A stable nodal integration method for explicit analysis of metal volume forming

Abstract

The application relates to a stable node integral calculation method for explicitly analyzing metal volume forming, which comprises the following steps: dividing a problem domain into a triangular or tetrahedral grid, and dividing a grid integral domain into a plurality of non-overlapping node integral domains based on nodes in the triangular or tetrahedral grid; performing node domain smoothing on the node integral domain based on triangular / tetrahedral strain components of the nodes, to obtain a smoothed domain; constructing an hourglass control between the nodes and adjacent nodes in the smoothed domain, and calculating hourglass forces between the nodes and the adjacent nodes; performing node integration on the nodes based on stress and deformation gradients of the nodes calculated by attributes of the nodes, to obtain original internal forces of the nodes; superimposing the hourglass forces borne by the nodes and the original internal forces of the nodes, to obtain internal forces of the stable node integral method; and solving discrete motion equations in explicit dynamics based on the internal forces of the stable node integral method, to obtain a dynamic response.

Description

Technical Field

[0001] This application relates to the field of high-performance numerical simulation analysis technology, and in particular to a stable node integral calculation method for explicit analysis of metal volume forming. Background Technology

[0002] Currently, while the spacetime stable nodal integration algorithm constructed by incorporating the strain gradient into the energy equation solves the numerical instability problem of nodal integration, it artificially introduces multiple integration points located on the coordinate axes during explicit dynamic calculations, leading to increased computational load and failing to realize the efficiency advantages of nodal integration. This limits its application and promotion in explicit dynamic calculations of large deformations. Therefore, how to ensure the numerical stability of stable nodal integration algorithms while maintaining the single-integration-point computational characteristics, and how to construct single-integration-point stable terms, are key scientific problems that urgently need to be solved. Summary of the Invention

[0003] Therefore, it is necessary to provide a method for calculating stable node integrals in explicit analysis of metal volume forming, the method comprising:

[0004] S1: Divide the problem domain into a triangular or tetrahedral mesh, and divide the mesh integration domain into multiple non-overlapping nodal integration domains based on the nodes in the triangular or tetrahedral mesh; the problem domain is a continuous metal billet; the node attributes include triangular / tetrahedral strain components, Young's modulus, Poisson's ratio, displacement vector, and acceleration vector;

[0005] In a triangular or tetrahedral mesh, by sequentially connecting the midpoints of the surrounding edges of a node to the midpoints of the surrounding triangular elements, multiple non-overlapping nodal integration domains centered on the node are obtained.

[0006] S2: Based on the triangular / tetrahedral strain components of the nodes, the nodal integration domain is smoothed to obtain the smooth domain;

[0007] S3: Construct hourglass control between nodes and adjacent nodes in a smooth domain, and calculate the hourglass force between nodes and adjacent nodes;

[0008] S4: Construct an elastic matrix based on the Young's modulus and Poisson's ratio of the nodes. Multiply the triangular / tetrahedral strain components of the nodes with the corresponding elastic matrix to obtain the stress of the nodes. Calculate the deformation gradient based on the displacement vector of the nodes. Perform nodal integration on the nodes based on the stress and deformation gradient to obtain the original internal forces of the nodes.

[0009] S5: The hourglass force on the node is superimposed with the original internal force of the node to obtain the internal force of the stable node integral method. Based on the internal force of the stable node integral method and the acceleration vector of the node, the discrete motion equation in the explicit dynamics is solved to obtain the dynamic response.

[0010] Preferably, in S2, the node-based triangular / tetrahedral strain component node-integral domain smoothing includes:

[0011] ;

[0012] in, Let denote the derivative of the smooth shape function in the nodal integration domain of the i-th node; Represents the area / volume of the i-th node; Let represent the derivative of the shape function in the nodal integration field of the i-th node; Represents the gradient; Represents the outward normal of the surrounding edges / surfaces of the i-th node; This represents the surrounding edges / surfaces of the i-th node.

[0013] Preferably, S3 includes:

[0014] Based on the deformation gradient increment between any node and its neighboring nodes in the smooth domain, calculate the theoretical relative positional relationship between the node and its neighboring nodes.

[0015] Based on the position of a node within a smooth domain and the positions of its neighboring nodes, calculate the actual relative positional relationship between the node and its neighboring nodes.

[0016] Calculate the relative positional deviation between the theoretical and actual relative positional relationships;

[0017] The elongation from the node to the adjacent node is calculated based on the relative position deviation.

[0018] Differentiating the potential energy function of the virtual spring with respect to its length yields the virtual spring force.

[0019] The internal forces between the node and its adjacent nodes are calculated based on the elongation and virtual spring force.

[0020] The internal forces between symmetrical nodes and adjacent nodes are used to obtain the hourglass force between nodes and adjacent nodes.

[0021] Preferably, the calculation of the elongation from the node to the adjacent node based on the relative position deviation includes:

[0022] ;

[0023] ;

[0024] in, Indicates the first i The node to its first j The elongation of each adjacent node; Indicates the first Time step i The relative positional deviation of each node; Indicates the first Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates L1 normal form; Indicates the first Time step node i and j The direction vector between nodes.

[0025] Preferably, the internal forces between the node and adjacent nodes are calculated based on the elongation and the virtual spring force, and the calculation formula is as follows:

[0026] ;

[0027] in, Indicates the first i Internal forces on each node side; Indicates the first i Young's modulus of each node; Indicates the first t Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Represents a node i The volume / area of ​​the smooth domain.

[0028] Preferably, the formula for calculating the hourglass force is:

[0029] ;

[0030] in, Indicates the first i The hourglass force between a node and all its neighboring nodes; Indicates the first i The node and its first j Hourglass force between adjacent nodes; Indicates the first i The number of neighboring nodes of each node; Indicates the first j Young's modulus of each node; Indicates the first i The node of the first j The elongation from each adjacent node to the i-th node; Indicates the hourglass coefficient; Represents a node j The volume / area of ​​the smooth domain.

[0031] Preferably, the formula for calculating the original internal forces at the nodes is:

[0032] ;

[0033] ;

[0034] ;

[0035] in, This represents the original internal force at the i-th node; Represents a node k The smooth region; Indicates node I pair j Partial derivatives of smooth shape functions in the direction; Represents a node j With nodes i The internal forces between them; k Represents a node k ; This represents the deformation of node k; Represents a node k Volume / area of ​​the smooth domain; Indicates node I pair j Partial derivatives of the shape function in the direction; The determinant representing the deformation gradient; Represents a node j With nodes k Deformation gradient between; Represents a node k With nodes i The stress between them.

[0036] Preferably, the formula for calculating the deformation gradient is:

[0037] ;

[0038] in, Indicates the first i The node and its first j The elongation between adjacent nodes; Indicates partial derivative; This represents the nodal integration field of the i-th node; Indicates the number of influential nodes around a node; Represents a node j The deformation; This represents the displacement vector of a node.

[0039] Preferably, the discrete motion equations in explicit dynamics for solving internal forces based on the stable nodal integration method include:

[0040] Calculate the external forces acting on the nodes using the stable nodal integration method;

[0041] The difference between the external forces and internal forces of a node under the stable node integration method is used as the external force vector in the discrete equation of motion.

[0042] The external force vector is the product of the lumped mass matrix and the acceleration vector, and the dynamic response is obtained from this.

[0043] Preferably, the formula for calculating the external force acting on a node using the stable nodal integration method is:

[0044] ;

[0045] in, Let represent the external force acting on the i-th node using the stable nodal integration method; This represents the nodal integration field of the i-th node; b represents the smooth region of nodes; b represents the body force. Indicates the boundary of the problem domain; Represents surface force.

[0046] Beneficial effects: This method divides the problem domain into triangular or tetrahedral meshes, and further divides the mesh integration domain into multiple non-overlapping nodal integration domains based on the nodes in the triangular or tetrahedral meshes; it smooths the nodal integration domain based on the triangular / tetrahedral strain components of the nodes to obtain a smooth domain; it constructs hourglass control between nodes and adjacent nodes in the smooth domain and calculates the hourglass force between nodes and adjacent nodes; it performs nodal integration on the nodes based on the stress and deformation gradients of the nodes calculated from the node properties to obtain the original internal forces of the nodes; it superimposes the hourglass force on the nodes with the original internal forces of the nodes to obtain the internal forces of the stable nodal integration method; and it solves the discrete motion equations in explicit dynamics based on the internal forces of the stable nodal integration method to obtain the dynamic response. This method constructs a stable nodal integration method based on unstructured triangular / tetrahedral meshes, which can adapt to complex geometric boundaries and greatly improve the computational accuracy based on unstructured meshes. At the same time, the calculation of hourglass force between nodes and adjacent nodes improves the computational stability of nodal integration, effectively maintaining the advantages of nodal integration with a small number of integration points and strong resistance to mesh distortion. It is a very efficient and stable explicit dynamic large deformation calculation method. Attached Figure Description

[0047] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0048] Figure 1 This is a flowchart of a stable node integral calculation method for explicit analysis of metal volume forming in an embodiment of this application.

[0049] Figure 2 This is a schematic diagram illustrating the application of stable node integral calculation to metal volume forming in the embodiments of this application.

[0050] Figure 3 This is a schematic diagram of the node integration domain in an embodiment of this application.

[0051] Figure 4 This is a schematic diagram of hourglass control between nodes and adjacent nodes in an embodiment of this application. Detailed Implementation

[0052] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the specific embodiments of this application are described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of this application. However, this application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of this application. Therefore, this application is not limited to the specific embodiments disclosed below.

[0053] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0054] In conventional finite element method (FEM) calculations for metal volume forming, the problem domain is divided into tetrahedrons before nodal integration. This results in relatively few integration points and high computational efficiency, a significant advantage of nodal integration methods in volume forming. However, nodal integration methods without stable terms can lead to instability in dynamic large deformation calculations like volume forming due to insufficient integration. Therefore, many researchers have proposed stable nodal integration, which constructs stable terms based on nodal integration to compensate for insufficient integration. While existing stable nodal integration methods effectively address the instability issue, their complex construction and the introduction of additional integration points for calculating stable terms in dynamic large deformation calculations result in additional computational load and cannot be considered true single-point integration methods. Therefore, this embodiment proposes a stable nodal integration calculation method for explicit analysis of metal volume forming. This method constructs truly single-point integrated stable terms, achieving stable calculation results without the need for additional integration points.

[0055] like Figure 1 As shown, this embodiment provides a method for calculating stable node integrals for explicit analysis of metal volume forming. The method includes:

[0056] S1: Divide the problem domain into triangular or tetrahedral meshes (e.g., ...). Figure 2 As shown in the figure, the mesh integration domain is divided into multiple non-overlapping node integration domains based on the nodes in the triangular or tetrahedral mesh; the problem domain is a continuous metal billet.

[0057] In this embodiment, the nodes not only carry field variable information such as displacement vector and acceleration vector, but also physical information such as Young's modulus, Poisson's ratio, and triangular / tetrahedral strain components, so as to achieve unified data storage without the need for mapping transformation from element integration points to nodes.

[0058] Specifically, in this step:

[0059] Dividing the mesh integration domain into multiple non-overlapping node integration domains based on nodes in a triangular mesh includes: for a triangular mesh divided for a two-dimensional problem, connecting the midpoints of the surrounding edges of a node and the midpoints of the surrounding triangular elements in sequence to obtain multiple non-overlapping node integration domains centered on the node.

[0060] Dividing the mesh integration domain into multiple non-overlapping node integration domains based on nodes in a tetrahedral mesh includes: for a tetrahedral mesh divided for a 3D problem, sequentially connecting the midpoints of the surrounding edges of a node, the midpoints of the surrounding triangles, and the body centers of the surrounding tetrahedral elements to obtain multiple non-overlapping node integration domains centered on the node.

[0061] The partitioned nodal integration domain is as follows Figure 3 As shown, this constructed nodal integration domain avoids region repetition or omission during integration. It should be noted that interpolation and integration in the finite element method are performed within the element, and the interpolation domain and integration domain are consistent, while interpolation and integration in nodal integration are performed within the element and within the constructed nodal smooth domain, respectively.

[0062] S2: Based on the triangular / tetrahedral strain components of the nodes, the nodal domain is smoothed to obtain the smooth domain.

[0063] Specifically, the formula for calculating nodal domain smoothing is:

[0064] ;

[0065] in, Let denote the derivative of the smooth shape function in the nodal integration domain of the i-th node; Represents the area / volume of the i-th node; Let represent the derivative of the shape function in the nodal integration field of the i-th node; Represents the gradient; Represents the outward normal of the surrounding edges / surfaces of the i-th node; This represents the surrounding edges / surfaces of the i-th node. This method is equivalent to reconstructing the gradient based on the node within the node integration domain, which can improve the "rigidity" of the results based on triangular / tetrahedral units.

[0066] S3: Construct hourglass control between nodes and adjacent nodes in a smooth domain, and calculate the hourglass force between nodes and adjacent nodes.

[0067] When calculating strain / deformation gradients, nodal integration can be simplified as homogenizing the strain / deformation gradients of the elements surrounding the node. This deformation mode is not unique; different deformation modes may yield the same strain / deformation gradient. Even slight perturbations can cause the mesh to arbitrarily transform between different deformation modes. This is very similar to the hourglass phenomenon caused by using reduced integrals on quadrilaterals / hexahedrons. This non-uniqueness leads to spurious zero-energy modes, making nodal integration highly unstable in explicit dynamic calculations and unsuitable for practical engineering problems. Nodal integration uses single-point integration, allowing relative movement between nodes and adjacent nodes. This embodiment introduces artificial stiffness between nodes and adjacent nodes to limit the transition between different deformation modes, thereby overcoming the hourglass phenomenon. Figure 4 As shown.

[0068] Specifically, the steps include:

[0069] Based on the deformation gradient increment between any node and its neighboring nodes within the smooth domain, the theoretical relative positional relationship between the node and its neighboring nodes is calculated using the following formula:

[0070] ;

[0071] in, Indicates the first Time step with the first i The theoretical relative positional relationship of each node; Indicates the first t Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates the time step increment; Indicates the first Time step i The node and its first j The deformation gradient increment between adjacent nodes.

[0072] Based on the positions of nodes within the smooth domain and the positions of their neighboring nodes, the actual relative positional relationship between a node and its neighboring nodes is calculated using the following formula:

[0073] ;

[0074] in, Indicates the first Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates the first Time step i The node of the first j The positions of the adjacent nodes; Indicates the first Time step i The position of each node.

[0075] The relative positional deviation between the theoretical and actual relative positional relationships is calculated using the following formula:

[0076] ;

[0077] in, Indicates the first Time step i The relative positional deviation of each node; Indicates the first Time step with the first i The theoretical relative positional relationship of each node; Indicates the first Time step i The node and its first jThe actual relative positional relationship between adjacent nodes.

[0078] This operation is equivalent to adding a spring between the node and its adjacent nodes, applying a certain stiffness to limit deformation. To ensure that the system maintains a linear displacement field after adding the spring, the elongation from the node to the adjacent node is calculated based on the relative position deviation. The calculation formula is as follows:

[0079] ;

[0080] ;

[0081] in, Indicates the first i The node to its first j The elongation of each adjacent node; Indicates the first Time step i The relative positional deviation of each node; Indicates the first Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates L1 normal form; Indicates the first Time step node i and j The direction vector between nodes.

[0082] Differentiating the potential energy function of the virtual spring with respect to its length yields the virtual spring force, calculated as follows:

[0083] ;

[0084] ;

[0085] in, f Represents virtual spring force; Represents elastic potential energy; L Indicates a length variable; E Indicates Young's modulus; Represents the deformation of the virtual spring; This indicates the initial length of the spring.

[0086] The internal forces between nodes and adjacent nodes are calculated based on the elongation and virtual spring force. The calculation formula is as follows:

[0087] ;

[0088] in, Indicates the first i Internal forces on each node side; EIndicates Young's modulus; Indicates the first t Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates L1 normal form; Indicates the first i The node to its first j The elongation of each adjacent node.

[0089] The internal forces between symmetrical nodes and adjacent nodes are used to obtain the hourglass force between a node and its adjacent nodes, calculated as follows:

[0090] ;

[0091] in, Indicates the first i The hourglass force between a node and all its neighboring nodes; Indicates the first i The node and its first j Hourglass force between adjacent nodes; Indicates the first i The number of neighboring nodes of each node; Indicates the first t Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates L1 normal form; Indicates the first i Young's modulus of each node; Indicates the first i The node to its first j The elongation of each adjacent node; Indicates the first j Young's modulus of each node; Indicates the first i The node of the first j The neighboring node to the first i The elongation of each node; Indicates the hourglass coefficient; Represents a node i Volume / area of ​​the smooth domain; Represents a node j The volume / area of ​​the smooth domain.

[0092] S4: Construct an elastic matrix based on the Young's modulus and Poisson's ratio of the nodes. Multiply the triangular / tetrahedral strain components of the nodes with the corresponding elastic matrix to obtain the stress of the nodes. Calculate the deformation gradient based on the displacement vector of the nodes. Perform nodal integration on the nodes based on the stress and deformation gradient to obtain the original internal forces of the nodes.

[0093] In this embodiment, the original internal force calculation formula for the node is:

[0094] ;

[0095] ;

[0096] ;

[0097] in, This represents the original internal force at the i-th node; Represents a node k The smooth region; Indicates node I pair j Partial derivatives of smooth shape functions in the direction; Represents a node j With nodes i The internal forces between them; k Represents a node k ; This represents the deformation of node k; Represents a node k Volume / area of ​​the smooth domain; Indicates node I pair j Partial derivatives of the shape function in the direction; The determinant representing the deformation gradient; Represents a node j With nodes k Deformation gradient between; Represents a node k With nodes i The stress between them.

[0098] Due to the characteristics of nodal integrals, the deformation gradient It can be calculated using the following formula:

[0099] ;

[0100] in, Indicates the first i The node and its first j The elongation between adjacent nodes; Indicates partial derivative; This represents the nodal integration field of the i-th node; Indicates the number of influential nodes around a node; Represents a node j The deformation; This represents the displacement vector of a node.

[0101] In this embodiment, the elasticity matrix constructed based on the Young's modulus and Poisson's ratio of the nodes is represented as follows:

[0102] ;

[0103] in, Represents the elasticity matrix. Indicates Young's modulus. It represents Poisson's ratio.

[0104] S5: The hourglass force on the node is superimposed with the original internal force of the node to obtain the internal force of the stable node integral method. Based on the internal force of the stable node integral method and the acceleration vector of the node, the discrete motion equation in the explicit dynamics is solved to obtain the product of the lumped mass matrix and the acceleration vector, and the dynamic response is obtained.

[0105] Furthermore, the discrete motion equations in explicit dynamics for solving internal forces based on the stable nodal integration method include:

[0106] Calculate the external forces acting on the nodes using the stable nodal integration method;

[0107] The difference between the external forces and internal forces of a node under the stable node integration method is used as the external force vector in the discrete equation of motion.

[0108] The external force vector is the product of the lumped mass matrix and the acceleration vector, and the dynamic response is obtained from this.

[0109] Furthermore, the formula for calculating the external force acting on a node using the stable nodal integration method is:

[0110] ;

[0111] in, Let represent the external force acting on the i-th node using the stable nodal integration method; This represents the nodal integration field of the i-th node; b represents the smooth region of nodes; b represents the body force. Indicates the boundary of the problem domain; Represents surface force.

[0112] The entire process is based on the original triangular / tetrahedral mesh, requiring no modification to the existing finite element calculation framework. It fully utilizes the advantage of fewer integration points in nodal integration to improve computational efficiency, and also has strong resistance to mesh distortion, significantly improving the calculation accuracy under low-quality meshes. It is very suitable for explicit analysis involving large material deformation, such as metal volume forming and contact collisions.

[0113] The stable node integral calculation method for explicit analysis of metal volume forming provided in this embodiment has the following advantages:

[0114] 1. Traditional linear elements based on unstructured meshes such as triangles / tetrahedrons struggle to guarantee computational accuracy, while structured meshes based on quadrilaterals / hexahedrons lack good discretization adaptability. The computational method presented in this embodiment constructs stable nodal integration elements based on unstructured meshes. While adapting well to complex geometric boundaries, it significantly improves computational accuracy, to some extent resolving the contradiction between computational accuracy and efficiency, and providing a new approach for solving practical engineering problems with complex boundaries.

[0115] 2. Based on the hourglass force constructed by the relative deformation of nodes and adjacent nodes within the nodal integration domain, a stable nodal integration method with single integration point characteristics is developed. This reduces the number of manual integration points required by the original stable nodal integration method. It can maintain the high accuracy and high resistance to mesh distortion of the stable nodal integration method while significantly improving computational efficiency, providing a new method for rapid prediction of explicit dynamic response of large dynamic deformation.

[0116] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0117] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A method for calculating stable node integrals for explicit analysis of metal volume forming, characterized in that, include: S1: Divide the problem domain into a triangular or tetrahedral mesh, and divide the mesh integration domain into multiple non-overlapping nodal integration domains based on the nodes in the triangular or tetrahedral mesh; the problem domain is a continuous metal billet; the node attributes include triangular / tetrahedral strain components, Young's modulus, Poisson's ratio, displacement vector, and acceleration vector; In a triangular or tetrahedral mesh, by sequentially connecting the midpoints of the surrounding edges of a node to the midpoints of the surrounding triangular elements, multiple non-overlapping nodal integration domains centered on the node are obtained. S2: Based on the triangular / tetrahedral strain components of the nodes, the nodal integration domain is smoothed to obtain the smooth domain; S3: Construct hourglass control between nodes and adjacent nodes in a smooth domain, and calculate the hourglass force between nodes and adjacent nodes; S4: Construct an elastic matrix based on the Young's modulus and Poisson's ratio of the nodes. Multiply the triangular / tetrahedral strain components of the nodes with the corresponding elastic matrix to obtain the stress of the nodes. Calculate the deformation gradient based on the displacement vector of the nodes. Perform nodal integration on the nodes based on the stress and deformation gradient to obtain the original internal forces of the nodes. S5: The hourglass force on the node is superimposed with the original internal force of the node to obtain the internal force of the stable node integral method. Based on the internal force of the stable node integral method and the acceleration vector of the node, the discrete motion equation in the explicit dynamics is solved to obtain the dynamic response.

2. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 1, characterized in that, In S2, the node-based triangular / tetrahedral strain component-based node-domain smoothing of the node integration domain includes: ； in, Let denote the derivative of the smooth shape function in the nodal integration domain of the i-th node; Represents the area / volume of the i-th node; Let represent the derivative of the shape function in the nodal integration field of the i-th node; Represents the gradient; Represents the outward normal of the surrounding edges / surfaces of the i-th node; This represents the surrounding edges / surfaces of the i-th node.

3. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 1, characterized in that, S3 includes: Based on the deformation gradient increment between any node and its neighboring nodes in the smooth domain, calculate the theoretical relative positional relationship between the node and its neighboring nodes. Based on the position of a node within a smooth domain and the positions of its neighboring nodes, calculate the actual relative positional relationship between the node and its neighboring nodes. Calculate the relative positional deviation between the theoretical and actual relative positional relationships; The elongation from the node to the adjacent node is calculated based on the relative position deviation. Differentiating the potential energy function of the virtual spring with respect to its length yields the virtual spring force. The internal forces between the node and its adjacent nodes are calculated based on the elongation and virtual spring force. The internal forces between symmetrical nodes and adjacent nodes are used to obtain the hourglass force between nodes and adjacent nodes.

4. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 3, characterized in that, The calculation of the elongation from the node to the adjacent node based on the relative position deviation includes: ； ； in, Indicates the first i The node to its first j The elongation of each adjacent node; Indicates the first Time step i The relative positional deviation of each node; Indicates the first Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Indicates L1 normal form; Indicates the first Time step node i and j The direction vector between nodes.

5. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 4, characterized in that, The internal forces between nodes and adjacent nodes are calculated based on the elongation and virtual spring force. The calculation formula is as follows: ； in, Indicates the first i Internal forces on each node side; Indicates the first i Young's modulus of each node; Indicates the first t Time step i The node and its first j The actual relative positional relationship between adjacent nodes; Represents a node i The volume / area of ​​the smooth domain.

6. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 5, characterized in that, The formula for calculating the hourglass force is: ； in, Indicates the first i The hourglass force between a node and all its neighboring nodes; Indicates the first i The node and its first j Hourglass force between adjacent nodes; Indicates the first i The number of neighboring nodes of each node; Indicates the first j Young's modulus of each node; Indicates the first i The node of the first j The elongation from each adjacent node to the i-th node; Indicates the hourglass coefficient; Represents a node j The volume / area of ​​the smooth domain.

7. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 1, characterized in that, The original internal force calculation formula for the node is: ； ； ； in, This represents the original internal force at the i-th node; Represents a node k The smooth region; Indicates node I pair j Partial derivatives of smooth shape functions in the direction; Represents a node j With nodes i The internal forces between them; k Represents a node k ; This represents the deformation of node k; Represents a node k Volume / area of ​​the smooth domain; Indicates node I pair j Partial derivatives of the shape function in the direction; The determinant representing the deformation gradient; Represents a node j With nodes k Deformation gradient between; Represents a node k With nodes i The stress between them.

8. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 7, characterized in that, The formula for calculating the deformation gradient is: ； in, Indicates the first i The node and its first j The elongation between adjacent nodes; Indicates partial derivative; This represents the nodal integration field of the i-th node; Indicates the number of influential nodes around a node; Represents a node j The deformation; This represents the displacement vector of a node.

9. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 1, characterized in that, The discrete motion equations in explicit dynamics for solving internal forces based on the stable nodal integration method include: Calculate the external forces acting on the nodes using the stable nodal integration method; The difference between the external forces and internal forces of a node under the stable node integration method is used as the external force vector in the discrete equation of motion. The external force vector is the product of the lumped mass matrix and the acceleration vector, and the dynamic response is obtained from this.

10. The method for calculating stable node integrals for explicit analysis of metal volume forming according to claim 1, characterized in that, The formula for calculating the external forces acting on a node using the stable nodal integration method is: ； in, Let represent the external force acting on the i-th node using the stable nodal integration method; This represents the nodal integration field of the i-th node; b represents the smooth region of nodes; b represents the body force. Indicates the boundary of the problem domain; Represents surface force.

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