Energy element numerical solution method for bending of two-dimensional medium-thick plate with arbitrary geometry

Abstract

The application discloses an energy element numerical solution method for bending of a two-dimensional medium-thick plate with an arbitrary geometric configuration. The method comprises the following steps: a hole is dug in a standard rectangular domain to simulate a two-dimensional medium-thick plate domain with an arbitrary geometric configuration; then, a discrete variable-thickness matrix and a discrete variable-stiffness matrix are formed based on Gaussian integral points; then, a structure global discrete variable-stiffness system is constructed to represent the distribution of energy in space; the geometric boundary is controlled through discrete energy; finally, the numerical solution of the bending problem of the two-dimensional medium-thick plate with the arbitrary geometric configuration is obtained by solving an energy functional under the control of a global trial function. The application can solve the problem that the Reissner method cannot solve the bending problem of the two-dimensional medium-thick plate with the arbitrary geometric configuration by using a global trial function.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of thin-walled structure, and particularly relates to an energy element numerical solution method for bending of a two-dimensional medium-thick plate with an arbitrary geometric configuration. BACKGROUND

[0002] Thin-walled structures are widely used in aerospace, automobile, ship and building engineering equipment fields, and the bending resistance as a typical mechanical characteristic of structure bearing is one of the most popular research fields in the last century. The deformation problem of thin-walled structures under distributed load is very common in engineering. For the bending problem of a two-dimensional medium-thick plate with an arbitrary geometric configuration, it is extremely difficult to directly solve the three equations of elasticity mechanics under the given boundary conditions. The approximate solution of the problem can be obtained by relaxing the requirements for the trial function by using the weighted residual method, which assumes that the high-order derivable trial function defined in the entire geometric domain satisfies the displacement boundary conditions and the natural boundary conditions. Based on the principle of minimum potential energy, the Ritz method relaxes the requirement for the smoothness of the trial function and only needs to satisfy the displacement boundary conditions without satisfying the natural boundary conditions. As a universal direct variational method, the Ritz method has been widely concerned and applied in the academic field. However, the Ritz method still cannot comprehensively solve complex engineering problems, especially the variational problem in a complex geometric domain. The core difficulty lies in finding a global trial function defined in a complex geometric domain that satisfies the displacement boundary conditions and the difficulty of integration in a complex geometric domain, which limits the development and application of the Ritz method. SUMMARY

[0003] In order to solve the above technical problems, the present application provides an energy element numerical solution method for bending of a two-dimensional medium-thick plate with an arbitrary geometric configuration. The method simulates the two-dimensional medium-thick plate domain with an arbitrary geometric configuration by digging holes in a standard rectangular domain, and constructs a global discrete variable stiffness system by combining the discrete elastic stiffness of the Gauss integral points to represent the distribution of energy in space. The geometric boundary is controlled by the discrete energy, and the solution is finally obtained under the control of the global trial function.

[0004] In order to achieve the above purpose, the technical scheme of the present application is as follows:

[0005] An energy element numerical solution method for bending of a two-dimensional medium-thick plate with an arbitrary geometric configuration, comprising the following steps:

[0006] Step 1, obtaining the material constants and load of the two-dimensional medium-thick plate with an arbitrary geometric configuration, defining the geometric boundary and boundary conditions of the arbitrary two-dimensional shape plate domain, covering the plate domain with a standard rectangular domain, and mapping to a dimensionless coordinate system;

[0007] Step 2, dividing the standard rectangular domain by a background network, and generating Gauss integral points in each grid by Gauss-Legendre integration to discretize the standard rectangular domain;

[0008] Step 3, each Gaussian integral point is determined to be located in the plate domain or the opening domain by the inclusion relationship operation of the Gaussian integral point and the boundary level set function;

[0009] Step 4, the elastic stiffness plate thickness h, the bending stiffness D and the shear modulus G of the isotropic arbitrary geometric configuration two-dimensional medium plate are expanded into matrices with the same dimension as the arranged Gaussian integral points, each element in the matrix corresponds to a Gaussian integral point; if the Gaussian integral point is located in the plate domain, the bending stiffness, the thickness and the shear modulus at the Gaussian integral point are set to 0; if the Gaussian integral point is located in the opening domain, the bending stiffness, the thickness and the shear modulus at the Gaussian integral point remain unchanged, forming a structure global discrete variable stiffness system, representing the strain energy of the arbitrary geometric configuration two-dimensional medium plate;

[0010] Step 5, the load potential energy acting on the arbitrary geometric configuration two-dimensional medium plate is numerically simulated, and the uniform load q is expanded into a matrix with the same dimension as the arranged Gaussian integral points;

[0011] Step 6, the global trial function is constructed in the standard rectangular domain dimensionless coordinates according to the boundary conditions of the arbitrary geometric configuration two-dimensional medium plate; the strain energy and the load potential energy are numerically integrated and the opening energy is removed therefrom; the numerical solution of the bending problem of the arbitrary geometric configuration two-dimensional medium plate is obtained by the Ritz method under the control of the global trial function.

[0012] Preferably, the step 4 specifically comprises the following steps:

[0013] Suppose that the arbitrary geometric configuration two-dimensional medium plate is made of isotropic material, E represents the elastic modulus of the isotropic material, v represents the Poisson's ratio, and the calculation formula of the bending stiffness D is D = Eh 3 / [12(1-v 2 )], and the calculation formula of the shear modulus G is G = E / [2(1+v)], in order to form a structure global discrete variable stiffness system,

[0014] The elastic stiffness plate thickness h, the bending stiffness D and the shear modulus G of the arbitrary geometric configuration two-dimensional medium plate are expanded into matrices with the same dimension as the arranged Gaussian integral points (ξ, η), i.e. h(ξ, η) matrix, D(ξ, η) matrix and G(ξ, η) matrix, each element in the matrix corresponds to a Gaussian integral point;

[0015] According to the results of the inclusion relationship operation of each Gaussian integral point in step 3, each element in the h(ξ, η), D(ξ, η) and G(ξ, η) matrix is globally discretely represented as variable stiffness, and the specific operation is as follows: if the Gaussian integral point (ξ, η) is located in the opening domain , the plate thickness, the bending stiffness and the shear modulus at the point are set to 0; if the Gaussian integral point (ξ, η) is located in the plate domain Ω, the plate thickness, the bending stiffness and the shear modulus at the point remain unchanged:

[0016] The plate thickness h, bending stiffness D, and shear modulus G are expressed as functions of the coordinates (ξ, η) of the Gauss integration points.

[0017] Preferably, the strain energy and the load potential energy are numerically integrated and the hole energy is removed therefrom, specifically including the following steps:

[0018] For the bending problem of the two-dimensional mid-thick plate of arbitrary geometry, the strain energy U is:

[0019]

[0020] Wherein κ is the shear correction factor, which is taken as 5 / 6; the load potential energy Q of the bending problem is determined by three kinds of loads: global uniform load, local uniform load, and concentrated force, and the corresponding load potential energy is in turn:

[0021] 1) Global uniform load:

[0022]

[0023] 2) Local uniform load:

[0024]

[0025] Wherein A is the minimum rectangular domain covering the local uniform load domain under the dimensionless coordinates Located within,

[0026] 3) Concentrated force load:

[0027] Q3 = P0w(ξ0, η0) (12)

[0028] Wherein P0 represents the concentrated force, (ξ0, η0) is the dimensionless coordinates of the concentrated force loading, and w(ξ0, η0) is the deflection value at the loading;

[0029] The discrete variable thickness matrix and the discrete variable stiffness matrix h(ξ, η), D(ξ, η), and G(ξ, η) of the two-dimensional mid-thick plate of arbitrary geometry are characterized by the strain energy in the plate domain by integrating in the standard rectangular domain R, while the load matrix The load potential energy is characterized by integrating in the minimum rectangular domain Rq, at this time, the elastic stiffness at the Gauss integration points in the hole domain is set to 0, and the load is numerically simulated based on the Gauss integration points.

[0030] Preferably, the numerical solution of the bending problem of the two-dimensional mid-thick plate of arbitrary geometry is obtained by solving the functional extreme value according to the Ritz method under the control of the global trial function, specifically including the following steps:

[0031] The total energy formula for the bending problem of a two-dimensional medium-thick plate with arbitrary geometric configuration can be expressed as:

[0032] Π=UQ k (13)

[0033] Based on the specific load conditions, the strain energy U and the load potential energy Qk are substituted into equation (13), where k=1 for a uniformly distributed load, k=2 for a locally distributed load, and k=3 for a concentrated force. Then, the stationary value of the functional is obtained:

[0034]

[0035] A set of A system of linear equations with unknown coefficient vectors:

[0036]

[0037] The formula for calculating the submatrix of the stiffness matrix K is as follows:

[0038]

[0039]

[0040]

[0041] Numerical integration is performed on the elements of the submatrix using Gaussian integrals:

[0042]

[0043]

[0044]

[0045]

[0046] In the formula χ i (ξ) and χ j (η) is the Legendre polynomial of the i-th and j-th terms. and It is the first Item and the Schlegel polynomial; element and Located in the corresponding Zhenhe The r-th row and s-th column of the array;

[0047] Load potential array Q i Each element Q in (i = 1, 2, 3) iψ Gaussian integrals are also used for numerical calculations:

[0048]

[0049] where ψ is the number of rows of the element in the array, and the expression is:

[0050] ψ = (j + 1) + im, i = 0, 1,..., m - 1; j = 0, 1,..., n - 1 (19)

[0051] From the above integral formula, since the elastic stiffness at the Gaussian integration point in the opening domain has been set to 0, and the load is numerically simulated based on the Gaussian integration point, finally when the Gaussian numerical integration is performed, the corresponding strain energy and load potential energy have been removed from the standard rectangular domain R and the minimum rectangular domain R q

[0052]

[0053] where g(ξ,η) represents the integrand in formula (17) and formula (18), is the weight coefficient;

[0054] After the energy removal is completed, the numerical solution of the bending problem can be obtained by solving the linear equation system in formula (15), wherein the bending moment value can be calculated according to the following formula:

[0055]

[0056] ​Based on the above technical scheme, the energy element method for solving the bending problem of a two-dimensional medium-thick plate with an arbitrary geometric configuration is provided, the material constants and the load of the two-dimensional medium-thick plate with an arbitrary geometric configuration are obtained, the geometric boundary and the boundary condition of the plate domain with an arbitrary two-dimensional shape are defined, the plate domain is covered with a standard rectangular domain, and is mapped to a dimensionless coordinate system; the standard rectangular domain is divided by a background network, and the standard rectangular domain is discretized by generating Gaussian integral points in each grid through Gaussian-Legendre integration; the inclusion relationship operation between the Gaussian integral points and the boundary level set function is performed to determine whether each Gaussian integral point is located in the plate domain or the opening domain; the elastic stiffness, the bending stiffness and the shear modulus of the isotropic two-dimensional medium-thick plate with an arbitrary geometric configuration are expanded into a matrix with the same dimension as the arranged Gaussian integral points, and each element in the matrix corresponds to a Gaussian integral point; if the Gaussian integral point is located in the plate domain, the bending stiffness, the thickness and the shear modulus at the Gaussian integral point are set to 0; if the Gaussian integral point is located in the opening domain, the bending stiffness, the thickness and the shear modulus at the Gaussian integral point remain unchanged, a structure global discrete variable stiffness system is formed, and the strain energy of the two-dimensional medium-thick plate with an arbitrary geometric configuration is represented; the load potential energy acting on the two-dimensional medium-thick plate with an arbitrary geometric configuration is numerically simulated, and the uniform load q is expanded into a matrix with the same dimension as the arranged Gaussian integral points; the global trial function is constructed in the dimensionless coordinates of the standard rectangular domain according to the boundary condition of the two-dimensional medium-thick plate with an arbitrary geometric configuration; the strain energy and the load potential energy are numerically integrated, and the opening energy is removed therefrom; the numerical solution of the bending problem of the two-dimensional medium-thick plate with an arbitrary geometric configuration is obtained by solving the functional extreme value according to the Ritz method under the control of the global trial function. The method simulates the plate domain of the two-dimensional medium-thick plate with an arbitrary geometric configuration by digging holes in the standard rectangular domain, then forms a discrete variable thickness matrix and a discrete variable stiffness matrix based on the Gaussian integral points, further constructs a structure global discrete variable stiffness system to represent the energy distribution in space, controls the geometric boundary through the discrete energy, and finally solves the energy functional under the control of the global trial function to obtain the numerical solution of the bending problem of the two-dimensional medium-thick plate with an arbitrary geometric configuration, thereby solving the problem that the Ritz method cannot solve the bending problem of the two-dimensional medium-thick plate with an arbitrary geometric configuration. BRIEF DESCRIPTION OF DRAWINGS

[0057] Figure 1 Fig. 1 is a geometric model and a discrete model of a two-dimensional medium-thick plate with an arbitrary geometric configuration in an embodiment, wherein (a) is a geometric model of a two-dimensional medium-thick plate with an arbitrary geometric configuration; (b) is a Gaussian integral point discrete model, the black Gaussian integral points are located in the plate domain Ω, and the gray Gaussian integral points are located in the opening domain ;

[0058] Figure 2is a global discrete variable stiffness representation method based on Gauss integration points in an embodiment, wherein (a) is a discrete variable stiffness matrix and variable thickness matrix without opening; (b) is a discrete variable stiffness matrix and variable thickness matrix with opening;

[0059] Figure 3 is a numerical simulation schematic diagram of uniform load in an embodiment;

[0060] Figure 4 is a modeling and solving flowchart of the energy element method in an embodiment;

[0061] Figure 5 is a geometric model and a discrete model of an irregular polygon plate example in an embodiment, wherein (a) is a geometric model; (b) is a geometric model and a CCSFFFS boundary condition in a standard rectangular domain dimensionless coordinate system; (c) is a discrete model;

[0062] Figure 6 is a deflection and bending moment deformation cloud atlas of an irregular polygon plate under the CCSFFFS boundary condition and global uniform load in an embodiment, wherein (a) is a deflection w deformation cloud atlas; (b) is a bending moment M x deformation cloud atlas; (c) is a bending moment M y deformation cloud atlas. DETAILED DESCRIPTION

[0063] The technical solutions in the embodiments of the present application will be clearly and completely described in combination with the drawings in the embodiments of the present application.

[0064] The present application proposes a numerical solving method based on global trial functions and global discrete variable stiffness of structures based on the Ritz method, and specifically is an energy element numerical solving method for bending problems of two-dimensional medium-thick plates with arbitrary geometric configurations. The technical solutions of the present application are as follows:

[0065] Step 1, geometric modeling of a two-dimensional medium-thick plate with arbitrary geometric configuration. A two-dimensional medium-thick plate with arbitrary geometric configuration is placed in a minimum rectangular domain that can cover it, and is defined as a standard rectangular domain R of the two-dimensional medium-thick plate with arbitrary geometric configuration, as shown in (a), wherein the gray area Omega represents the two-dimensional medium-thick plate domain with arbitrary geometric configuration. The two-dimensional medium-thick plate domain Omega with arbitrary geometric configuration can be obtained by opening a hole in the standard rectangular domain R, and is represented by Figure 1 , wherein the standard rectangular domain R is the union of the plate domain Omega and the opening domain . For convenience of calculation, the following dimensionless coordinates are defined:

[0066] ξ=2x / a,η=2y / b ξ,η∈[-1,1] (1)

[0067] ​Step 2: Discretization modeling using the energy element method based on the background mesh. Multiple background meshes are divided within a standard rectangular domain based on the geometric boundary of the 2D medium-thick plate with arbitrary geometry. Gaussian integration points are generated in each mesh using the Gauss-Legend integration method to discretize the standard rectangular domain. The energy concentration regions within the standard rectangular domain—i.e., the geometric boundary and stress concentration locations of the 2D medium-thick plate domain with arbitrary geometry—are defined as the critical regions of the standard rectangular domain. Gaussian integration points in the background mesh of the critical regions need to be refined, while those in non-critical regions are sparser. Figure 1 As shown in (b).

[0068] Step 3: Determining the Gaussian integral point based on the boundary level set function. Let the outer boundary of an arbitrary geometrically shaped two-dimensional medium-thick plate be Ξ0, and the inner boundary be Ξk, where k represents the k-th inner boundary. Then, the boundary level set function of this plate can be expressed as a function of coordinates ξ and η, that is, the boundary level set function can be expressed as:

[0069] Ξ0(ξ,η)=0,Ξ k (ξ,η)=0,k=1,2,3,… (2)

[0070] Therefore, whether a Gaussian integration point (ξ, η) is inside or outside the boundary can be determined by the following inclusion relation:

[0071]

[0072] Furthermore, the set of all boundary level set functions can be used to determine whether the Gaussian integral point lies within the arbitrary-shaped plate domain Ω or the open-hole domain. Inside, such as Figure 1 As shown in (b).

[0073] Step 4: Construction of a globally discrete variable stiffness system based on the energy element method. Assume an arbitrary geometric configuration two-dimensional medium-thick plate is manufactured using isotropic materials. Let E represent the elastic modulus of the isotropic material, v represent Poisson's ratio, and the formula for calculating the bending stiffness D is D = Eh. 3 / [12(1-v 2 The formula for calculating the shear modulus G is G=E / [2(1+v)]. To form a globally discrete variable stiffness system, the elastic stiffness plate thickness h, bending stiffness D, and shear modulus G of a two-dimensional medium-thick plate with arbitrary geometric configuration are expanded into matrices with the same dimension as the arranged Gaussian integration points (ξ,η), namely h(ξ,η) matrix, D(ξ,η) matrix, and G(ξ,η) matrix, where each element in the matrix corresponds one-to-one with a Gaussian integration point. Based on the result of the inclusion relation operation of each Gaussian integration point in step 3, a globally discrete variable stiffness characterization is performed on each element in the h(ξ,η), D(ξ,η), and G(ξ,η) matrices. The specific operation is as follows: if the Gaussian integration point (ξ,η) is located in the opening domain... If the Gauss integration point (ξ, η) is located in the plate domain Ω, the plate thickness, bending stiffness and shear modulus at this point remain unchanged:

[0074]

[0075] That is, the plate thickness h, bending stiffness D, and shear modulus G are expressed as functions of the Gauss integration point coordinates (ξ, η). To further illustrate this global discrete variable stiffness representation method, let the dimensionless coordinates ξ and η in the ξ and η directions take the number of Gauss integration points m = n = 30, respectively, and h = D = G = 1. First, assign all elements in this 30 x 30 matrix to 1, that is, assume that all Gauss integration points are located in the plate domain Ω, as shown in Figure 2 (a); then, according to the inclusion relationship operation of the Gauss integration point and the boundary level set functions Ξ0and Ξ k , i.e., equation (3), obtain the Gauss integration points in the open hole domain and set the elastic stiffness h, D, and G at the Gauss integration points in the open hole domain to 0, and the values of the elastic stiffness h, D, and G at the Gauss integration points in the plate domain Ω remain 1, as shown in Figure 2 (b). Finally, obtain two discrete variable stiffness matrices D(ξ, η) and G(ξ, η), and a discrete variable thickness matrix h(ξ, η), thereby forming a global discrete variable stiffness system of the structure, which can represent the discrete energy at each Gauss integration point and further represent the strain energy of a two-dimensional thick plate with an arbitrary geometric configuration in the subsequent numerical integration process.

[0076] Step 5: Numerical simulation of the potential energy of the load acting on the plate domain Ω, which is similar to step 4. Suppose that there is a global or local uniformly distributed load with a size of q acting on the plate domain Ω, and the load domain is Z. For any shape of the load domain Z, place a minimum rectangular domain R q that can cover it, then arrange appropriate number of background meshes and corresponding Gauss integration points in the minimum rectangular domain R q . Expand q into a load matrix with the same dimension as the arranged Gauss integration points , and perform numerical simulation of the load domain Z in the minimum rectangular domain R q by assigning different values to the elements in the matrix, as shown in Figure 3 .

[0077] When the Gauss integration point is located outside the uniformly distributed load scope Z but within the minimum rectangular domain R q , that is, in the domain , there is no uniformly distributed load at this point, Figure 3 which is represented by a gray Gauss integration point; if the Gauss integration point is located within the uniformly distributed load scope Z, then the uniformly distributed load remains unchanged, Figure 3 which is represented by a black Gauss integration point:

[0078]

[0079] The above equation expresses the uniform load q as a function of the coordinates of the Gauss integration point . Where is the complement of Z, the Gauss integration point may be the same as the Gauss integration point (ξ,η) in the plate domain Ω or different. If the standard rectangular domain R is the same as the minimum rectangular domain R q covering the load, the same Gauss integration point is taken to numerically integrate the strain energy and the potential energy of the load. If it is a local uniform load, different Gauss integration points are taken to numerically simulate the strain energy and the potential energy of the load. In particular, when a concentrated force acts on the plate domain Ω, no Gauss integration point is needed for simulation.

[0080] The above five steps can complete the characterization and discretization of the energy of the two-dimensional thick plate of arbitrary geometry and the numerical simulation of the strain energy and the potential energy of the load. Since the standard rectangular domain is divided into a plurality of background grids, the next step is to integrate the energy in each background grid and then superimpose it. In fact, each background grid contains part of the energy of the overall structure, so it is called an energy element. Superimposing the energy of all energy elements is the total energy possessed by the two-dimensional thick plate of arbitrary geometry. The energy element has: 1) storability; 2) cross-geometric boundary; 3) superposition. Therefore, the energy element method characterizes the geometric boundary by discretizing the stiffness system of the overall structure, and achieves the purpose of controlling the geometric boundary by discretizing the energy in the standard rectangular domain. Unlike other numerical methods that discretize displacement fields dominated by geometry, the energy element method is a numerical method that discretizes the overall elastic stiffness dominated by energy.

[0081] Step 6, solving the variational problem of the bending energy functional of the two-dimensional thick plate of arbitrary geometry based on the principle of minimum potential energy and the Ritz method.

[0082] Step 601, based on the thick plate theory, constructing the overall trial function in the dimensionless coordinates of the standard rectangular domain according to the boundary conditions:

[0083]

[0084] Where is the boundary condition function of the boundary Ξ k , N+1 is the total number of the inner and outer boundaries, is the unknown coefficient, χ i (ξ) and χ j (η) are the Legendre polynomials in the ξ and η directions, respectively, each taking m and n terms in the ξ and η directions, respectively. The Legendre polynomials can be obtained by the following recursive formula:

[0085]

[0086] where subscript i represents the highest power of the i-th Legendre polynomial. And the boundary condition function can be expressed by adding an exponential outside the boundary level set function , i.e. For free boundary (F), simply supported (S) or clamped (C), the boundary condition exponential is set as follows:

[0087]

[0088] When the boundary condition exponential of a side is taken as 0, the boundary condition function of the side is constant 1, i.e. no constraint is applied; after increasing the exponential, the multiplication of the N+1 boundary condition functions obtains the boundary condition function of the arbitrary geometry two-dimensional thick plate

[0089] Step 602, numerical integration is performed on the energy and the hole opening energy is removed from the strain energy and the load potential energy. For the bending problem of the arbitrary geometry two-dimensional thick plate, the strain energy U is:

[0090]

[0091] where κ is a shear correction factor, taken as 5 / 6. The load potential energy Q of the bending problem is determined by three kinds of loads: global uniform load, local uniform load and concentrated force, and the load potential energy of them is in turn:

[0092] 1) Global uniform load:

[0093]

[0094] 2) Local uniform load:

[0095]

[0096] where A is the minimum rectangular domain Rq covering the local uniform load domain in the dimensionless coordinate, is located in it,

[0097] 3) Concentrated force load:

[0098] Q3 = P0w(ξ0,η0) (12)

[0099] where P0 represents the concentrated force, (ξ0,η0) is the dimensionless coordinate of the concentrated force loading position, and w(ξ0,η0) is the deflection value at the loading position. It can be seen from the above formula that the discrete variable thickness matrix and the discrete variable stiffness matrix h(ξ,η), D(ξ,η) and G(ξ,η) of the arbitrary geometry two-dimensional thick plate represent the strain energy in the plate domain by integrating in the standard rectangular domain R, and the load matrix The load potential energy was characterized by integration over the smallest rectangular domain Rq, since the elastic stiffness at the Gaussian integration point in the open domain was set to 0 in Equations (4) and (5), and the load was numerically simulated based on the Gaussian integration point.

[0100] Step 603: Obtain the numerical solution to the bending problem by finding the extrema of the functional using the Ritz method. The total system energy formula for the bending problem of a two-dimensional medium-thick plate with arbitrary geometry can be expressed as:

[0101] Π=UQ k (13)

[0102] Based on the specific load conditions, the strain energy U and the load potential energy Q are... k Where the uniformly distributed load is applied over the entire domain, k=1; the uniformly distributed load is applied locally, k=2; and the concentrated force is applied, k=3. Substituting these values ​​into equation (13), we then find the stationary value of the functional:

[0103]

[0104] A set of A system of linear equations with unknown coefficient vectors:

[0105]

[0106] The formula for calculating the submatrix of the stiffness matrix K is as follows:

[0107]

[0108]

[0109]

[0110] Numerical integration is performed on the elements of the submatrix using Gaussian integrals:

[0111]

[0112]

[0113]

[0114]

[0115] In the formula χ i (ξ) and χ j (η) is the Legendre polynomial of the i-th and j-th terms. and It is the first Item and the Schlegel polynomial; element and Located in the corresponding the rth row and s th column of the matrix the rth row and s th column of the matrix

[0116] the load potential energy matrix Q i each element Q in (i = 1, 2, 3) iiψ Gauss integration is also used for numerical calculation:

[0117]

[0118] where ψ is the row number of the element in the matrix, and the expression is:

[0119] ψ = (j + 1) + im, i = 0, 1,..., m - 1; j = 0, 1,..., n - 1 (19)

[0120] From the above integral formula, since the elastic stiffness at the Gauss integration point in the opening domain has been set to 0, and the load has been numerically simulated based on the Gauss integration point. Finally, when performing Gauss numerical integration, the corresponding strain energy and load potential energy have been removed from the standard rectangular domain R and the minimum rectangular domain R q

[0121]

[0122] where g(ξ, η) represents the integrand in equations (17) and (18), is the weight coefficient.

[0123] Through the above three steps, the energy removal is completed, and finally solving the linear equations in equation (15) can obtain the numerical solution of the bending problem, where the bending moment value can be calculated according to the following formula:

[0124]

[0125] The flowchart of the energy element method is summarized in Figure 4 .

[0126] Embodiment

[0127] This embodiment is an example of an irregular polygonal plate under uniform distributed load.

[0128] The energy element method is applied to the bending analysis of an irregular polygonal plate under complex boundary conditions and uniform distributed load, and the deflection and bending moment of the marked points on the irregular polygonal plate are solved. The coordinates of the vertices of the irregular polygonal plate and the marked points 1, 2, and 3 are given in Figure 5 , and the marked points 4, 5, and 6 are the midpoints of the three free boundaries. The geometric and material dimensionless constants of the irregular polygonal plate are: a = b = 4, E = 1.092 x 10 7 , Poisson's ratio υ = 0.3, and shear modulus G = 4.2 x 10​6 , the bending stiffness D = 1, and the thickness of the plate h = 0.01. The geometric model and the discrete model of the irregular polygonal plate example are shown in Figure 5 : Figure 5 (a) is the geometric model thereof; Figure 5 (b) is the geometric model in the dimensionless coordinate system of the standard rectangular domain; Figure 5 (c) is the discrete model of the irregular polygonal plate. 20 x 20 = 400 background meshes are adopted, 10 x 10 Gauss integral points are arranged in the key area meshes of the geometric boundary, 4 x 4 Gauss integral points are arranged in the non-key area meshes, and the number of terms in the shape function in the ξ and η directions is m = n = 14 terms. The boundary conditions are shown in Figure 5 . The Gauss integral point set in the open domain can be defined by the boundary level set function as follows:

[0129]

[0130] where (ξ, η) is the coordinate of the Gauss integral point to be judged, the left lower corner vertex of the irregular polygonal plate is numbered as 1, and the remaining vertices are numbered counterclockwise, where (ξ1, η1) = (-1, -0.5), (ξ2, η2) = (1, -1), (ξ3, η3) = (1, 0), (ξ4, η4) = (0.5, 0), (ξ5, η5) = (0, 0.5), (ξ6, η6) = (0, 1) are the vertex coordinates of the irregular polygonal plate. It can be judged by formula (22) that the Gauss integral point (ξ, η) is located in the open domain .

[0131] Three kinds of boundary conditions, free boundary (F), simply supported (S) and clamped (C), are applied on the irregular polygonal plate, as shown in Figure 5 (b), and the boundary condition function CCSFFFS can be defined as:

[0132]

[0133] The specific implementation steps of the embodiment are as follows:

[0134] Step 1, geometric modeling of the irregular polygonal plate. Figure 5 (a) is the irregular polygonal plate domain Ω, and the thickness of the plate is h. First, a standard rectangular domain R is constructed, the length and width of which are a and b respectively, and the origin is located at the geometric center thereof, the open domain is represented by , and the standard rectangular domain R is the union of the irregular polygonal plate domain Ω and the open domain . And through the following formula:

[0135] ξ = 2x / a, η = 2y / b ξ, η ∈ [-1, 1] (24)

[0136] The standard rectangular domain is converted to dimensionless coordinates, as shown in Figure 5 (b);

[0137] Step 2, background grid-based energy element method discretization modeling. In the standard rectangular domain dimensionless coordinates, 400 square grids of the same size are divided, and the Gauss integral points and their weight coefficients in each background grid are generated by using the Gauss-Legendre quadrature formula. The Gauss integral point density of the key area grid and the non-key area grid is different, as shown in Figure 5 (c).

[0138] Step 3, Gauss integral point judgment based on boundary level set function. The Gauss integral points in the open hole domain are judged by the irregular polygon plate boundary level set function formula (22) and recorded. The black Gauss integral point represents the Gauss integral point in the plate domain Ω, and the gray Gauss integral point represents the Gauss integral point in the open hole domain , as shown in Figure 6 (c).

[0139] Step 4, structure global domain discretization variable stiffness system construction based on energy element method. The elastic stiffness plate thickness h, bending stiffness D and shear modulus G are expanded to the matrix with the same dimension as the arranged Gauss integral points. According to the judgment result of step 3, the global domain discretization variable stiffness representation is made to each element in the matrix according to formula (4), and each element in the matrix is one-to-one corresponding to the Gauss integral point.

[0140] Step 5, numerical simulation of global uniform load. When the irregular polygon plate is subjected to global uniform load, the standard rectangular domain R is the same as the minimum rectangular domain R q covering the load, so the same Gauss integral points as step 2 are taken to simulate the load.

[0141] Step 6, variational problem solution of irregular polygon plate bending analysis energy functional based on the principle of minimum potential energy and Ritz method. According to the above-mentioned construction of the discretization variable thickness matrix and the discretization variable stiffness matrix h(ξ,η), G(ξ,η), D(ξ,η) and the load matrix , the functional stationary value problem is solved, as shown in formulas (9)-(21), and the numerical solution of the bending problem of the irregular polygon plate can be obtained.

[0142] The bending problem of the irregular polygon plate under complex mixed boundary is solved by the proposed energy element method, and the results are shown in Table 1. Compared with the existing results, it can be seen that the energy element method analysis results are in good agreement with the existing results. ​ The deformation cloud of the deflection w and the bending moment M x , M y of the irregular polygon plate is given.

[0143] Table 1 deflection and bending moment values for labeled points on irregular polygonal plate

[0144]

[0145] The above merely provides preferred embodiments of the present application, and is not intended to limit the present application. For those skilled in the art, the present application can have various modifications and changes. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present application shall fall into the scope of protection of the present application.

Claims

1. An energy element numerical solution method for bending of a two-dimensional medium-thick plate of arbitrary geometry, characterized in that, It comprises the following steps: Step 1, obtaining the material constants and loads of the arbitrary geometry two-dimensional plate, defining the geometry boundary and boundary conditions of the arbitrary two-dimensional plate, covering the plate with standard rectangular domains and mapping to the dimensionless coordinate system; Step 2, dividing the standard rectangular domain by using the background network and generating the Gaussian integral points in each grid by Gauss-Legendre integral to discretize the standard rectangular domain; Step 3, judging whether each Gaussian integral point is located in the plate domain or the hole domain by the inclusion relationship operation between the Gaussian integral point and the boundary level set function; Step 4: Determine the elastic stiffness and thickness of the isotropic, arbitrarily geometrically configured two-dimensional medium-thick plate. h Bending stiffness D and shear modulus G The matrix is ​​expanded to have the same dimension as the Gaussian integration points, with each element in the matrix corresponding to a Gaussian integration point. If the Gaussian integration point is located within the plate domain, the bending stiffness, thickness, and shear modulus at the Gaussian integration point are simultaneously set to 0. If the Gaussian integration point is located within the opening domain, the bending stiffness, thickness, and shear modulus at the Gaussian integration point remain unchanged, forming a discrete variable stiffness system of the entire structure, which characterizes the strain energy of a two-dimensional medium-thick plate with arbitrary geometric configuration. Step 5. Numerical simulation of the potential energy of the load acting on the two-dimensional thick plate of arbitrary geometry, the uniform load q extended to a matrix of the same dimension as the arranged Gauss integration points; Step 6, constructing the global trial function under the dimensionless coordinate of the standard rectangular domain according to the boundary conditions of the arbitrary geometry two-dimensional plate, making numerical integration on the strain energy and the load potential energy and removing the hole energy therefrom, and obtaining the numerical solution of the bending problem of the arbitrary geometry two-dimensional plate according to the Ritz method under the control of the global trial function.

2. The energy element numerical solution method for bending of a two-dimensional plate of arbitrary geometry according to claim 1, wherein, The step 4 specifically comprises the following steps: Let's assume that an arbitrary geometry two-dimensional plate is made of isotropic material, with E representing the elastic modulus of the isotropic material, v representing the Poisson's ratio, the bending stiffness D The calculation formula is , the shear modulus G The calculation formula is is a discrete variable stiffness system formed throughout the structure The elastic stiffness and plate thickness of a two-dimensional medium-thick plate with arbitrary geometric configuration. h Bending stiffness D and shear modulus G Expanded to the arrangement of Gaussian integral points Matrices of the same dimension, i.e. Formation Zhenhe A matrix in which each element corresponds one-to-one with a Gaussian integration point; Based on the result of the relational operation for each Gaussian integration point in step 3, and Each element in the array is characterized by globally discrete variable stiffness. The specific operation is as follows: if the Gaussian integration point... Located in the aperture region If the plate thickness, bending stiffness, and shear modulus at that point are zero, then the Gaussian integral point... Located in the plate area If the plate thickness, bending stiffness, and shear modulus remain unchanged at that point: (4) imminent plate thickness h , bending stiffness D , shear modulus G expressed as a function of the coordinates of the gauss integration points ( ξ , η ).

3. The energy element numerical solution method for bending of a two-dimensional plate of arbitrary geometry according to claim 2, wherein Making numerical integration on the strain energy and the load potential energy and removing the hole energy therefrom specifically comprises the following steps: For the bending problem of a two-dimensional thick plate of any geometry, the strain energy U is: (9) wherein κ is the shear correction factor, taken as 5 / 6; the load potential energy for bending problems Q is determined by the following three kinds of loads: global uniform load, local uniform load, and concentrated force, and the corresponding load potential energy is in turn: 1) Global uniform load: (10) 2) Local uniform load: (11) where A is the minimum rectangular domain R covering the domain of the local uniform load under the dimensionless coordinate q , located within, 3) Concentrated load: (12) wherein P 0 represents the concentration force, ξ 0, η 0 is the dimensionless coordinate at the concentration force loading, w ξ 0, η 0 is the deflection value at the loading.​ Discrete variable thickness matrix and discrete variable stiffness matrix for arbitrary geometry two-dimensional mid-thickness plates , The strain energy in the plate domain is represented by integrating over the standard rectangular domain R, while the load matrix The load potential energy is represented by integrating over the minimum rectangular domain R q where the elastic stiffness at the Gauss integration points in the domain of the hole has been set to zero and the load has been numerically simulated based on the Gauss integration points.

4. The energy element numerical solution method for two-dimensional bending of plates of arbitrary geometry according to claim 3, characterized in that, Obtaining the numerical solution of the bending problem of the arbitrary geometry two-dimensional plate according to the Ritz method under the control of the global trial function specifically comprises the following steps: The system total energy formula of the bending problem of the two-dimensional middle-thick plate with arbitrary geometry can be expressed as: (13) According to the specific load conditions, the strain energy U and the load potential energy Q k , where the global uniform load loading takes k = 1, the local uniform load takes k = 2, the concentrated force takes k = 3, substitute equation (13), and then take the stationary value of the functional: (14) A set of linear equations with unknown coefficient vector can be obtained: (15) The sub-matrix of the stiffness matrix K is calculated according to the following formula: (16) The elements in the sub-matrix are numerically integrated using Gaussian integration: (17) In the formula and It is the first i Item and the j Schlegel polynomial, and It is the first Item and the Schlegel polynomial; element and Located in the corresponding Zhenhe The first r row and number s List; Load potential energy array Q i i each element in the array Gaussian integration is also used for numerical calculation: (18)​ wherein is the number of rows of the array for this element, expressed as: (19) From equation (17) and equation (18), it is known that the elastic stiffness at the Gauss integration points in the open hole domain has been set to zero, and the load has been numerically simulated based on the Gauss integration points; finally, when the Gauss numerical integration is performed, the corresponding strain energy and the load potential energy have been removed from the standard rectangular domain R and the minimum rectangular domain R q (20)​ wherein g ( ξ , η ) represents the integrand in formula (17) and formula (18), is a weight coefficient; After the energy removal, the numerical solution of the bending problem can be obtained by solving the linear equation system in the formula (15), wherein the bending moment value can be calculated according to the following formula: （21）。

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