A unified hourglass control method for low-order elements under one-point integration

Abstract

The present application relates to a unified hourglass control method for low-order units under one-point integration, comprising: setting the elastic modulus and Poisson's ratio of a metal elastic material, and performing interpolation processing on the entity units and displacement fields divided from the metal elastic material respectively, and calculating the covariant strain in a covariant coordinate system based on the shape functions of the interpolated entity units and the displacement field; performing tensor transformation on the covariant strain to obtain the overall strain in an overall Cartesian coordinate system; representing the elements in the covariant strain of each node with the gradient vector of the shape function, and unifying the format of each covariant strain to obtain a strain matrix; substituting the overall strain into the Galerkin format of the control equation of the linear elastic problem to obtain a discrete equation; 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 the entity unit; and calculating the 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:

[0004] 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.

[0005] 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;

[0006] S3: Substitute the global strain into the Galerkin format of the linear elastic problem governing equations to obtain the discretized equations; substitute the strain matrix into the stiffness matrix in the discretized equations and decompose the stiffness matrix into the constant term stiffness matrix and the unified hourglass stiffness matrix of the solid element;

[0007] S4: Based on the strain matrix of any solid element, the corresponding unified hourglass stiffness matrix is ​​calculated, and the corresponding stiffness matrix is ​​obtained by adding it to the constant term stiffness matrix. A load is applied to the metal elastic material, and the displacement vectors of all solid elements in the corresponding discrete equation are calculated based on the load and the stiffness matrix corresponding to any solid element. The displacement vector is used to reflect the deformation of the metal elastic material.

[0008] Preferably, performing interpolation processing on the entity element and the displacement field respectively includes:

[0009] ;

[0010] ;

[0011] 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.

[0012] Preferably, calculating the covariant strain in the covariant coordinate system based on the shape function of the interpolated solid element and the displacement field includes:

[0013] ;

[0014] in, Indicates the covariant strain in the covariant coordinate system. Row, No. Elements of the column, ; represents partial derivative; Represents the solid element in the generalized curvilinear coordinate system shape functions on ; Represents the displacement field 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 Curvilinear coordinates; Indicates the Curvilinear coordinates.

[0015] Preferably, performing tensor transformation on the covariant strain to obtain the global strain in the global Cartesian coordinate system includes:

[0016] ;

[0017] in, represents the first global strain in the global Cartesian coordinate system. Row, No. Elements of the column; Indicates the covariant strain in the covariant coordinate system. Row, No. Elements of the column; Indicates the The contravariant basis vectors of the curvilinear coordinates; Indicates the The contravariant basis vectors of the curvilinear coordinates.

[0018] Preferably, expressing the elements of the covariant strain of each node by the gradient vector of the shape function includes:

[0019] ; ; ; ;

[0020] in, Indicates the Covariant strain of nodes; Indicates the The gradient vector of the shape function of each node in the first curve coordinate; 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 The gradient vector of the shape function of each node in the second curve coordinate; Indicates the The gradient vector of the shape function of each node in the third curve coordinate; represents partial derivative.

[0021] Preferably, the formats of the covariant strains are unified to obtain the strain matrix including:

[0022] The covariant strains of all nodes are linearized to obtain the strain matrix corresponding to the solid element, which is expressed as:

[0023] ;

[0024] 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.

[0025] Preferably, the Galerkin format of the governing equations of the linear elastic problem is expressed as:

[0026] ;

[0027] 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;

[0028] Substituting the overall strain into the Galerkin format of the linear elastic problem governing equations, we obtain the discrete equations, which are expressed as:

[0029] ;

[0030] in, represents the stiffness matrix; Represents the displacement vector of all solid elements; Represents the load applied to metallic elastic materials.

[0031] Preferably, substituting the strain matrix into the stiffness matrix in the discretized equation and decomposing the stiffness matrix into a constant term stiffness matrix and a unified hourglass stiffness matrix of the solid element comprises:

[0032] ;

[0033] ;

[0034] ;

[0035] in, 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; The combined term representing the product of the coefficient matrix and the curve coordinates; represents transpose; represents the elasticity matrix; Indicates the The volume of a solid element.

[0036] Beneficial effects: The unified hourglass stiffness matrix constructed by dividing the metal elastic material into various solid units in this method fully utilizes the constant terms and linear terms in the unit deformation theory and removes the hourglass coefficient, so that the displacement vector of the solid unit is not affected by the inherent hourglass coefficient, thereby improving the stability of the simulation analysis and thus improving the design efficiency of the product; the unified hourglass stiffness matrix based on the unified format of various solid units greatly improves the computational efficiency and stability of the one-point integral in the numerical simulation analysis of metal elastic materials, and is applicable to hexahedral units, triangular prism units and pyramid units, greatly improving the computational stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0038] Figure 1 This is a flow chart of a unified hourglass control method for low-order units under one-point integration in an embodiment of the present application.

[0039] Figure 2a Schematic diagram of hexahedral units divided by metal elastic material in an embodiment of the present application.

[0040] Figure 2b Schematic diagram of triangular prism units divided by metal elastic material in an embodiment of the present application.

[0041] Figure 2c Schematic diagram of pyramid units divided by metal elastic material in an embodiment of the present application.

[0042] Figure 3 This is a displacement cloud diagram of all entity units divided from the cantilever beam in the embodiment of the present application. DETAILED DESCRIPTION

[0043] To make the above-mentioned objects, features, and advantages of the present application more clearly understood, the specific embodiments of the present application are described in detail below with reference to the accompanying drawings. The following description sets forth many specific details to facilitate a full understanding of the present application. However, the present application can be implemented in many other ways than those described herein, and those skilled in the art can make similar improvements without violating the scope of the present application. Therefore, the present application is not limited to the specific embodiments disclosed below.

[0044] 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 the technical features being referred to. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of such features. Throughout the description of this application, "plurality" means at least two, for example, two, three, etc., unless otherwise specifically defined.

[0045] like Figure 1 As shown, this embodiment provides a unified hourglass control method for low-order units under one-point integration, including:

[0046] S1: Set the elastic modulus and Poisson's ratio of the metal elastic material, and interpolate the entity elements and displacement fields divided by the metal elastic material. Each node on the entity element corresponds to any point on the metal elastic material. Calculate the covariant strain in the covariant coordinate system based on the shape function of the interpolated entity element and displacement field. Perform tensor transformation on the covariant strain to obtain the overall strain in the overall Cartesian coordinate system. The entity element includes hexahedral element, triangular prism element, and pyramid element. Its structure is as follows: Figure 2a 、 Figure 2b 、 Figure 2c shown.

[0047] Specifically, the interpolation processing of the entity element and the displacement field includes:

[0048] ;

[0049] ;

[0050] 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.

[0051] In this embodiment, although the unified hourglass control is limited to low-order units of elastic materials, the unified hourglass control can be replaced with low-order units of any other materials according to actual needs.

[0052] Furthermore, the covariant strain in the covariant coordinate system is calculated based on the shape function of the interpolated solid element and the displacement field, including:

[0053] ;

[0054] in, Indicates the covariant strain in the covariant coordinate system Row, No. Elements of the column, ; represents partial derivative; Represents the solid element in the generalized curvilinear coordinate system shape functions on ; Represents the displacement field 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 Curvilinear coordinates; Indicates the Curvilinear coordinates.

[0055] Furthermore, the covariant strain is transformed into a tensor, and the global strain in the global Cartesian coordinate system is obtained, which includes:

[0056] ;

[0057] in, represents the first global strain in the global Cartesian coordinate system. Row, No. Elements of the column; Indicates the covariant strain in the covariant coordinate system Row, No. Elements of the column; Indicates the The converse basis vectors of the curvilinear coordinates; Indicates the The contravariant basis vectors of the curvilinear coordinates.

[0058] S2: The elements of the covariant strain of each node are expressed by the gradient vector of the shape function, and the format of each covariant strain is unified to obtain the strain matrix.

[0059] Specifically, the elements of the covariant strain of each node are represented by the gradient vector of the shape function, including:

[0060] ; ; ; ;

[0061] in, Indicates the Covariant strain of nodes; Indicates the The gradient vector of the shape function of each node in the first curve coordinate; 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 The gradient vector of the shape function of each node in the second curve coordinate; Indicates the The gradient vector of the shape function of each node in the third curve coordinate; represents partial derivative.

[0062] Furthermore, the formats of the covariant strains are unified, and the strain matrix is ​​obtained including:

[0063] The covariant strains of all nodes are linearized to obtain the strain matrix corresponding to the solid element, which is expressed as:

[0064] ;

[0065] 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.

[0066] For example, set the integration point at the center point of the solid element and calculate the strain matrix at the center point:

[0067] The center point of the hexahedral element in the generalized curvilinear coordinate system is (0,0,0). Substituting the center point of the hexahedral element into the calculation formula of the strain matrix, the strain matrix at the center point of the hexahedral element is:

[0068] ;

[0069] in, represents the strain matrix at the center point of the hexahedral element; represents the first coefficient matrix;

[0070] The center point of the triangular prism element in the generalized curvilinear coordinate system is (1 / 3, 1 / 3, 0). Substituting the center point of the triangular prism element into the calculation formula of the strain matrix, the strain matrix at the center point of the triangular prism element is obtained as follows:

[0071] ;

[0072] in, 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;

[0073] The center point of the pyramid unit in the generalized curvilinear coordinate system is (0, 0, -3 / 5). Substituting the center point of the pyramid unit into the calculation formula of the strain matrix, the strain matrix at the center point of the pyramid unit is obtained as follows:

[0074] ;

[0075] in, represents the strain matrix at the center point of the pyramid unit; Represents the fourth coefficient matrix.

[0076] S3: Substitute the global strain into the Galerkin format of the linear elastic problem governing equations to obtain the discretized equations; substitute the strain matrix into the stiffness matrix in the discretized equations and decompose the stiffness matrix into the constant term stiffness matrix and the unified hourglass stiffness matrix of the solid element.

[0077] Specifically, the Galerkin format of the governing equations of the linear elastic problem is expressed as:

[0078] ;

[0079] 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;

[0080] Substituting the overall strain into the Galerkin format of the linear elastic problem governing equations, we obtain the discrete equations, which are expressed as:

[0081] ;

[0082] in, represents the stiffness matrix; Represents the displacement vector of all solid elements; Represents the load applied to metallic elastic materials.

[0083] Furthermore, the strain matrix is ​​substituted into the stiffness matrix in the discretization equation, and the stiffness matrix is ​​decomposed into the constant term stiffness matrix and the unified hourglass stiffness matrix of the solid element, including:

[0084] ;

[0085] ;

[0086] ;

[0087] in, 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; The combined term representing the product of the coefficient matrix and the curve coordinates; represents transpose; represents the elasticity matrix; Indicates the The volume of a solid element.

[0088] Furthermore, the stiffness matrix can be expressed as:

[0089] ;

[0090] Integrating the above formula based on the hexahedral element, the integral formula is:

[0091] ;

[0092] ;

[0093] Integrating the above formula based on the triangular prism unit, the integral formula is:

[0094] ;

[0095] ;

[0096] Integrating the above formula based on the pyramid unit, the integral formula is:

[0097] ;

[0098] ;

[0099] in, Represents an enumeration operation, which operates on each variable.

[0100] Expand the unified hourglass stiffness matrices of the three solid elements separately.

[0101] The unified hourglass stiffness matrix of the hexahedral element is expanded as:

[0102] ;

[0103] The unified hourglass stiffness matrix of the triangular prism element is expanded as follows:

[0104] ;

[0105] The unified hourglass stiffness matrix of the pyramid element is expanded as:

[0106] ;

[0107] in, represents the unified hourglass stiffness matrix of the hexahedral element; represents the unified hourglass stiffness matrix of the triangular prism element; Represents the unified hourglass stiffness matrix of the pyramid element.

[0108] The coordinate coefficients in the expansion of the unified hourglass stiffness matrix of each solid element are calculated according to the following integral:

[0109] ; ;

[0110] ; ;

[0111] ; ;

[0112] ; ;

[0113] ;

[0114] ;

[0115] S4: Based on the strain matrix of any entity element, the corresponding unified hourglass stiffness matrix is ​​calculated, and the corresponding stiffness matrix is ​​obtained by adding it to the constant term stiffness matrix. A load is applied to the metal elastic material. Based on the load and the stiffness matrix corresponding to any entity element, the displacement vectors of all entity elements in the corresponding discrete equation are calculated. The displacement vectors are used to reflect the deformation of the metal elastic material.

[0116] Taking the 3D cantilever beam as an example, the elastic modulus of the cantilever beam is set to 2.1×10 9 Pa, Poisson's ratio is 0.3, and the elastic matrix is ​​calculated based on the elastic modulus and Poisson's ratio set for the cantilever beam. , and through the elastic matrix , the first coefficient matrix and the combined product of the coefficient matrix and the curve coordinates , calculate the stiffness matrix corresponding to any solid element divided by the cantilever beam ; Apply 1N / m to the cantilever beam 2 The load and stiffness matrix are substituted into the discrete equation to obtain the displacement vectors of all solid elements divided by the cantilever beam. , its cloud map is as follows Figure 3 As shown in the figure, the red highlighted area is the maximum deformation area, and the blue area is the rigid area. The displacement vectors of all solid elements can directly reflect the deformation of the solid elements under load, which helps to intuitively evaluate the stiffness and stability of metal elastic materials and avoid failure due to excessive deformation. The unified hourglass stiffness matrix, which eliminates the influence of the hourglass coefficient, greatly improves the stability of displacement vector calculation. The calculated displacement vectors provide core data support for the simulation analysis of metal elastic materials and subsequent product design, safety assessment, and optimization.

[0117] The unified hourglass control method for low-order units under one-point integration provided in this embodiment has the following beneficial effects:

[0118] 1. A parameter-free hourglass control method is obtained based on a unified calculation format. It is applicable to hexahedron, triangular prism and pyramid elements. One-point integration greatly improves the calculation efficiency, and the parameter-free hourglass stability term greatly improves the calculation stability.

[0119] 2. This parameter-free hourglass control method makes full use of the constant and linear terms in the element deformation theory and removes the hourglass coefficient, so that the displacement vector of the solid element is not affected by the inherent hourglass coefficient, which improves the stability of the simulation analysis and thus improves the design efficiency of the product. It is a very effective method for handling hourglass control and is suitable for implicit and explicit calculations of structural simulation.

[0120] The technical features of the above-mentioned embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above-mentioned 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.

[0121] The above-described embodiments merely represent several implementation methods of the present application. 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 a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present patent application shall be determined by the appended claims.

Claims

1. A unified hourglass control method for low-order units 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 low-order units under one-point integration according to claim 1, characterized in that: 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 units under 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: ； in, Indicates the covariant strain in the covariant coordinate system. Row, No. Elements of the column, ; represents partial derivative; Represents the solid element in the generalized curvilinear coordinate system shape functions on ; Represents the displacement field 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 Curvilinear coordinates; Indicates the Curvilinear coordinates.

4. The unified hourglass control method for low-order units under 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: ； in, represents the first global strain in the global Cartesian coordinate system. Row, No. Elements of the column; Indicates the covariant strain in the covariant coordinate system. Row, No. Elements of the column; Indicates the The converse basis vectors of the curvilinear coordinates; Indicates the The contravariant basis vectors of the curvilinear coordinates.

5. The unified hourglass control method for low-order units under one-point integration according to claim 1, characterized in that: The elements of the covariant strain of each node are expressed as the gradient vector of the shape function, including: ； ； ； ； in, Indicates the Covariant strain of nodes; Indicates the The gradient vector of the shape function of each node in the first curve coordinate; 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 The gradient vector of the shape function of each node in the second curve coordinate; Indicates the The gradient vector of the shape function of each node in the third curve coordinate; represents 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 low-order units 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: ； in, represents the stiffness matrix; Represents the displacement vector of all solid elements; Represents the load applied to metallic elastic materials.

8. The unified hourglass control method for low-order units 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: ； ； ； in, 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; The combined term representing the product of the coefficient matrix and the curve coordinates; represents transpose; represents the elasticity matrix; Indicates the The volume of a solid element.

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