A mathematical modeling of FPC bending shape based on Jacobian matrix and size double iteration

Abstract

The invention relates to a mathematical modeling method based on Jacobian matrix and dual iteration of dimensions, which is used to predict the bending deformation of a flexible circuit board (FPC) under different conditions; the core steps include: establishing a finite element model of the FPC, performing simulation analysis, establishing a discrete point path along the surface of the FPC to obtain the Von Mises stress change trend, and appropriately correcting the stress value to obtain the accurate relationship between the curvature and length of the FPC; the invention adopts the Jacobian matrix iteration method to obtain unknown parameters in the linear relationship, such as the slope and the intercept, and solves the bending shape of the actual FPC through the dimension iteration method; this method not only improves the prediction accuracy, but also is suitable for rapid design iteration and prototype production; the invention helps to improve the performance and reliability of electronic products, while reducing the design and production costs; through this method, designers can achieve more innovative and complex product designs in a shorter time and at a lower cost.

Description

Technical field:

[0001] The present invention belongs to the field of electronic engineering, and in particular relates to a mathematical model for predicting the bending shape of a flexible printed circuit (FPC), specifically a mathematical modeling method for the bending shape of an FPC based on a Jacobian matrix and dual iteration of dimensions. Background technology:

[0002] (1) Importance and challenges of flexible printed circuits (FPC)

[0003] With the rapid development of electronic technology, flexible printed circuits (FPCs) play an increasingly important role in modern electronic products. With its excellent flexibility and foldability, FPCs are widely used in various portable electronic devices, wearable technologies, medical devices, and aerospace fields. However, FPCs will inevitably experience various degrees of bending and twisting during use, which poses new challenges to their electrical performance and mechanical stability.

[0004] (2) Existing technology: Finite element method (FEM)

[0005] Currently, deformation prediction of FPC mainly relies on the finite element method (FEM). FEM is a powerful computational tool that provides accurate stress and deformation analysis and is widely used in engineering and physics. In FPC design, FEM is used to simulate and predict the deformation behavior of circuit boards after being subjected to stress. However, the high computational cost and operational complexity of FEM in processing complex models make it inefficient in the rapidly developing and competitive electronic product market.

[0006] (3) Limitations of FEM

[0007] Although FEM is excellent in accuracy, it faces several major limitations in practical applications:

[0008] ① High computational cost: FEM analysis requires a lot of computing resources, especially when dealing with complex FPC structures.

[0009] ② The process is complex and time-consuming: FEM analysis requires professional knowledge and skills, and its setup and operation process is complex and time-consuming.

[0010] ③ Not suitable for rapid iteration: In the early stages of product design, when rapid iteration and design adjustment are required, the calculation speed and complexity of FEM become limiting factors.

[0011] (4) Requirement: Efficient and accurate FPC deformation prediction method

[0012] In the current market environment, fast and accurate prediction of FPC deformation behavior is crucial to speed up product development cycle, reduce costs and improve product performance. Therefore, developing a new FPC deformation prediction method that can maintain high accuracy and significantly improve efficiency has become an urgent need in the industry. Summary of the invention:

[0013] The present invention relates to a mathematical modeling method based on Jacobian matrix and dual iteration of size. The method provides a powerful tool for the design and manufacture of FPC by establishing the relationship between surface bending stress and FPC curvature and using finite element analysis to verify the accuracy and reliability of the mathematical model, so as to solve the problems of high calculation cost, complex and time-consuming process and unsuitability for rapid iteration of traditional finite element method.

[0014] To achieve the above object, the present invention adopts the following technical solutions:

[0015] A mathematical modeling method for FPC bending morphology based on Jacobian matrix and size double iteration includes the following steps:

[0016] S1. Based on the finite element tool, the corresponding FPC bending shape is modeled and simulated, and the relationship between its curvature change and the distance of the FPC along the length direction is obtained;

[0017] S2, replace the relationship between its curvature change and the distance along the length direction of FPC with a strict linear relationship;

[0018] S3, obtaining the unknowns in the corresponding linear relationship, namely the slope and intercept, through the Jacobi matrix iteration method;

[0019] S4. The actual bending shape of the FPC to be solved is solved by the dimension iteration method.

[0020] The step S1 specifically includes the following steps:

[0021] S1.1. Establish a finite element model of FPC;

[0022] S1.2, simulate and analyze its bending deformation process;

[0023] S1.3, establish a discrete point path along the surface of the FPC after bending to obtain its Von Mises stress change trend;

[0024] S1.4. Make appropriate corrections to the Von Mises stress, with positive values ​​for tensile stress and negative values ​​for compressive stress;

[0025] S1.5. According to the relationship between the FPC curvature and the bending stress, obtain the relationship between the FPC bending curvature and the distance along the length of the FPC.

[0026] In the steps S2 and S3, the relationship between the curvature change and the distance of the FPC along the length direction is replaced by a strict linear relationship, and two unknowns are obtained based on the corresponding linear relationship expression, namely the slope m and the intercept b;

[0027] The step S4 specifically includes the following steps:

[0028] S4.1. Initial parameter setting: Select the initial size parameters of the FPC, including the total length L0, the coordinate differences Δx0 and Δy0 in the X and Y directions at both ends, and the parameters m0 (slope) and b0 (intercept) in the preliminary estimated curvature expression.

[0029] S4.2, parameter segmentation: the difference between the actual size and the initial size of the FPC (including the total length L, the coordinate difference Δx and Δy at both ends) is divided into N equal segments, each segment is represented as dL, dΔx and dΔy.

[0030] S4.3. Initialization of the iterative process: set the number of iterations N, and set the parameters of the first FPC curve according to the initial parameters, such as L1=L0+dL, Δx1=Δx0+dΔx, Δy1=Δy0+dΔy.

[0031] S4.4, perform iterative calculation:

[0032] a) Using the initial curvature parameters m0 and b0, combined with the FPC size of the current iteration (such as L1, Δx1, Δy1), mathematical modeling and curvature prediction are performed;

[0033] b) Calculate the new curvature parameters m1 and b1 obtained in the current iteration.

[0034] S4.5, continuous iterative update: based on the curvature parameters (such as m1 and b1) obtained in the previous iteration, update the FPC size parameters of the next iteration, and repeat step 4 until N iterations are completed.

[0035] S4.6. Final curvature determination: After completing N iterations, the final curvature distribution of the FPC is determined using the curvature parameters mN and bN obtained from the last iteration.

[0036] S4.7. Based on the mN and bN obtained from the final iterative calculation, the corresponding FPC bending shape is calculated.

[0037] It mainly includes the following four parts

[0038] (1) Establishment of the relationship between bending stress and curvature

[0039] In order to predict the bending deformation of FPC under different conditions, it is first necessary to establish the relationship between the surface bending stress and the curvature of FPC. The core part of establishing this relationship includes the following steps:

[0040] S1. Analysis of initial state and bending state: First, the initial state and the state after bending of FPC are analyzed in detail. In the initial state, FPC appears as a straight line, and the neutral layer is represented by NN. After bending, the shape of FPC changes to form a curve, which includes arc segments in the neutral layer and straight segments on the curve.

[0041] S2. Relationship between elastic strain and bending stress: Through mathematical derivation, the relationship between the elastic strain of the material layer in the FPC and the distance from the neutral layer is established. This relationship shows that during the bending process, the elastic strain of the material layer is proportional to the distance from the neutral layer.

[0042] S3. Relationship between bending stress and elastic modulus: The relationship between bending stress and elastic modulus of the material is also established. This relationship shows that bending stress is proportional to elastic modulus, where bending stress is a function of the distance from the neutral layer and elastic modulus.

[0043] (2) Finite element modeling and stress analysis

[0044] Based on the established bending stress and curvature relationship, the finite element analysis method is used to verify the accuracy and reliability of the mathematical model. The following are the key steps of finite element modeling and stress analysis:

[0045] S1. Setting of FPC geometric parameters: In order to perform finite element analysis, it is necessary to define the geometric parameters of the FPC, including length, thickness and width. These parameters are used to construct the finite element model of the FPC.

[0046] S2. Construction of finite element model: The finite element model of FPC was constructed in Abaqus, which includes geometric parameters and material properties. The details of the model include settings such as unit size, element type, and analysis step.

[0047] S3. Apply boundary conditions: In the simulation, vertical and horizontal displacements were applied to the end points of the FPC model to simulate the bending conditions.

[0048] S4. Calculation of Von Mises stress: The Von Mises stress distribution on the FPC model was calculated through finite element analysis.

[0049] S5. Adjustment of stress values: In order to correctly represent the curvature trend of the FPC, a convention is adopted to treat compressive stress values ​​as negative and tensile stress values ​​as positive; this helps to obtain more accurate curvature results.

[0050] (3) Mathematical modeling and FPC curvature analysis

[0051] Based on the results of finite element analysis, mathematical modeling methods are used to infer the curvature distribution of FPC. The following are the key steps of mathematical modeling and FPC curvature analysis:

[0052] S1. Establishing the curvature equation of FPC: The present invention proposes a mathematical model for predicting the curvature of FPC, which is based on the relationship between bending stress and curvature. The model represents the complex relationship between curvature and the length, bending stress and material properties of FPC.

[0053] S2. Dimension iteration method: In order to improve the accuracy of the mathematical model, the dimension iteration method is introduced. This method iteratively adjusts the dimension parameters of the FPC, such as the total length and the coordinate difference between the two ends, to obtain more accurate curvature results. This enables the model to adapt to FPCs of different sizes.

[0054] S3. Generation of curvature diagram: Based on the mathematical model and the dimensional iteration method, the curvature diagram of the FPC can be generated to show the curvature values ​​of the FPC at different positions. These curvature diagrams provide detailed information on the bending state of the FPC.

[0055] (4) Prediction of multiple FPC curves

[0056] The method of the present invention is not only applicable to a single type of FPC, but can also predict a variety of different types of FPC curves according to different input conditions and parameters, which makes the technology widely applicable in various applications, including electronic equipment, communication equipment, medical equipment and other fields.

[0057] Compared with the prior art, the present invention has the following beneficial effects:

[0058] The present invention provides an advanced method to accurately predict the curvature deformation of flexible printed circuits (FPCs) under different conditions through mathematical modeling and finite element analysis. This technology helps to optimize the design and manufacture of FPCs, improve their performance and reliability, and provides a powerful tool for engineers and designers in multiple fields. Through this invention, the bending behavior of FPCs can be better understood and controlled, thereby improving the performance and reliability of electronic products. Description of the drawings:

[0059] Figure 1 Comparison of deformation before and after FPC bending; 1(a) is the initial state of FPC before bending, and 1(b) is the state after FPC bending;

[0060] Figure 2The finite element model of FPC and its dimension diagram; 2(a) is the original FPC model; 2(b) is the FPC model after bending; 2(c) shows the coordinate difference between points A' and B' in the X and Y directions;

[0061] Figure 3 is the FPC model surface tracking path; 3(a) is the entire FPC model surface tracking path; 3(b) is Figure 3 Magnified view of the blue circled area in (a);

[0062] Figure 4 The stress obtained along the tracing path and the stress trend adjusted based on the stress properties; 4(a) is the distribution of the Von Mises stress along the trajectory obtained from the finite element model; 4(b) is the distribution of the Von Mises stress value after adjustment;

[0063] Figure 5 is the relationship between the curvature of the deformed FPC and the real distance along its length;

[0064] Figure 6 It is a schematic diagram for analyzing the deformed FPC based on the principle of calculus;

[0065] Figure 7 It is a flowchart based on Jacobian matrix iteration;

[0066] Figure 8 It is a flow chart of the size iteration method;

[0067] Fig. 9 is an analysis of an "S"-shaped FPC curve; wherein 9(a) is an iterative analysis result; 9(b) is a comparison between a finite element method simulation curve and a mathematical modeling result based on the present invention;

[0068] Fig.10 Comparative analysis of FPC curvature: finite element model and mathematical model;

[0069] Fig.11 The analysis results of the Type 2 FPC curve are shown in Figure 11(a) and 11(b) respectively. The results of the finite element analysis are compared with those of the algorithm proposed in this paper.

[0070] Fig.12 The analysis results of the commonly used type 3 curve, namely Type 3FPC curve; 12(a) is the finite element simulation model, and 12(b) is the comparison between the finite element analysis results and the analysis results of the algorithm proposed in this paper;

[0071] Fig.13The analysis results of the Type 4 FPC curve are shown in Figure 13(a) and Figure 13(b) are the comparison between the finite element analysis results and the analysis results of the algorithm proposed in this paper. Specific implementation method:

[0072] In order to clarify the technical problems, technical solutions, implementation processes and performance demonstrations, the present invention is further described in detail below in conjunction with embodiments. It should be understood that the specific embodiments described herein are for explanation only. The present invention is not intended to limit the present invention. Various exemplary embodiments, features and aspects of the present disclosure will be described in detail below with reference to the accompanying drawings. The same reference numerals in the accompanying drawings represent elements with the same or similar functions. Although various aspects of the embodiments are shown in the accompanying drawings, the drawings do not have to be drawn to scale unless otherwise specified.

[0073] The word “exemplary” is used exclusively herein to mean “serving as an example, example, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments.

[0074] In addition, in order to better illustrate the present disclosure, numerous specific details are given in the following specific embodiments. It should be understood by those skilled in the art that the present disclosure can also be implemented without certain specific details. In some examples, methods, means, components and circuits well known to those skilled in the art are not described in detail in order to highlight the subject matter of the present disclosure.

[0075] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings.

[0076] 1. Exploration of the relationship between FPC curvature and surface bending stress

[0077] In order to predict the bending deformation of flexible printed circuits (FPC) under different conditions, it is necessary to figure out the changing trend of its curvature under different conditions. Therefore, it is very important to analyze the relationship between the curvature and the corresponding bending stress.

[0078] Figure 1 (a) shows the initial state of the flexible printed circuit (FPC) before bending, and the neutral layer is represented by NN. No deformation occurs at this time. Figure 1 (b) shows the state of the FPC after bending. Figure 1 The arc segment E'F' corresponding to the straight line segment EF in (a) is located in the neutral layer. Its length remains unchanged after bending and is a constant value δx. Figure 1 In (b), the length of arc segment E'F' can be expressed as R·θ.

[0079] And for Figure 1 The arc segment G'H' in (b) corresponds to Figure 1The length of the straight line segment GH in (a) is δx before bending, and becomes (R+y)·θ after bending. The change in length can be calculated by the following expression:

[0080]

[0081] Assuming that no plastic deformation is induced during the bending process, it can be deduced that the elastic strain induced in the material layer located at a distance 'y' from the neutral layer can be expressed as:

[0082]

[0083] The corresponding bending stress can be expressed as:

[0084]

[0085] Where E is the Young's modulus of the material. This expression describes the proportional relationship between the bending stress σ, the Young's modulus E of the material, and the strain ε.

[0086] In the finite element model used to analyze the flexible circuit board (FPC), some basic assumptions are adopted to simplify the analysis process. These assumptions include:

[0087] ① Uniform thickness (h): Assume that the thickness of the FPC remains consistent throughout the structure, represented by a constant 'h';

[0088] ② Neutral layer distance (h / 2): Another important assumption is that the distance from any surface point of the FPC to the neutral layer is exactly half of the aforementioned constant thickness 'h';

[0089] ③ Constant elastic modulus (E): The elastic modulus property 'E' of the FPC material is considered to be constant.

[0090] Based on these basic assumptions, the product of the elastic modulus ('E') of the material and the distance ('y') of a point on the surface from the neutral layer, as shown in equation (3), remains constant, leading to mathematical simplification. As a corollary, it can be concluded that within the structure of a flexible printed circuit (FPC), there is a linear relationship between bending stress ('σ') and curvature ('1 / R'), which can be expressed as:

[0091]

[0092] 2. Finite element modeling and stress analysis

[0093] Based on the relationship between surface bending stress and curvature, the change in curvature of the structure can be inferred from the change in surface bending stress. A finite element model of a flexible circuit board (FPC) was constructed in Abaqus with specific geometric parameters, including a length of 200 mm, a thickness of 0.1 mm, and a width of 5 mm. The global element size was set to 0.5, and the thickness direction was discretized into 10 elements. The element type selected was C3D8R, an 8-node linear octahedral element with reduced integration and wrinkle control. The analysis step used a static, general analysis step type with nonlinear geometry (Nlgeom) to consider large deformations and displacements. In the final model, a total of 40,000 elements were generated.

[0094] The original FPC model is as follows Figure 2 (a) shows that there is no internal stress at this time. The simulation process was carried out in two consecutive steps. In the first step, a vertical displacement of 20 mm and -20 mm was applied to the A and B ends of the FPC model along the Y axis respectively; in the second step, a horizontal displacement of 30 mm and -30 mm was applied to the A and B ends of the FPC model along the X axis respectively.

[0095] exist Figure 2 In the bent FPC model shown in (b), it can be seen that its shape resembles the letter 'S', with different stresses distributed at different locations. Figure 2 (c) shows the coordinate differences between points A' and B' in the X and Y directions, denoted as Δx and Δy, respectively.

[0096] In order to observe the trend of surface bending stress, a trajectory of tracking the FPC surface was established by generating a point sequence path, such as Figure 3 (a) shown. Figure 3 (b) Provided Figure 3 A magnified view of the blue circled area in (a). It can be clearly seen that the created path consists of a series of discrete points.

[0097] The distribution of Von Mises stress along the trajectory obtained from the finite element model is shown in Figure 4 (a). The Von Mises stress calculation formula is as follows:

[0098]

[0099] Among them, σ1, σ2 and σ3 represent the normal stresses in three mutually perpendicular directions obtained from the three-dimensional stress analysis, and τ1, τ2 and τ3 represent the shear stresses acting on the corresponding three perpendicular planes.

[0100] According to formula (5), it can be seen that the value of Von Mises stress is essentially non-negative. In order to correctly represent the curvature trend of FPC, the present invention adopts a convention that the compressive stress value is regarded as a negative value and the tensile stress value is regarded as a positive value. Based on this, the Von Mises stress value is adjusted, and the result is as follows: Figure 4 (b) as shown.

[0101] 3. Mathematical modeling and FPC curvature analysis

[0102] based on Figure 4 (b) It can be clearly seen that the corresponding interval can be divided into three segments, named S1, S2 and S3, as Figure 5 shown.

[0103] Within each region, there is a clear linear relationship between the surface bending stress and the actual distance along the FPC. Therefore, within these defined intervals, the relationship between the curvature of the FPC and the distance along the FPC can be considered to be linear.

[0104] For the S1 segment, the relationship between the curvature (c1) and the length ('l') on the FPC is:

[0105] c1=m·l+b1 (6)

[0106] Wherein, c1 represents the curvature value within segment S1, 'm' represents the slope of the linear relationship, 'l' represents the length along the FPC, and 'b1' represents the intercept of c1.

[0107] For the S2 segment, the relationship between the curvature (c2) and the length along the FPC can be expressed as:

[0108] c2=-m·l+b2 (7) Where c2 represents the curvature value in segment S2, and 'b2' is the corresponding intercept of c2. In view of the rotational symmetry of the FPC curve, equation (7) must pass through the point (L / 2,0), where 'L' represents the total length of the FPC. Therefore, equation (7) can be further expressed as:

[0109] c2=-m·l+m·L / 2 (8)

[0110] In addition, it can be deduced Figure 5 The coordinates of the midpoint C are (L / 4-b / 2m, mL / 4+b / 2).

[0111] To predict the deformation curve of an FPC based on the coordinates of points A and B and the total length of the FPC, two basic parameters need to be determined: the slope 'm' and the intercept 'b'. The distance between points A and B can be obtained by integrating the relationship between the curvature and the length along the FPC in different segments. Due to the rotational symmetry of the FPC curve, the corresponding analysis work is limited to half of the total FPC length.

[0112] Small increments in the size of the FPC in the X and Y directions can be expressed as:

[0113] dx=dl·cosθ (9)

[0114] dy = dl·sinθ (10) where 'dl' represents a small increment along the length of the FPC, and 'θ' represents the angle formed between the tangent line of the FPC curve and the positive X-axis at any given point, as Figure 6 shown.

[0115] It should be noted that 'θ' is a function of the distance l from its starting point to the specified point along the FPC, ie, it can be expressed as θ(l).

[0116] Therefore, the coordinate difference between points A and B can be expressed as:

[0117]

[0118]

[0119] like Figure 5 As shown, the relationship between the curvature of the FPC and the length along its curve needs to be distinguished in different segments. The complexity of this relationship requires that the corresponding integration be performed piecewise.

[0120] For the S1 segment, a small increment of the angle θ can be expressed as:

[0121] dθ S1 =-dl·c1 (13)

[0122] The angle between the tangent line through any point in the S1 segment and the X-axis can be expressed as:

[0123]

[0124] The small increments dx1 and dy1 of the dimensions in the X and Y directions within segment S1 can be described as follows:

[0125] dx1=cos(θ S1 )·dl (15)

[0126] dy1=sin(θ S1 )·dl (16)

[0127] Based on these relationships, the coordinate differences Δx1 and Δy1 at both ends of the FPC in segment S1 can be expressed as:

[0128]

[0129]

[0130] For Figure 5 Point C in the figure marks the end point of segment S1, and the value of θ at this point is θ C It can be expressed as:

[0131]

[0132] In segment S2, the small increment dθ of angle θ S2 It can be expressed as:

[0133] dθ S2 =-dl·c2=-dl·(-m·l+m·L / 2) (20)

[0134] The angle θ between the tangent line at any point in segment S2 and the X-axis S2 It can be expressed as:

[0135]

[0136] The small increments dx2 and dy2 of the dimensions in the X and Y directions within segment S2 can be expressed as:

[0137] dx2=cos(θ S2 )·dl (22)

[0138] dy2=sin(θ S2 )·dl (23)

[0139] Considering the rotational symmetry of the corresponding FPC curve, the corresponding computational analysis is limited to the first half of the S2 segment, extending to Figure 6 Point E in.

[0140] Therefore, based on the first half of the FPC curve in the S2 segment, it can be concluded that:

[0141]

[0142]

[0143] Wherein, Δx2 / 2 and Δy2 / 2 represent the X and Y coordinate differences between the first and last points in the first half of segment S2, respectively.

[0144] Due to the inherent rotational symmetry of the FPC curve, the overall dimensions of the FPC curve in the X and Y directions can be expressed as follows:

[0145]

[0146]

[0147] Among them, θ S1 and θ S2As defined in equations (14) and (21).

[0148] Δx and Δy represent the difference between the two end coordinates of the FPC curve in the X and Y directions, respectively. 'L' represents the total length of the FPC curve, while 'b' and 'm' are the intercept and slope of the 'c1' curve mentioned above.

[0149] The system of equations composed of equations (26) and (27) can be further expressed as equation (28), which has two unknown variables, namely 'b' and 'm', which need to be solved using nonlinear equation solving methods. The solution of these variables is crucial to predicting the shape of the FPC curve.

[0150]

[0151] Based on equations (14), (19) and (21), equation (28) is transformed into:

[0152]

[0153] Where F(u) represents a vector function, where u = [m, b], and includes only two unknown variables 'm' and 'b', which are the slope and intercept in the corresponding curvature expression. To solve this nonlinear equation system, we use the Jacobi matrix iteration method. The detailed method to solve this system is as follows:

[0154] (1) Initialization:

[0155] First, choose an initial estimate of the unknown variable, denoted as u (0) =[m0,b0].

[0156] (2) Iteration process:

[0157] a. Calculate the Jacobian matrix J(u (k) ):

[0158] In each iteration k, the Jacobian matrix J(u (k) ). Each element J of the Jacobian matrix ij (u (k) ) represents the partial derivative of the i-th equation with respect to the j-th unknown, which is expressed as follows:

[0159]

[0160] b. Calculate the equation value F(u (k) ):

[0161] Calculate and current estimate u (k) The corresponding function value vector F(u (k)). In mathematical terms, this involves evaluating the nonlinear equation on the current approximation to obtain a vector of function values. This process is commonly used in iterative numerical techniques, where the estimate of the unknown is improved over multiple iterations, helping to increase the accuracy of the solution.

[0162] c. Solve the linear equations:

[0163] Solve the linear equation system J(u (k) )Δu=-F(u (k) ) to obtain the update vector Δu. Where J(u (k) ) represents the Jacobian matrix, and Δu represents the correction vector. Solving this linear system is a key step in the iterative method and helps to accurately estimate u (k) Through iterative updates, it converges towards a more accurate approximation of the unknowns in the nonlinear system of equations.

[0164] d. Update the estimated value u (k+1) :

[0165] Use u (k+1) =u (k) +Δu to update the estimated value. In this expression, u (k+1) represents the refined estimate of the (k+1)th iteration, and u (k) represents the current estimate at the kth iteration. The update is achieved by incorporating the correction vector Δu into the current estimate, which is a key step in the iterative process. This process gradually improves the accuracy of the solution in the iterative solution of the nonlinear system of equations.

[0166] e. Iteration check:

[0167] Checking whether the termination condition is met is an important aspect of iterative numerical techniques. Usually, the corresponding iteration process will be terminated as long as any of the following conditions is met:

[0168] ① Termination criterion 1: Function value F(u (k+1) ) is below the predefined tolerance, indicating that the function value is close enough to zero.

[0169] ② Termination criterion 2: The norm of the update vector Δu is lower than the specified threshold, indicating that the solution has converged well.

[0170] ③Termination criterion 3: The maximum number of allowed iterations has been reached, indicating the end of the iteration process.

[0171] f. Continue or terminate the iteration:

[0172] If the specified termination condition is not met, the iterative calculation must be restarted from step 2a. This iteration cycle ensures further refinement of the corresponding solution in accordance with the defined accuracy criterion. Conversely, the iteration process ends as soon as the corresponding condition is met.

[0173] (3) Output:

[0174] After the iteration process is completed, the estimated value u (k+1) as the numerical solution to the specified nonlinear system of equations. This carefully obtained solution not only reflects the iterative refinement, but also indicates that the numerical algorithm has converged to an accurate approximation of the unknowns in the system of equations. The flowchart of the complete Jacobi matrix iteration method process is shown in Figure 7 shown.

[0175] In the context of solving nonlinear systems of equations based on Jacobi matrix iteration, it is crucial to choose appropriate initial estimates of the unknowns. The chosen estimates must be close to the actual values ​​so that the Jacobi matrix iteration method can converge quickly to the exact solution. If the initial values ​​are not approximate enough, it will be difficult to obtain the correct solution to the nonlinear system of equations.

[0176] To solve this problem, the "size iteration method" is introduced in the present invention. Initially, a fixed size of the FPC was selected, with a total length L0 = 50 mm, a coordinate difference of Δx0 = 20 mm in the X and Y directions at both ends of the FPC, and Δy0 = -20 mm, as well as accurate values ​​of m0 = 0.04571 and b0 = -0.40857. However, in actual situations, the actual total length L and the coordinate differences Δx and Δy at both ends of the FPC are often different from the above-mentioned predefined values ​​L0, Δx0 and Δy0, and even have large differences. Therefore, it is often difficult to solve and obtain the correct result using m0 and b0 as the initial values ​​of the iteration. In the present invention, a positive integer N is selected, and the differences in the parameters L, m and b are divided into N equal segments, as shown below:

[0177] dL=(L-L0) / N (31)

[0178] dΔx=(Δx-Δx0) / N (32)

[0179] dΔy=(Δy-Δy0) / N (33)

[0180] In the analysis, the initial FPC curve is defined by the following parameters: L1 = L0 + dL, Δx1 = Δx0 + dΔx, and Δy1 = Δy0 + dΔy. Using the values ​​of 'm0' and 'b0', the values ​​of 'm1' and 'b1' for the first FPC curve are determined using the Jacobi matrix iteration method. The subsequent second FPC curve parameters 'm2' and 'b2' values ​​are iteratively calculated based on the values ​​of 'm1' and 'b1'. This iterative process continues until the values ​​of the slope 'mN' and intercept 'bN' of the Nth FPC curve are obtained.

[0181] Based on the obtained values ​​of slope 'mN' and intercept 'bN', the coordinates of any point along the length of the FPC can be determined using equations (34) and (35):

[0182]

[0183]

[0184] Where (x0, y0) represents the coordinates of the initial point of the FPC, (x a ,y a ) represents the coordinates of any specific point on the FPC, and θ represents the angle between the tangent line through the selected point and the X-axis. If the point is located in the S1 segment, its expression corresponds to equation (14); if the point is located in the S2 segment, its expression corresponds to equation (21). The final curve, denoted as the Nth curve, represents the actual FPC curve that needs to be solved. The corresponding analysis process is as follows Figure 8 shown.

[0185] If the values ​​of 'm' and 'b' cannot be accurately iterated, it indicates that there is a large deviation between their estimated values ​​and the result values ​​obtained in the previous iteration. In this case, the value of N can be appropriately increased. Generally, N = 50 is sufficient to achieve accurate iterative calculation of these parameters in most cases.

[0186] 4. Iteration Analysis Results

[0187] Here N is set to 10, which means that the "size iteration method" iterates ten times, such as Fig. 9 (a) is shown. Curve 0 is used as the initial condition for the iterative process. After each iteration, a different FPC curve appears with a specific size and total length. The result of the tenth iteration, Curve 10, is the final solution. The entire iterative process proceeds smoothly, highlighting the robustness and reliability of the method. Fig. 9 (b) is a comparison between the finite element method simulation curve and the mathematical modeling result based on the present invention.

[0188] The overall curvature of the FPC can be determined using the slope (m) and intercept (b) values ​​obtained by iterative solution. Fig.10 A comparison of the curvature curves obtained by finite element simulation and the mathematical modeling proposed in the present invention is shown.

[0189] according to Fig.10 ,It is obvious that the differences between the curves are relatively small in most areas, except in the area near the inflection point, where the errors are somewhat ,obvious.

[0190] Using a similar method, we can also predict the FPC curves of several other commonly used curve types, such as Figure 11-13 As shown, Fig.11 (a), 12(a) and 13(a) are finite element simulation models. Fig.11 (b), 12(b), and 13(b) are the comparisons between the finite element analysis results and the analysis results of the algorithm proposed in this paper.

[0191] analyze Fig. 9 , 11 -13, it can be clearly seen that there is a slight difference between the finite element simulation results and the predicted results obtained by the mathematical modeling method proposed in the present invention, but for most cases, it is completely within an acceptable range. Although the mathematical model proposed in the present invention is based on the assumption that there is a linear relationship between the curvature of the FPC and the distance along its length, in fact, this relationship is not strictly linear, but approximately linear. However, the corresponding difference does not have much impact in most practical cases and can be ignored.

[0192] This paper proposes a new mathematical model to efficiently and accurately predict the curvature deformation of FPC. The core advantages of this model are:

[0193] ① Establish a simplified mathematical relationship: By establishing a linear relationship between the curvature of the FPC and its length, the traditional complex calculation process is simplified;

[0194] ② Improve computational efficiency: Compared with FEM, this model has a faster computation speed and is more suitable for rapid design iteration and prototyping;

[0195] ③Easy to apply: The model design is simple and intuitive, easy to understand and apply, and does not require deep professional knowledge or complex software support;

[0196] ④Wide applicability: Suitable for FPCs of various types and configurations, able to adapt to different design requirements and application scenarios.

[0197] The mathematical model of the present invention is not only of great significance in the field of electronic product design, but also has a profound impact on improving the reliability and performance of flexible electronic products and reducing design and production costs. It provides designers with a more efficient and flexible tool, enabling them to achieve more innovative and complex product designs in a shorter time and at a lower cost.

[0198] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. The above embodiments and descriptions are only preferred examples of the present invention and are not intended to limit the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A mathematical modeling method for FPC bending morphology based on Jacobian matrix and size double iteration, characterized in that: The following steps are involved: S1. Based on the finite element tool, the corresponding FPC bending shape is modeled and simulated, and the relationship between its curvature change and the distance of the FPC along the length direction is obtained; S2, replace the relationship between its curvature change and the distance along the length direction of FPC with a strict linear relationship; S3, obtaining the unknowns in the corresponding linear relationship, namely the slope and intercept, through the Jacobi matrix iteration method; S4. Solve the bending shape of the real FPC that needs to be solved by the size iteration method; The step S4 specifically includes the following steps: S4.

1. Initial parameter setting: Select the initial size parameters of the FPC, including the total length L 0. The coordinate difference between the two ends in the X and Y directions Δx 0 and Δy 0, and the initial estimated curvature expression parameters m 0: slope, and b 0: intercept; S4.2, Parameter segmentation: setting the number of iterations N , the difference between the actual size of the FPC and the initial size, including the total length L , the coordinate difference between the two ends Δx and Δy , divided into N equal segments, each segment is represented by dL , dΔx and dΔy ; S4.3, iterative process initialization: and set the parameters of the first FPC curve according to the initial parameters: L 1 = L 0 + dL , Δx 1 = Δx 0 + dΔx , Δy 1 = Δy 0 + dΔy ; S4.4, perform iterative calculation: a) Using initial curvature parameters m 0 and b 0, combined with the FPC size of the current iteration: L 1. Δx 1. Δy 1. Perform mathematical modeling and curvature prediction; b) Calculate the new curvature parameters obtained in the current iteration m 1 and b 1; S4.5, continuous iterative update: based on the curvature parameters obtained in the previous iteration, m 1 and b 1. Update the FPC size parameters for the next iteration and repeat step 4 until completed N Iterations; S4.

6. Final curvature determination: After completing N After iterations, the curvature parameters obtained from the last iteration are used. mN and Bn , determine the final curvature distribution of the FPC; S4.7, calculated according to the final iteration mN and Bn , calculate the corresponding FPC bending shape.

2. According to the FPC bending morphology mathematical modeling method based on Jacobian matrix and size double iteration according to claim 1, it is characterized in that: The step S1 specifically includes the following steps: S1.

1. Establish a finite element model of FPC; S1.2, simulate and analyze its bending deformation process; S1.3, establish a discrete point path along the surface of the FPC after bending to obtain its Von Mises stress change trend; S1.

4. Make appropriate corrections to the Von Mises stress, with positive values ​​for tensile stress and negative values ​​for compressive stress; S1.

5. According to the relationship between the FPC curvature and the bending stress, obtain the relationship between the FPC bending curvature and the distance along the length of the FPC.

3. According to the FPC bending morphology mathematical modeling method based on Jacobian matrix and size double iteration according to claim 1, it is characterized in that: In the steps S2 and S3, the relationship between the curvature change and the distance of the FPC along the length direction is replaced by a strict linear relationship, and two unknowns are obtained based on the corresponding linear relationship expression, namely, the slope m and the intercept b .

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