Ultra-thin sheet grinding deformation prediction method and system based on generalized finite difference method

Abstract

The invention discloses a method and system for predicting deformation of ultra-thin slice grinding based on a generalized finite difference method, which relates to the field of precision machining technology, and comprises the following steps: constructing a dynamic equation for ultra-thin slice grinding; discretizing the time domain of the dynamic equation, and obtaining a time discrete equation after parameter adjustment; discretizing the space domain of the dynamic equation, dividing the calculation domain into internal nodes, boundary nodes and supplementary nodes, and constructing an approximate equation of a local node group through Taylor expansion and weighted least squares method; substituting the equation into the time discrete equation, considering boundary conditions, and constructing a linear system of algebraic equations; obtaining grinding parameter data of an ultra-thin slice to be tested, substituting the data into the linear system of algebraic equations, and obtaining the displacement, velocity and acceleration at the node through solving the equation, so as to realize accurate prediction of dynamic deformation of the ultra-thin slice during grinding.

Description

Technical Field

[0001] The invention relates to the technical field of precision machining, and in particular to a method and system for predicting ultra-thin sheet grinding deformation based on a generalized finite difference method. Background Art

[0002] The statements in this section merely provide background information related to the present invention and do not necessarily constitute prior art.

[0003] Ultra-thin sheets with high performance and large diameter-to-thickness ratio have important applications in many fields, and grinding is a key process step in the manufacturing process of ultra-thin sheets. However, the current grinding of ultra-thin sheets mainly faces difficulties such as difficult flatness control, poor stress uniformity, prominent end jump problems, and difficulty in effectively suppressing deformation. In addition, the current production lacks the ability to predict the deformation of ultra-thin sheets during the grinding process. This deformation prediction is a prerequisite for feedback optimization of grinding process parameters. Therefore, it is difficult to dynamically adjust process parameters in production to adapt to the complex and changeable processing needs of ultra-thin sheets.

[0004] In recent years, the generalized finite difference method has been widely used in engineering mechanics, dynamics and other fields. Its advantages are that it can handle complex geometric shapes and non-uniform material properties, and has high computational efficiency and accuracy. However, for the deformation modeling and computational analysis of ultra-thin sheets during grinding, the direct application of the existing generalized finite difference method still has certain limitations: first, due to its large diameter-to-thickness ratio and low stiffness characteristics, ultra-thin sheets are prone to significant elastic deformation and nonlinear response during grinding, which puts higher requirements on the accuracy and stability of numerical modeling methods; second, the dynamic stress distribution and deformation behavior during grinding involve complex contact mechanics and multi-physics field coupling problems. The traditional generalized finite difference method often has difficulty in balancing computational efficiency and accuracy when dealing with these nonlinear dynamic problems; in addition, most of the existing grinding deformation prediction models are based on static or quasi-static assumptions, which are difficult to accurately reflect the transient response and dynamic deformation characteristics of ultra-thin sheets during grinding. Summary of the invention

[0005] In order to solve the shortcomings of the above-mentioned prior art, the present invention provides a method and system for predicting the grinding deformation of ultra-thin sheets based on the generalized finite difference method, which combines the generalized finite difference method with the Newmark-Beta method (i.e., the Newmark-Beta method, a kind of numerical method), and accurately describes the elastic mechanical behavior of the ultra-thin sheet through numerical calculation, handles complex boundary conditions and dynamic loads, and realizes accurate prediction of the dynamic deformation of the ultra-thin sheet during the grinding process.

[0006] In a first aspect, the present invention provides a method for predicting deformation of ultra-thin sheet grinding based on a generalized finite difference method.

[0007] A method for predicting deformation of ultra-thin sheet grinding based on generalized finite difference method, comprising:

[0008] Conduct dynamics theoretical analysis on the ultra-thin sheet grinding process and construct the dynamics equation of ultra-thin sheet grinding;

[0009] The time domain of the dynamics equation is discretized, and after parameter adjustment, the time discrete equation is obtained;

[0010] The spatial domain of the dynamic equation is discretized, and the computational domain corresponding to the ultra-thin sheet is divided into internal nodes, boundary nodes, and supplementary nodes. The approximate equation of the local node group is constructed by Taylor expansion and weighted least squares method.

[0011] Substitute the approximate equations of the local node group into the discrete time equations of the dynamics equations, consider the boundary conditions, and construct a linear system of algebraic equations in matrix form;

[0012] The grinding parameter data of the ultra-thin sheet to be tested is obtained and substituted into the linear system of algebraic equations for solution. The displacement, velocity and acceleration at any node on the ultra-thin sheet are obtained to predict the dynamic deformation of the ultra-thin sheet during the grinding process.

[0013] In a second aspect, the present invention provides an ultra-thin sheet grinding deformation prediction system based on a generalized finite difference method.

[0014] An ultra-thin sheet grinding deformation prediction system based on generalized finite difference method, comprising:

[0015] The dynamic equation construction module is used to conduct dynamic theoretical analysis on the ultra-thin sheet grinding process and construct the dynamic equation of ultra-thin sheet grinding;

[0016] The time domain discretization module is used to discretize the time domain of the dynamics equation. After parameter adjustment, the time discrete equation is obtained;

[0017] The spatial domain discretization module is used to discretize the spatial domain of the dynamic equation, divide the computational domain corresponding to the ultra-thin sheet into internal nodes, boundary nodes and supplementary nodes, and construct the approximate equation of the local node group through Taylor expansion and weighted least squares method;

[0018] The linear system of algebraic equations building module is used to substitute the approximate equations of the local node group into the discrete time equations of the dynamic equations, taking into account the boundary conditions, and building the linear system of algebraic equations in matrix form;

[0019] The ultra-thin sheet grinding deformation prediction module is used to obtain the grinding parameter data of the ultra-thin sheet to be tested, substitute it into the linear system of algebraic equations for solution, and obtain the displacement, velocity and acceleration at any node on the ultra-thin sheet, so as to realize the dynamic deformation prediction of the ultra-thin sheet during the grinding process.

[0020] In a third aspect, the present invention further provides an electronic device, comprising: a memory for storing executable instructions; and a processor for implementing the above-mentioned ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method when executing the executable instructions stored in the memory.

[0021] In a fourth aspect, the present invention further provides a computer-readable storage medium storing executable instructions for causing a processor to execute the executable instructions to implement the above-mentioned ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method.

[0022] In a fifth aspect, the present invention also provides a computer program product, which includes executable instructions stored in a computer-readable storage medium; wherein, when a processor of an electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, the above-mentioned ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method is implemented.

[0023] One or more of the above technical solutions have the following beneficial effects:

[0024] 1. The present invention provides a method and system for predicting the deformation of ultra-thin sheets during grinding based on the generalized finite difference method. The generalized finite difference method is combined with the Newmark-Beta method to accurately describe the elastic mechanical behavior of ultra-thin sheets through numerical calculation, handle complex boundary conditions and dynamic loads, and achieve accurate prediction of the dynamic deformation of ultra-thin sheets during grinding.

[0025] 2. In the present invention, the space domain is discretized by the generalized finite difference method, and its flexibility and high precision are used to effectively analyze the complex geometric shapes and non-uniform material properties of ultra-thin sheets. At the same time, the time domain is discretized in combination with the Newmark-Beta method, and the balance between accuracy and numerical stability is achieved by adjusting parameters. Through the combination of the above two methods, not only can the motion state of the ultra-thin sheet at a certain moment and a certain position during the processing of the ultra-thin sheet be efficiently solved, but also the complexity of establishing additional equations at the auxiliary nodes in the traditional method can be avoided, and the calculation efficiency can be significantly improved. Finally, by constructing a linear system of algebraic equations in matrix form, the physical quantities such as velocity and acceleration at the nodes are directly solved, so as to realize the accurate prediction of the dynamic deformation during the grinding process of the ultra-thin sheet, solve the problems of difficult flatness control, poor stress uniformity, prominent end jump problems in the current grinding process, and provide numerical predictions with higher accuracy and stability for the process optimization of ultra-thin sheet grinding.

[0026] Advantages of additional aspects of the present invention will be given in part in the following description, and in part will become obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] The accompanying drawings in the specification, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute improper limitations on the present invention.

[0028] Figure 1 It is an overall flow chart of the ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method in an embodiment of the present invention;

[0029] Figure 2 Schematic diagram of an ultra-thin sheet grinding deformation prediction system based on a generalized finite difference method in an embodiment of the present invention;

[0030] Figure 3 The diagram is a graph showing the node distribution in the computing domain and the neighborhood structure distribution of a single node in an embodiment of the present invention;

[0031] Figure 4 Schematic diagram of deformation analysis of a conventional grinding circular saw blade substrate in an embodiment of the present invention;

[0032] Figure 5 Schematic diagram of deformation analysis of a circular saw blade substrate after a soft gasket is used to change the grinding reference in an embodiment of the present invention;

[0033] Figure 6 Schematic diagram of single deformation analysis of a grinding circular saw blade substrate in an embodiment of the present invention. DETAILED DESCRIPTION

[0034] It should be noted that the following detailed descriptions are exemplary only, are intended to describe specific embodiments, are intended to provide further explanation of the present invention, and are not intended to limit exemplary embodiments according to the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those of ordinary skill in the art to which the present invention belongs. In addition, it should also be understood that when the terms "comprising" and / or "including" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0035] Embodiment 1

[0036] This embodiment provides a method for predicting the grinding deformation of ultra-thin sheets based on the generalized finite difference method. The spatial domain is discretized by the generalized finite difference method, and its flexibility and high precision are used to effectively analyze the complex geometric shapes and non-uniform material properties of the ultra-thin sheets. At the same time, the time domain is discretized in combination with the Newmark-Beta method, and the balance between accuracy and numerical stability is achieved by adjusting parameters. The above combination can not only efficiently solve the motion state of the ultra-thin sheet at a certain moment and position during the processing, but also avoid the complexity of establishing additional equations at auxiliary nodes in the traditional method, thereby significantly improving the calculation efficiency. Finally, by constructing a linear system of algebraic equations in matrix form, the physical quantities at the nodes of the ultra-thin sheet are directly solved, and the accurate numerical prediction of the grinding deformation of the ultra-thin sheet is realized, thereby providing a high-precision and high-stability numerical prediction method for the optimization of the ultra-thin sheet grinding process.

[0037] The ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method proposed in this embodiment is as follows: Figure 1 As shown, the specific steps include:

[0038] Step S1, performing a dynamics theoretical analysis on the ultra-thin sheet grinding process, and constructing a dynamics equation for ultra-thin sheet grinding, namely, a dynamic elastic mechanics model.

[0039] In this embodiment, in view of different processing and production requirements, in order to accurately describe the interaction between the grinding wheel and the ultra-thin sheet, the large diameter-to-thickness ratio and ultra-thin characteristics of the ultra-thin sheet as the workpiece being processed are considered, combined with Kirchhoff plate theory, comprehensive consideration of elastic deformation, time effect, acceleration change and damping factors, etc., combined with boundary conditions, a dynamic elastic mechanics model of the ultra-thin sheet during the grinding process is constructed. This model serves as the core control equation of the entire system and lays the foundation for subsequent calculation and analysis.

[0040] Step S1.1: First, a dynamics analysis is performed on the ultra-thin sheet grinding process. Specifically, according to the grinding process, the soft gasket used to assist in building the grinding reference in the grinding process is considered as an isotropic material, and its thickness is D , the elastic modulus is E ; The external force applied by the grinding wheel gradually increases with the grinding process, and the dynamic changes of the system are mainly reflected in the interaction between time, acceleration and external force; during the grinding process, as time goes by, the elastic force between the gasket, the ultra-thin sheet being ground and the grinding wheel changes dynamically.

[0041] Step S1.2: Based on the above dynamics theory analysis and combined with Kirchhoff's plate theory, boundary conditions are defined to construct the dynamics equation of the ultra-thin sheet grinding system, namely, the dynamic elastic mechanics model.

[0042] The constructed kinetic equation can be expressed as:

[0043] (1)

[0044] in, Indicates the influence of the bending stiffness of the circular saw blade base on its deformation; is a bi-tone operator, , , this bitonic operator reflects the comprehensive bending deformation of the plate, including the bending curvature and the corresponding mechanical response; It represents the Poisson's ratio of the ultra-thin sheet material, which is determined according to the material of the ultra-thin sheet (such as the substrate of the circular saw blade of the processed workpiece); Indicates the circular saw blade base z Point t The deflection at the moment z A dot represents the location of a point ( x,y ),( x,y ) are the horizontal and vertical coordinates of the position; represents the damping effect of the circular saw blade matrix, It represents the damping coefficient, which depends on the material properties of the ultra-thin sheet (i.e., the circular saw blade substrate), such as steel, aluminum alloy, etc. represents the inertia effect of the circular saw blade substrate, is the material density of the circular saw blade, For the circular saw blade substrate at the current time t The instantaneous thickness of It represents the distributed load generated by the grinding force applied by the grinding wheel on the saw blade base.

[0045] Furthermore, considering the strong adsorption effect of the grinding disk, the circular saw blade substrate is always in a fully adsorbed state regardless of whether a gasket is added, so its boundary should be classified as a completely fixed boundary. The corresponding boundary condition can be defined as:

[0046] ;

[0047] in, Indicates the boundary of the ultra-thin sheet, and in this embodiment, specifically refers to the boundary of the circular saw blade substrate; for The partial derivative in the normal direction indicates the function along the normal direction. The rate of change.

[0048] Step S2: Using the Newmark-Beta method, the time domain of the kinetic equation is discretized, and after parameter adjustment, a time discrete equation is obtained.

[0049] Considering that most of the current methods use the Houbolt method (i.e., Houbolt method, a kind of implicit calculation method) to discretize the first-order and second-order time derivatives in the dynamic equations, but this method may have problems such as insufficient numerical stability, limited ability to handle nonlinear problems, and difficulty in balancing accuracy and computational efficiency, this embodiment uses the Newmark-Beta method to discretize the time domain of the nonlinear dynamic equations, and adjusts the parameters introduced by it. and , in order to achieve a balance between accuracy and numerical stability, and then calculate the displacement, velocity and acceleration of the node position according to the adjusted parameters, which can ensure the numerical stability in nonlinear dynamic problems, making it more suitable for solving nonlinear or complex dynamic problems.

[0050] Step S2.1, considering the deformation of any node on the circular saw blade substrate at the previous and next moments, the discretized displacement, velocity and acceleration formulas of the node are calculated according to the Newmark-Beta method, which can be expressed as:

[0051] (1) The displacement formula (or position formula) can be expressed as:

[0052] (2)

[0053] Among them, the "position" in the above formula refers to the position of the deformed part of the original workpiece at the current moment after deformation. Since the deflection represents the size of the deformation, it can be used to represent the position of the circular saw blade substrate. z Point Deflection at moment to characterize the location.

[0054] (2) Taking the first-order derivative of the above displacement formula, the approximation of the first-order time derivative (i.e., velocity) can be expressed as:

[0055] (3)

[0056] (3) Taking the second-order derivative of the above displacement formula, the approximation of the second-order time derivative (i.e., acceleration) can be expressed as:

[0057] (4)

[0058] In the above formula, , These are two key parameters introduced by the Newmark-Beta method.

[0059] Step S2.2: Based on the above formula, adjust the parameters introduced by the Newmark-Beta method. and , determine the optimal parameter values ​​that balance computational accuracy and numerical stability.

[0060] Specifically, the Newmark-Beta method is an implicit time integration method, which is widely used to solve dynamic equations (such as motion equations in structural dynamics or earthquake engineering). Its core is to approximate continuous dynamic behavior by discretizing the relationship between displacement, velocity and acceleration. and These two key parameters, Controls the interpolation method of acceleration, affecting the stability of the numerical solution. The larger the value, the more stable the numerical solution, but the corresponding accuracy may be sacrificed; It is a parameter related to speed, which controls the interpolation method of speed and affects the damping and accuracy of the numerical solution. The larger the value, the stronger the damping effect of the solution and the better the stability, but the accuracy may be reduced. and A smaller value will increase the unconditional stability of the solution and reduce the accuracy of the dynamic response; conversely, a smaller value will increase the unconditional stability of the solution and reduce the accuracy of the dynamic response. and Values ​​of 0 and 1 retain more physical detail and accuracy, but may result in poor stability of the condition. and Adjust and optimize to achieve a balance between stability and accuracy.

[0061] In this embodiment, it is verified that when The time-stepping iteration process is unconditionally stable when is set to 0.5, and When it is set to 0.25, the Newmark-Beta algorithm not only has the highest accuracy, but also meets the unconditional stability condition. Therefore, this embodiment selects and as the optimal parameter combination.

[0062] Step S2.3, the discretized displacement, velocity and acceleration formulas after the above parameters are determined are substituted into the dynamic equation to obtain the kinematic equation in discretized form, i.e., the time discrete equation, which can be expressed as: (5)

[0063] in, represents the time step, Indicates the current time i time, Indicates the next moment i +1 time; for Abbreviation for thickness.

[0064] Preferably, in order to more conveniently establish a generalized finite difference method model with supplementary nodes, the above equation (5) is rearranged and re-expressed as the following formulas (6), (7) and (8):

[0065] (6)

[0066] in, ， (7)

[0067] (8)

[0068] Step S3: using the generalized finite difference method, discretize the spatial domain of the dynamic equation, divide the computational domain corresponding to the ultra-thin sheet into internal nodes, boundary nodes and supplementary nodes, and construct the approximate equation of the local node group through Taylor expansion and weighted least squares method.

[0069] Specifically, the entire ultra-thin workpiece is the computational domain. The generalized finite difference method is used to discretize the spatial domain and divide the computational domain into three types of node sets: internal nodes, boundary nodes, and supplementary nodes. For each node, a local support domain (i.e., a local node group) containing several neighboring nodes is constructed. Then, the approximate equation of the local node group is constructed through Taylor expansion and weighted least squares method to avoid establishing additional equations at auxiliary nodes and simplify the calculation process. Among them, the nodes include supplementary nodes, boundary nodes, and internal nodes, which are distributed in the area and Internal nodes, boundary nodes and supplementary nodes are divided according to the geometric distribution and physical constraints of the ultra-thin slice computational domain in the generalized finite difference method: internal nodes are located in the central area of ​​the computational domain to capture internal deformation behavior; boundary nodes are located at the geometric boundary to handle boundary effects based on physical boundary conditions (such as completely fixed boundaries); the number of supplementary nodes is consistent with that of boundary nodes, and the weight distribution is optimized by introducing virtual nodes to balance the influence of boundary conditions on the system equations, ensure numerical stability and computational efficiency, and avoid complex equation construction at auxiliary nodes. The introduction of supplementary nodes can help provide additional weight coefficients so that the system equations can be constructed with conditions that have deterministic solutions. The above The two-dimensional computational domain representing the ultra-thin sheet, i.e., the plane area of ​​the entire workpiece, is used to define the distribution range of the physical field; Represents the computational domain of boundaries, used to impose boundary conditions (such as completely fixed boundaries).

[0070] Furthermore, the process of discretization in the spatial domain using the improved generalized finite difference method is as follows:

[0071] Step S3.1: For each configuration node (excluding supplementary nodes), the central node and its recent m Neighboring internal nodes Together they form a local node group, such as Figure 3 As shown, the coordinates of each node can be calculated by MATLAB software.

[0072] Step S3.2: For any internal node At the central node The Taylor expansion is performed at , and the fourth order is taken to take into account the description of local high-order nonlinear effects and the balance between calculation accuracy and complexity, then the Taylor expansion is It can be expressed as:

[0073] (9)

[0074] in, , , ， , represents the remainder of Taylor expansion, and o is the symbol representing infinitesimal.

[0075] Through the above-mentioned Taylor expansion operation, the spatial variation of the high-order (such as 4th order) precise approximation function (such as deflection) near the reference point is obtained to obtain the Taylor approximation of the reference point, providing a mathematical basis for the differential relationship between nodes.

[0076] In addition, the selection of the above-mentioned Taylor expansion order is specifically determined based on the balance accuracy and complexity. The higher the Taylor expansion order, the more detailed the spatial variation of the function can be captured, thereby improving the accuracy of the numerical solution. For example, the 1st order Taylor expansion can only approximate linear variation, the 2nd order can capture quadratic variation, and the 3rd and 4th orders can respectively approximate higher-order nonlinear behaviors; however, the higher the order, the computational complexity also increases significantly, because more derivative terms and a larger local support domain (i.e., the number of neighboring nodes) are required. Therefore, in the generalized finite difference method, the choice of order needs to take into account accuracy, stability, efficiency, etc. In this embodiment, the 4th order is selected to meet the nonlinear effect requirements (such as the 3rd order cannot capture high-order nonlinear complex behaviors such as bending, and the 4th order can provide sufficient accuracy) and computational complexity restrictions (the 5th order or higher order will result in excessive computational burden).

[0077] Step S3.3: Based on the error between the Taylor approximation and the actual function value, combined with the defined weighting function, establish the residual function , the accuracy of the differential approximation can be optimized by minimizing the residual, ensuring that the calculation results are consistent with the authenticity of physical behavior. The established residual function can be expressed as:

[0078] (10)

[0079] Among them, the weighting function It is expressed as:

[0080] (11)

[0081] in, Representation Node With Node The distance between represents the maximum distance among all distances, .

[0082] By defining a weighting function, different weights can be assigned to adjacent nodes to balance the influence of boundary effects and non-uniform grids, thereby enhancing the coordination between stability and local accuracy.

[0083] Step S3.4, based on the above, define vector and matrix forms, and integrate the above results into a set of linear equations. At this time, the integrated set of linear equations is the approximate equation of the local node group, which is used to approximately describe the spatial deformation behavior of the ultra-thin sheet during the grinding process.

[0084] Here, we define a vector for:

[0085] (12)

[0086] By minimizing ,for The linear equations are:

[0087] (13)

[0088] in, for The abbreviation of , which represents the residual function; , represents the Taylor expansion, For example, it represents the node Taylor expansion of .

[0089] Furthermore, the above equation (13) can be rewritten as: (14)

[0090] in:

[0091] ;

[0092] Among them, SYM (Symmetric) is a symmetric parameter. Here, SYM is used to simplify the matrix A, that is, the matrix A takes the main diagonal as the symmetry axis, and the elements of the upper and lower parts are corresponding to each other. is the weighting parameter; ( x,y ) indicates the circular saw blade base z The location of the point, x , y Respectively represent the point z The horizontal and vertical coordinates of ( ) indicates the first j Points location, Respectively represent the point The horizontal and vertical coordinates of .

[0093] In this embodiment, the differential approximate expression of the node physical quantity is established through the above-mentioned Taylor expansion and weighted least squares method, which can eliminate the need for equation construction at the auxiliary node.

[0094] Preferably, when the unconditional stability condition ( and ), or when a higher-order derivative approximation is needed, or in order to improve the spatial resolution, the number of nodes in the local support domain can be increased to 14.

[0095] Step S4: Substitute the approximate equation of the local node group into the discrete time equation of the dynamic equation, consider the boundary conditions, and construct a linear system of algebraic equations in matrix form.

[0096] Step S5, obtaining grinding parameter data of the ultra-thin sheet to be tested, substituting the data into the linear system of algebraic equations for solving, and solving to obtain the displacement, velocity and acceleration at any node on the ultra-thin sheet, so as to realize the dynamic deformation prediction of the ultra-thin sheet during the grinding process.

[0097] Specifically, the equations obtained in step S2 and step S3 are substituted into the dynamic equation and the boundary conditions of the boundary nodes to form a linear algebraic equation in the form of a matrix AX=B, that is, a linear system of algebraic equations in the form of a matrix, wherein A is a coefficient matrix, X is a node displacement vector, and B is a load vector; then, the multi-dimensional grinding parameter data of the ultra-thin sheet to be tested is used as input, and the pre-processed conjugate gradient method is used for iterative solution to obtain and its derivatives at all matching nodes without establishing equations at auxiliary nodes. The obtained grinding parameters of ultra-thin sheets include: geometric parameters (such as shape, size, node distribution), material parameters (such as elastic modulus, Poisson's ratio, density, thickness, damping coefficient), load parameters (such as grinding force distribution, time step), numerical method parameters ( , , Taylor expansion order, weighting function parameters) and initial / boundary conditions (initial displacement, velocity, fixed boundary conditions), etc. These parameter data can be generated through experimental measurement, theoretical derivation or MATLAB simulation.

[0098] The above numerical method (i.e., GFDM + Newmark-Beta) discretizes the physical behaviors of the ultra-thin sheet, such as displacement, velocity, and acceleration, into mathematical equations. After solving these equations, the displacement field of each node that changes with time is obtained, and the physical quantities such as displacement, velocity, and acceleration at each node are obtained. Then, the deformation morphology of the ultra-thin sheet is reconstructed according to the displacement distribution, thereby predicting the dynamic deformation and stress state of the ultra-thin sheet during the grinding process. Preferably, the above method is applicable to the bending problem of thin elastic plates with simply supported boundary conditions or free boundary conditions, because the equations that satisfy these boundary conditions can be constructed by combining the corresponding derivatives of equation (14).

[0099] As an implementation method, according to the node displacement response obtained by solving step S4, the moving least square method is used to reconstruct the ultra-thin sheet free-form surface morphology, and the dynamic deformation and residual stress distribution of the grinding contact area are predicted by time series analysis (i.e., the dynamic deformation of the ultra-thin sheet is predicted), and a quantitative mapping model of deformation amount-process parameters is established. On this basis, by adjusting the input independent variable, i.e., the grinding parameter, the deformation distribution and deformation amount under different working conditions can be analyzed, and the dynamic deformation of the ultra-thin sheet can be controlled, providing a more intuitive and accurate visualization means for evaluating the grinding effect.

[0100] In this embodiment, the above calculation and analysis process relies on MATLAB software for calculation, simulation and analysis to achieve numerical simulation of deformation behavior during ultra-thin sheet grinding. Figure 3 As shown, the position coordinates of each node are calculated and output by MATLAB software based on the generalized finite difference method, which can be intuitively displayed in the workpiece simulation diagram.

[0101] Furthermore, the deformation cloud map and deformation ratio of the ultra-thin sheet can be obtained through MATLAB simulation analysis. According to the degree of deformation, the color of the cloud map can be set to change, where red indicates a large deformation and blue indicates a small deformation. Figure 4 and Figure 5 As shown, Figure 4 To analyze the machining parameters of circular saw blade substrates in various cases using conventional processes (without the use of soft pads to assist machining and grinding benchmarks) a~j The deformation cloud diagram under the following conditions: Figure 5 The various cases obtained by analyzing the machining parameters of the circular saw blade substrate for adding the reference grinding process ( a~j In addition, Figure 6This is a deformation cloud map of a single ground circular saw blade substrate, where the color bar on the right side of the map is used to represent the reference deformation degree. In addition, the deformation percentage (Total Deformation) is output below the cloud map each time the overall deformation of the workpiece is analyzed. The above deformation results can be used to feedback and optimize the grinding process parameters to ensure that the ultra-thin sheet maintains excellent flatness, stress uniformity and end jump control performance during processing, thereby significantly improving processing accuracy and surface quality.

[0102] Embodiment 2

[0103] This embodiment provides an ultra-thin sheet grinding deformation prediction system based on a generalized finite difference method, such as Figure 2 As shown, including:

[0104] The dynamic equation construction module is used to conduct dynamic theoretical analysis on the ultra-thin sheet grinding process and construct the dynamic equation of ultra-thin sheet grinding;

[0105] The time domain discretization module is used to discretize the time domain of the dynamics equation. After parameter adjustment, the time discrete equation is obtained;

[0106] The spatial domain discretization module is used to discretize the spatial domain of the dynamic equation, divide the computational domain corresponding to the ultra-thin sheet into internal nodes, boundary nodes and supplementary nodes, and construct the approximate equation of the local node group through Taylor expansion and weighted least squares method;

[0107] The linear system of algebraic equations building module is used to substitute the approximate equations of the local node group into the discrete time equations of the dynamic equations, taking into account the boundary conditions, and building the linear system of algebraic equations in matrix form;

[0108] The ultra-thin sheet grinding deformation prediction module is used to obtain the grinding parameter data of the ultra-thin sheet to be tested, substitute it into the linear system of algebraic equations for solution, and obtain the displacement, velocity and acceleration at any node on the ultra-thin sheet, so as to realize the dynamic deformation prediction of the ultra-thin sheet during the grinding process.

[0109] The above-mentioned time domain discrete module, space domain discrete module, and algebraic equation linear system construction module constitute the generalized wired difference method calculation and analysis module, and the above-mentioned ultra-thin sheet grinding deformation prediction module is Figure 2 The MATLAB simulation analysis and deformation prediction module in the system relies on MATLAB software for calculation, simulation and analysis.

[0110] Preferably, the system proposed in this embodiment also includes:

[0111] The process parameter optimization and deformation control module is used to establish a quantitative mapping model of deformation amount-process parameters according to the predicted dynamic deformation of the ultra-thin sheet, adjust the input grinding process parameters according to the quantitative mapping model, and control the dynamic deformation of the ultra-thin sheet.

[0112] Embodiment 3

[0113] This embodiment provides an electronic device, including: a memory, used to store executable instructions; and a processor, used to implement the above method provided in this embodiment when executing the executable instructions stored in the memory.

[0114] Embodiment 4

[0115] This embodiment also provides a computer-readable storage medium storing executable instructions. When the executable instructions are executed by a processor, the processor will be caused to execute the above method provided in this embodiment.

[0116] Embodiment 5

[0117] This embodiment provides a computer program product, which includes an executable instruction, which is a computer instruction; the executable instruction is stored in a computer-readable storage medium. When a processor of an electronic device reads the executable instruction from the computer-readable storage medium and the processor executes the executable instruction, the electronic device executes the above method provided in this embodiment.

[0118] The steps involved in the above embodiments 2 to 5 correspond to the method embodiment 1. For the specific implementation, please refer to the relevant description part of embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood to include any medium that can store, encode or carry an instruction set for execution by a processor and enable the processor to execute any method in the present invention.

[0119] Those skilled in the art should understand that the modules or steps of the present invention described above can be implemented by a general-purpose computer device, or alternatively, they can be implemented by a program code executable by a computing device, so that they can be stored in a storage device and executed by the computing device, or they can be made into individual integrated circuit modules, or multiple modules or steps therein can be made into a single integrated circuit module for implementation. The present invention is not limited to any specific combination of hardware and software.

[0120] The above description is only a preferred embodiment of the present invention. Although the specific implementation mode of the present invention is described in conjunction with the accompanying drawings, it is not a limitation of the protection scope of the present invention. Those skilled in the art should understand that on the basis of the technical solution of the present invention, various modifications or variations that can be made by those skilled in the art without creative work are still within the protection scope of the present invention.

Claims

1. A method for predicting deformation of ultra-thin sheet grinding based on generalized finite difference method, characterized in that: include: Conduct dynamics theoretical analysis on the ultra-thin sheet grinding process and construct the dynamics equation of ultra-thin sheet grinding; The Newmark-Beta method is used to discretize the time domain of the dynamic equation, and after parameter adjustment, the time discrete equation is obtained; The spatial domain of the dynamic equation is discretized by using the generalized finite difference method, and the computational domain corresponding to the ultra-thin sheet is divided into internal nodes, boundary nodes and supplementary nodes. The approximate equation of the local node group is constructed by Taylor expansion and weighted least squares method. Substitute the approximate equations of the local node group into the discrete time equations of the dynamics equations, consider the boundary conditions, and construct a linear system of algebraic equations in matrix form; The grinding parameter data of the ultra-thin sheet to be tested is obtained and substituted into the linear system of algebraic equations for solution. The displacement, velocity and acceleration at any node on the ultra-thin sheet are obtained to predict the dynamic deformation of the ultra-thin sheet during the grinding process.

2. The method for predicting ultra-thin sheet grinding deformation based on generalized finite difference method according to claim 1, characterized in that: Considering the large diameter-to-thickness ratio and ultra-thin characteristics of the ultra-thin workpiece, based on elastic deformation, time effect, acceleration change and damping factors, combined with Kirchhoff plate theory, boundary conditions are defined and the dynamic equation of ultra-thin sheet grinding is constructed, that is, the dynamic elastic mechanics model of ultra-thin sheets during grinding. The constructed kinetic equation is expressed as: ； in, Indicates the influence of the bending stiffness of the circular saw blade base on its deformation; is a bi-tone operator, , , this bitonic operator reflects the comprehensive bending deformation of the plate, including the bending curvature and the corresponding mechanical response; represents the Poisson's ratio of ultra-thin sheet materials; Indicates the circular saw blade base z Point t The deflection at the moment z A dot represents the location of a point ( x,y ),( x,y ) are the horizontal and vertical coordinates of the position; Represents the damping effect of the circular saw blade matrix; represents the damping coefficient; represents the inertia effect of the circular saw blade substrate, is the material density of the circular saw blade, For the circular saw blade substrate at the current time t The instantaneous thickness of It represents the distributed load generated by the grinding force applied by the grinding wheel on the saw blade base.

3. The method for predicting ultra-thin sheet grinding deformation based on generalized finite difference method as claimed in claim 2, characterized in that: The boundary conditions are defined as: ； ； in, represents the boundary of the ultra-thin sheet, Indicates the circular saw blade base z Point t The deflection at the moment, for Partial derivative in the normal direction.

4. The method for predicting ultra-thin sheet grinding deformation based on generalized finite difference method according to claim 1, characterized in that: The Newmark-Beta method is used to discretize the time domain of the dynamic equation. After parameter adjustment, the time discrete equation is obtained, including: Considering the deformation of any node on the circular saw blade substrate at the previous and next moments, the discretized displacement, velocity and acceleration formulas of the node are calculated according to the Newmark-Beta method; Adjust the key parameters introduced in the displacement, velocity and acceleration formulas and , determine the optimal parameter values ​​that balance computational accuracy and numerical stability; Substituting the discretized displacement, velocity and acceleration formulas after parameter determination into the dynamic equation, we can obtain the kinematic equation in discretized form, namely the time discrete equation.

5. The method for predicting ultra-thin sheet grinding deformation based on generalized finite difference method according to claim 1, characterized in that: The spatial domain is discretized using the generalized finite difference method, including: For each configuration node except the supplementary node, the central node and its recent Neighboring internal nodes Together they form a local node group and calculate the coordinates of each of the nodes; ; For any internal node At the central node Perform Taylor expansion at the node to obtain the Taylor approximation of the node; According to the error between Taylor's approximation and the actual function value, combined with the defined weighting function, a residual function is established; Minimize the residual function, and then integrate the formulas according to the defined vector and matrix forms to obtain a system of linear equations, that is, the approximate equations of the local node group.

6. The method for predicting ultra-thin sheet grinding deformation based on generalized finite difference method according to claim 1, characterized in that: Also includes: According to the predicted dynamic deformation of the ultra-thin sheet, a quantitative mapping model of deformation amount-process parameters is established, and the input grinding process parameters are adjusted according to the quantitative mapping model to control the dynamic deformation of the ultra-thin sheet.

7. An ultra-thin sheet grinding deformation prediction system based on generalized finite difference method, characterized in that: include: The dynamic equation construction module is used to conduct dynamic theoretical analysis on the ultra-thin sheet grinding process and construct the dynamic equation of ultra-thin sheet grinding; The time domain discretization module is used to discretize the time domain of the dynamic equation using the Newmark-Beta method. After parameter adjustment, the time discrete equation is obtained. The spatial domain discretization module is used to discretize the spatial domain of the dynamic equation using the generalized finite difference method, divide the computational domain corresponding to the ultra-thin sheet into internal nodes, boundary nodes, and supplementary nodes, and construct the approximate equation of the local node group through Taylor expansion and weighted least squares method; The linear system of algebraic equations building module is used to substitute the approximate equations of the local node group into the discrete time equations of the dynamic equations, taking into account the boundary conditions, and building the linear system of algebraic equations in matrix form; The ultra-thin sheet grinding deformation prediction module is used to obtain the grinding parameter data of the ultra-thin sheet to be tested, substitute it into the linear system of algebraic equations for solution, and obtain the displacement, velocity and acceleration at any node on the ultra-thin sheet, so as to realize the dynamic deformation prediction of the ultra-thin sheet during the grinding process.

8. An electronic device, characterized in that: include: A memory for storing executable instructions; The processor is used to implement the ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method as described in any one of claims 1 to 6 when executing the executable instructions stored in the memory.

9. A computer-readable storage medium, characterized in that: Executable instructions are stored, which are used to cause the processor to execute the executable instructions to implement the ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method as described in any one of claims 1 to 6.

10. A computer program product, characterized in that The computer program product includes executable instructions stored in a computer-readable storage medium; When the processor of the electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, the ultra-thin sheet grinding deformation prediction method based on the generalized finite difference method as described in any one of claims 1 to 6 is implemented.

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