Design and Accuracy Verification of Multi Curve Panel Adaptive Forming Mold Platform

JunjunWu1

ZhikunChen1

QingjiangHou1

HuamingZhou1

ZhigaoHuang1

1School of Materials Science and EngineeringHuazhong University of Science and Technology430070WuhanChina

Junjun Wu^a, Zhikun Chen^a, Qingjiang Hou^a, Huaming Zhou^a ,Zhigao Huang^a๡(corresponding author)

^a School of Materials Science and Engineering, Huazhong University of Science and Technology, Wuhan 430070, China

Abstract

This study proposes an adaptive deformable mold platform (AMP) for composite material surface forming, and solves its key design parameter optimization problem through theoretical modeling, numerical simulation, and experimental verification system. Based on Timoshenko beam theory, the optimal control point spacing was determined to be 100 mm, and under this condition, the forming error was reduced by 48.2%. By combining response surface analysis with the Mooney Rivlin hyperelastic material model, the optimal configuration of a 6 × 6 actuator array and a 50A hardness rubber panel was optimized. Based on the above parameters, an AMP prototype platform was constructed, and the forming accuracy was evaluated through 3D scanning. The results showed that under the condition of a target surface height difference of 88 mm, the maximum forming error was 4.11 mm, and the relative error was controlled within 4.6%, meeting the requirements of engineering applications.

Keywords:

Adaptive mold platform

Surface forming accuracy

Resin matrix composite

Control point spacing

Actuator dot matrix

Rubber panel

1. Introduction

Fiber reinforced resin based composite materials have been widely used in high-tech engineering fields such as aerospace, shipbuilding, automotive industry, and wind power generation due to their excellent mechanical properties, lightweight characteristics, and high structural designability^1–4. This class of materials not only possesses excellent specific strength and stiffness, but also demonstrates superior corrosion and fatigue resistance, making it a key material choice in the field of modern high-performance manufacturing.

However, with the increasing complexity of the geometric shapes of composite components, especially in small batch customized production and free-form surface structural component manufacturing scenarios, mold design and manufacturing costs have become important factors restricting their engineering applications^5–7. Traditional molds usually need to be developed separately for different curvatures and structural characteristics, resulting in extended production cycles, high manufacturing costs, and difficulty in meeting the practical needs of high efficiency, rapid iteration, and flexible manufacturing. Therefore, it is urgent to develop a deformable mold system with reconfigurable shape adjustment capability to significantly reduce mold types, lower manufacturing costs, and improve forming efficiency, thereby effectively enhancing the engineering adaptability and production feasibility of complex composite components in multiple scenarios.

With the advancement of science and technology, a series of flexible forming methods have emerged in the field of metal sheet processing, such as multi-point compression forming^{8, 9}, multi-point rolling forming^{10, 11}, multi-point stretching forming¹², single point incremental forming¹³, etc. The reliability and shaping accuracy of reconfigurable molds are the primary factors determining the quality of formed parts. Robinson¹⁴ designed and manufactured a reconfigurable die press for metal sheet forming, which mainly consists of two parts: the basic unit adjustment mechanism and the reconfigurable die forming mechanism. When forming, first adjust the shape adjustment mechanism to construct the mold surface required for forming; Then, the mold surface constructed by the shape adjustment mechanism is transferred to the reconfigurable mold through the forming surface transfer device, completing the reconstruction of the mold forming surface; Finally, the metal sheet is processed using the reconstructed mold.

Hardt et al^15–17. have conducted a series of studies on the mechanical structure and control principles of reconfigurable molds, and pointed out that using closed-loop control to adjust the basic unit can improve the dimensional accuracy of discrete molds. They conducted a series of metal sheet forming experiments using reconfigurable molds with variable geometric shapes, analyzing the effects of basic body control methods and forming process parameters on forming quality. Edwin et al¹⁸.developed a reconfigurable mold consisting of 36 basic body units and developed a specialized shape control system for the device. Through theoretical calculations and experimental tests, they evaluated the positioning accuracy of the basic body units, the load-bearing capacity and stability of the reconfigurable mold.

Gertner et al¹⁹. proposed a flexible forming technology for thermoplastic composite materials, which is similar to the rolling forming of metal sheets. It relies on a rotating work roll to produce continuous local deformation of the heated sheet metal. By adjusting the horizontal and vertical positions of the work roll, flexible processing of two-dimensional curved parts can be achieved. Kleespies and Crawford²⁰ applied a simple reconfigurable mold to the vacuum thermoforming of polymer sheets, but due to the fact that the basic unit of the reconfigurable mold used is not tightly arranged, the surface quality of the formed parts is poor. Franzen V. et al^{21, 22}. attempted to apply a commonly used flexible forming technology for metal sheet forming, single point incremental forming technology, to the flexible forming of polymer sheets. They conducted experimental research, summarized the forms and causes of defects, and explored the effects of forming tool size, forming speed, downforce, and sheet thickness on forming quality and accuracy. Simon et al^{23, 24}. developed a simple and reconfigurable mold that can be used for flexible processing of polymer sheets. The mold consists of 578 basic body units, with a distance of 30mm between adjacent basic body units and a maximum forming area of 510 × 1020mm. The author combined the reconfigurable mold with vacuum thermoforming equipment and conducted preliminary experimental research to analyze the influence of forming pressure on the surface quality and forming accuracy of the formed parts. In addition, the basic unit of the reconfigurable mold does not have an independent shape adjustment control mechanism, so its forming surface construction process is relatively complicated, and the shape adjustment accuracy is difficult to guarantee.

The above research indicates that the exploration of flexible mold technology mainly focuses on the field of multi-point forming of metal sheets, and some studies have extended and applied the principle of multi-point forming to the hot pressing forming of thermoplastic polymer sheets. However, the flexible mold technology suitable for liquid forming processes such as RTM and VARTM for fiber-reinforced resin based composites is still in the blank stage. To fill the research gap in this field, this paper independently designed and developed an Adaptive Molding Platform (AMP) for composite material molding. This study focuses on the structural design principles of the AMP platform. Based on finite element simulation and surface fitting algorithms, the optimal actuator control lattice scheme and rubber panel parameters were determined, and the platform construction was completed on this basis. Finally, the actual formed surface is digitally measured using a high-precision 3D scanner to evaluate the geometric errors between it and the target surface, thereby verifying the feasibility and forming accuracy of the platform design.

2. Principle of adaptive mold shape adjustment

The adaptive mold platform (AMP) is an innovative technology developed to solve the manufacturing difficulties of high-precision and complex curved resin based composite parts. It focuses on the research of material forming accuracy, surface deformability, and intelligent control, mainly including four modules: panel system, flexible connection system, electric actuator, and software control system. This platform can achieve high-precision fitting of mold surfaces and collaborative optimization of multiple physical fields through dynamic adjustment capabilities and intelligent control, enabling composite parts to have higher wettability, curing uniformity, and overall performance distribution during the forming process, significantly improving forming accuracy and efficiency.

Compared with traditional hot forming of polymer sheets, the biggest advantage of adaptive forming is the use of an adaptive mold with a reconfigurable forming surface instead of a single integral mold with a forming surface. The adaptive mold is composed of a first level control unit actuator (with regular arrangement and adjustable height) and a second level control body basic unit electromagnet (with regular arrangement and adjustable height following the movement of the actuator), and its forming surface is made of magnetic rubber that can be adsorbed. As shown in Fig. 1. By controlling the displacement of the actuator in the Z-axis direction, the adaptive forming platform can achieve flexible processing of polymer sheet metal parts with different three-dimensional shapes.

The shape adjustment of adaptive molds is a key step in the three-dimensional surface forming of polymer sheets, and the accuracy of shape adjustment determines the dimensional accuracy of the formed surface parts. Adaptive mold shaping is achieved by changing the relative height of the actuator in the Z-axis direction to drive the relative height of the electromagnet. If in a three-dimensional coordinate system, the x of each motion unit, The y-coordinate is invariant, therefore, accurately calculating the displacement of each actuator unit in the z-direction is the key to shape adjustment. When forming the adaptive mold, the rubber panel is tangent to the spherical surface at the end of the basic body unit. According to the above conditions, it is only necessary to find the common tangent point between the plate and each basic body spherical surface, calculate the z-coordinate value of the center of the sphere, and determine the relative height of each basic body unit. Based on this, the adaptive mold forming surface can be constructed.

Fig. 1

Forming surface of adaptive mold platform

Fig. 2

Schematic diagram of actuator unit height calculation method

As shown in Fig. 2, given the target surface P=(u, v), and x₀, y₀, calculate P_r+h=(u, v), thus obtaining the calculation method for Z_r+h as follows:

P_r+h(u,v) = P(u,v)-(r + h)n (1)

r is the radius of the top spherical surface of the electromagnet; h is the distance between the electromagnet and the actuator;

In the formula, n is the unit normal vector of the tangent plane of the target surface at point P (u, v), expressed as:

n(u,v)=

$\:\frac{{\text{P}}_{\text{u}}\left(\text{u},\text{v}\right)\times\:{\text{P}}_{\text{v}}\left(\text{u},\text{v}\right)}{\left|{\text{P}}_{\text{u}}\left(\text{u},\text{v}\right)\times\:{\text{P}}_{\text{v}}\left(\text{u},\text{v}\right)\right|}$

(2)

When the target surface is a regular surface, its parameter equation is expressed as:

x = u; y = v; z = f(u,v) (3)

$\:\left(\begin{array}{c}{f}_{u}^{{\prime\:}}/\sqrt{{{f}_{u}^{{\prime\:}}}^{2}+{{f}_{v}^{{\prime\:}}}^{2}+1}\\\:{f}_{v}^{{\prime\:}}/\sqrt{{{f}_{u}^{{\prime\:}}}^{2}+{{f}_{v}^{{\prime\:}}}^{2}+1}\\\:-1/\sqrt{{{f}_{u}^{{\prime\:}}}^{2}+{{f}_{v}^{{\prime\:}}}^{2}+1}\end{array}\right)$

(4)

By substituting x₀ and y₀ into equations (3) and (4), according to Eq. (1), can obtain

$\:\text{Z}\text{r}+\text{h}=\text{f}({x}_{0}+{y}_{0})-(\text{r}+\text{h})/\sqrt{{{f}_{{x}_{0}}^{{\prime\:}}}^{2}+{{f}_{{y}_{0}}^{{\prime\:}}}^{2}+1}$

When the target surface of the component is an irregular complex surface, NURBS surfaces can be used for construction. The expression for k × l NURBS surface is

P(u,v)=

$\:\frac{\sum\:_{0}^{m}\sum\:_{0}^{n}{W}_{i,j}{P}_{i,j}{N}_{i,k}\left(u\right){N}_{j,l}\left(v\right)}{\sum\:_{0}^{m}\sum\:_{0}^{n}{W}_{i,j}{N}_{i,k}\left(u\right){N}_{j,l}\left(v\right)}$

(6)

In the equation, P_i,jis the surface control point, W_i,j are corresponding weighting factors,N_i,k (u) and N_j,l (v) are k, l order B-spline basis functions, respectively.

For the surface constructed by Eq. (6), the formula for calculating its vector is

$\:{P}_{U}\left(u,v\right)=\frac{{A}_{u}\left(u,v\right)B\left(u,v\right)-A\left(u,v\right){B}_{u}\left(u,v\right)}{{B}^{2}(u,v)}$

$\:{P}_{V}\left(u,v\right)=\frac{{A}_{V}\left(u,v\right)B\left(u,v\right)-A\left(u,v\right){B}_{V}\left(u,v\right)}{{B}^{2}(u,v)}$

And:

A(u,v)=

$\:\sum\:_{0}^{m}\sum\:_{0}^{n}{W}_{i,j}{P}_{i,j}{N}_{i,k}\left(u\right){N}_{j,l}\left(v\right)$

B(u,v)=

$\:\:\sum\:_{0}^{m}\sum\:_{0}^{n}{W}_{i,j}{N}_{i,k}\left(u\right){N}_{j,l}\left(v\right)$

(8)

Substituting x₀,y₀ ,and Eq. (7) (8) into Eq. (1) yields a system of binary quadratic nonlinear equations:

P(u,v)+(r + h)n|_x=x0 =x₀

P(u,v)+(r + h)n|_y=y0 =y₀ (9)

By solving Eq. (9), z_r+h can be obtained, and the adaptive mold shaping system can determine the heights of each actuator based on this, constructing an adaptive mold forming surface with complex surfaces.

3. Factors Influencing the Accuracy of Adaptive Surface Forming

In the research of adaptive surface forming technology, the control of forming accuracy is a key issue. There are two main factors that affect the accuracy of adaptive surface forming: one is the error between the forming surface and the CAD drawing (target surface), which originates from the fact that the target surface interpolates between points in a different way from the drawing. Therefore, how to determine the actuator spacing and calculate the displacement of each actuator is the key to reducing this error. Secondly, the motion transmission and material selection in the overall system of the AMP can also cause errors. The accuracy of motion transmission directly affects the motion trajectory of the mold and the stability of the forming process, while the physical properties of the panel material and flexible rod, such as elastic modulus, Poisson's ratio, etc., will further affect the dimensional accuracy and surface quality after forming. Therefore, in order to improve the accuracy of adaptive surface forming, it is necessary to comprehensively consider various factors such as design, processing, and material selection, reduce errors through precise motion control and reasonable material selection, and achieve high-quality surface forming.

3.1 Derive the optimal interpolation point spacing based on spline interpolation theory

As shown in Fig. 2, based on the principle of deformable molds, the bending forming accuracy of flexible rods directly determines the forming accuracy of the final surface. The forming accuracy of flexible rods is mainly affected by the number of control points. In order to simplify the model, this paper derives the optimal number of control points for the flexible rod under the condition of minimizing the forming error through finite element simulation analysis. This derivation provides a theoretical basis for subsequent mold design and optimization.

As shown in the qualitative analysis of Fig. 3d, when the number of control points is 5–6, the best agreement between the simulation curve and the target curve is achieved. The quantitative analysis results in Fig. 3e indicate that the forming error shows a nonlinear trend of decreasing first and then increasing with the increase of the number of control points. It reaches the global minimum value when the number of control points N = 6 (the error is reduced by about 68%). This phenomenon can be attributed to: when there are insufficient control points (N < 6), discretization modeling is difficult to accurately describe complex curvature distributions, resulting in limited local deformation control capabilities; When there are too many control points (N > 6), the coupling effect between control points produces parasitic bending modes and local optimization leads to a decrease in overall forming coordination. Therefore, in practical applications, the bending characteristics of flexible rods are not simply proportional to the density of control points. In summary, when the number of control points N = 6 (i.e. the distance between each two points is 100 mm), the best molding accuracy can be achieved.

Fig. 3

a、Geometric contour of target forming curve;b、Fitting of target curve function and solving of control point coordinates (X, Y) based on MATLAB;c、The simulated forming curve obtained by controlling the displacement of control points; d、Qualitative comparative analysis of simulation curves and target curves under different numbers of control points;e、Analysis of overall error in forming accuracy corresponding to different numbers of control points.

In order to better match the actual adaptive mold platform, this study establishes a surface model to determine the optimal control point spacing, that is, the optimal spacing of actuators in adaptive surface forming. By using MATLAB software, a fitting surface S(u,v) that is close to the target surface can be generated. The fitting surface is based on the theoretical principle of polynomial fitting and constructed through a self written program. The polynomial fitting process not only ensures that S(u,v) can accurately approximate the target surface P(u,v), but also preserves the key geometric characteristics required for subsequent forming processes (as shown in Fig. 4). By systematically analyzing the deviation between S(u,v) and P(u,v), optimizing the spacing between actuators, a balance can be achieved between surface accuracy and actual system constraints. This method provides a solid computational foundation for optimizing actuator layout and improving the overall accuracy of adaptive surface forming systems. According to this method, the maximum error Max Error and mean square error MSE are calculated:

E(u,v) = P(u,v)- S(u,v) (10)

Max Error = max

$\:\left|\text{E}(\text{u},\text{v})\right|$

(11)

MSE = 1/N

$\:\sum\:{\left|\text{E}(\text{u},\text{v})\right|}^{2}$

(12)

Where N is the total number of sampling points.

Fig. 4

A、Comparison of target surface and 6×6 actuator fitting surfaces; B、 Comparison of cross-sections between two curved surfaces

Fig. 5

A、The error of each point on the diagonal between the target surface and the fitted surface in the Z-axis direction;B、Schematic diagram of the max-error and MSE between the target surface and the fitted surface under different numbers of actuators on the same surface area

Table.1 The specific values of the Max_error and MSE between two surfaces with different numbers of actuators on the same surface area

Actuator Error	4×4	5×5	6×6	7×7	8×8	9×9
MSE	27.9	31.5	25.3	29.8	20.6	25.4
Error_max	10.5	9.3	8.7	8.3	7.0	7.4

The Error_max provides an evaluation of the overall accuracy of the interpolation, serving as an indicator of the largest deviation between the interpolated surface and the target surface. On the other hand, the mean squared error (MSE) reflects the overall error distribution across the interpolation points, representing the average error level on the surface. As shown in Fig. 5B, it is evident that the maximum error decreases as the actuator density increases, indicating improved accuracy with higher actuator resolution. Interestingly, the analysis of MSE reveals a noteworthy pattern: when the number of actuators is odd, the MSE tends to be higher compared to even numbers of actuators. Regardless of whether the number of actuators is odd or even, the MSE consistently decreases as the actuator density increases, demonstrating that higher actuator density enhances the overall surface accuracy.This is because when the number of actuators is even, the layout of the entire actuator is more symmetrical, especially with respect to the symmetry of the central region. This symmetry makes the distribution of actuators more uniform across the entire surface, avoiding excessive or insufficient control at the center point, which helps reduce errors. When the number of actuators is odd, the layout lacks clear center symmetry, resulting in uneven control force in certain areas, especially in the center and edge areas, which may increase errors due to uneven distribution of controllers. The asymmetry of odd numbered layouts may introduce high errors in some local areas, thereby increasing the mean square error.

When interpolating and fitting on uniformly distributed control points or actuators, even numbered layouts are more likely to ensure smooth transitions and uniform interpolation effects. In even numbered layouts, there are symmetrically distributed control points at the boundaries of the surface, which can better fit the boundary curve and reduce errors. When the number of actuators is odd, interpolation points cannot be symmetrically distributed within the region, resulting in significant local errors in the fitted surface at the edges or center. This asymmetry deteriorates the fitting effect and leads to an increase in mean square error. In some control scenarios, boundary conditions and errors propagate towards the central region, and even numbered actuator layouts can better distribute on the boundaries, making error propagation smoother. However, odd numbered layouts may form uneven distribution of control points at the boundaries or central regions, leading to errors accumulating in certain local areas and increasing the overall mean square error. In practical physical systems, even numbered actuator layouts may have better local control capabilities because their layout is more uniform and more in line with symmetry requirements. In odd numbered layouts, the local control capability may deteriorate, especially when dealing with small surface adjustments. Odd numbered layouts are difficult to provide balanced control force, resulting in larger errors.

Based on the above calculation results, taking into account the project accuracy requirements and actual costs, the number of actuators selected in this study is 6 × 6, with a spacing of 100mm. As shown in Fig. 5, the fit between the 6 × 6 fitting surface and the target surface is relatively high, and in Table.1, the Error_max between the two surfaces is 8.7mm,and the MSE of 25.3mm is within a reasonable range of error.

To determine the theoretical range of surface curvatures that an adaptive forming platform can achieve under the given parameters, we first assume that the surface can be represented by a second-order polynomial. This assumption simplifies the mathematical model while capturing the essential geometric characteristics of the surface. Consequently, the surface equation can be expressed as follows:

Z = f(x,y)=

$\:{a}_{0}+{a}_{1}x+{a}_{2}y+{a}_{3}{x}^{2}+{a}_{4}xy+{a}_{5}{y}^{2}$

(13)

where a₀,a₁,a₂,a₃,a₄,a₅ are the coefficients that define the shape of the surface.

Therefore, for a binary function Z = f(x,y), the Gaussian curvature K and average curvature H of the surface are:

$\:\frac{{f}_{xx}{f}_{yy}-{f}_{xy}^{2}}{{(1+{f}_{x}^{2}+{f}_{y}^{2})}^{2}}$

(14)

$\:\frac{{(1+{f}_{y}^{2})f}_{xx}+(1+{f}_{x}^{2}){f}_{yy}-2{f}_{x}{f}_{y}{f}_{xy}}{2{(1+{f}_{x}^{2}+{f}_{y}^{2})}^{3/2}}$

(15)

By evaluating the curvature formula of the polynomial surface, it becomes evident that the actuator spacing and actuator stroke are critical factors in determining the range of surface curvatures that the platform can theoretically form. The actuator spacing directly influences the resolution of the deformation, as smaller spacing allows for finer adjustments to the surface shape, enabling the platform to replicate surfaces with higher curvature. Meanwhile, the actuator stroke determines the maximum displacement each actuator can achieve, thereby constraining the platform's ability to accommodate surfaces with large curvature variations.Based on these findings, it is possible to derive the theoretical range of curvatures that the platform can form under given system parameters(Fig. 6).

As shown in Fig. 6, under the same actuator stroke, a greater number of actuators enables the adaptive platform to achieve a wider curvature range. This is attributed to the fact that increasing the actuator count enhances the spatial resolution of the system, allowing it to approximate the target surface with greater precision. Similarly, under the same number of actuators, increasing the stroke of the actuators can significantly improve the deformation ability of the platform, thereby forming a larger curvature range. In this study, a layout of 6 actuator arrays was selected, with a single actuator stroke of 500mm. Under this parameter configuration, the maximum Gaussian curvature that the adaptive surface forming platform can theoretically form is 0.01, while the average curvature is 0.1. These results demonstrate that through rational configuration of actuator quantity and stroke parameters, the platform can achieve precise control over the geometric characteristics of surface formation within a defined range. This capability provides both theoretical support and a technical foundation for machining complex surfaces. Furthermore, this analysis establishes a basis for subsequent optimization of actuator layouts and enhancement of the platform’s forming accuracy.

Fig. 6

Relationship between the number of actuators and maximum curvature under different stroke conditions: A、Maximum Gaussian curvature; B、Maximum average curvature

3.2 Optimization of material parameters for panel and flexible rod structures using numerical simulation and surface fitting

3.2.1 Finite element analysis (FEA) model

AMP combines flexible formation with computer technology. In AMP, the reconfigurable mold consists of a series of electromagnets, each of which can be controlled collaboratively by an actuator, while each actuator is independently controlled by an electric motor to adjust its height to form a discrete three-dimensional surface. In the simulation process of AMP, the position data of the actuator should be determined first. Firstly, obtain the STP format of the target surface, and then input it into a special CAD software for AMP to obtain the specific position data of each actuator. Then, the position coordinates of the actuator are input into numerical simulation software to construct a surface. The finite element method (FEM) is widely used as a numerical method for simulating sheet metal stamping processes, and there are two typical time integration methods in FEM formulations. Dynamic explicit methods and static implicit methods. In terms of accuracy, static implicit methods are more advantageous than dynamic explicit methods, so the static implicit method was chosen in this simulation.

AMP is a complex plastic formation process characterized by large deformation, geometric nonlinearity, material nonlinearity, and contact nonlinearity. This article uses the commercial dynamic finite element software ABAQUS to simulate the process of AMP, the simulation analysis model is shown in Fig. 7,and the material properties of the rubber panel and the flexible rod are shown in Table 2 and Table 3.

Fig. 7

FEA model of AMP: a、the formed panel of AMP;b、the formed panel of the formed panel;c、the connection system of AMP༛d、the simplified FEA model of AMP

Table.2 Properties of rubber materials with different hardness

HARD(A)	Material constitutive	Material parameters(MPa)		Elastic modulus(MPa)
HARD(A)	Material constitutive	C₁₀	C₀₁	Elastic modulus(MPa)
45	Mooney-Rivlin	0.160	0.040	1.204
50	Mooney-Rivlin	0.201	0.041	1.452
55	Mooney-Rivlin	0.301	0.065	2.196
60	Mooney-Rivlin	0.381	0.102	2.898
65	Mooney-Rivlin	0.501	0.123	3.744
70	Mooney-Rivlin	0.622	0.152	4.644
75	Mooney-Rivlin	0.805	0.195	5.994

Table.3 Material properties of different material types applied to flexible rods

Material type	Elastic modulus(MPa)	Poisson's ratio	Density(g/cm³)	Elongation at break(%)
Spring steel	195	0.28	7.82	10
Glass fiber	83	0.23	2.55	5
Carbon fiber	410	0.15	1.75	3

3.2.2 Compare the continuity and error between the simulated surface and the target surface

Fig. 8

A、Schematic diagram of simulated surfaces of rubber with different hardness and the target surface as a whole; B、Schematic diagram of the overall error between simulated rubber surfaces with different hardness and the target surface in the same coordinate system; C、schematic diagram of the error and continuity between the simulated surface and the target surface on the X-Z plane in the same coordinate system

From the overall schematic diagram in Fig. 8A, it can be observed that although the hardness of the rubber panel is different, the simulated surface and the target surface remain consistent in the overall trend, indicating that the simulation method used in this study has certain reliability and applicability. This result validates the effectiveness of the deformation prediction model based on finite element analysis (FEM) in the simulation of deformable molds. In order to further quantify the accuracy error between the simulated surfaces of rubber panels with different hardness and the target surface, we compared and analyzed all simulated surfaces and the target surface in the same coordinate system in Fig. 8B. It can be clearly seen that there is still some error between the simulated surface and the target surface of rubber panels with different hardness, indicating that the hardness of rubber materials will affect the accuracy of mold forming. In addition, in Fig. 8C, we extracted the cross-sectional curves on the X-Z plane to more intuitively analyze the continuity and accuracy of the simulated surface. The results indicate that there is a significant deviation between the simulated surface and the target surface in local areas, and the continuity is poor within a certain range. This is because during the entire deformation process, the actuator drives the rubber panel to deform, but due to the elastic properties of the material itself, some of the input energy is absorbed and stored as elastic potential energy by the panel, rather than fully converted into the desired geometric deformation. This energy loss causes the simulated surface to not fully conform to the target surface in certain areas. And the displacement path of the actuator is calculated through surface fitting using MATLAB software. During the fitting process, it is inevitable to introduce certain errors, causing the surface actually driven by the actuator to deviate from the theoretically calculated surface, resulting in geometric errors between the target surface and the simulated surface.

Fig. 9

: A、Deformation accuracy evaluation of rubber panels with different hardness in the same coordinate system - error distribution between 50A hardness simulation surface and target surface; B、Comparative analysis of maximum error (Error_max) and mean square error (MSE) between simulated and target surfaces of rubber panels with different hardness; C、Error distribution between simulated surface and target surface of 50A hardness rubber panel - visualization of errors at each grid point

Rubber panels with lower hardness are prone to local collapse when subjected to force deformation, while rubber panels with higher hardness may not fully adhere to the target surface due to their greater rigidity. Therefore, at different hardness levels, simulated surfaces will exhibit different deviation patterns. In order to further quantify the error distribution between the simulated surface and the target surface, the maximum error and mean square error under various hardness conditions were compared and calculated, and detailed error evaluation results were presented in Fig. 9 to provide a scientific basis for optimizing the design of deformable molds.

From Fig. 9B, it can be observed that when the hardness of the rubber panel is 50A, both Error_max and MSE are at their lowest levels, indicating that the rubber panel at this hardness can adapt well to the target surface during deformation. When the hardness is low (such as A45), the flexibility of the rubber panel is greater, resulting in local collapse in the area between the actuators, which affects the forming accuracy. When the hardness is high (such as A60 and above), due to the increased rigidity of the material, the deformation force required for the panel increases, and some areas may be difficult to fully conform to the target surface, resulting in an increase in error. This indicates that the hardness of the material is crucial for the deformation adaptability of the mold, and an optimal balance point needs to be found between flexibility and rigidity. This phenomenon is further verified from Fig. 9A, where the simulated surface of the rubber panel with a hardness of 50A overlaps more with the target surface, indicating that it has better adaptability and higher molding accuracy in the current deformation environment. This result indicates that under the current loading conditions, 50A hardness can maximize the balance between deformation flexibility and support stability, resulting in the minimum overall surface error.

In Fig. 9C, further spatial analysis was conducted on the error distribution of the 50A rubber panel. It can be seen that the overall error trend is basically consistent with Fig. 9A, showing the characteristics of smaller errors in the middle region and larger errors in the edge region. This may be due to the rubber panel receiving less support constraints at the edges, resulting in a greater degree of deformation freedom and an increase in error. Further analysis reveals that in the area where each actuator comes into contact with the rubber panel, the error is relatively large, while the error in the surrounding area is relatively small. This indicates that during the deformation process, there may be a local deformation enhancement effect on the rubber panel at the actuator point, and due to the elastic properties of the material, some input energy is absorbed and not fully converted into the expected deformation, resulting in the generation of local errors.

Fig. 10

Analysis of the influence of flexible rod material characteristics on forming accuracy: A、 The distribution of forming errors of flexible rods with different elastic moduli under equivalent control point configuration; B、The error and continuity trend between the simulated surface and the target surface in the X-Z plane under the same coordinate system.

As shown in Fig. 10, under the condition of a fixed number of control points, the influence of elastic modulus on forming accuracy is not significant for both flexible rods and curved structures. This is mainly due to the fact that the forming accuracy of flexible structures depends more on geometric constraints rather than the stiffness properties of the material itself. In this study, the flexible rod is constrained by control points, and each control point is attached to the target curve through flexible connections. These control points essentially serve as local constraint supports, significantly limiting the free deformation space of the members. In an unconstrained state, flexible rods can theoretically exhibit an infinite number of bending forms, and once geometric constraints are introduced at specific locations, their feasible deformation modes are significantly compressed. Therefore, the main role of elastic modulus is reflected in the deformation force and final stress distribution during the loading process, rather than the dominant influence on geometric forming accuracy.

3.2.3 Simulation result analysis

Fig. 11

Under the condition of constant elastic modulus, the influence of different number of control points on the stress distribution induced by deformation of flexible rods: A、4 Control point configuration; B、5 Control point configuration; C、6 Control point configuration; D、7 Control point configuration;E、8 Control point configuration;

From the above analysis, it can be seen that the elastic modulus of the flexible rod has a relatively small impact on its forming accuracy, while the number of control points is the main factor determining its geometric forming accuracy. As shown in Fig. 11, with the increase of the number of control points, the maximum stress inside the flexible rod also increases accordingly. The essential reason for this phenomenon is that although increasing the number of control points can improve the accuracy of geometric constraints, it also introduces more frequent curvature changes, leading to intensified local curvature mutations between adjacent control points of the member. Due to the continuity constraint of the material, this high gradient deformation inevitably leads to the enhancement of stress concentration phenomenon. In addition, an increase in the number of control points also means further limitations on the free deformation ability of the flexible rod, making its deformation mode tend towards rigidity. Therefore, the material is forced to adapt to specific shapes, resulting in unnatural stress distribution, especially in the transition zone where significant internal force concentration is formed, leading to an overall increase in stress levels.

Fig. 12

Analysis of simulation results under the condition of rubber panel hardness of 50A:A、The relative elongation distribution of each area of the panel; B、The maximum reaction force under the action of each control point; C、The maximum internal stress distribution of the overall curved surface structure during the deformation process; D、The displacement range of the surface during the forming process.

Through comparative analysis of the continuity and error between the simulated surface and the target surface, it was found that the overall forming accuracy is optimal when the hardness of the rubber panel is 50A. In addition, the simulation results also revealed the internal stress distribution characteristics of the rubber panel during bending deformation under this hardness condition, providing important references for the optimization design of rubber panel material parameters in the future.

As shown in Fig. 12C, during the deformation process, the rubber panel with a hardness of 50A generates the maximum stress in the edge area, with a value of 0.98 MPa. Further observation of Fig. 12B reveals that the stress distribution differences at each control point are relatively small, indicating that the deformation force applied to the rubber panel by each control point is relatively uniform, thereby ensuring good continuity of the panel throughout the entire forming process. It can be seen from Fig. 12D that the maximum deformation stroke of the surface is 92 mm, and the maximum height difference within the surface reaches 88 mm, indicating that the target surface has significant curvature changes and good universality and representativeness.

4. Forming experiment and error analysis

4.1 Experiments and measurements

To evaluate the error performance of the optimal number of actuators and panel material combination obtained from simulation analysis and fitting calculation in the actual forming process, this paper conducted relevant experiments on the self-developed Adaptive Molding Platform (AMP). The accuracy of the formed surface is measured digitally and analyzed for errors using a 3D scanner to verify its fit with the target surface. The panel material used in the experiment is magnetic silicone rubber with a hardness of 50A, which has good magnetic response characteristics and flexibility, and can meet the dual requirements of active driving and shape adaptation. The panel size is 200mm x 200mm, with a thickness of 3mm. The spacing between the actuators is set to 100 mm, and the flexible driving rod is made of fiberglass material to achieve high structural stiffness and good response performance(Fig. 13A).

To obtain the shape of the formed parts after the experiment, a 3D laser measuring instrument was used. The high-precision laser measuring head scans the surface of the part point by point, and then obtains a series of three-dimensional coordinate values. Input the values into the error analysis software to generate a three-dimensional point cloud (Fig. 13C, D), as well as the surface formed by the measurement data. In the simulation results, the displacement in the Z direction is compared between the measurement results and the target shape, and the difference value can be obtained (Fig. 13B).

4.2 Comparison and analysis of errors

Fig. 13

Comparative analysis of experimental platform and surface forming accuracy:

A、Schematic diagram of AMP experimental platform and 3D scanner system; B、Qualitative comparison of target surface, simulated surface, and actual scanned surface in a unified coordinate system; C、Comparison of 3D point clouds between the target surface and the scanned surface; D、Comparison of 3D point clouds between simulated and scanned surfaces; E、The comparison results of the Error_max and MSE between the target surface、the simulated surface and the scanned surface

Figure 13B shows the error comparison between the target surface, simulated surface, and scanned surface (actual formed surface) in a unified coordinate system. From the figure, it can be observed that the degree of fit between the simulated surface and the scanned surface is significantly better than the fit between the target surface and the scanned surface. This is because the simulation surface is constructed based on the target surface and combined with the optimal actuator spacing and panel material obtained through fitting calculation for parameter setting, thus having stronger pertinence and feasibility. The AMP experimental platform further designed and built based on the simulation results inevitably introduces system errors during actual execution, such as residual stresses inside the material and actuator positioning errors. Therefore, the total error between the actual formed surface and the target surface can be regarded as the superposition of simulation error and system error, that is:

Error _{scanned surface− target surface}= Error _{simulated surface − target surface}+ Error_system

The analysis results of the error distribution point cloud map shown in Fig. 13C-D indicate that compared with the target surface, the simulated surface exhibits higher spatial uniformity in the error field between the simulated surface and the measured scanned surface. In contrast, the error distribution of the target surface exhibits obvious regional clustering characteristics. This difference indicates that the forming scheme based on simulation optimization has better performance in spatial consistency and can more effectively match the geometric features of actual forming surfaces.

The quantitative analysis of errors in Fig. 13E shows that the Error_max between the target surface and the scanning surface is 6.06 mm, with an MSE of 4.11 mm; The Error_max between the simulated surface and the scanned surface is 3.14 mm, with an MSE of 1.04 mm. This result indicates that parameter setting based on simulation optimization results is more helpful in improving the accuracy of actual forming compared to directly forming based on the target surface. The main sources of actual forming errors can be divided into two parts: simulation errors and system errors, among which system errors mainly include uncontrollable factors such as actuator positioning errors, uneven magnetic strain gradients, and interface contact stresses. In addition, it can be observed from Fig. 12D that under the working condition of forming height drop of 88mm, the actual error accounts for 4.6% (< 5% project threshold), which meets the preset engineering accuracy requirements. The above results verify the effectiveness and feasibility of the parameter optimization method based on simulation fitting proposed in this paper in improving the forming accuracy of deformable molds.

5. Conclusions

This study systematically investigated the key design parameter optimization problem of the adaptive mold platform for hyperbolic panels through a combination of theoretical modeling, numerical simulation, and experimental verification:

1.the deformation mechanism of the flexible rod: Based on the simplified model of the flexible rod, the simulation results show that the elastic modulus of the flexible rod material has limited impact on the forming accuracy. Its main influence is the load borne during the deformation process and the distribution of residual stress inside after deformation. The control point spacing becomes the dominant factor affecting the forming accuracy, and the forming error shows a trend of first decreasing and then increasing with the control point spacing. When the spacing is 100 mm, the forming error is the smallest, which is consistent with the characteristic bending length predicted by Timoshenko beam theory²⁵.

2.Optimization of actuator layout: The accuracy spacing relationship model established through response surface analysis shows that even numbered array layout is superior to odd numbered array layout, and a spacing of 100 mm achieves a good balance between forming accuracy and system complexity. The corresponding actuator array is 6 × 6, which improves surface fitting by 19.7% (compared to 5 × 5 lattice).

3. Panel material selection:Parameter optimization based on Mooney Rivlin hyperelastic model shows that 50A hardness rubber panel performs the best in forming accuracy and stress distribution uniformity, and its strain hardening characteristics can effectively suppress local buckling.

4. AMP platform construction

On the basis of the above simulation calculations, a prototype of an adaptive mold platform based on simulation optimization results was built. In actual experiments, a 3D scanner was used to measure the formed surface. The results showed that under the condition of a height difference of 88 mm, the actual forming error was 4.11mm (relative error within 4.6%), which meets the accuracy requirements for engineering applications.

The multi-objective parameter optimization method proposed in this study provides theoretical support for the design of complex surface deformable molds, and the constructed adaptive mold platform (AMP) demonstrates good feasibility and engineering promotion potential in achieving high-precision forming of large curvature and free-form surface components. Future research will focus on the following two directions: developing online compensation control algorithms for practical working conditions to further reduce system errors; explore the efficient preparation process of gradient hardness rubber panels to enhance the regional adaptability of panel performance. The relevant research results can provide important theoretical basis and engineering reference for the design of large and complex composite material forming molds in aerospace and other fields.

References:

Li C, Fei J, Zhang T, Zhao S, Engineering (2023) Relationship between surface characteristics and properties of fiber-reinforced resin-based composites

Kai Z, Yan MA, Shi-Quan Y, Materials P (2009) Corrosion-Resisting Properties of Carbon Fiber Composite Materials

Seo DK, Ha NR, Lee JH, Finishing (2015) Property Evaluation of Epoxy Resin based Aramid and Carbon Fiber Composite Materials. 27(1):11–17

Yue-Feng S, Mei-Qun (2017) Study on Tensile Toughness of Non-woven Fabrics of Fiber Composites

Vaidyanathan KRJJ (2000) Functional metal, ceramic, and composite prototypes by solid freeform fabrication. 52(12):30–30

Hamed H, Aghili S, Wüthrich R (2024) Electroforming of Personalized Multi-Level and Free-Form Metal Parts Utilizing Fused Deposition Modeling-Manufactured Molds. 15 (6)

Deden D, Gänswürger P, Buchheim A (2017) Industrial concept for the automated production of small batch series preforms for Carbon Fiber Reinforced Plastic (CFRP) components

Mingzhe Li; Shuzhong Su, Multi point forming technology for sheet metal without molds. (2000) (4), 18–20

Mingzhe Li (2000) Jianjun Chen; Multi point forming technology for three-dimensional curved surfaces of sheet metal. 000(010):27–29

10.

Mingzhe L, Zhiqing H (2007) Zhongyi Cai; Continuous and efficient plastic forming method for free-form surface workpieces. 37(3):6

11.

Cai ZY, Li MZ, Lan YW (2012) J. J. m. p. t., Three-dimensional sheet metal continuous forming process based on flexible roll bending: Principle and experiments. 212 (1), 120–127

12.

Cai ZY, Wang SH, Xu XD, Li MZ (2009) J. J. o. M. P. T., Numerical simulation for the multi-point stretch forming process of sheet metal. 209 (1), 396–407

13.

Jianhua Mo; Jie Liu; Digital Forming Technology for Large Automotive Panels (2001) 8 (2), 3

14.

Robinson (1987) Design of an automated variable configuration die and press for sheet metal forming

15.

Hardt DE, Gossard DC (1980) J. i., A variable geometry die for sheet metal forming: Machine design and control

16.

Hardt DE, Robinson RE, Webb RD (1985) Closed-loop control of die stamped sheet metal parts: Algorithm development and flexible forming machine design

17.

Hardt DE, Boyce MC, Ousterhout KB, Karafillis A, Eigen GM (1993) A CAD-Driven Flexible Forming System for Three-Dimensional Sheet Metal Parts

18.

Design (2008) and Test of a Reconfigurable Forming Die %J. J Manuf Process

19.

Gertner Y, Miller AKJJ, o. TCM (1996) Die-Less Forming of Large and Variable-Radii of Curvature in Continuous-Fiber Thermoplastic-Matrix Composite Materials. 9(2):151–182

20.

Kleespies HSI, Crawford RHJ (1998) J. o. M. S., Vacuum Forming of Compound Curved Surfaces with a Variable Geometry Mold. 17 (5), 325–337

21.

Franzen V, Kwiatkowski L, Martins PAF, Tekkaya AE (2009) J. J. o. M. P. T., Single point incremental forming of PVC. 209 (1), 462–469

22.

Martins PAF, Kwiatkowski L, Franzen V, Tekkaya AE, Kleiner MJC (2009) A.-M. T., Single point incremental forming of polymers. 58 (1), 229–232

23.

Simon D, Zitzlsberger S, Wagner J, Kern L, Reinhart GJ (2014) S. I. P., Forming Plastic Shields on a Reconfigurable Tooling System

24.

Gunther DSLKJW, CIRP RJP (2014) A Reconfigurable Tooling System for Producing Plastic Shields. 17 (1), 853–858

25.

Chsner A (2021) Timoshenko Beam Theory

Declarations

All the work in this article is done by me, and has not been published in other journals. We confirm that the manuscript has been submitted solely to this journal and is not published, in press, or submitted elsewhere.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.