A material removal simulation and workpiece deformation prediction method based on an extended finite element simulation model and application thereof

Abstract

The application belongs to the technical field of subtractive manufacturing, and discloses a material removal simulation and workpiece deformation prediction method based on an extended finite element simulation model and application. The method comprises the following steps: performing finite element mesh division on a three-dimensional model of a blank, and simultaneously performing mesh encryption processing around a preset virtual crack surface, so as to obtain a plurality of grid units; loading residual stress in the three-dimensional model of the blank, and calculating an overall load vector; calculating a global stiffness matrix; solving an overall displacement free vector; obtaining a displacement free vector of each unit from the overall displacement free vector, applying the displacement free vector of the unit on the grid nodes of each unit, so as to obtain the coordinates of the nodes after deformation, thereby realizing the prediction of deformation. Through the application, the problems of low calculation efficiency and poor calculation accuracy in the simulation and prediction of deformation are solved.

Description

Technical Field

[0001] This invention belongs to the technical field of subtractive manufacturing, and more specifically, relates to a method and application for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model. Background Technology

[0002] In the field of machining, during forming, heat treatment, and machining processes, complex residual stress fields can form inside the workpiece due to factors such as uneven plastic deformation, temperature gradients, or phase transformations. When the material is removed in subsequent processing, the original stress balance is disrupted, and the initial residual stress is redistributed, leading to significant elastoplastic deformation of the workpiece, and even causing problems such as dimensional deviations and assembly interference. For example, in the milling of thin-walled aluminum alloy parts for aerospace applications, the warping deformation caused by the release of residual stress seriously affects the surface accuracy and assembly compatibility of the parts.

[0003] Traditional deformation prediction methods mainly rely on empirical formulas or simplified mechanical models. However, due to the highly nonlinear distribution of residual stress and the dynamic characteristics of the material removal process, these methods often fail to accurately reflect the actual deformation law. In recent years, analyzing the workpiece machining mechanism and using finite element simulation to predict workpiece deformation caused by residual stress has become a major research direction in deformation prediction. Patent CN118690615A proposes a milling process simulation method considering multi-step stress release processes, simulating the material removal process through geometric re-meshment and mesh reconstruction. However, this method requires multiple mesh reconstructions, resulting in low computational efficiency, and is prone to introducing mesh distortion errors in multiple stress release iterations, leading to a decrease in cumulative accuracy. Furthermore, this method does not optimize for the measurement error of the initial residual stress distribution, making it difficult to adapt to high-precision machining scenarios. Patent CN113191050A proposes a workpiece deformation simulation prediction method considering initial residual stress. It uses actual measurement of the initial residual stress distribution, combined with multi-output support vector regression and genetic algorithms to correct measurement errors, thereby establishing a high-precision finite element prediction model. However, this method requires a large amount of measured data for model training, limiting its applicability to workpieces with complex geometries. Therefore, how to efficiently and accurately simulate and predict the deformation of the workpiece caused by the initial residual stress is an urgent problem to be solved. Summary of the Invention

[0004] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a method and application for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model, solving the problems of low computational efficiency and poor computational accuracy in deformation prediction simulation.

[0005] To achieve the above objectives, according to one aspect of the present invention, a method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model is provided, the deformation prediction method comprising the following steps: The 3D model of the blank is meshed using finite element methods, and the area around the preset virtual crack surface is further refined to obtain multiple mesh elements. Residual stress is applied to the three-dimensional model of the blank, and the load vector of each element is calculated to obtain the overall load vector. Calculate the strain-displacement matrix of each element, and use this to solve for the stiffness matrix of each element. Then, use the stiffness matrix of each element to obtain the global stiffness matrix. The overall displacement free vector is solved by utilizing the relationship between the overall displacement free vector, the overall stiffness matrix, and the overall load vector; the displacement free vector of each element is obtained from the overall displacement free vector, and the displacement free vector of each element is applied to the mesh nodes of each element to obtain the coordinates of the nodes after deformation, thereby realizing the prediction of deformation.

[0006] More preferably, the formula for calculating the overall load vector is as follows:

[0007]

[0008] in, This is the overall load vector, where e is the element number. It is the total number of units. It is the unit degree of freedom mapping matrix. is the element node load vector, and n is the total number of Gaussian points in the element; It is the weight coefficient of the i-th Gaussian point; It is the strain-displacement matrix at the i-th Gaussian point. Transpose of a matrix; It is the absolute value of the determinant of the Jacobian matrix at the i-th Gaussian point; It is the residual stress value at the i-th Gaussian point.

[0009] More preferably, the formula for calculating the Gaussian stress of each unit is as follows:

[0010] in, It is the Gaussian point stress within the element. These are the local coordinates of the Gaussian point within the cell. Let n be the shape function value at the Gaussian point, and n be the number of nodes in the element. Let be the initial residual stress value at node i.

[0011] More preferably, the formula for calculating the global stiffness matrix is ​​as follows:

[0012]

[0013] in, This is the global stiffness matrix, where e is the element number. It is the total number of units. It is the unit degree of freedom mapping matrix. The element stiffness matrix, The element stiffness matrix, It is an elastic-plastic matrix. It is the strain-displacement matrix; It is the strain-displacement matrix at the i-th Gaussian point; It is the weight coefficient of the i-th Gaussian point; It is the absolute value of the determinant of the Jacobian matrix at the i-th Gaussian point.

[0014] More preferably, before calculating the load vector and stiffness matrix of each element, it is determined whether the mesh element is a reinforcing element, and the reinforcing element satisfies the following conditions: The distances from all vertices of the element to the virtual crack surface can be positive or negative, and at least one vertex of the element has a perpendicular line drawn to the crack surface with its foot inside the virtual crack surface.

[0015] More preferably, when calculating the distance from the vertex to the virtual crack surface and the perpendicular line from the vertex to the crack surface, the virtual crack surface is first discretized into multiple crack segments. The distance from the vertex to the virtual crack surface is the distance from the vertex to any crack segment, and the perpendicular line from the vertex to the crack surface is the perpendicular line from the vertex to any crack segment.

[0016] More preferably, the formula for calculating the strain-displacement matrix is ​​as follows: When the mesh element is an enhanced element ]; When the mesh element is a non-reinforced element B = ; in, It is the derivative matrix of conventional shape functions. It is the derivative matrix of the enhancement function.

[0017] According to another aspect of the present invention, a material removal simulation and workpiece deformation prediction system based on an extended finite element simulation model is provided. The system includes an actuator that performs the above-described material removal simulation and workpiece deformation prediction method based on an extended finite element simulation model.

[0018] According to another aspect of the present invention, an application of a material removal simulation and workpiece deformation prediction method based on an extended finite element simulation model is provided in subtractive manufacturing, wherein the subtractive manufacturing includes three processes: roughing, semi-finishing, and finishing; the application steps are as follows: The blank is rough-machined, and the deformation prediction method described above is used to predict the deformation during rough machining. The workpiece deformed by rough machining is used as the blank for semi-finishing. The deformation prediction method described above is used to predict the deformation of the blank after semi-finishing. The workpiece deformed by semi-finishing is used as the blank for finishing. The deformation prediction method described above is used to predict the deformation of the blank after finishing, thereby obtaining the deformation after subtractive manufacturing.

[0019] More preferably, the process of obtaining the virtual crack surface during rough machining is as follows: Merge all final machining surfaces on the target model into a single plane. The plane is expanded to penetrate the target model, thereby obtaining a plane. As a virtual crack surface for finishing; The plane Offset according to the preset finishing allowance to obtain a plane. to the plane Extend it to penetrate the target model, thereby obtaining a plane. As a virtual crack surface in semi-finishing; The plane Offset according to the preset roughing allowance to obtain a plane. to the plane Extend it to penetrate the target model, thereby obtaining a plane. As a virtual crack surface for rough machining.

[0020] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art: 1. The deformation prediction method provided by this invention divides the blank model into finite element meshes, then applies residual stress to the blank model, and simulates the residual stress release process by fixing the mesh, thus avoiding the mesh reconstruction requirement of the traditional birth and death element method and significantly improving computational efficiency; the material is removed and the newly generated boundary is transformed into a virtual crack surface, and then the overall displacement degree of freedom vector is solved by using the overall load vector and the global stiffness matrix, accurately solving the overall displacement vector including the enhanced degree of freedom, thus solving the problem of poor computational accuracy.

[0021] 2. This invention obtains the unit load vector by solving Gaussian stress, and then assembles the total load vector using the unit load vector. The residual stress field is used as a tensor field. Compared with directly applying equivalent forces to the nodes, solving Gaussian stress can ensure mathematical integrity at the integration point (Gaussian point) and avoid physical distortion and accuracy collapse caused by nodal force equivalence.

[0022] 3. In solving the displacement-strain matrix, this invention calculates the elements separately as reinforced and non-reinforced elements, making the obtained overall load vector and global stiffness matrix more accurate. Reinforced elements refer to elements that are traversed by the virtual crack surface, which have discontinuous displacement interfaces inside. Therefore, an enhancement function (unit step function) is introduced to describe this discontinuity. Non-reinforced elements refer to elements that are not traversed by the virtual crack surface, whose internal material is continuous and does not require additional reinforcement. This distinction in calculation is to allow the simulation model to accurately match the medium characteristics after material removal—both continuous regions and discontinuous boundaries caused by material removal.

[0023] 4. When the deformation prediction method of this invention is applied in subtractive manufacturing, the material removal area is determined based on the machining allowance. A geometric offset technique is used to generate the material removal surface, which is regarded as a virtual crack surface in the extended finite element simulation model. The virtual crack surface is designed to adapt to the physical characteristics of the material removal process based on the extended finite element theory. Essentially, it describes the boundary characteristics after material removal by simulating the discontinuities on both sides of the "crack". The virtual crack surface provides an object for the enhancement function (element step function) in the extended finite element model. By defining the geometric position of the crack surface, the sign distance and perpendicular position from the mesh element node to the crack surface can be calculated to determine the enhancement element. This provides a basis for introducing displacement discontinuity terms in the subsequent displacement interpolation function. Setting the virtual crack surface can describe the material removal boundary using mathematical functions, which can ensure that the finite element mesh remains fixed during the material removal process. It does not require frequent mesh reconstruction due to discontinuous displacements inside the mesh elements. Furthermore, it can accurately capture the boundary stress singularity through the unit step function, thereby improving the prediction accuracy. Attached Figure Description

[0024] Figure 1This is a flowchart of a material removal simulation and workpiece deformation prediction method based on an extended finite element simulation model, constructed according to a preferred embodiment of the present invention. Figure 2 This is a schematic diagram illustrating the principle of generating a virtual crack surface according to a preferred embodiment of the present invention; Figure 3 This is a schematic diagram of the process for generating a virtual crack surface according to a preferred embodiment of the present invention; Figure 4 This is a schematic diagram showing the distance from a unit node to a virtual crack segment constructed according to a preferred embodiment of the present invention; Figure 5 This is a schematic diagram of a node constructed according to a preferred embodiment of the present invention, with a perpendicular line drawn from the node to the virtual crack surface; Figure 6 This is a schematic diagram of the enhanced node distribution constructed according to a preferred embodiment of the present invention. Detailed Implementation

[0025] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0026] like Figure 1 As shown, a method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model is described, with the following specific steps: S1 generates the grid.

[0027] (1) Use eight-node hexahedral elements or four-node tetrahedral elements to perform finite element mesh generation on the blank model, and set a localized refinement region near the roughing virtual crack surface: with the roughing virtual crack surface Centered on the crack surface, a specific geometric region is extended along its normal and tangential directions (the extension range is determined based on the model size and computational accuracy requirements). For regions ≤ a specified threshold away from the crack surface, mesh refinement is achieved by reducing element edge length and increasing node density.

[0028] (2) Distinguish between reinforced and non-reinforced elements in the mesh. like Figure 6 As shown, the rough-machined virtual crack surface The crack is discretized into several coplanar quadrilateral segments. Each segment is defined by the coordinates of its four coplanar endpoints, denoted as . , where k = 1,2,...,Nseg, and Nseg is the number of crack segments.

[0029] like Figure 3 and Figure 4 As shown, for any node P in the finite element mesh, calculate its signed distance to any crack segment. The algorithm determines whether the foot of the perpendicular line from any node to any crack segment falls within the crack surface. In material removal problems, the displacement on both sides of the cutting plane can exhibit discontinuities such as displacement jumps. Therefore, after expanding the cutting plane to obtain a virtual cutting surface, when the virtual crack surface passes through an element, it is identified as a reinforcement element, and a reinforcement function is introduced to solve the problem of discontinuous displacement on both sides of the crack surface.

[0030] An element is considered a reinforcement element if it meets the following two conditions: ① The nodes of the element are located on both sides of the crack surface, meaning that the sign of the distance from all vertices of the element to the crack surface can be positive or negative, such as... Figure 4 As shown; ② If at least one vertex of this element has its perpendicular foot to the crack surface lying within the crack surface, and both of the above conditions are satisfied, then it is a reinforcement element. Figure 5 As shown. Since the enhanced displacement degrees of freedom applied in the displacement interpolation function are applied to the enhanced nodes, it is also necessary to determine the enhanced nodes. It is important to note that all nodes of the enhanced element are enhanced nodes. An example of the enhanced node and enhanced element determination results is shown below. Figure 6 As shown.

[0031] S2 applies residual point stress to the blank model. In finite element analysis, directly calculating the stress and strain distribution throughout the entire element is extremely complex.

[0032] The Gaussian integral method calculates the mechanical properties of the entire element by selecting several specific points (Gaussian points) within the element and combining them with weighting coefficients and the coordinate transformation matrix (Jacobi matrix). This significantly reduces computational complexity while maintaining accuracy. Its core integral formula is: .in The Gaussian point weights are determined by the order of integration. Let these be the local coordinates of the Gaussian point. This is the determinant of the Jacobian matrix at the Gaussian point, used for the transformation from local to global coordinates. The residual stress at the element nodes is interpolated to the Gaussian point within the element using Gaussian point shape function interpolation. The interpolation formula is as follows: Gaussian point stress for each element.

[0033] in, These are the local coordinates of the Gaussian point; The shape function value at the Gaussian point is calculated from the local coordinates of the Gaussian point. For the residual stress at the nodes; is the Gaussian point stress; n is the number of element nodes.

[0034] S3 calculates the element load vector and element stiffness matrix. (1) Solve for the element load vector In the finite element method, the displacement distribution at any point within an element can be determined using the nodal displacements and displacement interpolation functions (shape functions). For reinforced elements, the displacement field within the element can be expressed using shape functions as follows:

[0035] In the formula, x is the coordinate of any position in any element. J represents the set of all nodes; J represents the set of nodes that need to be reinforced after being split by the crack. The regular finite element shape function representing node i; The vector representing the regular nodal displacement degrees of freedom of node i; This represents the vector of enhanced nodal displacement degrees of freedom; It is a unit step function used to account for displacement jumps on both sides of the crack surface; It is the value of the enhancement function at node j.

[0036] Based on the principle of virtual work, the Gaussian point stress is represented by the strain-displacement matrix (B matrix). Converted to external load of element node .

[0037] For non-enhanced elements, the B matrix is ​​derived solely from the derivatives of conventional finite element shape functions. = ( )constitute, B =

[0038] For the enhancement unit, B also includes the derivative of the enhancement function. = ( ),Right now ].

[0039] in,

[0040]

[0041] in, (i=1,2,3...) represents B The part of the regular displacement submatrix of the i-th node, express B The enhanced displacement submatrix of the i-th node. (The value of i depends on the element type. When the element type is an eight-node hexahedron, the value of i is 1~8. When the element type is a four-node tetrahedron, the value of i is 1~4.)

[0042] in, Let i represent the shape function at node i. Let represent the partial derivative of the shape function at node i with respect to coordinate x.

[0043]

[0044] in, Let i represent the shape function at node i. For a unit step function, based on the sign distance from coordinate x to the virtual crack surface. Confirmed, when ≤0, =1, when >0, =-1; Indicates node i The value of is subtracted to eliminate rigid body displacement and ensure the uniqueness of the solution. Let x represent the partial derivative of the enhancement function at node i with respect to x.

[0045] Calculate the element node load vector based on the B matrix. , The magnitude of the force load at the corresponding node is expressed as follows:

[0046] in, is the load vector of each node; n is the total number of Gaussian points in the cell; This represents the weight coefficient of the i-th Gaussian point; This represents the strain-displacement matrix at the i-th Gaussian point. Transpose of a matrix; The absolute value of the determinant of the Jacobian matrix at the i-th Gaussian point is the scaling factor for the coordinate transformation from the local coordinate system within the cell to the global coordinate system. This represents the residual stress value at the i-th Gaussian point.

[0047]

[0048] in, For the overall load vector, The total number of units, It is the element degree of freedom mapping matrix, which maps the local degrees of freedom of the element to the global degree of freedom number. It is automatically generated after mesh generation based on the global node numbering rules. It is the unit node load vector.

[0049] (2) Solve for the element stiffness matrix Set boundary conditions for the blank model. Boundary conditions refer to the conditions that need to be defined on the boundaries of the model when building a finite element model in order to simulate the boundary behavior of the real system. For the boundary nodes within the clamping region, the boundary conditions are expressed as follows: Fully fixed constraints (restricting displacement in all directions):

[0050] Partial degree of freedom constraints (restricting displacement in a specific direction): (or ) in, , , Define the displacement degrees of freedom components of node i in the X, Y, and Z directions of the global coordinate system. Label the degrees of freedom that need to be constrained and set the overall displacement degree of freedom vector. The corresponding constraint degree of freedom component is 0.

[0051] Simulation analysis is used to determine whether the force at the clamping point of the part during machining will exceed the clamping force, and the clamping force is adjusted or the clamping method is optimized based on the results.

[0052] Element stiffness matrix The expression is:

[0053] in, The element stiffness matrix; The elastic-plastic matrix is ​​constructed based on the material constitutive parameters (such as Young's modulus E and Poisson's ratio υ). The matrix is ​​the strain-displacement matrix; This represents the strain-displacement matrix at the i-th Gaussian point; This represents the weight coefficient of the i-th Gaussian point; Let represent the absolute value of the determinant of the Jacobian matrix at the i-th Gaussian point.

[0054] (3) Solve for the overall load vector and the global stiffness matrix. Based on the fundamental theory of elastic-plastic mechanics, the basic equations of the finite element method are obtained. .in, This represents the total displacement degrees of freedom vector, including the conventional displacement degrees of freedom and the enhanced displacement degrees of freedom, i.e. Excluding fixed degrees of freedom; F is the global load vector; K is the global stiffness matrix. F consists of the load vector of each element. It is assembled according to the degree of freedom number. K is the element stiffness matrix of each element. It is assembled based on the degree of freedom numbers in the global stiffness matrix.

[0055]

[0056] in, The total load vector; The total number of units; It is the element degree of freedom mapping matrix, which maps the local degrees of freedom of the element to the global degree of freedom number. It is automatically generated after mesh generation based on the global node numbering rules. It is the element node load vector; The global stiffness matrix. This is the element stiffness matrix.

[0057] S4 Solve for the deformation results. Based on the obtained global stiffness matrix K and global load vector... Through formula Solving the equations yields the global displacement degree of freedom vector. .Will By applying the coordinates to the mesh nodes, calculating the coordinates of the deformed nodes, and then visualizing them, the simulation results of the workpiece deformation can be obtained. The expression applied to the mesh nodes is as follows:

[0058]

[0059]

[0060] in, , , The initial global coordinates of the nodes (determined during mesh generation). , , For node i in The corresponding displacement degree of freedom components, , , The global coordinates of the material after removing node deformation.

[0061] The coordinates of all nodes after deformation are calculated, and the overall deformation of the workpiece can be displayed through finite element post-processing visualization technology, which can intuitively present the changes in the shape of the workpiece caused by the release of residual stress.

[0062] The application process of the above deformation method in actual subtractive manufacturing is as follows: Step 1: Place the workpiece model (target model) and the initial blank model with their centers and normals aligned to ensure that the blank model can be accurately mapped onto the workpiece model after material removal.

[0063] (1) Select all the final machined surfaces of the workpiece model. (i.e., the new surface generated after the removal of finishing materials), undergo topological fusion processing to form a single surface, eliminating tiny surface gaps and non-connected areas, ensuring a smooth fused surface. It satisfies the requirements of continuity without overlap or gaps, that is... ,in, This is the merged connected sub-face.

[0064] (2) such as Figure 2 As shown, based on the finishing allowance d, the normal equidistant offset technique is used to... Along its normal field Bias is applied to generate a surface The expression is: ,in, The cutting plane is used for roughing or semi-finishing. The cutting plane is defined as the newly exposed workpiece surface after the tool removes the material, that is, the new surface generated after material removal.

[0065] (3) After biasing, for Virtual crack surfaces are obtained by expanding the edges along their tangents to a preset width. To ensure that the virtual crack surface can penetrate the model, that is, to satisfy: virtual crack surface The intersection plane of the plane and the blank model satisfy .

[0066] (4) Similarly, for The surface is expanded according to a preset width to obtain a virtual crack surface for simulation of finishing deformation. Similarly, the preset width size must ensure the virtual crack surface is expanded. It can penetrate the blank model, that is, it satisfies The intersection of the plane and the blank model .like Figure 3 As shown, for multi-process machining, the process is deduced in reverse from finishing. The previous cutting plane is obtained based on the offset of the current process cutting plane, and virtual crack surfaces for each process are generated in sequence (the virtual crack surfaces exist independently of the workpiece model and the blank model).

[0067] (5) The plane The preset roughing allowance is offset to obtain a flat surface. to the plane Extend it to penetrate the target model, thereby obtaining a plane. As a virtual crack surface for rough machining.

[0068] Step 2: Perform rough machining on the blank, and use the deformation prediction method provided by this invention to predict the deformation during rough machining; The workpiece deformed by rough machining is used as a blank for semi-finishing. The deformation prediction method provided by this invention is used to predict the deformation of the blank after semi-finishing. The workpiece deformed by semi-finishing is used as the blank for finishing. The deformation prediction method provided by this invention is used to predict the deformation of the blank after finishing, thereby obtaining the deformation after subtractive manufacturing.

[0069] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for material removal simulation and workpiece deformation prediction based on an extended finite element model, characterized in that, The deformation prediction method includes the following steps: The 3D model of the blank is meshed using finite element methods, and the area around the preset virtual crack surface is further refined to obtain multiple mesh elements. Residual stress is applied to the three-dimensional model of the blank, and the load vector of each element is calculated to obtain the overall load vector. Calculate the strain-displacement matrix for each element, and use this to solve for the stiffness matrix of each element. Then, use the stiffness matrix of each element to obtain the global stiffness matrix. The overall displacement free vector is solved by utilizing the relationship between the overall displacement free vector, the overall stiffness matrix, and the overall load vector; the displacement free vector of each element is obtained from the overall displacement free vector, and the displacement free vector of each element is applied to the mesh nodes of each element to obtain the coordinates of the nodes after deformation, thereby realizing the prediction of deformation.

2. The method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model as described in claim 1, characterized in that, The formula for calculating the overall load vector is as follows: in, This is the overall load vector, where e is the element number. It is the total number of units. It is the unit degree of freedom mapping matrix. is the element node load vector, and n is the total number of Gaussian points in the element; It is the weight coefficient of the i-th Gaussian point; It is the strain-displacement matrix at the i-th Gaussian point. Transpose of a matrix; It is the absolute value of the determinant of the Jacobian matrix at the i-th Gaussian point; It is the residual stress value at the i-th Gaussian point.

3. The method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model as described in claim 2, characterized in that, The formula for calculating the Gaussian stress of each element is as follows: in, It is the Gaussian point stress within the element. These are the local coordinates of the Gaussian point within the cell. Let n be the shape function value at the Gaussian point, and n be the number of nodes in the element. Let be the initial residual stress value at node i.

4. The method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model as described in claim 3, characterized in that, The formula for calculating the global stiffness matrix is ​​as follows: in, This is the global stiffness matrix, where e is the element number. It is the total number of units. It is the unit degree of freedom mapping matrix. The element stiffness matrix, It is an elastic-plastic matrix. It is the strain-displacement matrix; It is the strain-displacement matrix at the i-th Gaussian point; It is the weight coefficient of the i-th Gaussian point; It is the absolute value of the determinant of the Jacobian matrix at the i-th Gaussian point.

5. A method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model as described in claim 1 or 4, characterized in that, Before calculating the load vector and stiffness matrix of each element, it is determined whether the mesh element is a reinforcement element. The reinforcement element satisfies the following conditions: The distances from all vertices of the element to the virtual crack surface can be positive or negative, and at least one vertex of the element has a perpendicular line drawn to the crack surface with its foot inside the virtual crack surface.

6. The method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model as described in claim 5, characterized in that, When calculating the distance from the vertex to the virtual crack surface and the perpendicular line from the vertex to the crack surface, the virtual crack surface is first discretized into multiple crack segments. The distance from the vertex to the virtual crack surface is the distance from the vertex to any crack segment, and the perpendicular line from the vertex to the crack surface is the perpendicular line from the vertex to any crack segment.

7. The method for material removal simulation and workpiece deformation prediction based on an extended finite element simulation model as described in claim 6, characterized in that, The formula for calculating the strain-displacement matrix is ​​as follows: When the mesh element is an enhanced element ; When the mesh element is a non-reinforced element B = ; in, It is the derivative matrix of conventional shape functions. It is the derivative matrix of the enhancement function.

8. A material removal simulation and workpiece deformation prediction system based on an extended finite element simulation model, characterized in that, The system includes an actuator that performs a material removal simulation and workpiece deformation prediction method based on an extended finite element simulation model as described in any one of claims 1-7.

9. A method for subtractive manufacturing based on material removal simulation and workpiece deformation prediction using an extended finite element model, characterized in that, The subtractive manufacturing process includes three steps: roughing, semi-finishing, and finishing; the method steps are as follows: The blank is rough-machined, and the deformation of the rough machining is predicted using the deformation prediction method according to any one of claims 1-7; The workpiece deformed by rough machining is used as a blank for semi-finishing. The deformation prediction method described in any one of claims 1-7 is used to predict the deformation of the blank after semi-finishing. The workpiece deformed by semi-finishing is used as the blank for finishing. The deformation prediction method described in any one of claims 1-7 is used to predict the deformation of the blank after finishing, thereby obtaining the deformation after subtractive manufacturing.

10. The method as described in claim 9, characterized in that, The process of obtaining the virtual crack surface in the roughing process is as follows: Merge all final machining surfaces on the target model into a single plane. The plane is then expanded to penetrate the target model, thus obtaining the plane. As a virtual crack surface for finishing; The plane Offset according to the preset finishing allowance to obtain a plane. to the plane Extend it to penetrate the target model, thereby obtaining a plane. As a virtual crack surface in semi-finishing; The plane Offset according to the preset roughing allowance to obtain a plane. to the plane Extend it to penetrate the target model, thereby obtaining a plane. As a virtual crack surface for rough machining.

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