Energy element numerical solution method for large deformation of thin plate in any shape

Abstract

The invention discloses an energy element numerical solution method for large deformation of a thin plate in any shape, which comprises the following steps of: forming a hole in a standard rectangular domain to simulate the shape of the thin plate in any shape, generating energy elements based on Gaussian integral points and variable stiffness in the standard rectangular domain to represent thin plate energy distribution, and constructing a discrete model of the thin plate in any shape; a global discrete variable stiffness energy system is formed; then capturing the geometric boundary of the thin plate through discrete energy based on a global trial function and removing energy in a hole domain to obtain a high-precision numerical integral of strain energy of the thin plate in any shape, and finally solving a nonlinear equation set based on a minimum potential energy principle and a Ritz method to complete large deformation analysis of the thin plate.

Description

Technical Field

[0001] The invention relates to the technical field of thin-walled structures, and in particular to a method for numerically solving an energy element of large deformation of a thin plate of arbitrary shape. Background Art

[0002] Thin-walled structures are widely used in engineering equipment fields such as aerospace, automobiles, ships and construction. The bending of thin plates is a classic problem in solid mechanics. When the deflection of a thin plate is much smaller than its thickness, the deformation of the thin plate is small and the in-plane deformation can be ignored. However, when the deflection exceeds half of the plate thickness or is equivalent to it, the deformation of the thin plate is large, the in-plane deformation cannot be ignored, and in-plane stress will be generated, which bears the external load together with the bending stress. Considering the geometric nonlinear effect of large deformation of thin plates on in-plane stress, von Karman's large deformation theory should be used for analysis. Large deformation theory contains two nonlinear partial differential equations to describe the large deformation response of thin plates. Due to their strong nonlinearity, it is very difficult to solve them directly. Therefore, the partial differential equations are usually converted into functional integral form through the minimum potential energy principle and solved by numerical methods. The Ritz method is one of the most widely used numerical methods. It uses the global trial function, and has the advantages of high accuracy, fast convergence and no mesh. It is often used to analyze large deformation problems of thin plates. However, the Ritz method cannot completely solve the large deformation problem of thin plates of arbitrary shapes. The core difficulty lies in the fact that the Ritz method needs to find a global trial function defined on the domain of thin plates of arbitrary shapes that satisfies the displacement boundary conditions, and the integration over complex geometric domains is difficult. Summary of the invention

[0003] In order to solve the above technical problems, the present invention proposes a method for numerically solving energy elements of large deformation of thin plates of arbitrary shapes. The shape of a thin plate of arbitrary shape is simulated by opening a hole in a standard rectangular domain, and then energy elements based on Gaussian integral points and variable stiffness are generated in the standard rectangular domain to characterize the energy distribution of the thin plate. A discrete model of a thin plate of arbitrary shape is constructed to form a global discrete variable stiffness energy system. Then, based on the global trial function, the geometric boundary of the thin plate is captured by discrete energy and the energy in the hole domain is removed to obtain a high-precision numerical integral of the strain energy of the thin plate of arbitrary shape. Finally, the large deformation analysis of the thin plate is completed by solving the nonlinear equations based on the minimum potential energy principle and the Ritz method.

[0004] In order to achieve the above object, the technical solution of the present invention is as follows:

[0005] A method for numerically solving energy elements of large deformation of a thin plate of arbitrary shape comprises the following steps:

[0006] Obtain material constants and loads of arbitrary-shaped thin plates, define geometric boundaries and boundary conditions of arbitrary-shaped thin plates, cover arbitrary-shaped thin plate domains with standard rectangular domains, and open holes in the standard rectangular domains according to the geometric boundaries of arbitrary-shaped thin plates to simulate the shape of arbitrary-shaped thin plate domains, and establish a dimensionless coordinate system;

[0007] According to the shape of the arbitrary-shaped thin plate, several energy elements are generated in the standard rectangular domain to cover the arbitrary-shaped thin plate domain;

[0008] According to the two-dimensional Gauss-Legendre quadrature rule, different numbers of Gaussian integration points are generated in each energy element based on the highest degree of the integrand polynomial function and the position of the hole domain.

[0009] Through the Boolean operation between the coordinates of each Gaussian integral point and the boundary level set function, the position of the Gaussian integral point in the energy element is determined, and the variable stiffness characterization is performed on each Gaussian integral point in turn according to the Boolean operation result to complete the discrete modeling of thin plates of arbitrary shapes;

[0010] Based on Legendre polynomials and boundary condition functions, a global trial function of a thin plate of arbitrary shape is constructed, the strain energy and load potential energy are integrated, and the energy in the hole domain is removed in combination with the global variable stiffness; based on the Ritz method, the stationary value of the functional is obtained, a nonlinear system of equations is derived, and the energy element is combined to complete the high-precision numerical integration of each element in the nonlinear system of equations; the nonlinear system of equations is solved, and the unknown coefficients are obtained and brought back to the global trial function to obtain the numerical solution of large deformation of thin plates of arbitrary shape.

[0011] Preferably, constructing a global test function of a thin plate of arbitrary shape based on Legendre polynomials and boundary condition functions specifically includes the following steps:

[0012]

[0013] In the formula (δ=u,v,w) is the boundary Ξ k The boundary condition function, T+1 is the total number of inner and outer boundaries, is the unknown coefficient, χ i (ξ) and χ j (η) are Legendre polynomials in the ξ and η directions, truncated at the mth and nth terms in the ξ and η directions respectively; boundary condition function Describe the boundary conditions imposed on the kth boundary, which is represented by the boundary level set function Ξ of the kth boundary k (ξ,η) is obtained by assigning boundary index, which is determined by the following three boundary conditions:

[0014]

[0015] Legendre polynomials are a set of orthogonal complete polynomials derived from the following recursive formula:

[0016]

[0017] Where i-1 represents the highest power of the i-th Legendre polynomial; after adding boundary exponents to the boundary level set function in sequence, multiplying T+1 boundary condition functions together, the boundary condition function of a thin plate of any shape can be obtained: By multiplying it by the Legendre polynomials in the ξ and η directions respectively, the global trial function construction of thin plates of arbitrary shapes can be completed.

[0018] Preferably, based on the Ritz method, a stationary value of the functional is obtained, a nonlinear equation group is derived, and the energy element is combined to complete the high-precision numerical integration of each element in the nonlinear equation group, which specifically includes the following steps:

[0019] According to the Ritz method, the total energy of the system of a thin plate of any shape can be expressed as:

[0020] Π=UQ (19)

[0021] The Ritz method requires the coefficient P to be determined ij Take the partial derivative to obtain the stationary value of the functional:

[0022]

[0023] A set of nonlinear equations can then be derived and written in matrix form:

[0024]

[0025] Where K l is the linear stiffness matrix containing 3mn×3mn elements; P is the undetermined coefficient array containing 3mn×1 elements, which is composed of the undetermined coefficients in the global trial function (7); K nl is the nonlinear stiffness matrix containing 3mn×1 elements, and the undetermined coefficients in brackets are contained in the nonlinear equation; Q is an array containing 3mn×1 elements derived from the load potential energy Q; the linear stiffness matrix K l The expressions for each element in are written as:

[0026]

[0027] The elements Located in the matrix The rth row and sth column of Located in the matrix The rth row and sth column of both contain the variable stiffness function c ij (ξ,η) and d ij (ξ,η), so c ij (ξ,η) and d ij (ξ,η) needs to be included in the integral:

[0028]

[0029] The expression of the integrand is:

[0030]

[0031] In the formula and Representative Item and Legendre polynomial, χ i (ξ) and χ j (η) are the ith and jth Legendre polynomials;

[0032] Nonlinear stiffness matrix K nl The expressions for each element in are written as:

[0033]

[0034] In the formula, the element Located in the array The rth row, element Located in the array The rth row of contains the variable stiffness function c ij (ξ,η), so c ij (ξ,η) needs to be included in the integral:

[0035]

[0036] Where α(ξ,η), β(ξ,η), γ(ξ,η) represent the global test function (7), and the expression of the integrand is:

[0037]

[0038] Calculate the function value at each Gaussian integral point in each rectangular element and non-rectangular element in turn, multiply it by the corresponding integral coefficient, and finally superimpose the integral value in each energy element:

[0039]

[0040] In the formula Corresponding to the integrand G is the total number of rectangular elements, H is the total number of non-rectangular elements, and are the integration domains of the kth rectangular element and the zth non-rectangular element respectively; and are the Gaussian integration points and quadrature coefficients within the kth rectangular element, and are the Gaussian integration points and quadrature coefficients within the zth non-rectangular element; and and Determine the tensile stiffness and bending stiffness of the Gaussian integration points in the kth rectangular element and the zth non-rectangular element respectively;

[0041] Numerically integrate the array derived from the load potential energy, where the elements Located in array Q w No. OK:

[0042]

[0043] Since the arbitrary shape thin plate domain is exactly the same as the global uniform load domain, the Gaussian integration points used for the numerical integration of strain energy can be directly used for the Numerical integration of :

[0044]

[0045] Based on the above technical solution, the beneficial effects of the present invention are as follows: the present invention proposes a method for numerically solving energy elements for large deformation of thin plates of arbitrary shapes, wherein the global trial function is constructed based on Legendre polynomials and boundary condition functions. In order to perform high-precision numerical simulation of thin plates of arbitrary shapes, the energy element method uses global trial functions, global Gaussian integral point discretization, global variable stiffness and extended interval integration to construct a discrete energy system on a standard rectangular domain, and characterizes the shape of a thin plate of arbitrary shape through the energy distribution of the global variable stiffness simulation plate, and then approximates the deformation of the thin plate through the global trial function. The so-called energy element is a unit that simulates the strain energy of the structure on a standard rectangular domain, wherein the displacement adopts the global trial function, and the stiffness adopts the global variable stiffness, and the geometric boundary of the structure is allowed to pass through the energy element. For two-dimensional problems, the energy element method first opens a hole in a standard rectangular domain to simulate the shape of a thin plate of arbitrary shape, and then generates energy elements based on Gaussian integral points and variable stiffness in the standard rectangular domain to characterize the energy distribution of the thin plate, constructs a discrete model of a thin plate of arbitrary shape, and forms a global discrete variable stiffness energy system; then, based on the global trial function, the geometric boundaries of the thin plate are captured through discrete energy and the energy in the hole domain is removed to obtain a high-precision numerical integral of the strain energy of a thin plate of arbitrary shape, and finally, the large deformation analysis of the thin plate is completed based on the minimum potential energy principle and the Ritz method to solve the nonlinear equations. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is the geometric model of a thin plate of arbitrary shape;

[0047] Figure 2 Generate examples of energy elements for thin plates of arbitrary shapes;

[0048] Figure 3 It is a discrete model of thin plates of arbitrary shape based on global variable stiffness;

[0049] Figure 4Examples of variable stiffness characterization are shown below: (a) is the variable stiffness characterization of the rectangular element 4; (b) is the variable stiffness characterization of the non-rectangular element 4;

[0050] Figure 5 It is a flow chart of a method for numerically solving energy elements for large deformation of a thin plate of arbitrary shape;

[0051] Figure 6 Schematic diagram of a circular plate model with a central elliptical hole under fully fixed boundary conditions: (a) is a geometric model diagram; (b) is a discrete model diagram;

[0052] Figure 7 Comparison of deformation cloud maps of the three displacement components u, v, and w of a circular plate with a central elliptical hole under fully fixed boundary conditions and globally uniformly distributed loads: (a) is the deformation cloud map in the u direction of the finite element method; (b) is the deformation cloud map in the v direction of the finite element method; (c) is the deformation cloud map in the w direction of the wavelet method; (d) is the deformation cloud map in the u direction of the energy element method; (e) is the deformation cloud map in the v direction of the energy element method; and (f) is the deformation cloud map in the w direction of the energy element method. DETAILED DESCRIPTION

[0053] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings in the embodiments of the present invention.

[0054] Based on the Ritz method, the present invention proposes a numerical solution method for simulating structural deformation and global variable stiffness discrete energy based on global trial function, specifically a numerical solution method for energy element of large deformation of arbitrary shape thin plate. The technical solution of the present invention is as follows:

[0055] Step 1: Geometric modeling of a thin plate of arbitrary shape. First, place the thin plate of arbitrary shape into its standard rectangular domain R, where R is defined as the minimum rectangular domain with a length a and a width b covering the plate, such as Figure 1 As shown. The gray area S represents a thin plate domain of arbitrary shape, which can be obtained by opening a hole in R according to the geometric boundary of S. Let Ω represent the hole domain, then the union of the hole domain Ω and the plate domain S is the standard rectangular domain R. 0 represents the outer boundary of the plate domain S, Ξ k Represents the kth inner boundary. To standardize the solution procedure, the following dimensionless coordinates are used for analysis:

[0056] ξ=2x / a,η=2y / b ξ,η∈[-1,1] (31)

[0057] Step 2, generate energy elements. The energy element method uses a global trial function to simulate structural deformation and a global variable stiffness to simulate structural energy distribution. Therefore, according to the principle of integral superposition, the strain energy of a thin plate of any shape can be divided into multiple rectangular subdomains, which are integrated and superimposed separately. Therefore, the energy represented by the rectangular subdomain is defined as an energy element. The purpose of generating energy elements is to capture the energy at local fine geometric boundaries (such as near complex opening domains) with high precision, while reducing the Gaussian integration points required for the discrete plate domain S and thus reducing the amount of calculation. The energy element is generated in the standard rectangular domain R according to the shape of the plate domain S and covers the entire plate domain S. Since Gaussian integrals can accurately integrate polynomial functions in rectangular domains, in order to reduce the amount of calculation, it is necessary to arrange rectangular energy elements with as large an area as possible in the plate domain S as much as possible, and at complex geometric boundaries, rectangular energy elements are used to cover the geometric boundaries and use more Gaussian integration points to perform high-precision numerical integration on the strain energy on the complex geometric domain: the energy element that does not contain the hole domain Ω is defined as a rectangular energy element, and the energy element that does not contain the hole domain Ω is a non-rectangular energy element, referred to as rectangular elements and non-rectangular elements. The intersection of the rectangular element and the hole domain Ω is empty and completely located in the plate domain S; however, the intersection of the non-rectangular element and the hole domain Ω is not empty and contains part of the hole domain Ω and part of the plate domain S. It should be noted that the energy element is different from the grid or unit in the traditional numerical method, because the energy element can contain any geometric boundary and be crossed by it, and its discretized object is the elastic stiffness rather than the displacement field. For the arbitrary shape thin plate shown in step 1, a total of 5 rectangular elements and 4 non-rectangular elements are used to discretize the plate domain S, see Figure 2 .

[0058] Step 3: Generate Gaussian integration points in the energy element. According to the two-dimensional Gauss-Legendre quadrature rule, if the highest degree of the integrand polynomial function is (2M–1)×(2N–1), an accurate integration result can be obtained by generating M×N Gaussian integration points in the rectangular element. However, for non-rectangular elements, more than M×N Gaussian integration points need to be generated in them to capture the energy near the irregular boundary of the plate domain S, so as to obtain a high-precision numerical integration of the strain energy on the plate domain S. The specific number of Gaussian integration points required in the non-rectangular element is determined according to the shape of the hole domain Ω it contains.

[0059] Step 4: Discrete modeling of arbitrary shape thin plates based on Gaussian integration points and global variable stiffness.

[0060] Step 401, determine the position of the Gaussian integration point based on the plate geometry boundary level set function. The discrete model of the arbitrarily shaped thin plate is composed of Gaussian integration points based on variable stiffness characterization: First, it is necessary to determine whether each Gaussian integration point is located in the plate domain S or the hole domain Ω, and then perform variable stiffness characterization on it. The position of the Gaussian integration point needs to be based on its coordinates (ξ,η) and the boundary level set function Ξ 0 (ξ,η),Ξ k Boolean operation judgment between (ξ,η), where Ξ 0(ξ,η) is the level set function of the outer boundary of the plate domain S, Ξ k (ξ,η) is the level set function of the kth inner boundary. T represents the total number of inner boundaries. If a Gaussian integral point (ξ,η) is inside all boundaries, then the point is located in the plate domain S:

[0061]

[0062] Otherwise the point is inside the hole domain Ω:

[0063]

[0064] Step 402: Based on the above Boolean operation results, the Gaussian integral points in the whole domain are characterized by variable stiffness. The stiffness of the large deformation problem of the thin plate can be divided into the tensile stiffness term c ij (i,j=1,2,3) and the bending stiffness term d ij Two parts, the tensile stiffness term c ij It can be derived through the following constitutive relation:

[0065]

[0066] Where E is the elastic modulus, h is the thickness of the plate, μ is the Poisson's ratio, and c ij is the element of the tensile stiffness matrix C, N is the mid-surface force vector containing three mid-surface force components N x ,N y ,N xy , ε 0 is the in-plane strain vector containing three in-plane strain components ε x0 ,ε y0 ,γ xy0 The bending stiffness term d ij It can be derived through the following constitutive relation:

[0067]

[0068] Where D = Eh 3 / 12(1–μ 2 ) is the bending stiffness, d ij is the element of the bending stiffness matrix D, M is the moment vector containing three moment components M x ,M y ,M xy , κ is the curvature vector containing three curvature components κ x ,κ y ,κ xy .

[0069] The variable stiffness characterization will be the tensile stiffness c ij and bending stiffness d ij Expressed as a variable stiffness function c about the coordinates of the Gaussian integration pointij (ξ,η) and d ij (ξ,η), and assign variable stiffness properties to each Gaussian integration point based on the Boolean operation results to capture the internal and external geometric boundaries of arbitrary-shaped thin plates. The tensile stiffness c at the Gaussian integration point in the plate domain S is ij and bending stiffness d ij The in-plane stiffness c at the Gaussian integration point in the hole domain Ω is constant. ij and bending stiffness d ij Set to 0:

[0070]

[0071] Finally, the discrete modeling of the arbitrary shape thin plate in step 1 can be completed, such as Figure 3 As shown, the gray Gaussian integration points are the points where the tensile stiffness and bending stiffness are 0, and the black Gaussian integration points are the points where the tensile stiffness and bending stiffness are constant.

[0072] To further clarify the discrete modeling process, the variable stiffness characterization process of Gaussian integration points in rectangular element 4 and non-rectangular element 4 is described in detail below. Assume that c ij (ξ,η)=d ij (ξ,η)=1, 7×7 Gaussian integration points are generated in rectangular element 4, and 10×10 Gaussian integration points are generated in non-rectangular element 4. Since rectangular element 4 is completely contained within all boundaries of the plate, it is located in the plate domain S, where the in-plane stiffness and bending stiffness at the Gaussian integration points are unchanged, see Figure 4 (a). For the non-rectangular element 4, it is necessary to determine whether each Gaussian integral point is inside all boundaries. If not, it means that the point is inside the hole domain Ω, and the in-plane stiffness and bending stiffness at the point need to be set to 0. The in-plane stiffness and bending stiffness of the other Gaussian integral points in the plate domain S remain unchanged, as shown in Figure 4 (b) as shown.

[0073] Step 5: Solve the large deformation variational problem of arbitrary shape thin plates based on the Ritz method.

[0074] Step 501: construct a global trial function based on thin plate theory and large deformation theory. The large deformation of a thin plate needs to consider the stretching of the mid-surface of the plate, so it is necessary to consider the two displacement components u and v of the mid-surface of the plate and the deflection displacement component w of the plate. Based on the above three displacement components, the global trial function of a thin plate of any shape can be constructed in the dimensionless coordinate system in step 1:

[0075]

[0076] In the formula (δ=u,v,w) is the boundary Ξ k The boundary condition function, T+1 is the total number of inner and outer boundaries, is the unknown coefficient, χi (ξ) and χ j (η) are Legendre polynomials in the ξ and η directions, truncated at the mth and nth terms in the ξ and η directions, respectively. Among them, the boundary condition function Describe the boundary conditions imposed on the kth boundary, which is represented by the boundary level set function Ξ of the kth boundary k (ξ,η) is obtained by assigning boundary index, which is determined by the following three boundary conditions:

[0077]

[0078] Legendre polynomials are a set of orthogonal complete polynomials derived from the following recursive formula:

[0079]

[0080] Where i-1 represents the highest power of the i-th Legendre polynomial. After adding boundary exponents to the boundary level set function in sequence, multiplying T+1 boundary condition functions together, the boundary condition function of a thin plate of any shape can be obtained. By multiplying it by the Legendre polynomials in the ξ and η directions respectively, the global trial function construction of thin plates of arbitrary shapes can be completed.

[0081] Step 502: Integrate the strain energy and load potential energy and remove the energy in the hole domain by combining the global variable stiffness. The strain energy U of a thin plate of arbitrary shape is calculated by the following formula:

[0082]

[0083] Where |J| is the Jacobian determinant and |J| = ab / 4. After completing the variable stiffness characterization, the elements in the tensile stiffness matrix C and the bending stiffness matrix D are both variable stiffness functions about the Gaussian integration point coordinates (ξ,η):

[0084]

[0085] From equations (10) and (11), it can be seen that the energy element method forms a global discrete variable stiffness system to discretize the strain energy of a thin plate of arbitrary shape, and combines the variable stiffness characterization to set the stiffness at the Gaussian integral point in the hole domain Ω to 0, so as to remove the discrete energy at the zero stiffness Gaussian integral point in the subsequent numerical integration process, and then obtain the high-precision numerical integration of the strain energy in the domain of a thin plate of arbitrary shape. Therefore, unlike the traditional numerical method of geometrically dominated discrete displacement fields, the energy element method is a meshless numerical method that controls the geometric boundaries of the structure through discrete energy. At the same time, the integration of the strain energy of a thin plate of arbitrary shape by the energy element method is performed in the standard interval [-1,1] in the dimensionless coordinate system, making the formula and calculation procedure of the energy element method completely standard, and it is a universal and standardized numerical method.

[0086] Similarly, when a uniformly distributed load q acts on a thin plate domain of arbitrary shape, its load potential energy Q can be numerically simulated in the same way as the strain energy integration:

[0087]

[0088] In the formula, the global uniformly distributed load q is expressed as the load distribution function q(ξ,η) about the coordinates of the Gaussian integration point (ξ,η).

[0089] Step 503: Based on the Ritz method, the stationary value of the functional is calculated to obtain a numerical solution to the large deformation problem. According to the Ritz method, the total system energy of a thin plate of any shape can be expressed as:

[0090] Π=UQ (43)

[0091] The Ritz method requires the coefficient P to be determined ij Take the partial derivative to get the stationary value of the functional:

[0092]

[0093] A series of nonlinear equations can then be derived and written in matrix form:

[0094]

[0095] Where K l is the linear stiffness matrix containing 3mn×3mn elements; P is the undetermined coefficient array containing 3mn×1 elements, which is composed of the undetermined coefficients in the global trial function (7); K nl is the nonlinear stiffness matrix containing 3mn×1 elements, and the undetermined coefficients in brackets are contained in the nonlinear equation; Q is an array containing 3mn×1 elements derived from the load potential energy Q. The linear stiffness matrix K l The expressions for each element in can be written as:

[0096]

[0097] The elements Located in the matrix The rth row and sth column of Located in the matrix The rth row and sth column of , both of which contain the variable stiffness function c ij (ξ,η) and d ij (ξ,η), so c ij (ξ,η) and d ij (ξ,η) needs to be included in the integral:

[0098]

[0099] The expression of the integrand is:

[0100]

[0101] In the formula and Representative Item and Legendre polynomial, χ i (ξ) and χ j (η) are the ith and jth Legendre polynomials.

[0102] Nonlinear stiffness matrix K nl The expressions for each element in can be written as:

[0103]

[0104] Similar to the numerical integration of linear matrices, the elements Located in the array The rth row, element Located in the array The rth row of , which all contain the variable stiffness function c ij (ξ,η), so c ij (ξ,η) needs to be included in the integral:

[0105]

[0106] Where α(ξ,η), β(ξ,η), γ(ξ,η) represent the global test function (7). The expression of the integrand in the above formula is:

[0107]

[0108] Based on the energy element, high-precision numerical integration of equations (17) and (20) can be achieved. It is necessary to calculate the function value at each Gaussian integral point in each rectangular element and non-rectangular element in turn, then multiply it by the corresponding integral coefficient, and finally superimpose the integral value in each energy element:

[0109]

[0110] In the formula Corresponding to the integrand G is the total number of rectangular elements, H is the total number of non-rectangular elements, and are the integration domains of the kth rectangular element and the zth non-rectangular element respectively; and are the Gaussian integration points and quadrature coefficients within the kth rectangular element, and are the Gaussian integration points and quadrature coefficients within the zth non-rectangular element; and and Determine the tensile stiffness and bending stiffness of the Gaussian integration points in the kth rectangular element and the zth non-rectangular element respectively.

[0111] Numerically integrate the array derived from the load potential energy, where the elements Located in array Q w No. OK:

[0112]

[0113] Since the arbitrary shape thin plate domain is exactly the same as the global uniform load domain, the Gaussian integration points used for the numerical integration of strain energy can be directly used for the Numerical integration of :

[0114]

[0115] Finally, the nonlinear equation system (15) is solved to obtain a set of unknown coefficients, which can be brought back to the global trial function to obtain the numerical solution of large deformation of thin plates of arbitrary shapes. The flowchart of the energy element method is summarized in Figure 5 middle.

[0116] Example

[0117] This embodiment is a calculation example of a circular plate with a central elliptical hole subjected to a uniformly distributed load over the entire area.

[0118] The energy element method is applied to the large deformation analysis of a circular plate with a central elliptical hole under a uniformly distributed load and fixed at both the inner and outer boundaries, and the deflection value of a point in the plate is calculated. The dimensionless constants of the geometry and materials of the circular plate with a central elliptical hole are: a=b=2, E=500, the radius of the circular plate r=1, the Poisson's ratio υ=0.3, and the thickness of the plate h=0.02. The dimensionless length of the semi-major axis of the elliptical inner hole is 0.5, and the semi-minor axis is 0.25. The geometric model and discrete model of the circular plate with a central elliptical hole are as follows: Figure 6 As shown: Figure 6 (a) is its geometric model; Figure 6 (b) is the discrete model.

[0119] Five non-rectangular elements are used to discretize the circular plate with a central elliptical hole. 65×65 Gaussian integration points are arranged in each non-rectangular element. The set of Gaussian integration points in the plate domain S can be defined by the boundary level set function:

[0120] S={(ξ, η)|(ξ 2 +η 2 -1)<0}∩{(ξ, η)|(4ξ 2 +16η2 -1)>0} (55)

[0121] Where (ξ,η) is the coordinate of the Gaussian integral point to be determined. 1 ,η 1 )=(0,0.5) are the coordinates of the deflection point to be determined.

[0122] The boundary conditions of the elliptical inner boundary and the circular outer boundary of the circular plate with a central elliptical hole are both fixed, so the corresponding boundary condition function can be defined as:

[0123]

[0124] The specific implementation steps of this embodiment are as follows:

[0125] Step 1: Geometric modeling of thin plates of arbitrary shapes. Figure 6 (a) The gray area shows a circular plate domain S with a central elliptical hole. First, construct its standard rectangular domain R, with a length and b width respectively, and the origin is located at its geometric center. The hole domain is represented by Ω. The circular plate domain S with a central elliptical hole can be obtained by opening a hole in the standard rectangular domain R and removing the hole domain Ω, and the following dimensionless coordinates are established:

[0126] ξ=2x / a, η=2y / b, ξ,η∈[-1,1] (57)

[0127] Step 2: Generate energy elements. In the standard rectangular domain dimensionless coordinates, according to the geometric shape of a circular plate with a central elliptical hole, five non-rectangular elements are used to cover the plate domain S to capture the energy within the fine geometric boundary near the hole domain Ω.

[0128] Step 3: Generate Gaussian integration points in the energy element. Considering the complex geometry of the plate, a high-order global trial function is required to obtain a converged solution. Combined with the position of the hole domain Ω contained in each non-rectangular element, 65×65 Gaussian integration points are generated in each non-rectangular element, of which 14176 Gaussian integration points are in the plate domain S. The zero stiffness Gaussian integration points have been removed from the discrete model, as shown in Figure 6 (b) as shown.

[0129] Step 4: Discrete modeling of arbitrary-shaped thin plates based on Gaussian integration points and global variable stiffness. Based on the Boolean operation between the coordinates of the Gaussian integration point (ξ,η) and the boundary level set function (25), it is determined whether the point is located in the plate domain S. If not, the point is in the hole domain Ω. The tensile stiffness term c is derived according to the constitutive relation: ij and the bending stiffness term d ij The expression of is shown in equations (4) to (5). Then, according to the result of Boolean operation at each Gaussian integral point, variable stiffness attribute is assigned to it, as shown in equation (6). Finally, the discrete modeling is completed, as shown in Figure 6(b).

[0130] Step 5: Solve the large deformation variational problem of arbitrary-shaped thin plates based on the Ritz method. According to the Legendre polynomials (9) and the boundary condition function formula (26) of the circular plate with a central elliptical hole, the global test function can be constructed. Then, the strain energy and load potential energy are integrated, and combined with the elastic variable stiffness function c ij (ξ,η) and d ij (ξ,η), and the load distribution function q(ξ,η), and the energy in the hole domain Ω is removed during the numerical integration process. Finally, the functional stationary value is obtained based on the Ritz method, and the high-precision numerical integration of the elements in the linear stiffness matrix and the nonlinear stiffness array is completed based on the energy element. The nonlinear equations are solved to obtain a set of undetermined coefficients, thereby obtaining the numerical solution to the large deformation problem of the circular plate with a central elliptical hole, as shown in formulas (7) to (24) for details.

[0131] The proposed energy element method is used to solve the large deformation problem of a circular plate with a central elliptical hole under different load concentrations. The results are shown in Table 1 and compared with the existing results. It can be seen that the analysis results of the energy element method are in good agreement with the existing results. In addition, Table 1 also shows the convergence process of the numerical solution of the energy element method, where the number of shape function terms m×n in the ξ and η directions is 13×13 to 15×15. Figure 7 The deformation cloud maps of the three displacement components u, v, and w of the circular plate with a central elliptical hole are given and compared with the deformation cloud maps in the finite element method and the literature, namely: (a) deformation cloud map in u direction by finite element method; (b) deformation cloud map in v direction by finite element method; (c) deformation cloud map in w direction by wavelet method; (d) deformation cloud map in u direction by energy element method; (e) deformation cloud map in v direction by energy element method; (f) deformation cloud map in w direction by energy element method.

[0132] Table 1 Deflection values ​​of the midpoint (0,0.5) of the circular plate with a central elliptical hole

[0133]

[0134] The above is only a preferred implementation of the energy element numerical solution method for large deformation of a thin plate of arbitrary shape disclosed in the present invention, and is not intended to limit the embodiments of the present application. For those skilled in the art, the embodiments of the present application may have various changes and variations, including the shape of the plate and the size and type of the applied load. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the embodiments of the present application shall be included in the protection scope of the embodiments of the present application.

Claims

1. A method for numerically solving energy elements of large deformation of thin plates of arbitrary shapes, characterized in that: The steps include: Obtain material constants and loads of arbitrary-shaped thin plates, define geometric boundaries and boundary conditions of arbitrary-shaped thin plates, cover arbitrary-shaped thin plate domains with standard rectangular domains, and open holes in the standard rectangular domains according to the geometric boundaries of arbitrary-shaped thin plates to simulate the shape of arbitrary-shaped thin plate domains, and establish a dimensionless coordinate system; According to the shape of the arbitrary-shaped thin plate, several energy elements are generated in the standard rectangular domain to cover the arbitrary-shaped thin plate domain; According to the two-dimensional Gauss-Legendre quadrature rule, different numbers of Gaussian integration points are generated in each energy element based on the highest degree of the integrand polynomial function and the position of the hole domain. Through the Boolean operation between the coordinates of each Gaussian integral point and the boundary level set function, the position of the Gaussian integral point in the energy element is determined, and the variable stiffness characterization is performed on each Gaussian integral point in turn according to the Boolean operation result to complete the discrete modeling of thin plates of arbitrary shapes; Based on Legendre polynomials and boundary condition functions, a global trial function of a thin plate of arbitrary shape is constructed, the strain energy and load potential energy are integrated, and the energy in the hole domain is removed in combination with the global variable stiffness; based on the Ritz method, the stationary value of the functional is obtained, a nonlinear system of equations is derived, and the energy element is combined to complete the high-precision numerical integration of each element in the nonlinear system of equations; the nonlinear system of equations is solved, and the unknown coefficients are obtained and brought back to the global trial function to obtain the numerical solution of large deformation of thin plates of arbitrary shape.

2. The method for numerically solving the energy element of large deformation of a thin plate of arbitrary shape according to claim 1 is characterized in that: Based on Legendre polynomials and boundary condition functions, a global test function for a thin plate of arbitrary shape is constructed, which specifically includes the following steps: In the formula (δ=u,v,w) is the boundary Ξ k The boundary condition function, T+1 is the total number of inner and outer boundaries, is the unknown coefficient, χ i (ξ) and χ j (η) are Legendre polynomials in the ξ and η directions, truncated at the mth and nth terms in the ξ and η directions respectively; boundary condition function Describe the boundary conditions imposed on the kth boundary, which is represented by the boundary level set function Ξ of the kth boundary k (ξ,η) is obtained by assigning boundary index, which is determined by the following three boundary conditions: Legendre polynomials are a set of orthogonal complete polynomials derived from the following recursive formula: Where i-1 represents the highest power of the i-th Legendre polynomial; after adding boundary exponents to the boundary level set function in sequence, multiplying T+1 boundary condition functions together, the boundary condition function of a thin plate of any shape can be obtained: By multiplying it by the Legendre polynomials in the ξ and η directions respectively, the global trial function construction of thin plates of arbitrary shapes can be completed.

3. The method for numerically solving the energy element of large deformation of a thin plate of arbitrary shape according to claim 2 is characterized in that: Based on the Ritz method, a nonlinear equation group is derived, and the energy element is combined to complete the high-precision numerical integration of each element in the nonlinear equation group, which specifically includes the following steps: According to the Ritz method, the total energy of the system of a thin plate of any shape can be expressed as: Π=UQ (4) The Ritz method requires the coefficient P to be determined ij Take the partial derivative to get the stationary value of the functional: A set of nonlinear equations can then be derived and written in matrix form: Where K l is the linear stiffness matrix containing 3mn×3mn elements; P is the undetermined coefficient array containing 3mn×1 elements, which is composed of the undetermined coefficients in the global trial function (7); K nl is the nonlinear stiffness matrix containing 3mn×1 elements, and the elements in brackets are the undetermined coefficients contained in the nonlinear equation; Q is an array of 3mn×1 elements derived from the load potential energy Q; the linear stiffness matrix K l The expressions for each element in are written as: The elements Located in the matrix The rth row and sth column of Located in the matrix The rth row and sth column of both contain the variable stiffness function c ij (ξ,η) and d ij (ξ,η), so c ij (ξ,η) and d ij (ξ,η) needs to be included in the integral: The expression of the integrand is: In the formula and Representative Item and Legendre polynomial, χ i (ξ) and χ j (η) are the ith and jth Legendre polynomials; Nonlinear stiffness matrix K nl The expressions for each element in are written as: In the formula, the element Located in the array The rth row, element Located in the array The rth row of contains the variable stiffness function c ij (ξ,η), so c ij (ξ,η) needs to be included in the integral: Where α(ξ,η), β(ξ,η), γ(ξ,η) represent the global test function (7), and the expression of the integrand is: Calculate the function value at each Gaussian integral point in each rectangular element and non-rectangular element in turn, multiply it by the corresponding integral coefficient, and finally superimpose the integral value in each energy element: In the formula Corresponding to the integrand G is the total number of rectangular elements, H is the total number of non-rectangular elements, and are the integration domains of the kth rectangular element and the zth non-rectangular element respectively; and are the Gaussian integration points and quadrature coefficients within the kth rectangular element, and are the Gaussian integration points and quadrature coefficients within the zth non-rectangular element; and and Determine the tensile stiffness and bending stiffness of the Gaussian integration points in the kth rectangular element and the zth non-rectangular element respectively; Numerically integrate the array derived from the load potential energy, where the elements Located in array Q w No. OK: Since the arbitrary shape thin plate domain is exactly the same as the global uniform load domain, the Gaussian integration points used for the numerical integration of strain energy can be directly used for the Numerical integration of :

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