A static analysis method for circular plates with holes based on circular energy elements on rectangular domains

Abstract

This application discloses a static analysis method for circular plates with holes based on circular energy elements on rectangular domains, belonging to the field of thin-walled structure technology. The method comprises: constructing a circular energy element based on global-local mapping technology; modeling the circular plate with holes based on the circular energy element and energy Boolean operations, and constructing boundary condition functions; constructing an energy functional for the static problem based on linear small deformation thin plate theory; and solving the static problem of the circular plate with holes based on global trial functions and the Ritz method. This method constructs a circular energy element capable of high-precision numerical integration based on global-local mapping technology, and proposes a new analytical solution algorithm for the static problem of two-dimensional circular plates with holes based on the circular energy element and energy Boolean operations within the framework of the Ritz method and global trial functions. This method establishes a standard energy functional and calculation procedure that is not affected by the plate's geometric configuration, thereby improving the accuracy and efficiency of the numerical solution of static problems within the domain of circular plates with holes.

Description

Technical Field

[0001] The present application relates to the technical field of thin-walled structures, and in particular to a static analysis method for circular perforated plates based on circular energy elements on a rectangular domain. Background Art

[0002] Solving the static problems of complex-shaped plates with holes using the Ritz method requires deriving the energy functional based on global trial functions and integrating the complex geometric domain. As the geometric configuration of the plate changes, the energy functional formula changes accordingly, making it difficult to standardize the solution process. Research has shown that by covering the plate domain with a rectangular domain, numerically integrating the true geometric domain of the plate within the rectangular domain and performing global interpolation based on complete orthogonal polynomials, it is possible to numerically simulate the deformation of arbitrarily shaped plates. However, the integral accuracy of generating Gaussian integral points within the entire rectangular domain to capture complex geometric boundaries is not high, which can easily lead to singularities and instabilities in the stiffness matrix. How to perform standardized high-precision numerical integration of complex geometric hole domains remains a challenge. In recent years, based on the Gaussian integral rule and mapping technology, relevant literature has studied numerical integration methods in arbitrary quadrilateral and regular polygonal domains. However, the geometric domain transformation process during numerical integration in these methods will cause the integral formula to change. Summary of the Invention

[0003] In response to the above-mentioned deficiencies in the prior art, the present application provides a static analysis method for circular perforated plates based on circular energy elements on rectangular domains, which solves the problem of standardization of the energy functional integral formula after the geometric domain of the circular perforated plate changes.

[0004] In order to achieve the above-mentioned invention objectives, the technical solutions adopted in this application are:

[0005] This application provides a static analysis method for a circular plate with a hole based on a circular energy element on a rectangular domain, including:

[0006] S1: Global mapping stage: construct a predefined rectangular domain to cover the circular aperture plate domain; map the predefined rectangular domain to a square solution domain, and map the circular aperture plate domain to a two-dimensional normalized aperture plate domain; local mapping stage: map the circular domain to an elliptical domain; map the elliptical domain to a normalized circular domain; map the normalized circular domain to a normalized square domain; generate Gaussian integral points and quadrature coefficients in the normalized square domain, and inversely solve the Gaussian integral points and quadrature coefficients to the dimensionless coordinate system of the global mapping stage to construct a circular energy element that simulates the energy on the circular area;

[0007] S2: Modeling the circular perforated plate based on the circular energy element and energy Boolean operation, and simulating boundary conditions;

[0008] S3: Constructing the total energy functional for static problems based on linear small deformation thin plate theory;

[0009] S4: Solve the static problem of a circular plate with a hole based on the global trial function and the Ritz method.

[0010] Furthermore, the S1 specifically includes:

[0011] S101: embed the circular perforated plate into a predefined rectangular domain, and perform global mapping on the predefined rectangular domain to obtain a square solution domain, wherein the global mapping is a dimensionless coordinate system transformation, that is, based on the following formula, the original x - y Coordinate system mapped to dimensionless coordinate system ζ-ζ :

[0012]

[0013] Where, and for xyz The length and width of the predefined rectangular domain in the coordinate system;

[0014] Correspondingly, after completing the coordinate transformation, the integrand The integral formula in the original elliptical hole plate domain Ω is transformed into:

[0015]

[0016] Where, for the reason xyz Coordinate system conversion to ζ-ζ Jacobian value of the coordinate system;

[0017] S102: xyz The circular domain in the coordinate system is transformed into ζ-ζ The elliptical domain in the coordinate system, where xyz The geometric equation of the circle in the coordinate system is:

[0018]

[0019] Where, is the radius, for xyz The center of the circular domain in the coordinate system;

[0020] After global mapping transformation, ζ-ζ The geometric equation of the ellipse in the coordinate system is:

[0021]

[0022] Summarize the geometric equation of the ellipse:

[0023] ;

[0024] Where, is the center of the ellipse, are the semi-major axis and semi-minor axis respectively;

[0025] S103: The x - or The ellipse domain mapping in the coordinate system is p - q The normalized circular domain in the coordinate system, the mapping relationship is:

[0026]

[0027] Depend on ζ-ζ Coordinate system conversion to pq The Jacobian of the coordinate system is:

[0028]

[0029] The numerical integral formula in the normalized circular domain after mapping is:

[0030]

[0031] Where, is the integrand, For the A circular domain;

[0032] S104: The p - q The normalized circular domain mapping in the coordinate system is α - β The normalized square domain in the coordinate system, the mapping relationship is:

[0033]

[0034] Depend on p - q Coordinate system conversion to α - β The Jacobian of the coordinate system is:

[0035]

[0036] S105: Substitute the mapping relationship formula in S104 into the mapping relationship formula in S103 to obtain a local mapping relationship formula:

[0037]

[0038] Based on the local mapping relationship formula, the numerical integral formula in the normalized circular domain is converted into a standard normalized square domain. Numerical integration expressions within:

[0039]

[0040] S106: Based on Gauss-Legendre integral rule, the normalized square domain can be Numerically integrate the integral within :

[0041]

[0042] Where, and represents the coordinates of the Gaussian integration point, and yes and The corresponding quadrature coefficient is, and For the counter, and The intervals are Inside and The number of Gaussian integration points in the direction;

[0043] S107: The Gaussian integral points and quadrature coefficients generated in the normalized square domain are transformed by inverse mapping to restore to x - or In the coordinate system, the inverse mapping relationship is:

[0044]

[0045] Where, ( , )and They are ζ-ζ Coordinate system Gaussian integration points and corresponding quadrature coefficients in circular energy elements, Representative A circular area in xyz The coordinates of the center of the circle in the coordinate system, and Representative The first Gaussian integration points, Representative x - y Coordinate system The radius of the circular domain; ( , )yes α-β The corresponding coordinate system Gaussian integration points generated in a normalized square domain, and yes and The corresponding quadrature coefficient;

[0046] Based on the circular domain and Gauss-Legendre integral, a circular energy element for numerically simulating the energy of the circular region is defined, wherein after the inverse mapping transformation, ζ-ζ Coordinate system The numerical integration formula within a circular domain is:

[0047] .

[0048] Furthermore, the circular perforated plate is modeled based on the circular energy element and energy Boolean operation in S2, specifically including:

[0049] Subtracting the smaller circular energy element from the circular energy element without a hole, we get the two-dimensional normalized hole plate domain. The corresponding Boolean operation is:

[0050]

[0051] Where, It is a circular energy element without a hole. For the i A smaller circular energy element, is the two-dimensional normalized aperture plate domain;

[0052] Among them, ζ-ζ In the coordinate system, the numerical integral expression in the two-dimensional normalized perforated plate domain is changed to:

[0053]

[0054] Where, for ζ-ζ Numerical integration in coordinate system, symbol Σ i Based on the circular energy element i Geometric Boolean operators for .

[0055] Furthermore, the simulation of boundary conditions in S2 specifically includes:

[0056] A1: Define the level set function to characterize the boundary contour of the circular opening plate. ζ-ζ Established in coordinate system:

[0057]

[0058] A2: By assigning different values ​​to the boundary index, different boundary conditions can be applied to the boundary profile of the circular opening plate. On the top, we get the boundary contour of the circular opening plate The boundary condition function of :

[0059]

[0060] Where, is the boundary outline of the circular opening plate The total number of boundaries, For outline Middle Boundary The boundary level set function of is the applied displacement component, 、 Along and The displacement component in the direction, is the boundary index, where .

[0061] Furthermore, in S3, a total energy functional is constructed based on the linear elastic small deformation thin plate theory, specifically including:

[0062] S301: According to the linear elastic small deformation thin plate theory, assuming that the in-plane displacement component is , strain components are derived based on the in-plane displacement components, and the strain components constitute a strain array:

[0063]

[0064] Where, 、 and is the strain component, To find the partial derivative operator;

[0065] S302: Define the elastic matrix of the isotropic plate:

[0066]

[0067] Where, is Young's modulus, is the material Poisson's ratio, is the stiffness coefficient, , ;

[0068] S303: According to the global mapping, Integrate the axis to obtain a square solution domain Inner two-dimensional normalized aperture plate domain The strain energy functional of :

[0069]

[0070] Where, is the thickness of the plate, is the elasticity matrix, is the strain array, Strain Array The transpose of

[0071] S304: In-plane load P Apply along the plate boundary contour Γ, and obtain the external force potential energy functional:

[0072]

[0073] Where, is the in-plane load P Along , The load components of the shaft, For the The load components of the shaft, For the The load components of the shaft, is the in-plane load P and The angle of the axis, For outline The borders;

[0074] S305: Based on the strain energy functional and the external force potential energy functional, the total energy functional of the circular perforated plate is obtained:

[0075] .

[0076] Furthermore, the static problem of the circular perforated plate is solved in S4 based on the global trial function and the Ritz method, specifically including:

[0077] S401: Constructing the displacement components of a circular perforated plate based on Legendre polynomials and boundary condition functions The global test function of :

[0078]

[0079] in, and Along and The number of Legendre polynomials used in the direction, is the displacement component The corresponding unknown coefficients; and They represent the first Item and Legendre polynomials, and They are the boundary contours of the circular opening plate Upper displacement component Boundary condition function;

[0080] S402: Calculate the total energy functional with respect to unknown coefficients based on the Ritz method The partial derivatives of , we get a set of linear equations:

[0081]

[0082] Write the linear equations in matrix form:

[0083]

[0084] Where, the stiffness matrix Contains four sub-matrices 、 、 、 , each submatrix contains MN×MN elements, unknown coefficient array and load array Then each contains a subarray 、 and 、 , each subarray contains MN× 1 element;

[0085] S403: Calculate the expressions of each element in the stiffness matrix and load matrix, and calculate the numerical integral of each element in the stiffness matrix and load matrix, where the stiffness matrix and load array The expressions for each element in are:

[0086]

[0087]

[0088] Where, is the in-plane load P The boundary contour of the effect, and Located in the load sub-array and No. OK, is the intermediate calculation matrix, where , ;

[0089] S404: Solve the matrix inversion problem in S402 to obtain the unknown coefficient array , and the resulting unknown coefficient array Substitute into the global test function and then substitute the strain components to obtain the strain array ;

[0090] S405: Calculate the stress field of the circular perforated plate based on the elastic matrix and the obtained strain array to complete the solution of the static problem of the circular perforated plate. The calculation formula of the stress field is:

[0091] .

[0092] Furthermore, the Legendre polynomial It can be derived through the following recursive relation:

[0093]

[0094] Where, is the first Legendre polynomial, is the second Legendre polynomial, It is Legendre polynomials, It is Legendre polynomials, It is Legendre polynomials The order of .

[0095] Furthermore, the stiffness matrix The intermediate calculation matrix in Chinese elements The numerical integration of can be obtained based on the circular energy element proposed in S107, and the calculation formula is:

[0096]

[0097] Where, for The integrand inside, and is the intermediate variable, , , , , and Representative i Item and j Legendre polynomials, and Representative Item and Legendre polynomials.

[0098] The beneficial effects of this application are:

[0099] The present application provides a static analysis method for circular plates with holes based on circular energy elements on rectangular domains. The method constructs a circular energy element that can perform high-precision numerical integration based on the global-local mapping technology, and proposes a new analysis and solution algorithm for the static problems of two-dimensional circular plates with circular holes within the framework of the Ritz method and global trial functions based on the circular energy element and energy Boolean operations. A standard energy functional and calculation procedure that are not affected by the geometric configuration of the plate are established, thereby improving the accuracy and efficiency of the analysis of the static problems of circular plates with holes. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] 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 embodiments can also be obtained based on these drawings.

[0101] Figure 1 A schematic diagram of a static analysis method for a circular perforated plate based on a circular energy element on a rectangular domain provided in an embodiment of the present application.

[0102] Figure 2 A global mapping diagram provided in an embodiment of the present application.

[0103] Figure 3 The embodiment of the present application provides a circular domain by xyz Coordinate system mapped to ζ-ζ Schematic diagram of the elliptical domain in the coordinate system.

[0104] Figure 4 A schematic diagram of a local mapping provided in an embodiment of the present application.

[0105] Figure 5 A schematic diagram of modeling a circular perforated plate based on circular energy elements and energy Boolean operations provided in an embodiment of the present application.

[0106] Figure 6 A schematic diagram of a model of a circular ring plate subjected to in-plane load under free boundary conditions provided in an embodiment of the present application.

[0107] Figure 7 The three stress components derived from the static problem analysis method proposed in this application and the finite element method s x , s y , t xy Schematic diagram of the comparison of stress distribution cloud maps. DETAILED DESCRIPTION

[0108] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field based on this application are within the scope of protection of this application.

[0109] Solving the static problems of plates with complex openings using the Ritz method requires deriving the energy functional based on global trial functions and integrating the complex geometric domain. As the geometric configuration of the plate changes, the energy functional formula changes accordingly, making it difficult to standardize the solution process. Research has shown that by covering the plate domain with a rectangular domain, numerically integrating the true geometric domain of the plate within the rectangular domain and performing global interpolation using complete orthogonal polynomials, it is possible to numerically simulate the deformation of plates of arbitrary shapes. However, the integral accuracy of generating Gaussian integral points within the entire rectangular domain to capture complex geometric boundaries is not high, which can easily lead to singularities and instabilities in the stiffness matrix. How to perform standardized high-precision numerical integration of complex geometric opening domains remains a challenge. In recent years, based on the Gaussian integral rule and mapping technology, relevant literature has studied numerical integration methods in arbitrary quadrilateral and regular polygonal domains. However, the geometric domain transformation process during numerical integration in these methods will cause the integral formula to change.

[0110] Based on this, the embodiment of the present application provides a static analysis method of a circular perforated plate based on a circular energy element on a rectangular domain. Figure 1 , Figure 1 FIG. 1 is a schematic diagram of a static analysis method for a circular plate with a hole based on a circular energy element on a rectangular domain, provided in an embodiment of the present application, including:

[0111] S1: Constructing a circular energy element based on global-local mapping technology.

[0112] Said S1 specifically includes:

[0113] S101: Embed the circular perforated plate Ω into a predefined rectangular domain Ω r , and globally map the predefined rectangular domain to obtain a square solution domain Ξ r .

[0114] like Figure 2 As shown, Figure 2 A global mapping diagram provided in an embodiment of the present application, wherein (a) is x - y The circular hole plate domain Ω in the coordinate system; (b) is x - or A circular hole plate domain Ξ in the coordinate system. First, consider a circular hole plate domain Ω, such as Figure 2As shown in (a) in the figure. Since the Legendre polynomials are defined on the interval When constructing the global test function based on it, its completeness and orthogonality must be guaranteed, so the solution of the circular hole plate must be in the interval To this end, the circular perforated plate domain Ω will first be embedded in a rectangular domain as small as possible, called the predefined rectangular domain Ω r , its length is a Width is b , the origin of the coordinate system is located in the predefined rectangular domain Ω r Geometric center, the hole domain is represented by Ω i ( i =1,2), the geometric boundary contour of the plate is represented by Afterwards, Ω is converted based on the dimensionless coordinate transformation formula. r Mapping to intervals The square solution domain Ξ r , and all subsequent calculations are in this interval and ζ-ζ coordinate system, see Figure 2 (b) in the Figure 2 (b) in the ζ-ζ The dimensionless coordinate transformation formula for the circular hole plate domain Ξ in the coordinate system is:

[0115] (1)

[0116] Where, and for x - y The length and width of the predefined rectangular domain in the coordinate system.

[0117] Let the integrand be F Defined in the original circular hole plate domain Ω, its integral value is express:

[0118] (2)

[0119] Then, after mapping through formula (1), the opening domain is represented by Ω i ( i =1,2) is mapped to Ξ i ( i =1,2), the circular hole plate domain Ω will be mapped to ζ-ζ The two-dimensional normalized perforated plate domain Ξ in the coordinate system, the corresponding symbol becomes: Ω r →Ξ r ,Ω→Ξ,Ω i →Ξ i , → Therefore, the integral formula (2) becomes:

[0120] (3)

[0121] Where, for the reason xyz Coordinate system conversion to ζ-ζ Jacobian value of the coordinate system.

[0122] It should be noted that the two-dimensional normalized aperture plate domain Ξ is replaced by the square solution domain Ξ r Covered, x , or The value range is in the interval Inside.

[0123] The global mapping aims to unify the computational domain of the circular hole plate domain Ω, so that the energy functional formula of the circular hole plate is established in the interval The square solution domain Ξ r Inside.

[0124] S102: Constructing a circular energy element based on the local mapping.

[0125] Assume that the integration region is a circular domain, then x - y In the coordinate system, the geometric equation of a circle can be written as:

[0126] (4)

[0127] The center of the circle is ( x 0, y 0), the radius is R After mapping based on formula (1), ζ-ζ The geometric equation of the ellipse in the coordinate system can be written as:

[0128] (5)

[0129] It can be further summarized as follows:

[0130] (6)

[0131] Where is is the center of the ellipse, and its semi-major axis and semi-minor axis are ,like Figure 3 As shown, Figure 3 The embodiment of the present application provides a circular domain by xyz Coordinate system mapped to ζ-ζ Schematic diagram of the elliptical domain in the coordinate system.

[0132] In this case, x - y The integral problem of the circular domain in the coordinate system is transformed into x- or Integration problems in elliptical domains under coordinate systems.

[0133] Next, to calculate x - y For numerical integration of circular domain in coordinate system, we need to x - or The elliptical domain in the coordinate system is then mapped to a normalized circular domain, and finally mapped to a normalized square domain, so that the Gaussian integral point is in the standard interval Internally generated, such as Figure 4 As shown, Figure 4 A local mapping diagram provided in an embodiment of the present application, wherein (a) is an elliptical domain, (b) is a normalized circular domain, and (c) is a normalized square domain. First, the elliptical domain in equation (6) needs to be mapped to a radius r =1 normalized circular domain ( p 2 + q 2 =1), see Figure 4 The mapping relationship between (a) and (b) is:

[0134] (7)

[0135] The normalized circular domain is p - q The unit circle in the coordinate system with its center at the origin, see Figure 4 (b) in the above equation is represented by x - or Coordinate system conversion to p - q The Jacobian of the coordinate system is:

[0136] (8)

[0137] Then, domain Ξ i Points It can be calculated by the following formula:

[0138] (9)

[0139] The second step is to further map the normalized circular domain to a domain defined in the standard interval The normalized square domain within Figure 4 (b) and (c) in:

[0140] (10)

[0141] The corresponding Jacobian can be calculated as follows:

[0142] (11)

[0143] Substituting formula (10) into formula (7), we can obtain the following relationship:

[0144] (12)

[0145] Where, for xyz The center of the circular domain in the coordinate system;

[0146] Finally, based on the above local mapping transformation, the integral in Equation (9) Finally, it is transformed into a standard normalized square domain The integral within:

[0147] (13)

[0148] Combined with Gauss-Legendre integral rule, the numerical integral expression of formula (13) is:

[0149] (14)

[0150] In the formula and represents the coordinates of the Gaussian integration point, and yes and The corresponding quadrature coefficient is, and For the counter, and The intervals are Inside and The number of Gaussian integration points in the direction.

[0151] Finally, the Gaussian integral points and quadrature coefficients in the normalized square domain need to be transformed through inverse mapping to restore them to x - or In the coordinate system:

[0152] (15)

[0153] Where, ( , )and They are ζ-ζ Coordinate system Gaussian integration points and corresponding quadrature coefficients in circular energy elements, Representative A circular area in xyz The coordinates of the center of the circle in the coordinate system, and Representative The first Gaussian integration points, Representative x - y Coordinate system The radius of the circular domain; ( , )yes α-β The corresponding coordinate system Gaussian integration points generated in a normalized square domain, and yes and The corresponding integral coefficient. Therefore, the numerical integration of formula (14) is still in the standard interval In:

[0154] (16)

[0155] In structural mechanics, when the integrand F When the integral of represents the strain energy, the integral within the unit circular domain represents a part of the strain energy contained in this domain. Therefore, Equation (16) defines a circular energy element Ξ for numerically simulating the energy of a circular region. i Based on this, combined with the global-local mapping technology, the integrand can be realized F High-precision numerical integration of polynomial energy functionals in a circular aperture domain. Based on the global-local mapping technique, the numerical integration schemes proposed in this paper are all in x - or Square solution domain Ξ in the coordinate system r The calculation formula for circular hole plates is completely consistent and is not affected by the geometric configuration of the plate, laying the foundation for the next step of constructing a standard energy functional and performing high-precision numerical integration of elements in the stiffness matrix and mass matrix.

[0156] S2: Modeling the circular perforated plate based on the circular energy element and energy Boolean operation, and simulating boundary conditions.

[0157] like Figure 2 As shown in (b), ζ-ζ In the coordinate system, the two-dimensional normalized perforated plate domain Ξ is replaced by the square solution domain Ξ r Covered with two circular holes i ( i=1,2), and the boundary contour of the two-dimensional normalized perforated plate is circular. Therefore, for this two-dimensional normalized perforated plate, a simulation can be performed based on circular energy elements and energy Boolean operations: by subtracting two smaller circular energy elements (Ξ1 and Ξ2) from a large circular energy element (Ξ0), the two-dimensional normalized perforated plate domain Ξ can be obtained. The corresponding Boolean operation formula is: Ξ=Ξ0–(Ξ1+Ξ2), as shown in Figure 5 As shown, Figure 5 A schematic diagram of modeling a circular perforated plate based on circular energy elements and energy Boolean operations provided in an embodiment of the present application.

[0158] Therefore, in x - or In the coordinate system, the integral formula (3) in the two-dimensional normalized perforated plate domain Ξ is changed to:

[0159] (17)

[0160] Substituting formula (16) into formula (17), we get the following calculation formula:

[0161] (18)

[0162] Where, for ζ-ζ Numerical integration in coordinate system, symbol Σ i Based on the circular energy element i The geometric Boolean operator of , whose sign needs to be determined according to the Boolean operation of deriving the plate geometry from the circular energy element.

[0163] In this way, the integral in the circular perforated plate domain Ω in Equation (2) can be realized in the normalized perforated plate domain Ξ based on circular energy elements and energy Boolean operations, and the numerical integration process is all done in the square solution domain Ξ established in the global stage. r Note that in formula (18), the symbol By the corresponding circular energy element Ξ i The geometric Boolean operation determines the circular energy element Ξ i The result of numerical integration can be a positive value "+" or a negative value "-".

[0164] For a circular plate with a hole, its boundary contour can be characterized by the following level set function:

[0165] (19)

[0166] For plate static analysis, different boundary conditions can be applied to the plate boundary contour Γ. j Boundary Γ j The boundary level set function is , by applying the displacement component ( = u , v ) corresponding to the boundary index On the level set function, the boundary condition function expression of the contour Γ can be given :

[0167] (20)

[0168] Where, is the boundary outline of the circular opening plate The total number of boundaries, For outline Middle Boundary The boundary level set function of is the applied displacement component, 、 Along and The displacement component in the direction, is the boundary index. Specifically, the boundary index It can be determined by the following formula:

[0169] (twenty one)

[0170] S3: Constructing the total energy functional for static problems based on linear small deformation thin plate theory.

[0171] According to the plane deformation theory, it is assumed that an in-plane load acts on the plate boundary contour Γ P When , no lateral displacement occurs, so only two in-plane displacement components need to be assumed , the strain components are derived according to the in-plane displacement components, and the strain components form a strain matrix:

[0172] (twenty two)

[0173] Where, 、 and is the strain component, To find the partial derivative operator.

[0174] For an isotropic plate, the elastic matrix can be defined as:

[0175] (twenty three)

[0176] Where, is Young's modulus, is the material Poisson's ratio, is the stiffness coefficient, , .

[0177] The stress field can be calculated using the following formula:

[0178] (twenty four)

[0179] To unify the energy functional and numerical integration formula, in formula (22) xyz The coordinate system needs to be converted to a dimensionless coordinate system through formula (1) ζ-ζ Then, calculate. Based on the global mapping proposed in S101, z After integrating the axis, the square solution domain Ξ can be derived r The strain energy functional calculation formula of the inner two-dimensional normalized perforated plate domain Ξ is:

[0180] (25)

[0181] Where, |J r |= ab / 4, is the thickness of the plate, is the elasticity matrix, is the strain array, Strain Array The transpose of .

[0182] In-plane load P It is applied along the boundary contour Γ of the plate, so the calculation formula of the external force potential energy functional is defined as follows:

[0183] (26)

[0184] In the formula is the in-plane load P Along , Load components of the shaft:

[0185] (27)

[0186] Where, For the The load components of the shaft, For the The load components of the shaft, is the in-plane load P and The angle of the axis, For outline The Border.

[0187] Finally, the total energy functional formula of the circular perforated plate can be expressed as:

[0188] (28)

[0189] S4: Solve the static problem of a circular plate with a hole based on the global trial function and the Ritz method.

[0190] Legendre polynomials are defined on the interval The complete orthogonal polynomials in can accurately simulate the local deformation of the structure and are often used to construct global test functions. The Legendre polynomials can be derived through the following recursive relationship:

[0191] (29)

[0192] Where, is the first Legendre polynomial, is the second Legendre polynomial, It is Legendre polynomials, It is Legendre polynomials, It is Legendre polynomials The order of .

[0193] Then, according to the Ritz method and the boundary condition function (20), the displacement component of the circular hole plate can be written as The global test function of :

[0194] (30)

[0195] In the formula and Along and The number of Legendre polynomials used in the direction, is the displacement component The corresponding unknown coefficients; and They represent the first Item and Legendre polynomials, and They are the boundary contours of the circular opening plate Upper displacement component The boundary condition function.

[0196] Substitute the global test function (30) into equation (22), then substitute it together with equation (23) into equation (25), and finally substitute it into equations (26)-(28) to calculate the total energy functional of the plate P .

[0197] Then, the total energy functional is obtained based on the Ritz method P About unknown coefficients The partial derivatives of , we can obtain a set of linear equations:

[0198] (31)

[0199] Write it in matrix form:

[0200] (32)

[0201] Where the stiffness matrix Contains four sub-matrices 、 、 、 , each submatrix contains MN×MN elements, unknown coefficient array and load array Then each contains a subarray 、 and 、 , each subarray contains MN× 1 element. Specifically, the expressions of each element in the stiffness matrix K are as follows:

[0202] (33)

[0203] The expressions for each element in the load array W are as follows:

[0204] (34)

[0205] In the formula is the in-plane load P The boundary contour of the effect, and Located in the load sub-array and No. OK, is the intermediate calculation matrix, where , For formula (33), the matrix Chinese elements It can be calculated by the following formula:

[0206] (35)

[0207] in,

[0208] (36)

[0209] Where, and is the intermediate variable, and Representativei Item and j Legendre polynomials, and Representative Item and Legendre polynomials.

[0210] In the local stage, the two-dimensional normalized perforated plate domain Ξ is contained in the square solution domain Ξ r Internally, the elements in the stiffness matrix K are The high-precision numerical integration of can be realized based on the circular energy element proposed in S107:

[0211] (37)

[0212] In the formula F represent The integrand inside, , and They are i The Gaussian integration points and their integration coefficients in the circular energy element are shown in equation (15); the symbol It represents the circular energy element Ξ i The geometric Boolean operator of , whose sign needs to be determined according to the Boolean operation of deriving the plate geometry from the circular energy element.

[0213] After completing the numerical integration of each element in the stiffness matrix K and the load array W, the assumed unknown coefficient array C can be solved by solving the matrix inversion problem of Equation (32) and substituted back into the global trial function Equation (30); then, the strain field is calculated by Equation (22), and finally, the stress field of the circular hole plate can be derived based on Equation (24), completing the solution of the static problem of the circular hole plate.

[0214] In one embodiment of the present application, the static analysis method of the present invention is applied to the static problem of a circular ring plate subjected to an in-plane load, the stress field of the plate is solved, and a stress distribution cloud diagram is drawn. The geometric and material constants of the circular ring plate are: , , , , radius of the large circle , radius of the small circle When performing static analysis, the inner and outer boundaries of the plate are free, and the uniformly distributed in-plane load is applied inward along the outer boundary normal of the plate. In addition, the geometric model and discrete model of the plate are as follows Figure 6 As shown, Figure 6 A schematic diagram of a model of a circular ring plate subjected to in-plane load under free boundary conditions provided in an embodiment of the present application, wherein (a) is xyz The geometric model in the coordinate system, (b) is ζ-ζGeometric model under the coordinate system, (c) is the circular energy element discrete model.

[0215] The specific implementation steps of this embodiment are as follows:

[0216] Step 1: Construction of circular energy element based on global-local mapping technology. Figure 6 The gray area in (a) is the annular plate domain Ω subjected to in-plane load. First, place it into the predefined rectangular domain Ω r , the coordinate origin is located in the predefined rectangular domain Ω r The geometric center of the circular opening is Indicate, use Represents the boundary contour of the annular plate, and then based on the dimensionless coordinate transformation formula, Ω r Mapping to a square solution domain Ξ in the interval [-1,1] r :

[0217] (38)

[0218] After the above mapping transformation, the circular ring plate domain Ω mapping is ζ-ζ Normalized circular plate domain Ξ in the coordinate system, all subsequent solution calculation processes are performed in the square solution domain Ξ r In-house, see Figure 6 Next, we construct a circular energy element based on the local mapping. First, we transform xyz The circular domain is mapped to the coordinate system ζ-ζ The elliptical domain in the coordinate system, and then map this elliptical domain to a radius r = 1, and finally the normalized circular domain is mapped to a normalized square domain defined in the standard interval [-1,1]. For the convenience of calculation, the Gaussian integral points and corresponding quadrature coefficients generated in the normalized square domain are restored to ζ-ζ In the elliptical domain of the coordinate system, based on the principles of structural mechanics and A circular energy element is defined to achieve high-precision numerical integration of strain energy within the circular plate domain. In this embodiment, two circular energy elements are used to discretize the circular plate domain, and 1764 Gaussian integration points are arranged in each circular energy element.

[0219] Step 2: Model the circular perforated plate based on the circular energy element and energy Boolean operation, and construct the boundary condition function. Figure 6 As shown in (b) in ζ-ζ In the coordinate system, the normalized circular ring plate domain Ξ is replaced by the square solution domain Ξ rThe plate is covered by a circular hole Ξ1, and the boundary contour of the normalized circular plate is circular. Therefore, based on the energy Boolean operation, the geometric model of the circular plate can be simulated by subtracting a small circular energy element from a large circular energy element, as shown in the following example: Figure 6 As shown in (c) in .

[0220] Then according to the boundary conditions imposed on the plate, based on the boundary index The expression of determines the corresponding boundary index. Since the inner and outer boundaries of the annular plate are free and no boundary conditions are applied, the boundary condition function expression of the boundary profile Γ is:

[0221] (39)

[0222] Step 3: Construct the energy functional for the static problem based on the linear small deformation thin plate theory. According to the linear elastic small deformation thin plate theory, assume that the two in-plane displacement components , and then derive the strain component expression. xyz The coordinate system is transformed into a dimensionless coordinate system through a global mapping relationship x - or Then, the strain energy functional of the circular plate and the external force potential energy functional can be calculated by combining the elastic matrix, and finally the square solution domain Ξ can be derived. r The total energy functional formula of the inner normalized circular ring plate domain Ξ is shown in Equation (28).

[0223] Step 4: Solve the static problem of the circular plate with a hole based on the global trial function and the Ritz method. Based on the Legendre polynomials and the boundary condition function of the circular plate, the displacement component can be constructed. The global test function; then the total energy functional is obtained according to the Ritz method About unknown coefficients The partial derivative of derives a linear equation system, which is written in matrix form; then, according to the expressions of each element in the stiffness matrix and the load matrix and The expression of the circular energy element and energy Boolean operation is used to complete the stiffness matrix and load array Numerical integration of each element in, solve the matrix inversion problem in the matrix form of equation (28), and derive the assumed unknown coefficient array , and finally based on Calculate the stress field of the annular plate, draw the stress distribution cloud diagram, and complete the solution.

[0224] The stress field distribution of a circular ring plate with uniformly distributed in-plane load at a fully free boundary is solved by the proposed circular energy element based on global-local mapping. The results are shown in Figure 2. Figure 7 As shown, Figure 7The three stress components derived from the static problem analysis method proposed in this application and the finite element method s x , s y , t xy Comparative schematic diagram of stress distribution cloud map, where: Figure 7 The upper part is the three stress components derived from the static problem analysis method proposed in this application s x , s y , t xy Stress distribution cloud diagram, the bottom is the three stress components derived by the finite element method s x , s y , t xy Stress distribution cloud diagram. It can be seen that the results of the proposed static analysis method for circular plates with holes based on circular energy elements on rectangular domains using global-local mapping technology and energy Boolean operations are in good agreement with the results of the finite element method.

[0225] This application provides a static analysis method for circular plates with a perforated hole based on circular energy elements on rectangular domains. By placing the circular plate with a perforated hole in a predefined rectangular domain, a standard energy functional and integral formula are constructed that are independent of the plate's geometric configuration. In the global phase, the energy functional and integral formula are established on the predefined rectangular domain. In the local phase, each circular domain is first mapped into an elliptical domain, then mapped into a normalized square domain, and then the circular energy element is constructed in combination with the Gauss-Legendre integral rule. Furthermore, based on the geometric configuration of the circular plate with a perforated hole, the circular plate is modeled within the constructed rectangular domain based on circular energy elements and energy Boolean operations. The numerical integral results corresponding to each circular energy element are calculated separately, and the positive and negative signs are assigned to the Boolean operations derived from the circular energy elements based on the plate geometry. This method achieves high-precision numerical integration of the elements in the stiffness matrix within the circular plate domain, thereby solving the static problem. Furthermore, because the energy functional is based on the constructed predefined rectangular domain, the Ritz method for solving the static problem of circular plates with a perforated hole, which is based on the global trial function, is fully standardized.

[0226] It should be noted that those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of this application, and it should be understood that the scope of protection of this application is not limited to such specific descriptions and embodiments. Those skilled in the art can make various other specific variations and combinations based on the technical teachings disclosed in this application without departing from the essence of this application, and such variations and combinations are still within the scope of protection of this application.

Claims

1. A static analysis method for circular perforated plates based on circular energy elements on rectangular domains, characterized in that: include: S1: Global mapping stage: construct a predefined rectangular domain to cover the circular perforated plate domain; map the predefined rectangular domain to a square solution domain, and map the circular perforated plate domain to a two-dimensional normalized perforated plate domain; Local mapping stage: map the circular domain to an elliptical domain; and then map the elliptical domain to a normalized circular domain; Mapping the normalized circular domain to a normalized square domain; generating Gaussian integral points and quadrature coefficients in the normalized square domain, and inversely solving the Gaussian integral points and quadrature coefficients to a dimensionless coordinate system in a global mapping stage to construct a circular energy element that simulates the energy on the circular area; S2: Modeling the circular perforated plate based on the circular energy element and energy Boolean operation, and simulating boundary conditions; S3: Constructing the total energy functional for static problems based on linear small deformation thin plate theory; S4: Solve the static problem of a circular plate with a hole based on the global trial function and the Ritz method.

2. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 1 is characterized in that: Said S1 specifically includes: S101: embed the circular perforated plate into a predefined rectangular domain, and perform global mapping on the predefined rectangular domain to obtain a square solution domain, wherein the global mapping is a dimensionless coordinate system transformation, that is, based on the following formula, the original x - y Coordinate system mapped to dimensionless coordinate system ξ-η : Where, and for xy The length and width of the predefined rectangular domain in the coordinate system; Correspondingly, after completing the coordinate transformation, the integrand The integral formula in the original elliptical hole plate domain Ω is transformed into: Where, for the reason xy Coordinate system conversion to ξ-η Jacobian value of the coordinate system; S102: xy The circular domain in the coordinate system is transformed into ξ-η The elliptical domain in the coordinate system, where xy The geometric equation of the circle in the coordinate system is: Where, is the radius, for xy The center of the circular domain in the coordinate system; After global mapping transformation, ξ-η The geometric equation of the ellipse in the coordinate system is: Summarize the geometric equation of the ellipse: ； Where, is the center of the ellipse, are the semi-major axis and semi-minor axis respectively; S103: The ξ - η The ellipse domain mapping in the coordinate system is p - q The normalized circular domain in the coordinate system, the mapping relationship is: Depend on ξ-η Coordinate system conversion to pq The Jacobian of the coordinate system is: The numerical integral formula in the normalized circular domain after mapping is: Where, is the integrand, For the A circular domain; S104: The pq The normalized circular domain mapping in the coordinate system is α - β The normalized square domain in the coordinate system, the mapping relationship is: Depend on pq Coordinate system conversion to α - β The Jacobian of the coordinate system is: S105: Substitute the mapping relationship formula in S104 into the mapping relationship formula in S103 to obtain a local mapping relationship formula: Based on the local mapping relationship formula, the numerical integral formula in the normalized circular domain is converted into a standard normalized square domain. Numerical integration expressions within: S106: Based on Gauss-Legendre integral rule, the normalized square domain can be Numerically integrate the integral within : Where, and represents the coordinates of the Gaussian integration point, and yes and The corresponding quadrature coefficient is, and For the counter, and The intervals are Inside and The number of Gaussian integration points in the direction; S107: The Gaussian integral points and quadrature coefficients generated in the normalized square domain are transformed by inverse mapping to restore to ξ - η In the coordinate system, the inverse mapping relationship is: Where, ( , )and They are ξ-η Coordinate system Gaussian integration points and corresponding quadrature coefficients in circular energy elements, Representative A circular area in xy The coordinates of the center of the circle in the coordinate system, and Representative The first Gaussian integration points, Representative x - y Coordinate system The radius of the circular domain; ( , )yes α-β The corresponding coordinate system Gaussian integration points generated in a normalized square domain, and yes and The corresponding quadrature coefficient; Based on the circular domain and Gauss-Legendre integral, a circular energy element for numerically simulating the energy of the circular region is defined, wherein after the inverse mapping transformation, ξ-η Coordinate system The numerical integration formula within a circular domain is: 。 3. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 2 is characterized in that: The circular perforated plate is modeled based on the circular energy element and energy Boolean operation in S2, specifically including: Subtracting the smaller circular energy element from the circular energy element without a hole, we get the two-dimensional normalized hole plate domain. The corresponding Boolean operation is: Where, It is a circular energy element without a hole. For the i A smaller circular energy element, is the two-dimensional normalized aperture plate domain; Among them, ξ-η In the coordinate system, the numerical integral expression in the two-dimensional normalized perforated plate domain is changed to: Where, for ξ-η Numerical integration in coordinate system, symbol Σ i Based on the circular energy element i Geometric Boolean operators for .

4. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 3 is characterized in that: The simulation of boundary conditions in S2 specifically includes: A1: Define the level set function to characterize the boundary contour of the circular opening plate. ξ-η Established in coordinate system: A2: By assigning different values ​​to the boundary index, different boundary conditions can be applied to the boundary profile of the circular opening plate. On the top, we get the boundary contour of the circular opening plate The boundary condition function of : Where, is the boundary outline of the circular opening plate The total number of boundaries, For outline Middle Boundary The boundary level set function of is the applied displacement component, 、 Along and The displacement component in the direction, is the boundary index, where .

5. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 4 is characterized in that: In S3, the total energy functional is constructed based on the linear elastic small deformation thin plate theory, specifically including: S301: According to the linear elastic small deformation thin plate theory, assuming that the in-plane displacement component is , strain components are derived based on the in-plane displacement components, and the strain components constitute a strain array: Where, 、 and is the strain component, To find the partial derivative operator; S302: Define the elastic matrix of the isotropic plate: Where, is Young's modulus, is the material Poisson's ratio, is the stiffness coefficient, , ; S303: According to the global mapping, Integrate the axis to obtain a square solution domain Inner two-dimensional normalized aperture plate domain The strain energy functional of : Where, is the thickness of the plate, is the elasticity matrix, is the strain array, Strain Array The transpose of S304: In-plane load P Apply along the plate boundary contour Γ, and obtain the external force potential energy functional: Where, is the in-plane load P Along , The load components of the shaft, For the The load components of the shaft, For the The load components of the shaft, is the in-plane load P and The angle of the axis, For outline The borders; S305: Based on the strain energy functional and the external force potential energy functional, the total energy functional of the circular perforated plate is obtained: 。 6. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 5 is characterized in that: In S4, the static problem of the circular perforated plate is solved based on the global trial function and the Ritz method, specifically including: S401: Constructing the displacement components of a circular perforated plate based on Legendre polynomials and boundary condition functions The global test function of : in, and Along and The number of Legendre polynomials used in the direction, is the displacement component The corresponding unknown coefficients; and They represent the first Item and Legendre polynomials, and They are the boundary contours of the circular opening plate Upper displacement component Boundary condition function; S402: Calculate the total energy functional with respect to unknown coefficients based on the Ritz method The partial derivatives of , we get a set of linear equations: Write the linear equations in matrix form: Where, the stiffness matrix Contains four sub-matrices 、 、 、 , each submatrix contains MN×MN elements, unknown coefficient array and load array Then each contains a subarray 、 and 、 , each subarray contains MN× 1 element; S403: Calculate the expressions of each element in the stiffness matrix and load matrix, and calculate the numerical integral of each element in the stiffness matrix and load matrix, where the stiffness matrix and load array The expressions for each element in are: Where, is the in-plane load P The boundary contour of the effect, and Located in the load sub-array and No. OK, is the intermediate calculation matrix, where , ; S404: Solve the matrix inversion problem in S402 to obtain the unknown coefficient array , and the resulting unknown coefficient array Substitute into the global test function and then substitute the strain components to obtain the strain array ; S405: Calculate the stress field of the circular perforated plate based on the elastic matrix and the obtained strain array to complete the solution of the static problem of the circular perforated plate. The calculation formula of the stress field is: 。 7. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 6 is characterized in that: The Legendre polynomial It can be derived through the following recursive relation: Where, is the first Legendre polynomial, is the second Legendre polynomial, It is Legendre polynomials, It is Legendre polynomials, It is Legendre polynomials The order of .

8. The static analysis method of a circular plate with a hole based on a circular energy element on a rectangular domain according to claim 7 is characterized in that: The stiffness matrix The intermediate calculation matrix in Chinese elements The numerical integration of can be obtained based on the circular energy element proposed in S107, and the calculation formula is: Where, for The integrand inside, and is the intermediate variable, , , , , and Representative i Item and j Legendre polynomials, and Representative Item and Legendre polynomials.

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