A method for fast prediction of composite curing process based on transformation optics

Abstract

This invention proposes a rapid prediction method for the curing process of composite materials based on transformation optics. By introducing transformation optics theory and utilizing equivalent transformations of material parameters, the method maintains the stability of electromagnetic fields, temperature fields, and stress-strain fields when the thickness of the composite material in the simulation model is artificially increased. This reduces the spatial scale differences of components in the simulation model, decreases the number of meshes and computational complexity, and shortens the simulation time. While ensuring computational accuracy, it achieves rapid prediction of the composite material curing process (curing parameters).

Description

Technical Field

[0001] This invention belongs to the field of simulation and calculation methods for composite material molding processes, and relates to a method for predicting the curing process of composite materials, specifically a method for rapid prediction of the curing process of composite materials based on transformation optics. Background Technology

[0002] Composite materials possess advantages such as light weight, high specific strength, high specific modulus, corrosion resistance, good fatigue resistance, and designable material properties, leading to their widespread application in aerospace, military, transportation, and energy fields. The manufacturing process of composite materials generally consists of four steps: shaping, curing, processing, and assembly. Among these, the curing process is the key step determining the final performance and shape accuracy of composite components. During curing, the resin undergoes a cross-linking reaction, and the fibers and resin together form the final structure. Simultaneously, complex physical phenomena such as temperature changes, curing degree evolution, stress accumulation, and component deformation occur, directly affecting the final quality of the composite component. For example, uneven temperature distribution during curing can easily lead to uneven curing, increased curing defects, high residual stress, and severe deformation in composite components. Therefore, accurately predicting the curing process of composite materials is an indispensable fundamental task in the manufacturing of composite components, serving as a crucial basis for optimizing the curing process and ultimately ensuring the molding quality of composite components.

[0003] In recent years, numerical simulation analysis methods have become an important and effective means of predicting the curing process of composite materials due to their high prediction accuracy and wide applicability. However, composite material components are usually thin-walled structures made up of multiple layers of prepreg, with thicknesses typically on the order of millimeters and lengths on the order of meters, often resulting in significant spatial scale differences in the geometric dimensions of the components. Since the finite element method relies on mesh elements to discretize the simulation object, these significant spatial scale differences lead to a substantial increase in the number of mesh elements in the simulation model, resulting in high computational complexity and long simulation times, making it difficult to meet the needs of rapid evaluation of composite material curing parameters, efficient optimization of curing processes, and real-time guidance for manufacturing. Therefore, achieving rapid prediction of the composite material curing process remains a major challenge.

[0004] To address the aforementioned issues, the inventors introduced transformation optics theory into the prediction of composite material curing processes. This method, through equivalent transformation of material parameters, expands the thickness of the composite material in the simulation model (reducing the spatial scale differences of composite component parts) while maintaining unchanged electromagnetic, thermodynamic, and stress-strain behaviors, thereby enabling rapid prediction of key curing parameters such as temperature, degree of curing, and curing deformation during the composite material curing process. Summary of the Invention

[0005] The purpose of this invention is to address the problems of large mesh size, long computation time, and low computational efficiency in the simulation and prediction of composite material curing processes. This invention proposes a fast prediction method for composite material curing processes based on transformation optics, so as to achieve rapid prediction of composite material curing processes while ensuring computational accuracy.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows:

[0007] A rapid prediction method for the curing process of composite materials based on transformation optics includes the following steps:

[0008] 1) Composite material thickness magnification: Obtain the geometric shape and size parameters of the actual composite material component, and based on the transformation optics theory, magnify the thickness of the composite material, construct the spatial coordinate transformation relationship, and obtain the spatial transformation Jacobian matrix corresponding to the composite material thickness before and after magnification.

[0009] 2) Obtaining equivalent material parameters of composite materials: Obtain the original material parameters of the actual composite material. Based on the Jacobian matrix obtained in step 1), calculate the equivalent material parameters of the composite material after thickness magnification according to the parameter equivalence relationship in transform optics.

[0010] 3) Construction of Curing Process Simulation Model: Establish a numerical simulation model of the curing process after the thickness of the composite material is enlarged, and assign the equivalent material parameters obtained in step 2) to the enlarged composite material component;

[0011] 4) Numerical solution and prediction of curing process parameters: The simulation model is meshed and numerically solved under the constraints of the control equation and boundary conditions to obtain the prediction results of at least one curing process parameter during the curing process of composite material components.

[0012] When the composite material curing process takes place within a closed cavity, a transformation layer is disposed on the outside of the composite material to compensate for changes in the electromagnetic field, temperature field, and / or mechanical field in the surrounding free space caused by the increased thickness of the composite material. The method for disposing of the transformation layer includes:

[0013] 1) Introduce a continuous medium region as a transformation layer on the outside of the composite material;

[0014] 2) Keep the outer boundary of the transformation layer unchanged before and after the coordinate transformation, and make its inner boundary change accordingly as the thickness of the composite material is increased. Use a spatial coordinate system consistent with the composite material to construct the spatial coordinate transformation relationship of the transformation layer, and calculate the spatial transformation Jacobian matrix of the transformation layer after the thickness of the composite material is increased.

[0015] 3) Obtain the original medium parameters of the transformation layer, and based on the Jacobian matrix, perform equivalent transformation on the medium parameters of the transformation layer according to the parameter equivalence relationship in transformation optics.

[0016] The thickness magnification factor n is related to the composite material thickness t and the minimum in-plane dimension L of the component. min The relationship satisfies the following formula:

[0017]

[0018] Where, n e n represents the thickness magnification factor limited by engineering tolerance requirements. u This is an empirical parameter, with a value of 8.2.

[0019] The curing process parameters include temperature, temperature field, degree of curing, degree of curing field, stress, stress field, and curing deformation.

[0020] The beneficial effects of this invention are:

[0021] This invention introduces transformation optics theory and utilizes equivalent transformations of material parameters to maintain the stability of electromagnetic, temperature, and stress-strain fields when the thickness of the composite material in the simulation model is artificially increased. This reduces spatial scale differences among components in the simulation model, decreases the number of meshes and computational complexity, and shortens simulation time. While ensuring computational accuracy, it achieves rapid prediction of the composite material curing process (curing parameters), providing an effective technical means for curing process optimization and final component quality assurance. Attached Figure Description

[0022] Figure 1 (a) A schematic diagram of a fast simulation prediction method based on transformation optics and (b) A schematic diagram of coordinate transformation of a cylindrical composite material structure model before and after applying transformation optics.

[0023] Figure 2 These are schematic diagrams of (a) scenario O1 in the oven curing process simulation model and (b) scenario O2 in the oven curing process simulation model.

[0024] Figure 3 These are temperature distribution diagrams of cylindrical composite material structures at the final heating moment under different magnifications in (a) scenario O1 and (b) scenario O2.

[0025] Figure 4 It shows the location distribution of temperature monitoring points (P1-P7) in scene O1 and their corresponding temperature change curves at different magnifications.

[0026] Figure 5(a) A schematic diagram of the interfaces defined on the composite material structure and (b) the distribution curves of the average relative error (MRE) in the thickness direction at the five interfaces of the composite material structure at different magnifications in scenarios O1 and (c) O2.

[0027] Figure 6 The images show (a) the mesh size distribution and (b) the statistical results of mesh size and quantity for the composite material structure in scene O1 at different magnifications.

[0028] Figure 7 These are schematic diagrams of (a) scenario M1 in the microwave curing process simulation model and (b) scenario M2 in the microwave curing process simulation model.

[0029] Figure 8 These are simulation results of the final heating time at different magnifications in the microwave curing model. (a) Electric field distribution in the microwave heating cavity in scenario M1; (b) Temperature distribution of the cylindrical composite material structure in scenario M1; (c) Electric field distribution in the microwave heating cavity in scenario M2; (d) Temperature distribution of the cylindrical composite material structure in scenario M2.

[0030] Figure 9 It shows the location distribution of temperature monitoring points (P1-P5) in scene M1 and their corresponding temperature change curves at different magnifications.

[0031] Figure 10 These are the thickness average relative error (MRE) distribution curves at five interfaces of the composite material structure under different magnifications in scenarios M1 (a) and M2 (b).

[0032] Figure 11 The diagram shows (a) the mesh size distribution and (b) the statistical results of mesh size and quantity for the composite material structure in scene M1 at different magnifications.

[0033] Figure 12 These are schematic diagrams of (a) scenario M3 in the microwave curing process simulation model, (b) scenario M4 in the microwave curing process simulation model, and (c) scenario M5 in the microwave curing process simulation model.

[0034] Figure 13 These are simulation results for scenarios M3 and 4 in the microwave curing model, with a heating time of 1 min at different magnifications. (a) Electric field distribution in the microwave heating cavity in scenario M3; (b) Temperature distribution of the T-shaped composite material structure in scenario M3; (c) Electric field distribution in the microwave heating cavity in scenario M4; (d) Temperature distribution of the porous variable thickness composite material structure in scenario M4.

[0035] Figure 14These are simulation results of the final heating time in scenarios M3 and 4 of the microwave curing model at different magnifications. (a) Electric field distribution in the microwave heating cavity in scenario M3; (b) Temperature distribution of the T-shaped composite material structure in scenario M3; (c) Electric field distribution in the microwave heating cavity in scenario M4; (d) Temperature distribution of the porous variable thickness composite material structure in scenario M4.

[0036] Figure 15 These are simulation results of scene M5 in the microwave curing model when the heating time is 1 min at different magnifications. (a) Electric field distribution in the microwave heating cavity; (b) Temperature distribution of the hemispherical composite material structure.

[0037] Figure 16 These are simulation results of the final heating moment in scenario M5 of the microwave curing model at different magnifications. (a) Electric field distribution inside the microwave heating cavity; (b) Temperature distribution of the hemispherical composite material structure.

[0038] Figure 17 This is a schematic diagram showing the sampling point locations of composite material structures in (a) scenario M3, (b) scenario M4 and (c) scenario M5. Detailed Implementation

[0039] The method of the present invention will now be clearly and completely described and illustrated with reference to the accompanying drawings and specific embodiments. Obviously, the described embodiments are merely a part of the embodiments of the invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0040] The following detailed description of the invention uses the prediction of temperature field distribution during the curing process of composite materials as an example. Other curing process parameters (such as degree of cure distribution, residual stress distribution, and curing deformation) can also be determined with reference to this invention.

[0041] Transformation optics theory:

[0042] The theoretical foundation of transformation optics lies in the invariance of the form of Maxwell's equations under coordinate transformation. Since the heat conduction equation and the hydrostatic equilibrium equation also possess form invariance, this theory can be naturally extended and applied to thermodynamics and solid mechanics to achieve equivalent transformations and control of temperature, stress, and strain fields. For transformation optics, this can be explained as follows: when an original space is transformed to a new space through coordinate transformation, its electromagnetic, thermodynamic, and mechanical behaviors remain consistent, but the relevant physical parameters of the medium in the new space need to be transformed accordingly to compensate for the effects of the spatial transformation on the electromagnetic, temperature, and stress-strain fields. The fields obtained in the new space can be mapped back to the original space through inverse coordinate transformation, thereby obtaining the corresponding electromagnetic, temperature, or stress-strain field distributions in the original space. Specifically, assuming the coordinate system of the original space is (x, y, z), the coordinate system of the new space obtained after spatial coordinate transformation is (x′, y′, z′).

[0043] The electromagnetic and temperature fields behave consistently in the original space and the transformed space. The relative permittivity ε′, relative permeability μ′, electrical conductivity σ′, thermal conductivity k′, density ρ′, and specific heat capacity C in the transformed space remain the same. ρ The relative permittivity ε, relative magnetic permeability μ, electrical conductivity σ, thermal conductivity k, density ρ, and specific heat capacity C in the original space are compared with those in the original space. ρ The relationship can be given by transformation optics theory:

[0044] (1)

[0045] (2)

[0046] (3)

[0047] (4)

[0048] (5)

[0049] Where J is the Jacobian matrix, J T Let J be the transpose of J. The Jacobian matrix J can be expressed as:

[0050] (6)

[0051] The Jacobian matrix reflects the geometric transformation from the original spatial coordinate system to the transformed new spatial coordinate system. Therefore, by artificially enlarging the thickness scale of the composite material in the simulation model and using transformation optics to calculate the equivalent material parameters of the enlarged composite material, while keeping the electromagnetic field distribution and temperature field distribution in the model unchanged, the spatial scale difference of the composite material components can be reduced. This solves the problems of large mesh number and low computational efficiency, and enables rapid prediction of the composite material curing process.

[0052] A rapid prediction method for the curing process of composite materials based on transformation optics is proposed. The key lies in magnifying the thickness of the composite material and obtaining its equivalent material parameters after magnification. As a specific embodiment, a cylindrical composite material structure composed of four layers of prepreg is used as an example to illustrate the process of magnifying the thickness of the composite material and obtaining its corresponding equivalent parameters. Figure 1 (a) schematically demonstrates how introducing transformation optics to scale the model space can reduce the spatial scale differences of a cylindrical composite structure. It is important to note that the air layer compensates for the transformation domain of the internal composite structure and the surrounding free space, maintaining the continuity and stability of the physical fields within and around the composite structure. Its outer boundary typically remains unchanged, while the inner boundary varies with the composite structure. Figure 1 (b) Further, two-dimensional dimensional diagrams of the cylindrical composite material structure before and after the application of transformation optics and the corresponding coordinate transformations are given. As shown in the figure, the mold ( No transformation is needed, while the prepreg in the original space ( ) is enlarged to a thicker prepreg in the transformation space ( Air layer ( ) is compressed, its outer boundary ( The original spatial coordinates (r, θ, z) and the transformed spatial coordinates (r′, θ′, z′) of the cylindrical composite material structure model remain unchanged. If we consider the coordinate transformation in cylindrical coordinates, the original spatial coordinates (r, θ, z) and the transformed spatial coordinates (r′, θ′, z′) of the cylindrical composite material structure model satisfy the following transformation relationship:

[0053] (7)

[0054] The Jacobian matrix J of the cylindrical composite material structure model, from the original space to the transformed space in cylindrical coordinates, can be expressed as:

[0055] (8)

[0056] Based on the Jacobian matrix obtained above, the equivalent material parameters of the mold, prepreg, and air layer in the transformed space can be calculated using equations (1)-(5). Specifically, taking the relative permittivity as an example, assuming the anisotropic relative permittivity ε in the original space in cylindrical coordinates is:

[0057] (9)

[0058] Subsequently, the relative permittivity ε′ in the transformed space under cylindrical coordinates can be calculated as:

[0059] (10)

[0060] It is important to note that when performing simulations in rectangular coordinates, the material parameters need to be converted from cylindrical coordinates to rectangular coordinates. Therefore, the relative permittivity ε′ in the transformed space in rectangular coordinates can be expressed as:

[0061] (11)

[0062] The following are several examples based on the mechanism and method steps of the present invention:

[0063] Example 1.

[0064] Oven curing is widely used in composite material manufacturing due to its mature technology and wide applicability. The curing process involves complex heat transfer behavior. This embodiment takes a cylindrical composite material structure composed of four layers of prepreg stacked as an example, and uses oven curing to construct a corresponding numerical simulation model to achieve rapid prediction of temperature distribution during the oven curing process of the composite material. Specifically, this embodiment includes two calculation scenarios, namely scenario O1 and scenario O2.

[0065] Scene O1, for example Figure 2 As shown in (a), a cylindrical composite structure consisting of four layers of unidirectional carbon fiber reinforced resin matrix (CFRP) prepreg is laid on an aluminum mold and placed at the center of a rectangular oven. The layup of the composite structure is [0° / 90° / 0° / 90°], where 0° and 90° correspond to the circumferential and axial directions of the cylinder, respectively. The initial single-layer thickness of the prepreg is 0.4 mm (length-to-thickness ratio of 375), the inner and outer radii of the mold are 28 mm and 38 mm, respectively, and the initial air layer thickness is 25 mm. The material parameters of the mold, CFRP prepreg, and air layer are shown in Table 1. The thermal parameters of the CFRP prepreg (including specific heat capacity and thermal conductivity) are assumed to vary with temperature. In the heat transfer analysis, convection, conduction, and radiation heat transfer are considered, with the airflow velocity and heat flux density distribution at the oven inlet set to 0.3 m / s and 60,000 W / m, respectively. 2The emissivity of the oven wall, prepreg, and aluminum mold were set to 0.1, 0.8, and 0.2, respectively. Furthermore, during the curing process of the composite material, the cross-linking polymerization reaction of the resin generates an exothermic effect, constituting an important internal heat source. Therefore, a corresponding exothermic term was introduced into the model. The initial temperature was set to 20 °C, and the heating time was 40 min. The scenario O2 was as follows... Figure 2 As shown in (b), the oven inlet size is reduced compared to scenario O1, while the other parameters and conditions remain unchanged.

[0066] Numerical simulations of the oven curing process were performed in scenarios O1 and O2. The thickness of the prepreg was magnified in multiples of n=1, 1.5, and 2. Based on transformation optics theory, the equivalent thermal parameters of the mold, CFRP prepreg, and air layer in the magnified composite structure model were obtained according to the aforementioned calculation steps for equivalent material parameters after transformation. These parameters included thermal conductivity, specific heat capacity, and density. Based on these initial conditions, boundary conditions, and equivalent material parameters, corresponding numerical models were constructed to simulate and predict the temperature distribution during the curing process of the composite structure in both scenarios. The simulations were performed using COMSOL Multiphysics 6.3 software on a 64-bit Windows 10 workstation configured with an AMD EPYC 975412 8-core processor, 352 GB of RAM, and a 1 TB SSD. The model's mesh is automatically and adaptively generated by COMSOL using tetrahedral meshes based on the following parameters: maximum mesh size of 20.0 mm, minimum mesh size of 0.4 mm, maximum cell growth rate of 1.8, curvature factor of 0.6, and narrow region distribution rate of 0.5.

[0067] After the simulation, the temperature distribution at the final heating moment, the temperature change process at different locations, the calculation error, the number of grids, and the calculation time obtained from the numerical simulation model were evaluated. First, Figure 3 The temperature distribution of the cylindrical composite material structure at the final heating moment is shown in scenarios O1 and O2 at different magnifications. It can be observed that the temperature distribution inside the composite material structure is relatively uniform in the oven curing simulation model. More importantly, in both scenarios, the temperature distribution of the composite material structure after thickness magnification shows a high degree of consistency with the predicted results in the original space. Before and after the transformation, the spatial temperature gradient and overall thermal response of the composite material structure remain almost unchanged. This indicates that the proposed fast prediction method based on transformation optics does not change the inherent thermodynamic behavior inside the model, demonstrating the feasibility of this method for predicting the temperature distribution during the curing process of composite material structures, and exhibiting high accuracy and good versatility.

[0068] Furthermore, using scenario O1 as an example, the temperature changes of seven feature points (P1-P7, as shown in Figure 4) inside the cylindrical composite material structure during oven curing simulation were monitored under different composite material structure thickness magnification conditions. The corresponding temperature change curves are shown in Figure 4, indicating that the temperature change trends at the seven feature points remained largely consistent before and after the transformation. In addition, the difference between the transformation-based fast prediction method and the traditional method in the oven curing simulation model was quantitatively evaluated by calculating the mean relative error (MRE). Specifically, 80 sampling points were selected on the cylindrical composite material structure, arranged as follows: four positions were selected along the circumference (0°, 90°, 180°, and 270°, defined in the coordinate system shown in Figure 4), four evenly distributed positions were selected along the axial direction, and five radial positions were selected along the thickness direction. Subsequently, the mean relative error of the temperature at the final heating moment was calculated to quantify the prediction accuracy of the proposed method compared to the traditional method. The MRE results at different magnifications are summarized in Table 2. The results show that as the magnification increases, the MRE generally shows a slight upward trend. However, under the conditions of n = 1.5 and n = 2.0, the MRE of both scenarios O1 and O2 remains below 10%, and this error level is considered suitable for rapid prediction simulation in the curing process of composite materials.

[0069] Furthermore, the prediction error distribution along the thickness direction of the cylindrical composite structure was evaluated by calculating the MRE at five interfaces in the thickness direction of the composite structure. Figure 5 As shown in (a), the five interfaces along the thickness direction of the composite material are labeled 1-5, where interface 1 corresponds to the mold-composite material interface, interfaces 2-4 are the interlayer interfaces between adjacent layups, and interface 5 corresponds to the composite material-air interface. Sixteen sampling points are selected at each interface, arranged as follows: four positions are selected along the circumferential direction (0°, 90°, 180°, and 270°, defined as follows). Figure 4 Four uniformly distributed positions were selected along the axial direction, consistent with the aforementioned sampling strategy. The thickness-direction MRE distribution results for scenes O1 and O2 at different magnifications are shown below. Figure 5 (b) and Figure 5 As shown in (c), it can be observed that the prediction error along the thickness direction remains relatively uniform in both scenarios under different magnifications, indicating that the proposed method can maintain stable prediction accuracy across the entire thickness range of the composite material structure. These results further demonstrate the reliability and accuracy of this rapid prediction method.

[0070] Figure 6This paper presents the mesh size distribution of the composite material structure in scene O1 of the oven-cured model at different magnifications, along with the corresponding statistical results of mesh size and element number. For quantitative analysis, the mesh size is divided into 20 intervals to clearly characterize the distribution characteristics and variation patterns of the element number. It can be seen that as the transformation scale increases, the overall mesh size of the composite material structure gradually increases, while the number of mesh elements decreases significantly. During this process, the mesh size in the transition regions between the composite material structure and the air layer, and between the composite material structure and the mold, also increases, resulting in a corresponding decrease in the number of elements in these regions. Simultaneously, statistical analysis was performed on the mesh number and computation time of scenes O1 and O2 in the oven-cured model at different magnifications, and the results are summarized in Table 2. It is evident that both indicators significantly decrease after thickness scaling. Specifically, when the prepreg thickness is magnified to 2.0 times the original thickness, the mesh number in scenes O1 and O2 decreases by approximately 73.08%, and the corresponding computation time decreases by approximately 76.87% and 74.88%, respectively. Furthermore, based on the simulation results, it can be inferred that as the magnification factor of the prepreg thickness further increases, the calculation time can continue to be shortened, but at the same time, it will cause a decrease in simulation accuracy.

[0071] In summary, the rapid prediction method for composite material curing processes based on transformation optics proposed in this invention effectively solves the problems of large mesh count, long computation time, and low computational efficiency caused by large spatial scale differences in the simulation prediction of composite material components. This method significantly accelerates the prediction speed of temperature distribution in oven curing of composite material components with large scale differences while ensuring prediction accuracy.

[0072] Example 2.

[0073] In recent years, novel curing technologies have attracted widespread attention due to their potential for efficient and low-cost manufacturing of composite material components. Among them, microwave heating has become a research hotspot due to its unique advantages such as selective heating and volumetric heating. Given the complex multiphysics coupling characteristics between the electromagnetic and thermodynamic fields during microwave curing, microwave curing is further selected as another typical example. Similarly, this embodiment takes a cylindrical composite material structure composed of four layers of prepreg stacked as an example, employing microwave heating curing to construct a corresponding numerical simulation model to achieve rapid prediction of the temperature distribution during the microwave curing process of the composite material. Specifically, this embodiment includes two types of calculation scenarios, namely scenario M1 and scenario M2.

[0074] Scene M1, for example Figure 7As shown in (a), a cylindrical CFRP composite material structure with a stacking order of [90°]4 is laid on an aluminum mold at the center of a rectangular microwave heating cavity, where the 90° direction is aligned with the cylinder axis. Microwaves are fed in through a WR340 waveguide port. The initial thickness of the CFRP prepreg is 0.4 mm, the inner and outer radii of the mold are 28 mm and 38 mm, respectively, and the initial air layer thickness is 30 mm. The corresponding material parameters are listed in Table 1, where it is assumed that the relative permittivity of the CFRP prepreg varies with the degree of curing. The real and imaginary parts of the relative permittivity are expressed as a function of the degree of curing using cubic spline interpolation based on the discrete data points given in Table 3. The thermal parameters are consistent with the aforementioned definitions and are assumed to vary with temperature. The microwave frequency is set to 2.45 GHz, the input power is 400 W, the initial temperature is 20 °C, and the heating time is 10 min. Similarly, heat conduction in the air domain and heat radiation between solid surfaces are considered in the heat transfer analysis. Furthermore, as Figure 7 As shown in (b), in scenario M2, compared to scenario M1, the waveguide direction and structural position were adjusted, the input power was set to 300 W, and the inner and outer radii of the mold were 24 mm and 34 mm, respectively.

[0075] Numerical simulations of the microwave curing process were performed in scenarios M1 and M2. The thickness of the prepreg was magnified in multiples of n=1, 1.5, 2.0, and 2.5. Based on transformation optics theory, the equivalent thermal and electromagnetic parameters of the mold, CFRP prepreg, and air layer in the magnified composite structure model were obtained according to the aforementioned calculation steps for equivalent material parameters after transformation. These parameters included thermal conductivity, specific heat capacity, density, relative permittivity, relative magnetic permeability, and electrical conductivity. Based on these parameters, and according to the initial conditions, boundary conditions, and equivalent material parameters, corresponding numerical models were constructed to simulate and predict the temperature distribution during the curing process of the composite structure in both scenarios. The simulations were performed using COMSOL Multiphysics 6.3 software on a 64-bit Windows 10 workstation configured with an AMD EPYC 9754 128-core processor, 352 GB of RAM, and a 1 TB SSD. The model's mesh is automatically and adaptively generated by COMSOL using tetrahedral meshes based on the following parameters: maximum mesh size of 20.0 mm, minimum mesh size of 0.4 mm, maximum cell growth rate of 1.8, curvature factor of 0.6, and narrow region distribution rate of 0.5.

[0076] Similarly, after the simulation, the electric field and temperature distributions at the final heating moment, the temperature evolution at different locations, the S11 parameters at the final heating moment, as well as the calculation errors, mesh count, and calculation time in the microwave curing numerical simulation model were analyzed and evaluated. First, Figure 8The electric field distribution within the heating cavity and the temperature distribution of the cylindrical composite structure at the final heating moment are shown at different magnifications in scenarios M1 and M2. It can be observed that after transformation, the electric field and temperature distributions during the heating process in both scenarios are basically consistent with the original prediction results. The results demonstrate that the proposed fast prediction method can effectively preserve the electromagnetic and thermodynamic behavior of the model while possessing high accuracy and reliability.

[0077] Taking scenario M1 as an example, during the microwave curing simulation under different magnification conditions, five monitoring points (P1-P5, such as...) inside the cylindrical composite material structure were monitored. Figure 9 Temperature changes were monitored (as shown in the image). The corresponding temperature change results are as follows: Figure 9 As shown in the figure, the temperature change trends at all monitoring points remained largely consistent before and after the transformation. Furthermore, the reflection coefficient S11 at the final heating moment in scenarios M1 and M2 of the microwave curing model at different magnifications was evaluated. The corresponding S11 values ​​are summarized in Table 4, showing that the changes are only minor. Specifically, when the prepreg thickness is magnified to 2.5 times the original thickness, the S11 difference between scenarios M1 and M2 is 0.1596 dB and 0.5567 dB, respectively. The corresponding differences in the equivalent energy loss rate within the microwave cavity are 2.1% and 6.2%, respectively, both within acceptable ranges.

[0078] Furthermore, the average relative error and thickness direction error distribution of scenes M1 and M2 at different magnifications were calculated using the same steps described in Example 1. For scene M2, the four circumferential monitoring positions were set to 45°, 135°, 225°, and 315° (e.g., ...). Figure 9 (Coordinate system shown). The calculated MRE is summarized in Table 4. It can be seen that the MRE increases slightly with the increase of the transformation scale. When the prepreg thickness is magnified to 2.5 times the original thickness, the MRE of scene M1 and scene M2 reach 4.51% and 6.49% respectively, both still within the acceptable range. The thickness-direction MRE distribution of scenes M1 and M2 at different magnifications is shown in the figure. Figure 10 As shown, the prediction error in the thickness direction of the composite material structure is relatively uniform, and no significant error accumulation occurs at any interface, further demonstrating that the method has good robustness and reliability in the entire thickness direction.

[0079] Similarly, Figure 11The table shows the mesh size distribution of the composite material structure in scene M1 of the microwave curing model at different magnifications, along with the corresponding mesh size and element count statistics. As the transformation scale increases, the overall mesh size of the composite material structure increases, leading to a significant reduction in the element count. This trend indicates that transformation optics-based methods can effectively magnify the mesh size in the model. Furthermore, Table 4 summarizes the mesh count and computation time for scenes M1 and M2 in the microwave curing model at different magnifications, clearly showing a significant decrease in both metrics after magnification. Specifically, when the prepreg thickness is magnified to 2.5 times the original thickness, the mesh count for scenes M1 and M2 decreases by approximately 82.00% and 81.93%, respectively, with corresponding computation time reductions of approximately 79.42% and 79.11%.

[0080] In summary, the rapid prediction method for composite material curing processes based on transformation optics proposed in this invention effectively solves the problems of large mesh count, long computation time, and low computational efficiency caused by large spatial scale differences in the simulation prediction of composite material components. This method significantly accelerates the prediction speed of temperature distribution in microwave-cured composite material components with large scale differences while ensuring prediction accuracy.

[0081] Example 3.

[0082] Composite material components used in practical applications often have complex geometries, such as reinforcing ribs, stiffening plates, variable thickness structures, inserts, holes, and hyperboloid structures. This leads to more complex temperature and electromagnetic field distributions during the curing process. This embodiment uses T-shaped structures, perforated variable thickness structures, and hemispherical structures as representative structures, employing microwave curing to construct corresponding numerical simulation models for rapid prediction of temperature distribution during microwave curing of composite materials. The calculation scenarios for these three representative structures in this embodiment are scenarios M3, M4, and M5, respectively.

[0083] Scene M3, etc. Figure 12 As shown in (a), a T-shaped CFRP composite structure was laid on a 5 mm thick quartz glass mold, located at the center of a rectangular microwave heating cavity. The layup order of the flange and web was [0°]4 and [90°]4, respectively, where the 0° direction was defined along the x-axis and the 90° direction was defined along the z-axis. The corresponding material parameters are listed in Table 1, and the emissivity of the mold was 0.8. The microwave input power was set to 50 W. For scenario M4 (see... Figure 12(b) A CFRP composite structure with varying thickness and pores is laid on a 5 mm thick quartz glass mold located at the center of a rectangular microwave heating cavity. The layup order for the thin and thick regions is [0°]4 and [0°]8, respectively, with the 0° direction defined along the x-axis. The radius of the pores is 15 mm, and the microwave input power is set to 30 W. Scene M5 is shown. Figure 12 As shown in (c), a hemispherical CFRP composite material structure with a layup sequence of [0°]4 is laid on a quartz glass mold located at the center of a rectangular microwave heating cavity. The 0° direction is defined as along the circumference of the hemisphere. The inner and outer radii of the mold are 65 mm and 70 mm, respectively, and the initial air layer thickness is 30 mm. The microwave input power is set to 40 W. Furthermore, all other material properties, boundary conditions, input parameters, and simulation settings in scenarios M3, M4, and M are the same as those in scenarios M1 and M2 described in the examples. Similarly, microwave curing simulations are performed with the prepreg thickness scaled up to n=1, 1.5, 2, and 2.5 times, and the calculation process is consistent with Example 2.

[0084] After the simulation, for each scenario, the electric field and temperature distribution were recorded at different heating times, and the reflection coefficient S11 was recorded at the final heating time. Simultaneously, the computational error, mesh count, and computation time were calculated. First, Figure 13-16 The electric field distribution and temperature distribution of the composite material components within the heating cavity of scenes M3, M4, and M5 are shown at different magnifications in the microwave curing model. The results are presented at two heating time points: 1 minute (Figures 13 and 15) and the final heating time (…). Figure 14 and Figure 16 As can be seen, in the three scenarios with complex geometries, the electric field and temperature distributions during the transformed heating process are highly consistent with the original prediction results, which proves that the proposed method has high accuracy and good robustness under complex structural configurations.

[0085] Furthermore, the reflection coefficient S11 of scenarios M3, M4, and M5 at the final heating moment under different magnifications in the microwave curing model was analyzed. Table 5 summarizes the corresponding S11 results, showing that the changes in S11 are relatively small under different magnifications. When the prepreg thickness increases to 2.5 times the original thickness, the differences in S11 for scenarios M3, M4, and M5 are 0.2464 dB, 0.2005 dB, and 0.3951 dB, respectively. These changes correspond to changes in the equivalent energy loss rate within the microwave cavity of 2.7%, 2.2%, and 2.2%, respectively, all within acceptable ranges. Meanwhile, the mean relative error (MRE) for the three scenarios under different magnifications was also calculated. 87, 94, and 85 sampling points were strategically selected on the composite material components in scenarios M3, M4, and M5, respectively. The sampling points are shown in Table 5. Figure 16As shown in Table 5, the calculated MRE (summarized in Table 5) remained low under different transformation conditions, increasing only slightly with increasing magnification scale, and all values ​​were below 10%, which is within a reasonable range.

[0086] Furthermore, statistical analysis was conducted on the mesh count and computation time of scenes M3, M4, and M5 at different magnifications in the microwave curing model, and the results are summarized in Table 5. Both indicators decreased significantly after the thickness scale was increased. Specifically, when the prepreg thickness was changed to 2.5 times the original thickness, the mesh count of scenes M3, M4, and M5 decreased by approximately 81.20%, 82.37%, and 81.75%, respectively, and the corresponding computation times also decreased by approximately 77.21%, 77.86%, and 81.52%, respectively.

[0087] In summary, the rapid prediction method for composite material curing processes based on transformation optics proposed in this invention effectively solves the problems of large mesh count, long computation time, and low computational efficiency caused by large spatial scale differences in the simulation prediction of composite material components. Even under complex geometric conditions, the proposed transformation optics-based method significantly improves computational efficiency while ensuring prediction accuracy, achieving rapid prediction of temperature distribution during the curing process.

[0088] Table 1: Material parameters for mold, prepreg, and air layer.

[0089] *(local 1-2-3): Direction 1 represents the fiber direction, direction 2 corresponds to the in-plane transverse direction, and direction 3 represents the thickness direction. The unit of T is °C, and ε1(α), ε2(α), and ε3(α) represent functions related to the degree of curing.

[0090] Table 2: Simulation results of scenes O1 and O2 in the oven curing model at different magnifications.

[0091]

[0092] Table 3: Discrete data of the real and imaginary parts of the relative permittivity of CFRP prepreg along the fiber direction and perpendicular to the fiber direction.

[0093]

[0094] Table 4: Simulation results of scenes M1 and M2 in the microwave solidification model at different magnifications.

[0095]

[0096] Table 5: Simulation results of scenes M3, M4 and M5 in the microwave solidification model at different magnifications.

[0097]

[0098] All parts not covered in this invention are the same as or can be implemented using existing technologies.

Claims

1. A method for rapid prediction of the curing process of composite materials based on transformation optics, characterized in that, Includes the following steps: 1) Composite material thickness magnification: Obtain the geometric shape and size parameters of the actual composite material component, and based on the transformation optics theory, magnify the thickness of the composite material to construct the spatial coordinate transformation relationship, and obtain the spatial transformation Jacobian matrix corresponding to the composite material thickness before and after magnification. 2) Obtaining equivalent material parameters of composite materials: Obtain the original material parameters of the actual composite material. Based on the Jacobian matrix obtained in step 1), calculate the equivalent material parameters of the composite material after thickness magnification according to the parameter equivalence relationship in transform optics. 3) Construction of Curing Process Simulation Model: Establish a numerical simulation model of the curing process after the thickness of the composite material is enlarged, and assign the equivalent material parameters obtained in step 2) to the enlarged composite material component; 4) Numerical solution and prediction of curing process parameters: The simulation model is meshed and numerically solved under the constraints of the control equation and boundary conditions to obtain the prediction results of at least one curing process parameter during the curing process of composite material components.

2. The rapid prediction method according to claim 1, characterized in that: When the composite material curing process takes place within a closed cavity, a transformation layer is disposed on the outside of the composite material to compensate for changes in the electromagnetic field, temperature field, and / or mechanical field in the surrounding free space caused by the increased thickness of the composite material. The method for disposing of the transformation layer includes: 1) Introduce a continuous medium region on the outside of the composite material as a transformation layer; 2) Keep the outer boundary of the transformation layer unchanged before and after the coordinate transformation, and make its inner boundary change accordingly as the thickness of the composite material is increased. Use a spatial coordinate system consistent with the composite material to construct the spatial coordinate transformation relationship of the transformation layer, and calculate the spatial transformation Jacobian matrix of the transformation layer after the thickness of the composite material is increased. 3) Obtain the original medium parameters of the transformation layer, and based on the Jacobian matrix, perform equivalent transformation on the medium parameters of the transformation layer according to the parameter equivalence relationship in transformation optics.

3. The rapid prediction method according to claim 1, characterized in that: The thickness magnification factor n is related to the composite material thickness t and the minimum in-plane dimension L of the component. min The relationship satisfies the following formula: Where, n e n represents the thickness magnification factor limited by engineering tolerance requirements. u This is an empirical parameter, with a value of 8.

2.

4. The rapid prediction method according to claim 1, characterized in that: The curing process parameters include temperature, temperature field, degree of curing, degree of curing field, stress, stress field, and curing deformation.

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