Non-contact control and curvature optimization method of FPC layout based on iterative trial and discrete point analysis

Abstract

The invention relates to a non-contact control and curvature optimization method for FPC layout based on iterative attempts and discrete point analysis; the method aims to optimize the curvature of an FPC by constructing a mathematical model and performing iterative attempts, while avoiding contact with adjacent components, thereby improving the performance and reliability of the FPC; specifically, the invention iteratively adjusts the length of the FPC, analyzes discrete points, determines whether it contacts with adjacent components, accurately calculates its maximum curvature value at different lengths, and determines its length range according to the curvature limit value; the invention is suitable for the design and optimization of FPCs in electronic products requiring high precision and reliability, especially in applications requiring miniaturization and strict performance; through the invention, engineers and designers can optimize the FPC layout more effectively and accurately, significantly improve the overall performance and reliability of the product, and reduce the design complexity and production cost.

Description

Technical field:

[0001] This technical field focuses on the design and application of flexible printed circuits (FPC) in electronic engineering, especially in optimizing the layout of FPC, improving its wear resistance, maintaining electrical performance stability, and coping with mechanical and thermal stress management. The research and development of FPC is crucial to achieving miniaturization, reliability and high performance of electronic equipment. Background technology:

[0002] Flexible printed circuits (FPCs) are extremely important in modern electronic products due to their layout flexibility, thin and light features, and bending capabilities. They face challenges in avoiding contact with other components and controlling curvature, all of which have a significant impact on the wear, damage, electrical performance, thermal management, and mechanical stress of FPCs, directly related to their reliability and performance.

[0003] Effectively avoiding contact and excessive curvature between FPC and other components is the key to improving its performance and reliability. Contact may cause wear and damage, while excessive curvature may cause wire breakage and interlayer short circuits, increase stress concentration, and thus affect the overall function and durability of FPC. Therefore, the study of FPC curvature and bending stress is crucial to optimize its design and function.

[0004] In view of the practical problems in FPC design, studying the relationship between FPC bending curvature and bending stress has direct practical value. Through finite element simulation and mathematical modeling analysis, the curvature and length of FPC can be accurately predicted and controlled, which is extremely critical to improving the performance and reliability of FPC. The development of this method not only solves specific engineering problems, but also provides important support for the theory and practice of FPC design.

[0005] With the miniaturization of electronic devices and the improvement of performance requirements, FPC technology is facing continuous innovation pressure. From material selection to design methods, to the application of simulation technology, innovation in all aspects is crucial to improving the performance and reliability of FPC. Especially in terms of curvature and stress analysis, new mathematical models and simulation methods are of great significance for optimizing FPC design.

[0006] In the face of the electronics industry's continuous pursuit of high performance and high reliability, in-depth research and improvement of FPC curvature and stress response is the key to achieving these goals. By accurately controlling and optimizing the design of FPC, its performance and reliability in various applications can be effectively improved, while also paving the way for future technological development and innovation.

[0007] The finite element method in the existing technology has the disadvantages of being cumbersome and time-consuming. It also requires engineers to have a high level of professional knowledge. The calculation efficiency is low when dealing with large or complex simulations, and convergence difficulties may occur during the analysis process. Summary of the invention:

[0008] The present invention relates to a non-contact control and curvature optimization method for FPC layout based on iterative attempts and discrete point analysis. The method first obtains the curvature change trend of the corresponding FPC based on finite elements, and then analyzes the corresponding FPC curve through mathematical modeling. Under the mathematical model, the shape of the FPC under specified conditions can be easily obtained. Then, the length of the FPC is iterated to obtain the shape of the FPC at different lengths. Based on the discrete point analysis, it is judged whether the corresponding FPC layout is in contact with adjacent components, and the maximum curvature value of the FPC is obtained by calculating the curvature at the key point, and it is judged whether the corresponding FPC curvature meets the conditions.

[0009] In order to overcome the many shortcomings of the finite element method in the prior art, such as cumbersome operation, long time consumption, the need for engineers to have a high level of professional knowledge, low computational efficiency when dealing with large or complex simulations, and possible convergence difficulties during the analysis process, the present invention proposes improvement measures, which are aimed at simplifying the operation process, improving the processing speed, and ensuring high efficiency and stable convergence performance even in the face of complex simulation tasks, ultimately providing an efficient tool for the design and optimization of FPC.

[0010] To achieve the above object, the present invention adopts the following technical solutions:

[0011] The non-contact control and curvature optimization method of FPC layout based on iterative trial and discrete point analysis is characterized by comprising the following steps:

[0012] S1. Based on the finite element tool, the corresponding FPC deformation is modeled, and a corresponding tracking path is created along its curved deformation surface, which is composed of a series of discrete points;

[0013] S2. Based on the tracking path, the changing trend of the curvature of the FPC after deformation is obtained, and based on this, a corresponding mathematical model of FPC deformation is established;

[0014] S3, taking the straight-line distance between the two end points of the FPC as the initial distance, and analyzing the deformation of the FPC based on the mathematical model established in the early stage, first observe whether the corresponding nonlinear equation group has a correct solution;

[0015] S4. If there is no correct solution, gradually increase the length of the FPC, and obtain a more accurate minimum length value through the coarse iteration value and the fine iteration value;

[0016] S5. After correctly obtaining the FPC curve, the curve is evenly discretized into 1001 isolated points;

[0017] S6. Analyze whether each isolated point is in the adjacent component region based on its coordinate value. If so, determine that it is in contact with the component. Otherwise, determine that it is not in contact.

[0018] S7, gradually and iteratively increasing the length value of the FPC, and analyzing whether it contacts with adjacent components;

[0019] S8, through continuous iteration and judgment process, finally obtaining the length value range in which the FPC does not contact the adjacent components;

[0020] S9, for the obtained FPC length range, using the refined iteration value as the iteration value, analyzing the maximum curvature value of the FPC under different length values ​​and recording them as the analysis object;

[0021] S10, according to a given curvature limit value (curvature allowable value), obtaining a length range of the maximum curvature of the FPC within a given curvature range as a final optimized length range of the FPC length;

[0022] S11. In the final optimized length range, obtain the length value corresponding to the minimum maximum curvature value of the FPC as the optimal length value L of the FPC best .

[0023] In step S2, it is determined whether the end point of the FPC curve obtained by mathematical modeling coincides with the FPC end point specified before modeling to determine whether the corresponding equation group has a correct solution.

[0024] The step S3 specifically includes the following steps:

[0025] S3.1, gradually increase the coarse iteration value for the FPC length, and determine whether the equation system has a correct solution;

[0026] S3.2, when the equation system has a correct solution, gradually reduce the fine iteration value of the FPC length, and determine whether the equation system has a correct solution;

[0027] S3.3. When the equation system does not have a correct solution for the first time, a fine iteration value is added to FPC again. The FPC length value at this time is the minimum value obtained after preliminary optimization.

[0028] The step S10 specifically includes the following steps:

[0029] S10.1. For the solution of the maximum curvature value of the curve under different length values ​​of the FPC, first solve the corresponding nonlinear equation group to obtain the slope and intercept of the curvature change of the FPC in different segments;

[0030] S10.2, respectively calculating the curvature values ​​of the FPC at the starting point, the end point and the turning point of the curvature change curve;

[0031] S10.3, record the maximum absolute value of the curvature values ​​at these locations;

[0032] S10.4, based on this, a series of length values ​​of the FPC and a corresponding series of maximum curvature values ​​are obtained;

[0033] S10.5. Based on the given curvature limit value (curvature allowable value), further optimize the FPC length range.

[0034] Compared with the prior art, the present invention has the following beneficial effects:

[0035] The present invention effectively optimizes FPC layout under various conditions by introducing an innovative method that combines mathematical modeling, iterative attempts and discrete point analysis. This technology is crucial to improving the design and manufacturing process of FPCs and can significantly enhance their performance and reliability. It provides a powerful tool to help engineers and designers in multiple industries optimize FPC layouts more accurately and efficiently, thereby significantly improving the overall quality and reliability of electronic products.

[0036] In addition, the mathematical model used in the present invention is simple and easy to use, has a fast calculation speed and high accuracy, and is particularly suitable for rapid design iteration and prototype development. This method reduces the reliance on professional skills and allows more engineers and designers to directly participate in the design and optimization of FPC. This advantage not only shortens the product development cycle and reduces costs, but also plays an important role in improving the reliability and innovation of the design. Description of the drawings:

[0037] Figure 1 Comparative analysis of the deformation of FPC before and after bending;

[0038] Figure 2 Figure 2 is a schematic diagram of the bell-shaped FPC finite element model and its dimensions; 2(a) is the FPC model without deformation; 2(b) is the FPC model after bending deformation; 2(c) is the FPC dimensions after bending deformation;

[0039] Figure 3 The path created along the curved surface of the bell-shaped FPC; 3(a) is a curved FPC and the path created along its surface; 3(b) is an enlarged view of a portion of the curved FPC;

[0040] Figure 4 The stress variation trend and result adjustment obtained on the created tracking path; 4(a) is the Mises stress distribution on the tracking path; 4(b) is the result of adjusting it based on the stress properties;

[0041] Figure 5 is the relationship between the curvature of the bent FPC and the length of the path along its surface;

[0042] Figure 6 A schematic diagram of analyzing the bell-shaped FPC curve after bending based on calculus;

[0043] Figure 7 This is the flowchart of Jacobi matrix iteration method;

[0044] Figure 8 Flow chart for the size iteration process;

[0045] Fig. 9 It is a comparison between the size iteration process and the finite element method (FEM) simulation and mathematical modeling results; 9(a) is a series of curves obtained from the size iteration process; 9(b) is a comparison between the results obtained from the finite element simulation and mathematical modeling;

[0046] Fig.10 This is a schematic diagram of the FPC's inappropriate length interfering with adjacent components;

[0047] Fig.11 Flowchart of the process of optimizing the FPC length range to avoid contact with adjacent components;

[0048] Fig.12 Flow chart of the process to obtain the coordinates of 1001 discrete points on FPC;

[0049] Fig.13 is a bell-shaped FPC curve within the optimized length range;

[0050] Fig.14 The ideal FPC curvature variation trend used in the analytical model;

[0051] Fig.15 Schematic diagram of FPC length range optimization based on curvature limitation (bell curve); 15(a) is the curvature variation trend of points A, B and T; 15(b) is the FPC length range optimization based on contactless and curvature value limit optimization;

[0052] Fig.16 The curvature and contactless verification of the optimal length bell-shaped FPC curve; 16(a) is a comparison between the bell-shaped FPC curve and the corresponding arc with the curvature limit value; 16(b) is a three-dimensional view of the finite element model of the bell-shaped FPC curve; 16(c) is a two-dimensional view of the finite element model of the bell-shaped FPC curve in the Z direction;

[0053] Fig.1717(a) is the deformed FPC and the tracking path created along its surface; 17(b) is the relationship between the FPC curvature and the length along the surface;

[0054] Fig.18 The 'U'-shaped FPC curve and its adjacent components within the optimized length range;

[0055] Fig.19 Schematic diagram of FPC length range optimization based on curvature limitation ('U' shaped curve); 19(a) shows the curvature variation trend of points A, B and T; 19(b) shows the FPC length range optimization based on contactless and curvature value optimization;

[0056] Fig. 20 The curvature and contactless verification of the optimal length 'U'-shaped FPC curve; 20(a) is a comparison between the 'U'-shaped FPC curve and the corresponding arc with curvature limit value; 20(b) is a two-dimensional view of the finite element model of the 'U'-shaped FPC curve in the Z direction;

[0057] Fig.21 'S' shaped FPC finite element model and the FPC curvature variation trend obtained therefrom; 21(a) is the deformed FPC and the path created along its surface; 21(b) is the relationship between the FPC curvature and the length along the surface;

[0058] Fig. 22 Schematic diagram of FPC length range optimization based on curvature limitation ('S' shaped curve); 22(a) is the curvature variation trend of points A, B and T; 22(b) is the FPC length range optimization based on non-contact and curvature value optimization;

[0059] Fig.23 The 'S'-shaped FPC curve and its adjacent components within the optimized length range;

[0060] Fig.24 The curvature and contact-free verification of the optimal length 'S'-shaped FPC curve; 24(a) is a comparison between the 'S'-shaped FPC curve and the corresponding arc with curvature limit value; 24(b) is a two-dimensional view of the finite element model of the 'S'-shaped FPC curve in the Z direction. Specific implementation method:

[0061] 1Relationship between FPC bending curvature and bending stress

[0062] Understanding the deformation behavior of FPC is of vital importance, especially regarding their curvature response under different conditions. Here, we can first analyze the relationship between bending stress and curvature to lay a theoretical foundation for further work.

[0063] exist Figure 1 In the figure, the state of the FPC before and after bending is shown. In the unbent state, the neutral layer is NN and the FPC material is not deformed. After bending, the neutral layer is N'-N', and the curve segment E'F' in the neutral layer (corresponding to the straight line EF in the unbent state) maintains its length δx unchanged. Therefore, in the bent state, the length of the curve segment E'F' is equal to the product of the radius R and the angle θ.

[0064] On the contrary, in the unbent state, the segment GH corresponding to the bent arc G'H' is extended from a length of δx to (R+y)·θ after bending. The change in length Δl can be expressed as:

[0065]

[0066] Assuming that no plastic deformation occurs, the elastic strain ε in the material layer at a distance 'y' from the neutral layer can be calculated:

[0067]

[0068] The corresponding bending stress σ can be expressed as:

[0069]

[0070] Where E represents Young's modulus, which is the inherent elasticity of the material. This expression illustrates the relationship between bending stress σ, Young's modulus E, strain ε, radial distance y, and radius of curvature R.

[0071] In finite element analysis, three main assumptions are incorporated to simplify the study of FPC:

[0072] ① Uniform thickness (h): Assuming that the thickness of C is uniform 'h', the material is considered to be of uniform thickness;

[0073] ② Neutral layer distance (h / 2): Assume that the distance from any point on the surface to the neutral layer is half the thickness 'h';

[0074] ③ Constant elastic modulus (E): The elastic modulus of the FPC is assumed to be stable to ensure consistent elastic behavior in the analysis.

[0075] These assumptions simplify the mathematical model, and ultimately the relationship between bending stress (σ) and curvature (1 / R) can be obtained as shown in equation (4):

[0076]

[0077] This expression emphasizes the important connection between FPC curvature and surface bending stress.

[0078] 2 Finite element simulation and stress analysis

[0079] In order to analyze the changing trend of FPC curvature, a finite element model was constructed using Abaqus 2020. The dimensions of the model are 60 mm in length, 2 mm in width, and 0.1 mm in thickness. The global element size is selected as 0.2 mm, and the thickness direction is divided into 10 elements to improve the resolution. The C3D8R element type is selected, that is, an 8-node linear octahedron element. The nonlinear geometry (Nlgeom) setting is turned on, and the final model generates 30,000 elements.

[0080] In the initial unbent state, such as Figure 2 As shown in (a), the FPC model is in a straight line. A displacement of (25mm, 0) and a rotation angle of 90° are applied to end A. At the same time, a displacement of (25mm, -5mm) and a rotation angle of -90° are applied to end B. The subsequent bending state of the FPC is bell-shaped, as shown in Figure 2 (b) shows the coordinate difference (Δx and Δy) between point A' and point B', as Figure 2 (c) as shown.

[0081] In order to analyze the surface bending stress distribution, a path is created along the curved surface of the FPC, such as Figure 3 As shown in (a), Figure 3 When the area within the blue circle in (a) is enlarged, Figure 3 As shown in (b), it can be seen that the path consists of a series of discrete points.

[0082] The corresponding Von Mises stress distribution is extracted along the path, such as Figure 4 (a). The calculation expression of Von Mises stress is:

[0083]

[0084] Among them, σ1, σ2 and σ3 represent the three principal stresses, and τ1, τ2 and τ3 are the corresponding shear stresses.

[0085] It is worth noting that the Von Mises stress value is usually non-negative and needs to be modified accordingly to more accurately depict the curvature trend of the FPC. Specifically, compressive stress is considered as a negative value and tensile stress is considered as a positive value. This adjustment helps to more accurately illustrate the stress distribution related to curvature, such as Figure 4 (b) as shown.

[0086] 3 Mathematical modeling and analysis of FPC non-contact curve

[0087] 3.1 Mathematical modeling

[0088] right Figure 4The analysis of the results shown in (b) shows that the variation trend of the FPC curvature can be systematically divided into two discrete segments, marked as S1 and S2. Figure 5 shown.

[0089] In each segment (S1 and S2), it can be observed that the curvature of the FPC is basically linearly related to its length. In the S1 segment, the curvature is denoted as c1 and the length is denoted as l, as shown below:

[0090] c1=m·l+b1 (6)

[0091] Here, c1 represents the curvature in segment S1, m represents the slope in the linear relationship, l represents the length along the FPC, and b1 represents the intercept of c1. In segment S2, a similar but opposite relationship is observed, expressed as:

[0092] c2=-m·l+b2 (7)

[0093] Where c2 represents the curvature in segment S2 and b2 represents its intercept. The coordinates of the intersection point T, where c1 and c2 intersect, can be determined as

[0094] To predict the deformation curve of an FPC based on the coordinates of points A and B and the total length of the FPC, three key parameters need to be known: the slope 'm' and the intercepts 'b1' and 'b2'. Determining these parameters involves a calculus-based analysis of the relationship between the curvature and the spatial distribution of the FPC, specifically focusing on the different segments previously determined.

[0095] The differential increment along the FPC in the X and Y directions can be expressed mathematically as:

[0096] dx=dl·cosθ (8)

[0097] dy=dl·sinθ (9)

[0098] Where 'dl' represents the small length increment along the FPC and 'θ' is the tangent angle at any given point on the FPC curve, e.g. Figure 6 shown.

[0099] The tangent angle 'θ' fluctuates with the distance 'l' from the initial point to the specified point along the FPC curve. Therefore, the displacement along the X and Y axes between points A' and B' can be calculated using calculus:

[0100]

[0101]

[0102] like Figure 5As shown, the relationship between the curvature of the FPC and its length is not consistent in different segments, and the variation of the curvature in different segments needs to be solved. This requires analyzing the function θ(l) and its variation in these segments.

[0103] For segment S1, the small increment dθ of the tangent angle S1 , which can be expressed as:

[0104] dθ S1 =-dl·c1 (12)

[0105] The tangent angle at any given point on segment S1 is:

[0106]

[0107] In the S1 segment, the micro-increments dx1 and dy1 in the X and Y directions can be expressed as:

[0108] dx1=cos(θ S1 (l))·dl (14)

[0109] dy1=sin(θ S1 (l))·dl (15)

[0110] The changes in the FPC end coordinates Δx1 and Δy1 in the S1 segment can be expressed as:

[0111]

[0112]

[0113] Figure 5 The end point T of the middle S1 segment, the corresponding tangent angle θ T It can be expressed as:

[0114]

[0115] Transitioning to S2 segment, the micro-increment of the tangent angle can be expressed as follows:

[0116] dθ S2 =-dl·c2=-dl·(-m·l+m·L / 2) (19)

[0117] The tangent angle corresponding to any point in segment S2 can be expressed as follows:

[0118]

[0119] The micro increments dx2 and dy2 of the X and Y dimensions within the S2 segment can be expressed as:

[0120] dx2=cos(θ S2(l))·dl (21)

[0121] dy2=sin(θ S2 (l))·dl (22)

[0122] The change in the coordinates of the two ends of the S2 segment FPC, represented by Δx2 and Δy2, can be expressed as follows:

[0123]

[0124]

[0125] The total size of the FPC curve in the X and Y directions can be expressed as:

[0126]

[0127]

[0128] Among them, θ S1 and θ S2 As shown in equations (13) and (20), Δx and Δy represent the dimensions of the entire FPC curve along the X and Y axes, respectively. 'L' represents the total length of the FPC curve, while 'b1', 'b2' and 'm' are the intercept and slope, respectively, as described above.

[0129] for Figure 3 The tangent angle of the starting point A' of the bending curve shown in can be expressed as:

[0130] θ A' =0 (27)

[0131] The corresponding tangent angle at the end point B' is:

[0132]

[0133] The difference between the tangent angles at the starting point A' and the end point B' can be expressed as:

[0134]

[0135] Equations (25), (26) and (29) emphasize the need for three undetermined variables, clearly labeled as 'm', 'b1' and 'b2'. Based on equations (13), (18) and (20), equations (25), (26) and (29) can be recombined into the corresponding nonlinear equation system as follows:

[0136]

[0137] Here, F(u) is defined as a vector function, and u = [m, b1, b2], consisting of only three undetermined variables, ‘m’, ‘b1’, and ‘b2’. Precisely determining the values ​​of these variables is crucial to accurately predict the curvature of the FPC.

[0138] 3.2 Solving nonlinear equations

[0139] In order to solve this nonlinear equation system, the Jacobi matrix iteration method is used. The detailed process is as follows:

[0140] (1) Initialization:

[0141] First, establish the initial estimate of the unknown variable, expressed as

[0142] (2) Iteration process:

[0143] a. Calculate the Jacobian matrix J(u (k) ):

[0144] In each iteration step k, the Jacobian matrix J(u (k) ). In this matrix, each component element, represented by J ij (u (k) ), represents the partial derivative of the ith equation with respect to the jth unknown. It can be expressed as follows:

[0145]

[0146] b. Calculate the equation F(u (k) )value:

[0147] Calculate and current estimate u (k) The corresponding function value vector F(u (k) ). This involves evaluating the nonlinear equations at the current level of approximation to derive a vector of function values. In the field of iterative numerical methods, this process is crucial because it gradually improves the solution by gradually increasing the accuracy of the estimate of the unknown variables.

[0148] c. Solve nonlinear equations:

[0149] To determine the correction vector Δu, solve J(u (k) )Δu=-F(u (k) ) is a linear system, where J(u (k) ) represents the Jacobian matrix. This process is crucial in the iterative method because it can improve the current approximation u (k) , improving accuracy. By iteratively updating the solution, these methods gradually converge to a more accurate solution for the unknown variables in the system of nonlinear equations.

[0150] d. Update the estimated value u(k+1) :

[0151] Based on u (k+1) =u (k) +Δu to improve the estimate, where u (k+1) represents the improved estimate after the (k+1)th iteration, u (k) represents the previous estimate at the kth iteration. Integrating the correction vector Δu into the current estimate is a key step in the iterative process, which can gradually improve the accuracy of the solution.

[0152] e. Iteration termination condition:

[0153] Iteration termination condition, which is a key element in iterative numerical methods. Typically, termination is determined by one of the following conditions:

[0154] ① Convergence criterion: Check whether the change between the current solution estimate and the previous estimate is less than a predetermined threshold. If the change is small enough, the iteration can be terminated.

[0155] ② Maximum number of iterations: Specify a maximum number of iterations. If this number is reached and other termination criteria are not met, the iteration will stop.

[0156] ③ Convergence of function value: Check whether the function value has reached a predetermined target or fluctuates within a certain range. If the function value meets the termination condition, the iteration can stop.

[0157] f. Iteration continues or terminates:

[0158] If the specified termination criteria are not met, then the iterative sequence must be restarted from step 2a. This repetitive cycle ensures that the solution is continually improved to the desired accuracy criterion. Conversely, once the predetermined criteria are met, the iterative process ends. This rigorous process is essential to ensure the accuracy and reliability of the numerical solution.

[0159] (3) Output:

[0160] At the end of the iterative sequence, the estimated value u (k+1) is the final numerical solution to the specified system of nonlinear equations. This result, achieved through careful iterative refinement, represents the convergence of the numerical algorithm toward an approximation of the exact solution for the unknown variables. Figure 7 The complete method of Jacobi matrix iteration is explained in detail.

[0161] When using the Jacobian iteration method to solve a nonlinear system of equations, it is crucial to choose an appropriate initial guess for the unknown variables. This initial guess should be as close to the actual value as possible in order to efficiently converge to the exact solution. If the initial guess is not accurate enough, then it becomes challenging or even impractical to find the correct solution to the nonlinear equations. Therefore, careful selection of the initial guess is key to effectively apply the Jacobian iteration method.

[0162] To solve this problem, an innovative strategy called "dimension iteration method" was introduced. Initially, a fixed size was defined for the FPC, with a total length of L0 = 50 mm, a coordinate difference of Δx0 = 20 mm and Δy0 = 0 at both ends, and the corresponding initial value of the unknown variable was m. (0) =-0.00638, and However, in actual situations, the actual total length L and the coordinate differences Δx and Δy at the ends of the FPC often do not conform to these predetermined values ​​L0, Δx0, and Δy0. (0) , and The values ​​do not accurately reflect the actual parameter values.

[0163] In our analysis method, a positive integer N is selected to discretize the changes of parameters L, m, b1 and b2 into N segments. The process is defined as follows:

[0164] dL=(L-L0) / N (32)

[0165] dΔx=(Δx-Δx0) / N (33)

[0166] dΔy=(Δy-Δy0) / N (34)

[0167] The first FPC curve is defined by the parameters L1 = L0 + dL, Δx1 = Δx0 + dΔx, and Δy1 = Δy0 + dΔy. (0) , and Combined with the Jacobian matrix iteration method, the parameter m of the FPC curve solved in the first iteration is determined. (1) , and Secondly, based on the previously calculated m (1) , and values, and systematically derive the parameters of subsequent FPC curves (such as the m of the second iteration). (2) , and And so on). This iterative process will continue until the slope m of the Nth FPC curve (N) and the intercept and The values ​​of the three unknown parameters 'm', 'b1' and 'b2' of the FPC curve that needs to be solved are obtained.

[0168] Using the calculated values ​​of slope 'm' and intercepts 'b1' and 'b2', the coordinates of any given point on the FPC can be determined according to equations (35) and (36):

[0169]

[0170]

[0171] Wherein, the symbol (x0, y0) represents the coordinates of the initial point of the FPC, and (x, y) represents the coordinates of any specified point on the FPC. The variable θ represents the tangent angle of the specified point on the FPC. If the point in question is located in the S1 segment, it is calculated based on equation (13). Alternatively, if the point is located in the S2 segment, it is calculated based on equation (20).

[0172] The final curve, the Nth curve, represents the actual FPC curve that needs to be analyzed. Figure 8 shown.

[0173] When the values ​​of 'm', 'b1', and 'b2' are difficult to determine, as evidenced by the significant difference between their estimates and actual values ​​in early iterations, an effective strategy is to increase the value of N. Typically, 50 is sufficient for most occasions.

[0174] 3.3 Iterative analysis and result comparison

[0175] Here N is set to 10, which means that the calculation process goes through ten size iterations, such as Fig. 9 (a). Curve 0 describes a constant FPC configuration and serves as the starting point for the iterative process. After each iteration, a new FPC curve emerges, each with different dimensions and total length. It is worth noting that Curve 10, produced by the tenth iteration, represents the key result of the required analysis. The entire iterative process went smoothly, highlighting the reliability of our method. Fig. 9 (b) shows the comparison between the curves simulated using the finite element method (FEM) and the curves obtained by the mathematical modeling proposed in the present invention.

[0176] 4 Design Optimization Strategy

[0177] 4.1 Contactless Optimization

[0178] Fig.10The shaded areas are shown, representing components close to the FPC. The FPCs with lengths of 100 mm and 117 mm interfere with different components. To ensure that the FPC does not come into contact with these adjacent components, it is important to optimize its length.

[0179] The coordinates of point R are expressed as (x R ,y R ), the coordinates of point S are expressed as (x S ,y S ), where y R Equal to y S The y coordinate of the upper component is denoted by y top For any point M on the FPC, its coordinates are (x M ,y M ) if its coordinates satisfy any of the following conditions:

[0180] ①y M >y top ,

[0181] ②x R <x M <x S And y M <y S ,

[0182] It can be concluded that interference occurs between adjacent components of the FPC.

[0183] The present invention adopts an iterative incremental method to determine the suitable length range of the FPC. This method involves gradually increasing the length of the FPC. After each increase, any point on the FPC is analyzed to see whether it meets the previously specified conditions.

[0184] The initial length of the FPC is set to the geometric distance between points A' and B'. However, in most cases, this FPC length will result in no solution to equation (30). Therefore, the length should be increased step by step until equation (30) has a correct solution.

[0185] In order to optimize time efficiency and improve accuracy, the large step size and small step size are specified as ΔL = 5 mm and δL = 0.1 mm, respectively. Fig.11 shown.

[0186] In order to determine whether a point on the FPC is within the specified range, the length of the FPC is systematically divided into 1000 equal segments, each of which has a length of sl = L / 1000. This division produces 1001 points on the FPC. For each point, its corresponding coordinates must be determined. The process of determining these points and their corresponding coordinates is as follows Fig.12 shown.

[0187] Based on the above principle, the length range of the FPC is determined to be: 104.1 mm ≤ L ≤ 113 mm. Fig.13 The FPC curves of different lengths are shown in FIG. 1 , which are L1 = 104.1 mm, L2 = 109 mm and L3 = 113 mm.

[0188] It can be seen that all of these curves successfully avoid contact with adjacent components, with the curve with a length of L2 = 109 mm being particularly noteworthy in this regard.

[0189] 4.2 Curvature Optimization

[0190] Optimizing the curvature of the FPC is critical to mitigate a range of harmful consequences. Excessive bending of the FPC can lead to a variety of complex problems, such as mechanical stress within the circuit, which may cause intermittent or permanent damage to the circuit's functionality. In addition, poor heat dissipation due to excessive bending may shorten the life of the circuit and compromise the reliability of the device. In addition, excessive curvature increases the risk of physical damage to the FPC during the assembly process and during its operation. From a manufacturing perspective, maintaining a moderate curvature of the FPC is critical to optimizing the assembly process and ensuring compatibility with other electronic components.

[0191] The curvature trend of the FPC obtained from the finite element model is shown in Figure 5 As shown in the figure, the ideal curvature trend used in the analytical model is as follows Fig.14 As shown. Point A and point B represent the starting and ending positions of the FPC, respectively, while point T represents the position where the slope of the curvature on the FPC begins to change. It can be inferred that the limit value of the curvature of the FPC is limited to the area defined by these three points.

[0192] The curvature trend of FPC is represented by the linear equations c1 = m·l + b1 and c2 = -m·l + b2. The coordinates of points A, B and T are defined as (0, b1), (L, -m·l + b2) and Under various specified conditions of FPC, if the equation group (30) is solved correctly, the coordinate values ​​of points A, B and T can be obtained.

[0193] As the length of the FPC increases from 50.1 mm to 120 mm, the corresponding curvature trends at points A, B, and T are as follows Fig.15 As shown in (a), it can be seen that as the length of the FPC increases, the curvature values ​​at points A, B and T rise rapidly, reaching a peak at about 70 mm, and then the curvature values ​​at these points begin to gradually decrease.

[0194] Maximum curvature c max is defined as follows:

[0195] c max =max(|c A|,|cB|,|c T |) (37)

[0196] c max The trend of the value is Fig.15 (b) The curvature requirement stipulates that the curvature at any point on the FPC shall not exceed 0.115mm -1 Therefore, the allowable range of FPC length is between 107.6mm and 113mm. Fig.15 Indicated by the shaded area in (b).

[0197] The optimal value of FPC length is denoted as L best , is defined as the case corresponding to the minimum value of the maximum curvature within a certain range. In this case, L best It is determined to be 113 mm, which can be found in Fig.15 (b) can be easily found.

[0198] The curve of FPC with length L = 113 mm is as follows Fig.16 (a) For comparative analysis, three curves with a radius of R = 8.7 mm (corresponding to a curvature value of c = 1 / R = 0.115 mm) are superimposed at points A, B and T. -1 ) circle. In addition, corresponding finite element simulations were also performed, such as Fig.16 As shown in (b) and (c).

[0199] It is observed that the curvature values ​​of the FPC at points A, B, and T are all less than the determined curvature limit c = 0.115 mm -1 This shows that when the length is optimized, the maximum curvature of the FPC meets the curvature requirements, thus verifying the practical reliability of this optimization. In addition, the finite element simulation results show that the FPC curve with the optimal length effectively avoids contact with adjacent components. These evidences jointly confirm the reliability of the FPC curve with the optimized length.

[0200] 5 Analysis of two other common FPC curves

[0201] 5.1 “U” shaped FPC curve

[0202] Fig.17 For the "U" shaped FPC curve shown in (a), a similar method can be used to analyze and predict its curve. In the same way, a path is constructed along the trajectory of the FPC curve, consisting of a series of discrete points. By using a similar method, the slope of the FPC curvature curve is successfully determined, such as Fig.17 (b) as shown.

[0203] Analyzing the "U"-shaped FPC curve, the relevant equations are as follows:

[0204]

[0205] The difference between equations (38) and (30) is that the tangent angles at their start and end points are different. Specifically, for the U-shaped curve, the tangent angle starts at π / 2 and ends at -π / 2. In contrast, the tangent angles at the start and end points of the bell-shaped curve are both 0. Therefore, the tangent angle of the U-shaped curve changes from the start to the end point by -π, while the tangent angle of the bell-shaped curve changes by 0.

[0206] For the neighboring components marked by points P, Q, R, and S, the coordinates are (-9.5, 29), (10.5, 29), (-7, 20), and (7, 20), respectively. A designated area is allocated in this geometric configuration for the routing of the FPC. The FPC starts from its starting point A'(-7.05, 0) and extends to its end point B'(-7.05, -2).

[0207] based on Fig.11 The theoretical framework shown in , determines the optimal range of FPC length to be 62.4mm≤L≤70.3mm. The applicability of the FPC curve is based on any given point Pi (x i ,y i ) meets at least one of the following criteria:

[0208] ①x R <x i <x S And y i <y R ;

[0209] ②x i <x P ;

[0210] ③x i >x Q ;

[0211] ④y i >y P .

[0212] The FPC curve corresponding to the length L1 = 70.3mm and L2 = 62.4mm is as follows Fig.18 shown.

[0213] The curvature trend at points A, B and T is as follows Fig.19 (a) In order to optimize the FPC curvature, the focus is on the maximum curvature, i.e., c max , in this case, is the curvature at point T, such as Fig.19 As shown in (b).

[0214] like Fig.19(b) shows the length range of the FPC determined by the optimization process to avoid contact with adjacent components, marked by two dashed lines. The curvature limit for this optimization is c = 0.14 mm -1 ,like Fig.19 The horizontal dotted line in (b) shows that the FPC length range is finally determined to be 66.9mm≤L≤70.3mm. Within this range, the FPC length corresponding to the FPC curve with the smallest absolute curvature value is 70.3mm.

[0215] The curve of FPC length of 70.3mm is as follows Fig. 20 (a) shows a radius of R = 7.143 mm (corresponding to a curvature value of c = 1 / R = 0.14 mm). -1 ) is superimposed at point T. In addition, the corresponding finite element simulation was carried out, and the results are shown in Fig. 20 (b) as shown.

[0216] The observation results show that the curvature of the FPC at point T is lower than the set curvature limit of 0.14mm-1. In addition, the FPC curve cleverly avoids contact with adjacent components. These factors together indicate that the optimal FPC length determined through the above process is both effective and practical.

[0217] 5.2 “S” shaped FPC curve

[0218] Regarding the analysis of the 'S' shaped FPC curve, the finite element model of the creation path and the FPC curvature change trend are shown as follows: Fig.21 As shown in (a) and (b).

[0219] In contrast to the previous bell-shaped and 'U'-shaped FPC curves, the 'S'-shaped FPC curve in this case needs to divide its curvature into three parts, such as Fig.21 (b) As shown. Since the 'S'-shaped curve has rotational symmetry, Fig.21 The curvature change curve shown in (b) must pass through the point (L / 2,0). This requires the condition -m·l+b2=0 to be satisfied. In order to analyze the 'S' shaped FPC curve, the corresponding set of equations is shown below:

[0220]

[0221] Among them, θ S1 ,θ S2 and θ S3 are the tangent angles in the S1, S2 and S3 segments, respectively, which can be expressed as:

[0222]

[0223]

[0224]

[0225] Among them, θ T1 and θ T2 are the tangent angles at point T1 and point T2, respectively, expressed as:

[0226]

[0227]

[0228] The tangent angle of the 'S'-shaped curve changes from the starting point to the end point to 0, that is, Δθ=0.

[0229] For the adjacent components of the FPC determined by points R(-4,18) and S(4,-18), a restricted space is allocated for the routing of the FPC. The starting point of the FPC is at point A'(-20,30) and the end point is at point B'(20,-30), as shown in Fig.23 shown.

[0230] and Fig.11 The theoretical model shown in Figure 2 is consistent with the theoretical model, which determines that the ideal length range of FPC is 79.4mm to 95.4mm, and these two values ​​are respectively given by Fig. 22 The two dashed lines in (b) indicate that the applicability of the FPC curve is determined by checking whether any specific point Pi(x i ,y i ) is evaluated based on whether it meets one of the following conditions:

[0231] ①x i <x R And y i <y R ;

[0232] ②x i >x S And y i >y S ;

[0233] As the length of FPC increases, the curvature of FPC at points A, B, T1 and T2 changes as shown in Fig. 22 (a) In order to optimize the curvature of the FPC, the curvature value with the largest absolute value is given priority, corresponding to the curvature change trend of point T1. The curvature threshold for this optimization is c = 0.08 mm -1 ,like Fig. 22 Indicated by the horizontal dotted line in (b).

[0234] The previously determined FPC length range is used to ensure that it avoids contact with adjacent components, defined as 79.4mm≤L≤95.4mm. Within this range, a more specific interval of 79.4mm≤L≤93mm was established to keep the curvature of the FPC within the specified threshold. This interval corresponds to Fig. 22 (b) The shaded part. It is worth noting that at a length of 93 mm, the FPC reaches its minimum curvature, so this length is identified as the optimal length L best .

[0235] The FPC curves corresponding to the lengths L1 = 79.4 mm, L2 = 81.5 mm, L3 = 93 mm and L4 = 95.4 mm are as follows Fig.23 Adjacent components are shaded in the figure.

[0236] The optimal length for this case is determined to be L best =81.5mm flexible printed circuit (FPC) curve, the curvature is verified as follows Fig.24 Here, four circles with radius R = 12.5 mm (corresponding to curvature value c = 1 / R = 0.08 mm) -1 ) are superimposed on points A, B, T1 and T2 respectively. In addition, Fig.24 (b) Verification of the non-contact state of the curve with adjacent components is shown through finite element modeling.

[0237] It can be seen that at the specified four points, the curvature values ​​of the FPC meet the specified curvature limits. In addition, the FPC curve successfully avoids contact with adjacent components. This result confirms the reliability and practicality of the 'S' shaped FPC curve optimization.

[0238] The above content describes the basic principle, main features and advantages of the present invention. It should be understood by those skilled in the art 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 to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A non-contact control and curvature optimization method for FPC layout based on iterative attempts and discrete point analysis, characterized in that: The following steps are involved: S1. Based on the finite element tool, the corresponding FPC deformation is modeled, and a corresponding tracking path is created along its curved deformation surface, which is composed of a series of discrete points; S2. Based on the tracking path, the changing trend of the curvature of the FPC after deformation is obtained, and based on this, a corresponding mathematical model of FPC deformation is established; S3, taking the straight-line distance between the two end points of the FPC as the initial distance, and analyzing the deformation of the FPC based on the mathematical model established in the early stage, first observe whether the corresponding nonlinear equation group has a correct solution; S4. If there is no correct solution, gradually increase the length of the FPC, and obtain a more accurate minimum length value through the coarse iteration value and the fine iteration value; S5. After correctly obtaining the FPC curve, the curve is evenly discretized into 1001 isolated points; S6. Analyze whether each isolated point is in the adjacent component region based on its coordinate value. If so, determine that it is in contact with the component. Otherwise, determine that it is not in contact. S7, gradually and iteratively increasing the length value of the FPC, and analyzing whether it contacts with adjacent components; S8, through continuous iteration and judgment process, finally obtaining the length value range in which the FPC does not contact the adjacent components; S9, for the obtained FPC length range, using the refined iteration value as the iteration value, analyzing the maximum curvature value of the FPC under different length values ​​and recording them as the analysis object; S10, according to a given curvature limit value, that is, a curvature allowable value, obtaining a length range of the maximum curvature of the FPC within a given curvature range as a final optimized length range of the FPC length; S11. In the final optimized length range, obtain the length value corresponding to the minimum maximum curvature value of the FPC as the optimal length value L of the FPC best ; Mathematical modeling and analysis of FPC non-contact curve; In each segment S1 and S2, a linear relationship is observed between the curvature of the FPC and its length; in the S1 segment, the curvature is denoted as c1 and the length is denoted as l, as shown below: c1=m·l+b1 (6) Here, c1 represents the curvature in segment S1, m represents the slope in the linear relationship, l represents the length along the FPC, and b1 represents the intercept of c1; in segment S2, there is an opposite relationship, which is expressed as: c2=-m·l+b2 (7) Where c2 represents the curvature in segment S2, b2 represents its intercept; the coordinates of the intersection point T, that is, the point where c1 and c2 intersect, are determined as To predict the deformation curve of the FPC based on the coordinates of points A and B and the total length of the FPC, three key parameters need to be known: the slope 'm' and the intercepts 'b1' and 'b2'; determining these parameters involves a calculus-based analysis of the relationship between the curvature and the distribution of the FPC along the space, especially focusing on the different segments previously determined; The differential increment along the FPC in the X and Y directions is mathematically expressed as: dx=dl·cosθ (8) dy=dl·sinθ (9) Where, 'dl' represents a small length increment along the FPC, and 'θ' is the tangent angle at any given point on the FPC curve; The tangent angle 'θ' fluctuates with the distance 'l' from the initial point to the specified point along the FPC curve; therefore, calculus is used to calculate the displacement between points A' and B' along the X and Y axes: The relationship between the curvature of the FPC and its length is not consistent in different segments, and the variation of curvature in different segments needs to be solved; this requires analyzing the function θ(l) and its variation in these segments; For segment S1, the small increment dθ of the tangent angle S1 , expressed as: dθ S1 =-dl·c1 (12) The tangent angle at any given point on segment S1 is: In the S1 segment, the micro-increments dx1 and dy1 in the X and Y directions are expressed as: dx1=cos(θ S1 (l))·dl (14) dy1=sin(θ S1 (l))·dl (15) The changes in the FPC end coordinates Δx1 and Δy1 in the S1 segment are expressed as: S1 segment end point T, corresponding tangent angle θ T It is expressed as: Transitioning to S2 segment, the micro increment of the tangent angle is expressed as follows: dθ S2 =-dl·c2=-dl·(-m·l+m·L / 2) (19) The tangent angle corresponding to any point in segment S2 is expressed as follows: The micro increments dx2 and dy2 of the X and Y dimensions within the S2 segment are expressed as: dx2=cos(θ S2 (l))·dl (21) dy2=sin(θ S2 (l))·dl (22) The change in the coordinates of the two ends of the FPC in the S2 segment is represented by Δx2 and Δy2, as follows: The total size of the FPC curve in the X and Y directions is expressed as: Among them, θ S1 and θ S2 As shown in equations (13) and (20), Δx and Δy represent the dimensions of the entire FPC curve along the X and Y axes, respectively; 'L' represents the total length of the FPC curve, and 'b1', 'b2' and 'm' are the intercept and slope mentioned above, respectively; The tangent angle of the bending curve starting point A' is expressed as: i A′ =0 (27) The corresponding tangent angle at the end point B′ is: The difference between the tangent angles at the starting point A' and the end point B' is expressed as: Equations (25), (26), and (29) emphasize the need for three undetermined variables, clearly labeled as 'm', 'b1', and 'b2'; based on equations (13), (18), and (20), equations (25), (26), and (29) are recombined into the corresponding nonlinear equations as follows: Here, F(u) is defined as a vector function, and u = [m, b1, b2], consisting of only three undetermined variables 'm', 'b1' and 'b2'; precisely determining the values ​​of these variables is crucial to accurately predict the curvature of the FPC.

2. The method for non-contact control and curvature optimization of FPC layout based on iterative trial and discrete point analysis according to claim 1, characterized in that: In step S2, it is determined whether the end point of the FPC curve obtained by mathematical modeling coincides with the FPC end point specified before modeling to determine whether the corresponding equation group has a correct solution.

3. The FPC layout contactless control and curvature optimization method based on iterative trial and discrete point analysis according to claim 1 is characterized in that: The step S3 specifically includes the following steps: S3.1, gradually increase the coarse iteration value for the FPC length, and determine whether the equation system has a correct solution; S3.2, when the equation system has a correct solution, gradually reduce the fine iteration value of the FPC length, and determine whether the equation system has a correct solution; S3.

3. When the equation system does not have a correct solution for the first time, a fine iteration value is added to FPC again. The FPC length value at this time is the minimum value obtained after preliminary optimization.

4. The method for non-contact control and curvature optimization of FPC layout based on iterative trial and discrete point analysis according to claim 1, characterized in that: The step S10 specifically includes the following steps: S10.

1. For the solution of the maximum curvature value of the curve under different length values ​​of the FPC, first solve the corresponding nonlinear equation group to obtain the slope and intercept of the curvature change of the FPC in different segments; S10.2, respectively calculating the curvature values ​​of the FPC at the starting point, the end point and the turning point of the curvature change curve; S10.3, record the maximum absolute value of the curvature values ​​at these locations; S10.4, based on this, a series of length values ​​of the FPC and a corresponding series of maximum curvature values ​​are obtained; S10.

5. Further optimize the FPC length range based on the given curvature limit value, i.e., the curvature allowable value.

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