Unified hourglass control method for low-order unit under one-point integration

Abstract

The invention relates to a unified hourglass control method for a low-order unit under one-point integration, and the method comprises the steps: setting the elastic modulus and Poisson's ratio of a metal elastic material, and carrying out the interpolation processing of an entity unit and a displacement field divided from the metal elastic material, calculating covariant strain under a covariant coordinate system based on the interpolated entity unit and the shape function of the displacement field; performing tensor conversion on the covariant strain to obtain overall strain under an overall Cartesian coordinate system; representing elements in the covariant strain of each node by using a gradient vector of a shape function, and unifying the format of each covariant strain to obtain a strain matrix; substituting the overall strain into a Galerkin format of a linear elasticity problem control equation to obtain a discrete equation; substituting the strain matrix into a stiffness matrix in the discrete equation, and decomposing the stiffness matrix into a constant term stiffness matrix and a unified hourglass stiffness matrix of an entity unit; and calculating a corresponding unified hourglass stiffness matrix based on the strain matrix of any entity unit.

Description

Technical Field

[0001] The present application relates to the technical field of numerical simulation analysis, and in particular to a unified hourglass control method for low-order units under one-point integration. Background Art

[0002] The essence of the hourglass phenomenon is insufficient element integration, which causes different deformation modes of the element to correspond to the same deformation gradient. This allows the element to switch arbitrarily between different deformation modes under the same deformation gradient, resulting in computational instability. Hourglass control suppresses hourglass deformation by adding artificial stiffness, ensuring that any deformation gradient of the element corresponds to only one deformation mode. Simply put, in the numerical simulation analysis of metal elastic materials, after applying one-point integration to the quadrilateral / hexahedral elements that divide the metal elastic material, the internal gradient field of the element is constant, allowing arbitrary linear variations in all directions. This leads to instability in the numerical simulation analysis of metal elastic materials and, in turn, instability in the analysis of the actual internal forces of the metal elastic material, affecting product design efficiency. For example, when hourglassing occurs in a local solid element of an automotive suspension spring, it distorts the stress and strain distribution, especially in high shear or bending regions. This leads to underestimation of actual stress and miscalculation of spring life, ultimately affecting the design efficiency of the automotive suspension spring product. Furthermore, existing solid element hourglass control is influenced by the hourglass coefficient, which directly affects simulation analysis results. Improper setting of the hourglass coefficient can lead to simulation analysis distortion, seriously affecting product design. Therefore, it is urgent to uniformly control the hexahedral elements, triangular prism elements and pyramid elements without introducing the hourglass coefficient, so as to improve the stability of simulation analysis and thus improve the efficiency of product design. This is a scientific problem that needs to be solved urgently. Summary of the Invention

[0003] Based on this, it is necessary to provide a unified hourglass control method for low-order units under one-point integration, including: S1: Set the elastic modulus and Poisson's ratio of the metal elastic material, and interpolate the solid elements and displacement fields divided from the metal elastic material. Each node on the solid element corresponds to an arbitrary point on the metal elastic material. Calculate the covariant strain in the covariant coordinate system based on the shape function of the interpolated solid element and displacement field. Perform tensor transformation on the covariant strain to obtain the global strain in the global Cartesian coordinate system. The solid elements include hexahedral elements, triangular prism elements, and pyramid elements. S2: The elements of the covariant strain of each node are expressed as the gradient vector of the shape function, and the format of each covariant strain is unified to obtain the strain matrix; S3: Substitute the overall strain into the Galerkin form of the governing equations for linear elastic problems to obtain the discrete equations; substitute the strain matrix into the stiffness matrix in the discrete equations, and decompose the stiffness matrix into the stiffness matrix of the constant term and the unified hourglass stiffness matrix of the solid elements. S4: Calculate the corresponding unified hourglass stiffness matrix based on the strain matrix of any solid element, add it to the stiffness matrix of the constant term to obtain the corresponding stiffness matrix, apply a load to the metallic elastic material, and calculate the displacement vectors of all solid elements in the corresponding discrete equations based on the load and the stiffness matrix corresponding to any solid element. The displacement vectors are used to reflect the deformation of the metallic elastic material.

[0004] Preferably, the interpolation processing of the solid elements and the displacement field respectively includes: ; ; wherein, represents the shape function of the solid element in the generalized curvilinear coordinate system ; represents the number of nodes on the solid element; represents the -th node's shape function in the generalized curvilinear coordinate system ; represents the first curvilinear coordinate; represents the second curvilinear coordinate; represents the third curvilinear coordinate; represents the abscissa of the -th node in the overall Cartesian coordinate system; represents the ordinate of the -th node in the overall Cartesian coordinate system; represents the vertical coordinate of the -th node in the overall Cartesian coordinate system; represents the shape function of the displacement field in the generalized curvilinear coordinate system ; represents the displacement component of the -th node on the horizontal axis; represents the displacement component of the -th node on the vertical axis; represents the displacement component of the -th node on the vertical axis.

[0005] Preferably, calculating the covariant strain in the covariant coordinate system based on the shape functions of the interpolated solid elements and the displacement field includes: ; wherein, represents the -th row and the The elements of the column ; Indicates the partial derivative; Indicates the shape function of the solid element in the generalized curvilinear coordinate system ; Indicates the shape function of the displacement field in the generalized curvilinear coordinate system ; Indicates the first curvilinear coordinate; Indicates the second curvilinear coordinate; Indicates the third curvilinear coordinate; Indicates the th curvilinear coordinate; Indicates the th curvilinear coordinate.

[0006] Preferably, a tensor transformation is performed on the covariant strain to obtain the overall strain in the global Cartesian coordinate system, including: ; wherein, Indicates the element in the th row and th column of the overall strain in the global Cartesian coordinate system; Indicates the element in the th row and th column of the covariant strain in the covariant coordinate system; Indicates the contravariant basis vector of the th curvilinear coordinate; Indicates the contravariant basis vector of the th curvilinear coordinate.

[0007] Preferably, expressing the elements in the covariant strain of each node in terms of the gradient vector of the shape function includes: ; ; ; ; wherein, Indicates the covariant strain of the th node; Indicates the gradient vector of the shape function of the th node with respect to the first curvilinear coordinate; Indicates the shape function of the th node in the generalized curvilinear coordinate system ; Indicates the first curvilinear coordinate; Indicates the second curvilinear coordinate; Indicates the third curvilinear coordinate; Indicates the gradient vector of the shape function of the th node with respect to the second curvilinear coordinate; Denote the gradient vector of the shape function of the th node on the third curvilinear coordinate; Denote the partial derivative.

[0008] Preferably, unify the format of each covariant strain to obtain a strain matrix including: Linearly organize the covariant strains of all nodes to obtain the strain matrix corresponding to the solid element, denoted as: ; Wherein, Denote the strain matrix of any solid element in the generalized curvilinear coordinate system ; Denote the first coefficient matrix; Denote the second coefficient matrix; Denote the third coefficient matrix; Denote the fourth coefficient matrix; Denote the fifth coefficient matrix; Denote the sixth coefficient matrix; Denote the seventh coefficient matrix; Denote the first curvilinear coordinate; Denote the second curvilinear coordinate; Denote the third curvilinear coordinate; Denote the combined term of the product of the coefficient matrix and the curvilinear coordinate.

[0009] Preferably, the Galerkin format of the control equation for the linear elastic problem is expressed as: ; Wherein, Denote the volume of the solid element; Denote the trial strain; Denote trial functions of nodes; Denote the concentrated force acting on the th node; Denote the body force of nodes; Denote the element in the th row and th column of the overall stress in the overall Cartesian coordinate system; ; Wherein, Denote the stiffness matrix; Denote the displacement vector of all solid elements; Denote the load applied to the metal elastic material.

[0010] Preferably, substituting the strain matrix into the stiffness matrix in the discrete equation and decomposing the stiffness matrix into a constant-term stiffness matrix and a unified hourglass stiffness matrix of solid elements includes: ; ; ; Among them, represents the stiffness matrix; represents the constant-term stiffness matrix; represents the unified hourglass stiffness matrix of solid elements; represents the first coefficient matrix; represents the combined term of the product of the coefficient matrix and the curvilinear coordinates; represents the transpose; represents the elastic matrix; represents the th volume of the solid element.

[0011] Beneficial effects: In this method, the unified hourglass stiffness matrix constructed for various solid elements divided from the metal elastic material makes full use of the constant term and linear term in the element deformation theory, removes the hourglass coefficient, so that the displacement vector of the solid element is not affected by the inherent hourglass coefficient during calculation, improving the stability of the simulation analysis, and further improving the design efficiency of the product; based on the unified hourglass stiffness matrix with a unified format for various solid elements, the calculation efficiency and stability of the one-point integration in the numerical simulation analysis of the metal elastic material are greatly improved, which is applicable to hexahedron elements, triangular prism elements and pyramid elements, greatly improving the calculation stability. Brief Description of the Drawings

[0012] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0013] Figure 1 is a flowchart of the unified hourglass control method for low-order elements under one-point integration in the embodiments of the present application.

[0014] Figure 2a is a schematic diagram of a hexahedron element divided from a metal elastic material in the embodiments of the present application.

[0015] Figure 2b is a schematic diagram of a triangular prism element divided from a metal elastic material in the embodiments of the present application.

[0016] Figure 2c Schematic diagram of the pyramid unit divided by the metal elastic material in the embodiment of the present application.

[0017] Figure 3 Displacement nephogram of all solid elements divided by the cantilever beam in the embodiment of the present application. Detailed implementation manners

[0018] To make the above objects, features, and advantages of the present application more obvious and understandable, the following will describe the detailed implementation manners of the present application with reference to the accompanying drawings. Many specific details are set forth in the following description to fully understand the present application. However, the present application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without departing from the connotation of the present application. Therefore, the present application is not limited by the specific embodiments disclosed below.

[0019] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be construed as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include at least one of such features. In the description of the present application, "a plurality" means at least two, such as two, three, etc., unless otherwise specifically defined.

[0020] As Figure 1 shown, this embodiment provides a unified hourglass control method for low-order elements under one-point integration, including: S1: Set the elastic modulus and Poisson's ratio of the metal elastic material, and perform interpolation processing on the solid elements and displacement fields divided by the metal elastic material respectively. Each node on the solid element corresponds to an arbitrary point on the metal elastic material; calculate the covariant strain in the covariant coordinate system based on the shape functions of the interpolated solid elements and displacement fields; perform tensor transformation on the covariant strain to obtain the overall strain in the overall Cartesian coordinate system. The solid elements include hexahedron elements, triangular prism elements, and pyramid elements, and their structures are as Figure 2a , Figure 2b , Figure 2c shown.

[0021] Specifically, performing interpolation processing on the solid elements and displacement fields respectively includes: ; ; Among them, represents the shape function of the solid element in the generalized curvilinear coordinate system ; represents the number of nodes on the solid element; represents the The shape functions of a node in the generalized curvilinear coordinate system ; Denotes the first curvilinear coordinate; Denotes the second curvilinear coordinate; Denotes the third curvilinear coordinate; Denotes the abscissa of the th node in the global Cartesian coordinate system; Denotes the ordinate of the th node in the global Cartesian coordinate system; Denotes the applicate of the th node in the global Cartesian coordinate system; Denotes the shape functions of the displacement field in the generalized curvilinear coordinate system ; Denotes the th displacement component of the node on the horizontal axis; Denotes the th displacement component of the node on the vertical axis; Denotes the th displacement component of the node on the applicate axis.

[0022] In this embodiment, although only the unified hourglass control for low-order elements of elastic materials is defined, it can be replaced with the unified hourglass control for low-order elements of any other material according to actual needs.

[0023] Furthermore, calculating the covariant strain in the covariant coordinate system based on the shape functions of the interpolated solid elements and the displacement field includes: ; where, Denotes the element in the th row and th column of the covariant strain in the covariant coordinate system, ; Denotes the partial derivative; Denotes the shape functions of the solid elements in the generalized curvilinear coordinate system ; Denotes the shape functions of the displacement field in the generalized curvilinear coordinate system ; Denotes the first curvilinear coordinate; Denotes the second curvilinear coordinate; Denotes the third curvilinear coordinate; Denotes the th curvilinear coordinate; Denotes the th curvilinear coordinate.

[0024] Even further, performing a tensor transformation on the covariant strain to obtain the global strain in the global Cartesian coordinate system includes: ; Among them, represents the element in the th row and th column of the overall strain in the overall Cartesian coordinate system; represents the element in the th row and th column of the covariant strain in the covariant coordinate system; represents the contravariant basis vector of the th curvilinear coordinate; represents the contravariant basis vector of the th curvilinear coordinate.

[0025] S2: Express the elements in the covariant strain of each node using the gradient vector of the shape function, and unify the format of each covariant strain to obtain the strain matrix.

[0026] Specifically, expressing the elements in the covariant strain of each node using the gradient vector of the shape function includes: ; ; ; ; Among them, represents the covariant strain of the th node; represents the gradient vector of the shape function of the th node in the first curvilinear coordinate; represents the shape function of the th node in the generalized curvilinear coordinate system ; represents the first curvilinear coordinate; represents the second curvilinear coordinate; represents the third curvilinear coordinate; represents the gradient vector of the shape function of the th node in the second curvilinear coordinate; represents the gradient vector of the shape function of the th node in the third curvilinear coordinate; represents the partial derivative.

[0027] Furthermore, unifying the format of each covariant strain to obtain the strain matrix includes: Linearly organize the covariant strains of all nodes to obtain the strain matrix corresponding to the solid element, expressed as: ; Among them, represents the strain matrix of any solid element in the generalized curvilinear coordinate system ; represents the first coefficient matrix; represents the second coefficient matrix; represents the third coefficient matrix; represents the fourth coefficient matrix; represents the fifth coefficient matrix; represents the sixth coefficient matrix; represents the seventh coefficient matrix; represents the first curve coordinate; represents the second curve coordinate; represents the third curve coordinate; represents the combined term of the product of the coefficient matrix and the curve coordinate.

[0028] For example: Set the integration point at the center point of the solid element, and calculate the strain matrix at the center point: The center point of the hexahedron element in the generalized curve coordinate system is (0, 0, 0). Substitute the center point of the hexahedron element into the calculation formula of the strain matrix, and the strain matrix at the center point of the hexahedron element is obtained as: ; where, represents the strain matrix at the center point of the hexahedron element; represents the first coefficient matrix; The center point of the triangular prism element in the generalized curve coordinate system is (1 / 3, 1 / 3, 0). Substitute the center point of the triangular prism element into the calculation formula of the strain matrix, and the strain matrix at the center point of the triangular prism element is obtained as: ; where, represents the strain matrix at the center point of the triangular prism element; represents the second coefficient matrix; represents the third coefficient matrix; represents the fifth coefficient matrix; The center point of the pyramid element in the generalized curve coordinate system is (0, 0, -3 / 5). Substitute the center point of the pyramid element into the calculation formula of the strain matrix, and the strain matrix at the center point of the pyramid element is obtained as: ; where, represents the strain matrix at the center point of the pyramid element; represents the fourth coefficient matrix.

[0029] S3: Substitute the overall strain into the Galerkin form of the governing equation of the linear elastic problem to obtain the discrete equation; substitute the strain matrix into the stiffness matrix in the discrete equation, and decompose the stiffness matrix into the constant term stiffness matrix and the unified hourglass stiffness matrix of the solid element.

[0030] Specifically, the Galerkin form of the governing equations for the linear elastic problem is expressed as: ; Wherein, represents the volume of the solid element; represents the trial strain; represents trial functions at nodes; represents the concentrated force acting on the th node; represents the body force at nodes; represents the element in the th row and th column of the global stress in the global Cartesian coordinate system; ; Wherein, represents the stiffness matrix; represents the displacement vector of all solid elements; represents the load applied to the metallic elastic material.

[0031] Furthermore, substituting the strain matrix into the stiffness matrix in the discrete equation, the stiffness matrix is decomposed into a constant-term stiffness matrix and a unified hourglass stiffness matrix of the solid element, including: ; ; ; Wherein, represents the stiffness matrix; represents the constant-term stiffness matrix; represents the unified hourglass stiffness matrix of the solid element; represents the first coefficient matrix; represents the combined term of the product of the coefficient matrix and the curvilinear coordinates; represents the transpose; represents the elastic matrix; represents the th solid element volume.

[0032] Even further, the stiffness matrix can also be expressed as: ; Based on the hexahedral element, the integral of the above equation is: ; ; Integrate the above formula based on the triangular prism element, and the integral formula is: ; ; Integrate the above formula based on the pyramid element, and the integral formula is: ; ; Among them, represents the enumeration operation, and each variable therein is operated on.

[0033] Expand the unified hourglass stiffness matrices of the three types of solid elements respectively, The unified hourglass stiffness matrix of the hexahedron element expands to: ; The unified hourglass stiffness matrix of the triangular prism element expands to: ; The unified hourglass stiffness matrix of the pyramid element expands to: ; Among them, represents the unified hourglass stiffness matrix of the hexahedron element; represents the unified hourglass stiffness matrix of the triangular prism element; represents the unified hourglass stiffness matrix of the pyramid element.

[0034] Calculate the coordinate coefficients in the expansion formula of the unified hourglass stiffness matrix of each solid element according to the following integral: ; ; ; ; ; ; ; ; ; ; S4: Calculate the corresponding unified hourglass stiffness matrix based on the strain matrix of any one of the solid elements, add it to the constant term stiffness matrix to obtain the corresponding stiffness matrix, apply a load to the metal elastic material, and use the load and the stiffness matrix corresponding to any one of the solid elements to calculate the displacement vectors of all solid elements in the corresponding discrete equation. The displacement vectors are used to reflect the deformation of the metal elastic material.

[0035] Taking a 3D cantilever beam as an example, set the elastic modulus of the cantilever beam to 2.1×10 9 Pa, and the Poisson's ratio to 0.3. Calculate the elastic matrix according to the set elastic modulus and Poisson's ratio of the cantilever beam , and through the elastic matrix , the first coefficient matrix , and the combined term of the product of the coefficient matrix and the curvilinear coordinates , calculate the stiffness matrix corresponding to any kind of solid element divided from the cantilever beam ; Apply a load of 1 N / m 2 to the cantilever beam, substitute the load and the stiffness matrix into the discrete equation to obtain the displacement vector of all the solid elements divided from the cantilever beam. Its nephogram is as shown in Figure 3 . The red highlighted part in the figure is the maximum deformation area, and the blue part is the rigid area. The displacement vectors of all solid elements can directly reflect the deformation of the solid elements under the action of the load, which helps to visually evaluate the stiffness and stability of the metal elastic material and avoid failure caused by excessive deformation; The unified hourglass stiffness matrix that eliminates the influence of the hourglass coefficient greatly improves the stability of the displacement vector calculation. The calculated displacement vector provides core data support for the simulation analysis of metal elastic materials and subsequent product design, safety assessment, and optimization.

[0036] The unified hourglass control method for low-order elements under one-point integration provided in this embodiment has the following beneficial effects: 1. A parameterless hourglass control method is obtained based on a unified calculation format, which is applicable to hexahedron, triangular prism, and pyramid elements. One-point integration greatly improves the calculation efficiency, and the parameterless hourglass stabilization term greatly improves the calculation stability.

[0037] 2. This parameterless hourglass control method makes full use of the constant term and linear term in the element deformation theory, removes the hourglass coefficient, so that the calculation of the displacement vector of the solid element is not affected by the inherent hourglass coefficient, improves the stability of the simulation analysis, and then improves the design efficiency of the product. It is a very effective method for dealing with hourglass control and is applicable to implicit and explicit calculations of structural simulation.

[0038] The technical features of the above-described embodiments can be combined arbitrarily. For the sake of brevity of description, 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, it should be considered to be within the scope described in this specification.

[0039] The above-described embodiments merely represent several implementation manners of the present application. The description thereof is relatively specific and detailed, but it should not be construed as a limitation to the scope of the patented application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all fall within the protection scope of the present application. Therefore, the protection scope of the patent of the present application shall be subject to the appended claims.

Claims

1. A unified hourglass control method for low-order elements under one-point integration, characterized in that, include: S1: Set the elastic modulus and Poisson's ratio of the metal elastic material, and interpolate the solid elements and displacement fields divided from the metal elastic material. Each node on the solid element corresponds to an arbitrary point on the metal elastic material. Calculate the covariant strain in the covariant coordinate system based on the shape function of the interpolated solid element and displacement field. Perform tensor transformation on the covariant strain to obtain the global strain in the global Cartesian coordinate system. The solid elements include hexahedral elements, triangular prism elements, and pyramid elements. S2: The elements of the covariant strain of each node are expressed as the gradient vector of the shape function, and the format of each covariant strain is unified to obtain the strain matrix; S3: Substitute the global strain into the Galerkin format of the governing equations of the linear elastic problem to obtain the discretized equations; Substitute the strain matrix into the stiffness matrix in the discretization equation and decompose the stiffness matrix into the constant term stiffness matrix and the unified hourglass stiffness matrix of the solid element; S4: Based on the strain matrix of any solid element, the corresponding unified hourglass stiffness matrix is calculated and added to the constant term stiffness matrix to obtain the corresponding stiffness matrix; A load is applied to the metal elastic material, and the displacement vectors of all entity elements in the corresponding discrete equation are calculated based on the load and the stiffness matrix corresponding to any entity element. The displacement vector is used to reflect the deformation of the metal elastic material.

2. The unified hourglass control method for the lower-order element under one-point integration according to claim 1, wherein The interpolation processing of the solid element and displacement field includes: ； ； in, Represents the solid element in the generalized curvilinear coordinate system shape functions on ; Indicates the number of nodes on the solid element; Indicates the Nodes in the generalized curvilinear coordinate system shape functions on ; represents the first curvilinear coordinate; represents the second curvilinear coordinate; represents the third curvilinear coordinate; Indicates the first The horizontal coordinate of each node; Indicates the first The vertical coordinate of each node; Indicates the first The vertical coordinate of each node; Represents the displacement field in the generalized curvilinear coordinate system shape functions on ; Indicates the The displacement component of each node on the horizontal axis; Indicates the The displacement component of each node on the vertical axis; Indicates the The displacement component of a node on the vertical axis.

3. The unified hourglass control method for low-order elements with one-point integration according to claim 2, characterized in that, The covariant strains in the covariant coordinate system are calculated based on the shape functions of the interpolated solid elements and the displacement field, including: ； Among them, represents the element in the th row and th column of the covariant strain in the covariant coordinate system; ; represents the partial derivative; represents the shape function of the solid element in the generalized curvilinear coordinate system ; represents the shape function of the displacement field in the generalized curvilinear coordinate system ; represents the first curvilinear coordinate; represents the second curvilinear coordinate; represents the third curvilinear coordinate; represents the th curvilinear coordinate; represents the th curvilinear coordinate.

4. The unified hourglass control method for low-order elements with one-point integration according to claim 3, characterized in that, The tensor transformation of the covariant strain to obtain the global strain in the global Cartesian coordinate system includes: ； Among them, represents the element in the th row and th column of the total strain in the overall Cartesian coordinate system; represents the element in the th row and th column of the covariant strain in the covariant coordinate system; represents the contravariant basis vector of the th curvilinear coordinate; represents the contravariant basis vector of the th curvilinear coordinate.

5. The unified hourglass control method for low-order elements under one-point integration according to claim 1, wherein The elements of the covariant strain of each node are expressed as the gradient vector of the shape function, including: ； ； ； ； Among them, represents the covariant strain of the th node; represents the gradient vector of the shape function of the th node in the first curvilinear coordinate; represents the shape function of the th node in the generalized curvilinear coordinate system ; represents the first curvilinear coordinate; represents the second curvilinear coordinate; represents the third curvilinear coordinate; represents the gradient vector of the shape function of the th node in the second curvilinear coordinate; represents the gradient vector of the shape function of the th node in the third curvilinear coordinate; represents the partial derivative.

6. The unified hourglass control method for low-order units under one-point integration according to claim 1, characterized in that: Unify the format of each covariant strain and obtain the strain matrix including: The covariant strains of all nodes are linearized to obtain the strain matrix corresponding to the solid element, which is expressed as: ； in, Represents any entity element in the generalized curvilinear coordinate system The strain matrix on ; represents the first coefficient matrix; represents the second coefficient matrix; represents the third coefficient matrix; represents the fourth coefficient matrix; represents the fifth coefficient matrix; represents the sixth coefficient matrix; represents the seventh coefficient matrix; represents the first curvilinear coordinate; represents the second curvilinear coordinate; represents the third curvilinear coordinate; A combination of the coefficient matrix and the curve coordinates.

7. The unified hourglass control method for the lower-order element under one-point integration according to claim 6, characterized in that, The Galerkin format of the governing equations of linear elasticity is expressed as: ； in, Represents the volume of the solid element; Indicates test strain; express The test function of each node; Indicates that it acts on Concentrated force on a node; express The physical strength of each node; The first stress in the global Cartesian coordinate system is represented by Row, No. Elements of the column; Substituting the overall strain into the Galerkin format of the linear elastic problem governing equations, we obtain the discrete equations, which are expressed as: ； Among them, represents the stiffness matrix; represents the displacement vector of all solid elements; represents the load applied to the metallic elastic material.

8. The unified hourglass control method for the lower-order element under one-point integration according to claim 7, characterized in that, Substituting the strain matrix into the stiffness matrix in the discretization equation, the stiffness matrix is decomposed into the constant term stiffness matrix and the unified hourglass stiffness matrix of the solid element, including: ； ； ； Among them, represents the stiffness matrix; represents the constant term stiffness matrix; represents the unified hourglass stiffness matrix of the solid element; represents the first coefficient matrix; represents the combined term of the product of the coefficient matrix and the curvilinear coordinates; represents the transpose; represents the elastic matrix; represents the volume of the

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