thermal properties of quasi-isotropic CFRP and studied the influence of parameters under uniform induction heating. ²¹ Xiong et al. employed finite element and principal component analyses to evaluate the effects of coil parameters and process factors on heating efficiency and uniformity, verifying the optimal parameter combination via orthogonal tests. ²²

In the field of multi-objective optimization, the introduction of intelligent algorithms has markedly enhanced efficiency and global search capability. Kranjc et al. developed a finite element model and applied a genetic algorithm to optimize coil position, current amplitude, and frequency, achieving efficient heating of steel materials. ²³ Park et al. integrated experimental design with numerical simulation, employing RSM and a genetic algorithm to optimize voltage and frequency in the induction heating process. ²⁴ Chen et al. proposed a hybrid optimization approach combining neural networks and genetic algorithms, proving its superiority over conventional orthogonal methods. ²⁵ Wang et al. further improved induction heating quality and efficiency through joint optimization of process parameters using a neural network predictor coupled with a genetic algorithm. ²⁶

Overall, research on CFRP induction heating has evolved into a technically integrated framework linking mechanism modeling, coil topology design, and process optimization. Mechanistically, unified modeling of three heat-generation modes has been experimentally validated; structurally, conformal coils, magnetic-core coupling, and multi-coil frequency modulation have markedly improved temperature uniformity; and in optimization, data-driven multi-objective algorithms have replaced empirical tuning. Typical strategies involve constructing surrogate models through finite element analysis and Design of Experiments (DoE), followed by global optimization using algorithms such as NSGA-II, GA, and the Whale Optimization Algorithm. However, most existing studies focus on planar or welding scenarios. Systematic research on multi-objective synergistic optimization for rotational CFRP tubular structures under external induction heating remains scarce, and algorithm diversity is limited, making it challenging to simultaneously balance peak temperature, temperature uniformity, and energy efficiency.

To address these gaps, this study focuses on the process parameter optimization of induction heating curing for CFRP-wound circular tubes. Two external coil topologies are first modeled through an electromagnetic–thermal coupled finite element framework and validated experimentally. Then, with temperature uniformity, peak temperature, and maximum temperature differential within the effective heating zone as optimization objectives, output current, coil turns, and rotational speed are selected as design variables to construct a surrogate model via RSM. Finally, a multi-objective optimization is achieved through the ESOA, yielding the optimal parameter set that ensures sufficient curing temperature while significantly enhancing temperature uniformity. The results provide both theoretical and data support for the efficient and energy-saving design of induction heating curing processes for thermosetting CFRP tubes.

Mathematical model of electromagnetic inductive heating

Governing equations of the electromagnetic field

During the induction heating process, an alternating current flowing through the coil generates an alternating magnetic field, which induces eddy currents in the CFRP circular tube, thereby realizing internal heating of the material. This process is governed by Maxwell’s equations, with the core mechanism being the coupling among the electric field, magnetic field, and current density. First, Ampère’s circuital law describes the relationship between the magnetic field and the current density:

$\nabla \times H=J$

(1)

where $\nabla$ denotes the curl operator; $H$ represents the magnetic field intensity vector (A/m); and $J$ denotes the current density vector (A/m²). This equation indicates that the curl of the magnetic field intensity $H$ is proportional to the current density $J$ .

The alternating magnetic field induces an electric field according to Faraday’s law of electromagnetic induction, which can be expressed by equation (2):

$\nabla \times E=-\frac{\partial B}{\partial t}$

(2)

where $E$ denotes the induced electric field (V/m), and $B$ represents the magnetic flux density.

The magnetic flux density $B$ of the carbon fiber reinforced composite can be determined by equation (3):

$B={\mu }_0{\mu }_{r}H$

(3)

where ${\mu }_0$ is the vacuum permeability of the carbon fiber reinforced composite (H/m), with ${\mu }_0$ = 4π × 10⁻⁷ H/m; and ${\mu }_r$ is the relative permeability of the carbon fiber reinforced composite.

In solving electromagnetic field problems, directly computing the induced electric field $E$ and the magnetic field intensity $H$ is often challenging. Therefore, an auxiliary physical quantity, the vector magnetic potential $A$ (Wb/m), is typically introduced. It is defined by equation (4):

$B=\nabla \times A$

(4)

According to Ohm’s law, the total current density J generated in the carbon fibers is given by equation (5):

$J_f=\sigma E$

(5)

where $\sigma$ denotes the electrical conductivity of the composite material (S/m).

The relationship between the induced electric field $E$ and the displacement field $D$ is given by equation (6):

$D={\epsilon }_0{\epsilon }_{r}E$

(6)

where ${\epsilon }_0$ is the vacuum permittivity of the carbon fiber reinforced composite (F/m); ${\epsilon }_0=8.85\times {10}^{-12}\left(\mathrm{F}/\mathrm{m}\right)$ and ${\epsilon }_r$ is the relative dielectric constant of the composite.

Governing equation of heat conduction

During the inductive heating process of carbon fibers, Joule heat serves as the primary heating mechanism. The heat generation rate $Q$ (W/m³) can be calculated using equation (7):

$Q=\frac{J_f^2}{\sigma }$

(7)

During the inductive heating process of the CFRP wound circular tube, heat originates from within the material or is transferred between different materials. The governing equation for this process is given in equation (8), where $\rho$ is the material density $\left(\text{kg}/{\mathrm{m}}^3\right)$ , $C_p$ is the specific heat capacity $\left(\mathrm{J}/\left(\text{kg}·\mathrm{K}\right)\right)$ , $\lambda$ is the thermal conductivity tensor, $\nabla T$ is the temperature difference $\left(°\mathrm{C}\right)$ , and $\mu$ is the resin flow velocity $\left(\mathrm{m}/\mathrm{s}\right)$ , The term $\rho C_p\frac{\partial T}{\partial t}$ represents the temperature variation over time, $\nabla ·\left(-\lambda \nabla T\right)$ denotes the heat conduction term, and $\rho C_p\mu ·\nabla T$ represents the thermal convection term.

$\rho C_p\frac{\partial T}{\partial t}+\nabla ·\left(-\lambda \nabla T\right)+\rho C_p\mu ·\nabla T=Q$

(8)

Boundary conditions

As the temperature increases, the presence of a temperature gradient leads to heat exchange between the composite material and its surrounding environment. In this study, since the composite is surrounded by air, convective heat transfer between the composite and the air must be taken into consideration.

${\varphi }_c=hA\left(T-T_0\right)$

(9)

where ${\varphi }_c$ denotes the thermal convection flux (W), $A$ is the surface area at temperature $T$ (m²), $T$ represents the surface temperature of the composite ( $°\mathrm{C}$ ), $T_0$ is the ambient air temperature ( $°\mathrm{C}$ ), and $h$ is the convective heat transfer coefficient between the composite and the air (W/(m²·K)).

When there is a significant temperature difference between the composite material and its surrounding environment, heat is also transferred through thermal radiation. This process is governed by equation (10).

${\varphi }_r=\epsilon \gamma A\left(T^4-T_0^4\right)$

(10)

where ${\varphi }_r$ denotes the heat flux due to thermal radiation, $\epsilon$ is the surface emissivity of the composite material, and $\gamma$ is the Stefan–Boltzmann constant.

Establishment of the geometric model

This study established a three-dimensional electromagnetic–thermal coupled model of a carbon fiber reinforced polymer wound circular tube using COMSOL Multiphysics 6.3. Two coil geometries for external induction heating were considered: the first is a cover-shaped coil, semicircular in form and positioned on one side of the tube, as shown in Figure 1; the second is a copying coil, whose configuration conforms to the outer wall of the tube to provide a larger heating coverage, as illustrated in Figure 2. In Figure 1, the carbon fiber reinforced polymer wound circular tube rotates, whereas in Figure 2 the tube remains stationary. This modeling approach enables a direct comparison of the effects of cover-shaped and copying coil topologies on the temperature field distribution of the tube under identical external heating conditions.

Figure 1.

Macroscopic geometric model of cover-shaped coil induction heating.

Figure 2.

Macroscopic geometric model of copying coil induction heating.

This study’s induction heating model defines the CFRP-wound circular tube as a thin-walled structure, 150 mm in length, with a wall thickness of 3 mm and an inner diameter of 30 mm. The heating coil was uniformly specified with a thickness of 2 mm, 50 turns, an operating frequency of 13 kHz, and an input current of 15 A. The ambient temperature was maintained at 20°C, and the tube rotated at 5 r/min during heating. The cover-shaped coil was designed as a semi-circular arc covering 180° of the tube’s outer wall, axially aligned over a length of 50 mm and coaxial with the tube axis. For the copying coil, the input current was set at 18 A, with all other parameters identical to those of the cover-shaped coil, enabling a comparative analysis of temperature field distributions under different external heating configurations.

Material parameter settings

The material parameters of the CFRP-wound circular tube can be determined using the mixture rule and the Springer-Tsai model. ²⁷ Its density is calculated based on the fiber volume fraction $V_f$ , fiber density ${\rho }_f$ , and resin density ${\rho }_r$ , as defined by the following equation:

${\rho }_m=V_f{\rho }_f+\left(1-V_f\right){\rho }_r$

In this equation, ${\rho }_m$ represents the overall density of the composite material. The specific heat capacity of the CFRP is calculated using the following formula:

$C_m=\frac{V_f{\rho }_f{C}_f+\left(1-V_f\right){\rho }_r{C}_r}{{\rho }_m}$

In this equation, $C_m$ denotes the specific heat capacity of the composite material, while $C_f$ and $C_r$ represent the specific heat capacities of the fiber and the resin, respectively.

In terms of thermal conductivity calculation, the Springer–Tsai model can be used to account for the anisotropic effects caused by fiber orientation. The thermal conductivity ${\lambda }_{22}$ of the composite in the direction perpendicular to the fibers is given by the following equation:

$\frac{{\lambda }_{22}}{{\lambda }_r}=\left(1-2\sqrt{\frac{V_f}{\pi }}\right)+\frac{1}{B}\left(\pi -\frac{4}{\sqrt{1-B^2{V}_f/\pi }}\arctan \left(\frac{\sqrt{1-B^2{V}_f/\pi }}{1+B\sqrt{V_f/\pi }}\right)\right)$

The parameter $B$ is defined as:

$B=2\left(\frac{{\lambda }_r}{{\lambda }_f^T}-1\right)$

In this equation, ${\lambda }_r$ represents the thermal conductivity of the resin, and ${\lambda }_f^T$ denotes the thermal conductivity of the fiber in the direction perpendicular to the fiber orientation.

The thermal conductivity ${\lambda }_{11}$ in the fiber direction can be expressed using the following mixture rule:

${\lambda }_{11}={\lambda }_r\left(1-V_f\right)+{\lambda }_f^L{V}_f$

In this equation, ${\lambda }_f^L$ represents the thermal conductivity of the fiber in the fiber direction, and ${\lambda }_{11}$ denotes the effective thermal conductivity of the composite in that direction.

This study focuses on a thermosetting carbon fiber reinforced polymer wound circular tube with an alternating ±45° lay-up to investigate the effect of fiber orientation on heating behavior. To accurately characterize the influence of the winding structure on the temperature field distribution, the spatial coordinate system in the model was transformed into a cylindrical coordinate system, and anisotropic thermal conductivity was defined in diagonal matrix form to differentiate heat conduction along different directions; the remaining physical parameters were specified in the original spatial coordinate system. Most of the simulation input data were obtained from technical documentation provided by the company and from the material library embedded in COMSOL software, with the specific parameter settings summarized in Table 1. To reduce computational complexity, the material parameters were assumed to remain constant within the studied temperature range and not vary with temperature.

Table 1.

Simulation model parameters.

	Coils	Air	CFRP wound circular tube
Thermal conductivity (W/(m·K))			(2,8,8)
Electrical conductivity (S/m)	5.9 × 107	0	(30,1800,1800)
Specific heat capacity (J/(kg·K))			1000
Density (kg/m^3)	8960		1500
Relative permittivity	1	1	6.8
Relative permeability	1	1	1

In finite element analysis, proper mesh generation has a significant impact on computational accuracy and convergence. To ensure the stability and reliability of the electromagnetic–thermal coupled analysis, the geometric model was reasonably meshed. Specifically, the coil and tube models were discretized using free tetrahedral elements, with local refinement applied in key regions to improve the accuracy of the electromagnetic and thermal field calculations, while a relatively coarse mesh was used for the air domain to reduce computational cost. Figure 3 shows the mesh generation of the two coil configurations, and Figure 4 presents the corresponding mesh quality distribution. As shown in the figure, the mesh quality of the geometric model is mainly distributed between 0.5 and 0.9, where a value closer to one indicates better element quality. Therefore, the established model exhibits high mesh quality and can accurately represent the physical characteristics of the actual induction heating process, providing a reliable basis for subsequent electromagnetic–thermal coupled simulations.

Figure 3.

Mesh generation diagram.

Figure 4.

Mesh quality distribution diagram.

Analysis of temperature field distribution and experimental validation

Figure 5 shows the eddy current distributions in the CFRP wound circular tube under different coil configurations. Clear differences are observed between the cover-shaped coil and the copying coil in terms of their effective heating regions. Figure 5(a) presents the eddy current distribution of the cover-shaped coil, where the eddy current intensity is mainly concentrated in the area directly facing the coil, forming a distinct annular high-intensity zone that rapidly decreases toward both ends. Figure 5(b) displays the eddy current distribution of the copying coil, where the eddy current intensity is more continuous along the axial direction of the tube and covers a wider region. The high-intensity zone gradually transitions from the middle of the tube toward both ends. These results indicate that the cover-shaped coil tends to achieve localized concentrated heating, whereas the copying coil generates eddy current effects over a broader region, enabling a wider heating range.

Figure 5.

Eddy current field distribution.

Figure 6 shows the heat generation distributions of the CFRP wound circular tube under different coil configurations. The distributions of heat generation closely correspond to the eddy current field patterns. Figure 6(a) presents the static heat generation of the cover-shaped coil, where the heat source is mainly concentrated in the region directly facing the coil, forming a distinct annular concentration effect that rapidly attenuates toward both ends. Figure 6(b) displays the heat generation of the copying coil, where the high-intensity region is more continuous along the axial direction of the tube, covering a larger area and gradually extending toward both ends, thereby producing a broader heating region. These results are consistent with the eddy current distributions: the cover-shaped coil achieves localized heating through intense eddy currents in a confined area, whereas the copying coil induces eddy currents over a wider range, leading to a more distributed heat source region.

Figure 6.

Heat generation distribution.

Figure 7 shows the temperature field distribution of the CFRP wound circular tube heated by the cover-shaped coil. The evolution pattern corresponds closely to the previously described eddy current and heat generation distributions. The cover-shaped coil induces strong eddy currents in the region directly opposite the coil, leading to concentrated heat generation and a rapid temperature rise in the early stage, thus forming a distinct annular high-temperature zone. Electromagnetic induction heating operates by generating eddy currents in CFRP under an alternating magnetic field, which causes the material itself to undergo Joule heating. As the entire CFRP structure participates in heat generation and interlaminar heat conduction occurs within the material, the initially localized heating gradually diffuses outward over time. Consequently, the central region continues to increase in temperature, and the heat propagates toward both ends. At 60 s, the tube’s middle section has largely reached a high-temperature state, while the ends remain relatively cool, showing a “hot center and cool ends” distribution. With extended heating to 300 s and 600 s, the temperature field tends to stabilize, and the high-temperature zone consistently remains in the region directly facing the cover-shaped coil. These results indicate that cover-shaped coil heating is characterized by an initial localized heating effect, which subsequently spreads through thermal conduction, leading to global heating dominated by a concentrated heat source.

Figure 7.

Temperature field distribution diagram.

Figure 8 shows the temperature field distribution of the CFRP wound circular tube heated by the copying coil. The evolution pattern corresponds to the previously described eddy current and heat generation distributions. In the initial stage, a localized temperature rise appears in the central region of the tube. Within 5–10 s, the temperature increases markedly, forming a continuous high-temperature zone near the center. As the heating time extends to 60 s, the central region undergoes a rapid temperature rise, with heat propagating toward both ends and progressively expanding the coverage area. At 300 s and 600 s, the high-temperature region further expands and stabilizes, with the overall temperature level higher than that achieved with the cover-shaped coil. These results indicate that the copying coil, owing to its wide eddy current distribution, generates heat over a broader area, resulting in rapid temperature rise and extensive coverage in the evolution of the temperature field, consistent with its uniform eddy current and heat generation patterns.

Figure 8.

Temperature field distribution diagram.

Figure 9 shows the temperature–time profiles of the inner and outer surfaces of the CFRP wound circular tube under cover-shaped and copying coil heating. In both cases, the inner and outer surface temperatures increase progressively over time, while the temperature difference remains within 2°C throughout the heating process. This minimal radial difference is attributed to the alternating magnetic field inducing eddy currents in the CFRP, which directly generate heat within the material, and to the excellent interlaminar thermal conductivity that promotes rapid heat transfer across the thickness, thereby suppressing significant radial gradients. The combined effect of volumetric heat generation and interlaminar conduction ensures efficient and uniform heating across the entire tube wall thickness. These results indicate that the outer surface temperature can serve as a reliable indicator of internal temperature evolution, providing a practical reference for predicting inner-wall temperature and assessing whether curing conditions have been met.

Figure 9.

Temperature rise curves of the inner and outer surfaces.

To verify the accuracy of the established finite element model, an experimental study on the temperature field distribution of the carbon fiber wound circular tube under the cover-type coil heating configuration was conducted. Due to the obstruction caused by the coil, it is difficult to obtain a complete temperature field map during the copying coil heating process; therefore, corresponding experiments were not performed. It should be noted that the finite element model in this study was developed based on the fundamental physical coupling between the electromagnetic and thermal fields, and its reliability has been validated through the cover-type coil experiments. The model can accurately capture the mechanism characteristics of the induction heating process and can thus be applied to the numerical simulation and mechanism analysis of the copying coil configuration.

Figure 10 shows the experimental setup, which mainly consists of a rotating mechanism, an induction heating coil, a power supply, a CFRP wound circular tube, an infrared thermal imager, and an image display system. The steel tubes at both ends of the rotating mechanism serve only as clamps, while the mandrel supporting the CFRP wound circular tube is made of fiberglass. As a nonmetallic material with small thickness, low thermal conductivity, and negligible magnetic permeability, the fiberglass mandrel does not generate eddy current losses during heating and has little effect on the magnetic field distribution; thus, its thermo-magnetic effect can be ignored. Temperature measurement was performed using a FOTRIC 628C infrared thermal imager, with a resolution of 640 × 480, thermal sensitivity of 30 mK, and a measurement range from −20°C to 2000°C, with an accuracy of ±2°C or ±2%. These specifications meet the requirements for real-time monitoring of the temperature field during CFRP induction heating. The image display system was used to visualize the temperature distribution in real time, facilitating analysis and validation of the heating performance.

Figure 10.

Experimental platform.

Figure 11 shows the experimental results of electromagnetic induction heating of the CFRP wound circular tube. The experimental and simulation results exhibit strong agreement in the evolution of the temperature field: at 10 s, the temperature begins to rise, mainly concentrated in the region directly facing the coil; at 300 s, the temperature further increases and gradually becomes more uniform; at 600 s, the temperature field reaches near stability, with the peak temperature closely matching the simulation. These results indicate that the developed model can accurately predict the heating process. It should be noted that, since the cover-shaped coil was arranged on only one side of the rotating tube, the thermal imaging data were acquired from the outer surface opposite the coil. Furthermore, the influence of the fiber winding structure led to certain local discrepancies between the experimental and simulated temperature distributions.

Figure 11.

Experimental temperature field distributions.

Key process parameter modeling and experimental analysis of induction heating

In practical induction heating processes, the presence of the mold makes it difficult to place the coil inside the mold cavity, while the copying coil wrapped around the outer surface of the composite not only complicates mold disassembly but also prevents real-time observation of the temperature field distribution through infrared thermography during heating, which is unfavorable for experimental validation and process analysis. In contrast, the cover-type coil offers greater installation flexibility, enabling stable heating and temperature monitoring without affecting the mold structure, thus providing higher experimental feasibility and observability. Moreover, in actual industrial manufacturing scenarios involving simultaneous rotation and heating of CFRP components, the cover-type coil is more widely applied due to its simple structure, stable electromagnetic coupling, and ease of integration with rotational mechanisms. Therefore, the subsequent process parameter optimization in this study focuses primarily on the heating mode of the cover-type coil.

In electromagnetic induction heating, the uniformity of the temperature field is a critical criterion for evaluating forming quality and process stability. Traditionally, the maximum temperature difference or temperature gradient has been employed for characterization; however, both metrics are strongly influenced by the absolute temperature level. To address this limitation, the Relative Standard Deviation (RSD) is introduced in this study as the evaluation metric for temperature field uniformity. RSD characterizes the consistency of temperature distribution through the ratio of standard deviation to average temperature, thereby eliminating dimensional effects and providing a clearer measure of uniformity within the heating region. Its applicability and comparability for such analyses have been confirmed in previous research. ²⁸ The RSD is calculated as follows:

$RSD=\frac{1}{\overline{T}}\sqrt{\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(T_i-\overline{T}\right)}^2}$

$\overline{T}=\frac{1}{N}{\displaystyle \sum }_{i=1}^N{T}_i$

where $T_i$ is the temperature at the i measuring point, $\overline{T}$ is the average temperature, and N is the total number of measuring points. A smaller RSD value indicates that the temperature distribution is closer to uniform and the heating process is more stable; conversely, a larger RSD value suggests significant variations in the temperature field and poorer uniformity.

In the electromagnetic induction heating of CFRP wound circular tubes, rotational speed, number of coil turns, and output current are the three principal parameters that most directly reflect process control effectiveness. Rotational speed governs the heating distribution and heat diffusion rate within the tube, thereby determining the uniformity of the temperature field. The number of coil turns defines the strength and distribution of the magnetic field, influencing the magnitude of the induced current and the heating range, and thus directly affecting peak temperature and temperature gradients. Output current regulates the induction power and heating rate, serving as the key factor determining the energy absorbed by the material. Accordingly, this study selects rotational speed, number of coil turns, and output current as the optimization variables, while adopting RSD, maximum temperature, and maximum temperature difference in the heating zone as the optimization objectives. Process parameters are optimized from three dimensions—temperature uniformity, heating intensity, and temperature difference control—with the aim of improving both the quality of the temperature field and the stability of the curing process.

This study considered a rotational speed range of 5–35 r/min, a coil turn range of 35–65 turns, and an output current range of 10–20 A. The BBD method was applied, with rotational speed, number of coil turns, and output current selected as the design variables, while the RSD within 600 s of heating, the maximum tube temperature, and the maximum temperature difference in the heating zone were taken as the response variables. Table 2 summarizes the factor levels of the BBD experimental design, and Table 3 reports the results of the response surface analysis.

Table 2.

Response surface design factors and levels.

Factors	Levels
	−1	0	1
Rotational speed (r/min)	5	20	35
Coil turn (N)	35	50	65
Output current (A)	10	15	20

Table 3.

Response surface experimental results.

Experiment No.	Parameter selection			Experimental results
	Rotational speed (r/min)	Coil turns (N)	Output current (A)	RSD	$T_{\max }$ (°C)	$\mathrm{\Delta }{T}_{\max }$ (°C)
1	−1	−1	0	2.1231	134.549	34.849
2	1	−1	0	2.0234	134.582	36.746
3	−1	1	0	3.1561	189.11	68.993
4	1	1	0	3.1593	189.23	66.987
5	−1	0	−1	1.6170	102.128	30.892
6	1	0	−1	1.3171	102.256	29.763
7	−1	0	1	3.1603	197.63	57.075
8	1	0	1	3.1604	197.54	56.843
9	0	−1	−1	1.0851	68.133	12.258
10	0	1	−1	2.1372	144.949	32.741
11	0	−1	1	3.1421	164.6	50.456
12	0	1	1	5.1689	223.85	82.23
13	0	0	0	3.1463	143.467	50.894
14	0	0	0	3.1463	142.345	49.256
15	0	0	0	3.3464	144.567	52.346

Multivariate nonlinear regression fitting was applied to the response values in Table 3 to develop response surface models that describe the relationships of the tube’s RSD, maximum temperature, and maximum temperature difference in the effective heating zone with rotational speed, number of coil turns, and output current, as shown below:

$\begin{array}{l}\text{RSD}=3.2130-0.04954v+0.65597n+1.05941I+0.02573vn+0.07500vI\\ +0.24368nI-0.58357v^2-0.01395n^2-0.31573I^2\end{array}$

$\begin{array}{l}{T}_{\max }=143.4597+0.02387v+30.65938n+45.76925I+0.02175vn-0.05450vI\\ -4.39150nI+8.95679v^2+9.45129n^2-2.52796I^2\end{array}$

$\begin{array}{l}\mathrm{\Delta }{T}_{\max }=50.8320-0.18375v+14.58025n+17.61875I-0.97575vn+0.22425vI\\ +2.82275nI+0.14188v^2+0.91988n^2-7.33062I^2\end{array}$

The fitting degrees of the above models were 0.966, 0.995, and 0.963, respectively, all close to 1, indicating good fitting performance. This demonstrates that the multivariate nonlinear regression models developed in this study are statistically significant and can effectively capture the relationships between the influencing factors and the response variables.

In the process of response surface modeling, each factor was first coded as −1, 0, and one to facilitate multifactor analysis of variance and regression modeling. The quadratic polynomial model derived from this coded format clearly represents the statistical significance of both main effects and interaction effects. However, the coded variables lack direct engineering and physical meaning, making them inconvenient for practical substitution and prediction of process parameters. Therefore, in addition to presenting the coded form equations, this study further decoded them into actual process parameters—rotational speed (V), number of coil turns (N), and output current (I)—as shown below. This approach ensures that the models maintain statistical rigor while also being directly applicable to engineering optimization and process design.

$\begin{array}{l}\text{RSD}=-3.26868+0.07973V-0.00109N+0.40830I+0.0001143VN+0.001000VI\\ +0.003249NI-0.0025937V^2-0.0000620N^2-0.012629I^2\end{array}$

$\begin{array}{l}{T}_{\max }=-41.92825-1.58466V-1.28025N+15.12960I+0.0000967VN-0.0007267VI\\ -0.058553NI+0.039808V^2+0.042006N^2-0.101118I^2\end{array}$

$\begin{array}{l}\mathrm{\Delta }{T}_{\max }=-81.09482+0.13451V+0.08537N+10.37887I-0.0043367VN+0.002990VI\\ +0.037637NI+0.0006306V^2+0.004088N^2-0.293225I^2\end{array}$

An analysis of variance (ANOVA) was conducted to evaluate the validity of the regression models, as shown in Table 4. All three response models were found to be statistically significant, confirming the reliability of the regression equations. Among the investigated factors, the number of coil turns and the output current exerted the greatest influence on RSD, $T_{\max }$ , and $\mathrm{\Delta }{T}_{\max }$ , whereas the effect of rotational speed was comparatively minor. This highlights current and coil turns as the dominant parameters governing the temperature field distribution. The predictive accuracy of the models was further verified by comparing predicted and experimental values, as illustrated in Figure 12. Strong agreement was observed across all three response variables, with most data points lying close to the ideal diagonal line and only minor deviations. These results demonstrate that the models effectively capture the nonlinear relationships between process parameters and response variables, yielding low fitting errors and high predictive precision. Overall, the established response surface models exhibit strong reliability and generalization capacity, providing a robust foundation for subsequent multi-objective optimization analyses.

Table 4.

ANOVA table for response surface model.

Source	F-value			p-value
	RSD	$T_{\max }$	$\mathrm{\Delta }{T}_{\max }$	RSD	$T_{\max }$	$\mathrm{\Delta }{T}_{\max }$
Model	37.02	153.26	58.94	<0.0001	<0.0001	<0.0001
V	0.1973	0.00019	0.00788	0.67548	0.98964	0.93273
N	34.59618	307.32167	49.58559	0.00202	<0.0001	0.00089
I	90.23682	684.87952	72.4062	0.00022	<0.0001	0.00037
V2	0.0266	<0.0001	0.11104	0.87682	0.99332	0.75248
N 2	0.22612	0.00049	0.00586	0.65446	0.98327	0.94193
I2	2.38696	3.15255	0.92927	0.18301	0.13597	0.37933
VN	12.63732	12.10541	0.00217	0.0163	0.01767	0.96467
VI	0.00722	13.47897	0.09109	0.93558	0.01442	0.77494
NI	3.69896	0.9643	5.78516	0.11246	0.3712	0.06122

Figure 12.

Comparison of actual and predicted values of different response variables.

Figure 13 presents the single-factor analysis curves derived from the response surface models, showing the effects of each process parameter on the responses. The results reveal that increasing current leads to substantial rises in both RSD and $\mathrm{\Delta }{T}_{\max }$ , as well as a marked increase in $T_{\max }$ , confirming current as the dominant factor influencing temperature uniformity and peak temperature. By contrast, rotational speed exerts only minor effects, with slight fluctuations across all responses. The number of coil turns contributes notably to the enhancement of $T_{\max }$ but has limited influence on RSD and $\mathrm{\Delta }{T}_{\max }$ . Overall, current is identified as the primary determinant, followed by coil turns, with rotational speed exerting the weakest effect. These insights provide valuable guidance for subsequent multi-objective optimization.

Figure 13.

Single-factor influence analysis based on the response surface model.

Figure 14 illustrates the interaction effects of process factors on the RSD. Under the combined influence of current and coil turns, $T_{\max }$ increases markedly with both parameters, while RSD and $\mathrm{\Delta }{T}_{\max }$ also rise, indicating that excessive current compromises temperature uniformity. In the interaction between current and rotational speed, current remains the dominant factor, with speed adjustments improving uniformity only within a limited range. For the interaction between rotational speed and coil turns, $T_{\max }$ increases with coil turns, whereas the effect of speed on uniformity is comparatively minor. Overall, current exerts the strongest influence in two-factor interactions, followed by coil turns, with rotational speed having the weakest effect.

Figure 14.

Interaction effects of process factors on RSD.

Figure 15 shows that the interactions among process factors exert a considerable influence on the maximum temperature. The interaction between current and coil turns is the most pronounced, with $T_{\max }$ rising substantially when both parameters increase simultaneously, indicating a strong synergistic effect. The interaction between current and rotational speed is of secondary importance: $T_{\max }$ increases primarily with current, while changes in speed have only a limited impact. The interaction between rotational speed and coil turns is relatively weak; $T_{\max }$ increases gradually with the number of turns, whereas the fluctuations induced by speed variations remain minor.

Figure 15.

Interaction effects of process factors on $T_{\max }$ .

Figure 16 shows that the interactions among process factors significantly affect the maximum temperature difference in the effective heating zone. The interaction between current and coil turns is the most pronounced: $\mathrm{\Delta }{T}_{\max }$ rises sharply with increasing current, and additional coil turns further aggravate non-uniformity, indicating that excessive current–turns combinations enlarge temperature disparities. In the interaction between current and rotational speed, current dominates, while speed variations only partially mitigate the growth of $\mathrm{\Delta }{T}_{\max }$ . The interaction between rotational speed and coil turns is relatively weak; $\mathrm{\Delta }{T}_{\max }$ increases gradually with the number of turns, whereas the effect of speed remains limited.

Figure 16.

Interaction effects of process factors on $\mathrm{\Delta }{T}_{\max }$ .

The integrated analysis of single- and two-factor interactions highlights distinct disparities in the influence of process parameters on the response variables. Current consistently emerges as the dominant factor, directly governing the magnitude of peak temperature rise while also being the principal variable responsible for non-uniform temperature distribution and the enlargement of $\mathrm{\Delta }{T}_{\max }$ in the effective heating zone. Coil turns also exert a notable positive effect on peak temperature, though their influence on temperature uniformity is secondary to that of current. Rotational speed has only a marginal effect, contributing limited improvements to uniformity within a narrow range. Further interaction analysis shows that the synergistic effect of current and coil turns produces the most pronounced increase in $T_{\max }$ , but at the cost of aggravating the expansion of $\mathrm{\Delta }{T}_{\max }$ . The interaction between current and rotational speed primarily reflects the dominance of current, with speed providing only minor adjustments to uniformity. The interaction between rotational speed and coil turns is comparatively weak. Overall, current is identified as the key determinant in balancing heating efficiency and temperature uniformity, followed by coil turns, while rotational speed plays the least significant role. These insights provide a theoretical basis for prioritizing parameter adjustments in subsequent multi-objective optimization.

Multi-objective parameter optimization based on the Egret Swarm Optimization Algorithm

Principle of the Egret Swarm Optimization Algorithm

The Egret Swarm Optimization Algorithm (ESOA), proposed by Chen et al., ²⁹ is a swarm intelligence optimization method inspired by the predatory behavior of egret populations. It features advantages such as a small number of parameters, simple implementation, and rapid convergence. The underlying principle is derived from the contrasting foraging strategies of snowy egrets and great egrets: the snowy egret typically adopts a low-energy “sit-and-wait” strategy to obtain stable rewards, while the great egret favors continuous pursuit, which consumes more energy but generally yields higher returns. In the algorithm, these strategies are abstracted into a waiting strategy, an aggressive strategy, and a discriminant condition. The egret swarm is divided into several subgroups, each consisting of three egrets assigned distinct roles: egret a applies the waiting-guidance strategy, egret b performs random wandering before prey detection, and egret c executes encirclement and pursuit once the prey is identified. Through this cooperative division of roles, the algorithm achieves efficient global exploration and local exploitation, thereby approaching the optimal solution.

(1) Waiting Strategy

Suppose the position of the ith egret subgroup is $x_i\in R^{\mathrm{n}}$ , where n represents the dimensionality of the search space. Egret a evaluates the probability of prey existence at the current position, where A denotes the evaluation method used by egret a to assess prey presence at its current position. The evaluation result is expressed as ${\widehat{y}}_i$ , which can be parameterized as:

${\widehat{y}}_i=A\left(x_i\right)=w_i·x_i$

where $w_i\in R^{\mathrm{n}}$ denotes the weight of the evaluation method, and the estimation error $e_i$ can be expressed as:

$e_i={\Vert {\widehat{y}}_i-y_i\Vert }^2$

For the error $e_i$ , the gradient in the search space yields an n-dimensional vector ${\widehat{g}}_i$ , and its direction can be denoted as ${\widehat{d}}_i$ .

${\widehat{g}}_i=\frac{\partial {\widehat{e}}_i}{\partial w_i}=\frac{\partial {\Vert {\widehat{y}}_i-y_i\Vert }^2/2}{\partial w_i}=\left({\widehat{y}}_i-y_i\right)·x_i$

${\widehat{d}}_i=\frac{{\widehat{g}}_i}{\Vert {\widehat{g}}_i\Vert }$

During the foraging process, egret a adjusts its position by combining the optimal position of its subgroup and the optimal positions of all subgroups, thereby correcting toward the optimal direction. The correction terms with respect to the subgroup position and the global optimal position are denoted as $d_{h,i}$ and $d_{g,i}$ , respectively. The correction formula can be expressed as:

$d_{h,i}=\frac{x_{hbest}-x_i}{\Vert x_{hbest}-x_i\Vert }\frac{f_{hbest}-f_i}{\Vert x_{hbest}-x_i\Vert }+d_{hbest}$

$d_{g,i}=\frac{x_{gbest}-x_i}{\Vert x_{gbest}-x_i\Vert }\frac{f_{gbest}-f_i}{\Vert x_{gbest}-x_i\Vert }+d_{gbest}$

After the correction, the gradient is denoted as ${\mathrm{g}}_i\in R^{\mathrm{n}}$ , expressed as follows:

$g_i=\left(1-r_h-r_g\right)·{\widehat{d}}_i+r_h·d_{h,i}+r_g·d_{g,i}$

In the formula, $r_h$ and $r_g$ take values within the range [0,0.5), serving as balancing coefficients for different search strategies. The Adam optimizer is employed to update the gradient $g_i$ . The optimization method is expressed as follows:

$m_i={\beta }_1·m_i+\left(1-{\beta }_1\right)·g_i$

$v_i={\beta }_2·v_i+\left(1-{\beta }_2\right)·g_i$

$g_i=g_i-\frac{m_i}{\sqrt{v_i}}$

In the formula, ${\beta }_1$ and ${\beta }_2$ are set to 0.9 and 0.99, respectively. $m_i$ and $v_i$ represent the moving averages of the gradient and the squared gradient, respectively. Based on egret a judgment of the current situation, the next sampling point is taken as $x_{a,i}$ , expressed as:

$x_{a,i}=x_i+\exp \left(-\frac{t}{0.1·t_{\mathit{max}}}\right)·0.1·hop·g_i$

$y_{a,i}=f\left(x_{a,i}\right)$

In the formula, $t$ and $t_{\mathit{max}}$ represent the current iteration number and the maximum number of iterations, respectively, while hop denotes the difference between the upper and lower bounds in the solution space. $y_{a,i}$ represents the fitness at position $x_{a,i}$ .

• (2) Aggressive Strategy

Egret b adopts a behavior inclined toward random prey searching, which can be described as

$x_{b,i}=x_i+\hspace{0.17em}\tan \left(r_{b,i}\right)·\frac{hop}{1+t}$

$y_{b,i}=f\left(x_{b,i}\right)$

Egret c tends to actively pursue prey, and therefore employs an encirclement mechanism to update its position:

$x_{c,i}=\left(1-r_h-r_g\right)·x_i+r_h·D_h+r_g·D_g$

$D_h=x_{best}-x_i\hspace{0.5em};\hspace{0.5em}{D}_g=x_{gbest}-x_i$

$y_{c,i}=f\left(x_{c,i}\right)$

$D_h$ and $D_g$ represent the differences between the current position of the egret subgroup and its historically optimal position, as well as the global optimal position among all subgroups. $r_g$ and $r_h$ are random numbers within the range [0,0.5).

• (3) Discriminant Condition

After each member of the egret subgroup determines its own strategy, the subgroup will select the best plan and act together. $x_{s,i}$ denotes the solution matrix of the ith subgroup:

$x_{s,i}=\left[x_{a,i},x_{b,i},x_{c,i}\right]$

$y_{s,i}=\left[y_{a,i},y_{b,i},y_{c,i}\right]$

$c_i=\arg \hspace{0.17em}\min \left(y_{s,i}\right)$

$x_i=\{\begin{array}{ll}{x}_{s,i|c_i},& \text{if} y_{s,i|c_i}<y_i\hspace{0.5em}\text{or}\hspace{1em}r<0.3\\ x_i,& \text{else}\end{array}$

If the fitness $y_{s,i}$ of $x_{s,i}$ is superior to the current fitness $y_i$ , the subgroup will accept this option. Meanwhile, a random number $r\in \left(0,1\right)$ is introduced. When $r<0.3$ , the subgroup will also accept this option, implying a 30% probability of adopting a less optimal strategy.

Egret Swarm–based multi-objective optimization integrating weighted decomposition and non-dominated sorting

This research introduces a multi-objective optimization framework based on the Egret Swarm, integrating weighted decomposition with non-dominated sorting. Building upon the single-objective Egret Swarm Optimization Algorithm, the method employs a stochastic weight decomposition strategy to decompose the multi-objective optimization problem into a series of single-objective subproblems, while non-dominated sorting is applied to approximate the Pareto front. This approach, referred to as a scalarization-based Multi-Objective Egret Swarm Optimization Algorithm (MOESOA), differs from conventional multi-objective optimization methods that rely solely on non-dominated sorting. The optimization workflow of MOESOA is illustrated in Figure 17. First, a multi-objective optimization model is constructed using response surface models, with clearly defined parameter ranges and constraints. Latin hypercube sampling is then applied to initialize the population, ensuring sufficient diversity of candidate solutions. The multi-objective problem is subsequently decomposed into single-objective problems through weight allocation, which are solved using the Egret Swarm Optimization Algorithm. Non-dominated sorting is employed to generate the Pareto-optimal solution set, followed by constraint filtering to verify process feasibility. Finally, a weighted evaluation method is adopted to select the best solution from the Pareto set, achieving a balanced optimization of temperature uniformity and heating efficiency.

Figure 17.

Flowchart of the MOESOA optimization algorithm.

In solving the optimization problem based on the response surface models established from the above experimental data, it is necessary to impose certain restrictions on the process parameters, namely the optimization constraints. As the preceding work involved the induction-heating curing of CFRP-wound circular tubes, the feasible domain of the objective functions in the optimization model was constructed according to the experimental conditions. The constraints are summarized as follows:

(1) Rotational speed constraint:

$5\mathrm{r}/\hspace{0.17em}\min \le v\le 35\mathrm{r}/\hspace{0.17em}\min$

• (2) Coil turn constraint:

$35\le n\le 65$

• (3) Output current constraint:

$10A\le I\le 20A$

In summary, the optimization model for the electromagnetic induction heating curing of CFRP-wound circular tubes is formulated as follows:

$Y=\mathit{Min}\left[\text{RSD}\left(v,n,I\right),\mathrm{\Delta }{T}_{\max }\left(v,n,I\right),-T_{\max }\left(v,n,I\right)\right]$

$\{\begin{array}{l}5\mathrm{r}/\hspace{0.17em}\min \le v\le 35\mathrm{r}/\hspace{0.17em}\min \\ 35\le n\le 65\\ 10A\le I\le 20A\end{array}$

The proposed multi-objective process parameter optimization model was solved using the MOESOA algorithm to obtain the optimal Pareto solution set. The algorithm parameters were configured as follows: population size of 60, maximum iterations of 600, foraging aggregation factor of 1.2, waiting factor of 0.2, elite ratio of 15%, and random disturbance probability of 0.5. To ensure sufficient global search capability, the initial population was generated via Latin hypercube sampling, and representative Pareto solutions were identified through weight decomposition combined with non-dominated sorting. After 400 generations of evolution, the final Pareto solution set was obtained, as shown in Figure 18. The results demonstrate clear trade-offs among the objectives: pursuing higher $T_{\max }$ is typically accompanied by increases in RSD and $\mathrm{\Delta }{T}_{\max }$ , whereas optimizing uniformity metrics constrains the improvement of $T_{\max }$ .

Figure 18.

Pareto-optimal solution set.

Theoretically, the desired objective is to maximize $T_{\max }$ while minimizing both RSD and $\mathrm{\Delta }{T}_{\max }$ . In practice, however, this ideal scenario is difficult to achieve due to the mutual constraints among the influencing factors. Therefore, a method combining constraint-based filtering and weighted evaluation is employed to identify the optimal solution. The specific steps are as follows:

(1) Obtain the Pareto solution set containing the three objectives (RSD, $T_{\max }$ , and $\mathrm{\Delta }{T}_{\max }$ ) through the multi-objective Egret Swarm Optimization Algorithm.

(2) Apply process-specific constraints to filter the Pareto solution set, eliminating solutions that do not meet the requirements and retaining only feasible candidates.

(3) Perform dimensionless processing of the objective values of the candidate solutions to ensure comparability among indicators with different units. The commonly used method is min–max normalization:

$f_{i^{\prime }}\left(x\right)=\frac{f_i\left(x\right)-\min \left(f_i\right)}{\max \left(f_i\right)-\min \left(f_i\right)}\hspace{1em}.$

• (4) In the standardized solution set, the weighted scoring method is applied, where weights are assigned according to the importance of the process objectives, and a comprehensive score is calculated.

$S\left(x\right)=w_1{f}_{1^{\prime }}\left(x\right)+w_2{f}_{2^{\prime }}\left(x\right)+w_3{f}_{3^{\prime }}\left(x\right)$

• (5) Based on the evaluation results, the solution with the highest comprehensive score is selected as the final optimization scheme, achieving a balance between temperature uniformity and heating efficiency.

As shown in Table 5, the optimized process parameters were applied to the finite element model, yielding a maximum temperature of 112.46°C, a tube RSD of 1.145, and a maximum temperature differential of 29.6°C in the effective heating zone, which are highly consistent with the optimization results. These findings demonstrate that the proposed multivariate nonlinear regression model can effectively predict the relationships among rotational speed, coil turns, and output current with the overall temperature field’s RSD, maximum temperature, and maximum temperature differential, thereby verifying the reliability and practical value of the model.

Table 5.

Optimization results of ESOA.

	Rotational speed	Coil turn	Output current	RSD	$T_{\max }$	$\mathrm{\Delta }{T}_{\max }$
Optimization range	5-35	35-65	10-20	5.168-1.085	68.13-223.85	12.258-82.23
Optimization results	19.2	54.7	10.2	1.092	109.5	28.7

Conclusion

This study investigated the induction heating curing process of CFRP-wound circular tubes by establishing an electromagnetic–thermal coupled finite element model and validating its accuracy through experiments. The eddy current distribution, heat generation characteristics, and temperature field evolution under two external heating configurations—the cover-shaped coil and the copying coil—were systematically analyzed. The results revealed notable differences in heating performance and temperature distribution between the two methods.

Among the process parameters, output current exerted the most significant influence on peak temperature and temperature uniformity, followed by the number of coil turns, while the effect of rotational speed was comparatively minor. The synergistic interaction between current and coil turns markedly increased the peak temperature, although excessively high combinations led to larger temperature differentials and reduced uniformity.

Furthermore, multivariate nonlinear regression models were developed with peak temperature, Relative Standard Deviation, and maximum temperature differential in the effective heating zone as optimization objectives. By integrating the Egret Swarm Optimization Algorithm into a multi-objective optimization framework, an optimal parameter scheme was identified that balanced heating efficiency and uniformity. The strong agreement between finite element predictions and experimental results confirmed the reliability of the proposed modeling and optimization approach.

Overall, the proposed parameter optimization framework effectively improves the temperature field uniformity while ensuring sufficient curing levels, providing both a robust theoretical foundation and practical engineering reference for the efficient and energy-saving design of CFRP induction heating curing processes.